Research ArticleAdaptive Robust Backstepping Control ofPermanent Magnet Synchronous Motor Chaotic System withFully Unknown Parameters and External Disturbances
Yang Yu Xudong Guo and Zengqiang Mi
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power UniversityBaoding 071003 China
Correspondence should be addressed to Yang Yu ncepu yy163com
Received 15 December 2015 Revised 16 March 2016 Accepted 30 March 2016
Academic Editor Shengjun Wen
Copyright copy 2016 Yang Yu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The chaotic behavior of permanent magnet synchronous motor is directly related to the parameters of chaotic system Theparameters of permanentmagnet synchronousmotor chaotic system are frequently unknownHence chaotic control of permanentmagnet synchronous motor with unknown parameters is of great significance In order to make the subject more general andfeasible an adaptive robust backstepping control algorithm is proposed to address the issues of fully unknown parametersestimation and external disturbances inhibition on the basis of associating backstepping control with adaptive control Firstlythe mathematical model of permanent magnet synchronous motor chaotic system with fully unknown parameters is constructedand the external disturbances are introduced into the model Secondly an adaptive robust backstepping control technology isemployed to design controller In contrastwith traditional backstepping control the proposed controller ismore concise in structureand avoids many restricted problems The stability of the control approach is proved by Lyapunov stability theory Finally theeffectiveness and correctness of the presented algorithm are verified through multiple simulation experiments and the resultsshow that the proposed scheme enablesmaking permanentmagnet synchronousmotor operate away from chaotic state rapidly andensures the tracking errors to converge to a small neighborhood within the origin rapidly under the full parameters uncertaintiesand external disturbances
1 Introduction
In recent years the permanent magnet synchronous motor(PMSM) is utilized widely in various industrial fields dueto its constantly dropping production cost simple structurehigh torque and high efficiency However Hemati foundthat PMSM would generate chaotic behavior with systemparameters entering into a certain region [1] Previous studieshave shown that the chaotic movement of PMSM willproduce irregular oscillations of torque and speed exacerbatecurrent noise and worsen operation performance and mayeven damage the entire drive system Therefore research onPMSM chaos phenomenon has attracted extensive attentionworldwide [2ndash5] and further studying on the controlmethodof PMSM chaos is of extreme significance [6ndash8]
The nonlinear characteristics of PMSM such as multi-variability strong coupling and high dimension make it dif-ficult to control for traditional linear control theory Hence
a variety of modern and nonlinear control algorithms areintroduced to suppress PMSM chaotic behavior In terms ofthese control algorithms whether or not relying on the modelparameters the previous control methods can primarily bedivided into two categories The first type is on the basisof accurate model parameters such as entrainment andmigration control [9] exact feedback linearization control[10] and decoupling control [11] However the accuracy ofthese control methods directly depends on PMSM modelparameters if the system parameters deviate from the ratedvalues the control performance will go bad The second typeis based on unknown parameters which have become theresearch focus of PMSM chaos suppression recently mainlyincluding sliding mode variable structure control [12 13]fuzzy control [14] and 119867
infincontrol [15] However sliding
mode variable structure control requires uncertain parame-ters to satisfy certain matching conditions fuzzy control is
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3690240 14 pageshttpdxdoiorg10115520163690240
2 Mathematical Problems in Engineering
dependent on the fuzzification of Takagi-Sugeno and 119867infin
control is inclined to ignore the operating states under specialconditions [16] In essence PMSM chaotic system is highlysensitive to initial states and parameters and PMSM modelparameters are susceptible to the temperature and humidityof the surrounding environment Therefore PMSM chaoticrepression with unknownmodel parameters has applicabilityto a broader field and ismore in linewith reality [17] Actuallythe adaptive control (AC) provides a natural routine forPMSM chaotic control with unknown parameters which hasbeen presented in literatures [12 13 18]
Backstepping control (BC) is one of the most popu-lar nonlinear control methods newly proposed to addressparameter uncertainty specifically the uncertainty not sat-isfying matching condition which has been successfullyapplied to many engineering fields such as motor drivetemperature control of boiler main steam and rocket loca-tion tracking The core idea of BC is that complex high-dimensional nonlinear systems are decomposed into manysimple low-dimensional subsystems and virtual control vari-ables are introduced to backstepping process to design con-crete controllers In addition BC has been successfullyapplied to suppress Liu chaotic system [19] and Chen chaoticsystem [20] Therefore the idea of combining BC with ACprovides a useful and feasible train of thought to controlPMSM chaotic systemwith unknown parameters Literatures[21 22] have exactly practiced this idea
However the conventional backstepping approach is con-fronted with two major problems of solving complicatedldquoregression matrixrdquo [23] and encountering ldquoexplosion oftermsrdquo [24] In [25] the complexity of regression matrix issufficiently manifested which almost occupies one full pageNevertheless explosion of terms is an inherent shortcomingand is induced by repeated differentiations of virtual vari-ables particularly in design of adaptive backstepping con-troller [26] Additionally integration of BC with AC is fre-quently facedwith the singularity arising from any estimationterm emerging as a denominator of any control input Theoverparameterization caused by the number of estimationslarger than actual system parameters hinders the conven-tional adaptive backstepping control
In addition to the above problems to the extent of ourknowledgemostly existing literatures on PMSMchaotic con-trol only concentrate on the cases of single unknown param-eter and partial unknown parameters [21 22] and there isno way to address the issue of fully unknown parametersFurthermore the existing researches mainly aim at the sit-uation of sudden power failure during PMSM operation [16]the existing conclusions lack the generality Hence throughcombination of BC and AC not only does this paper studythe control issue of PMSM chaos suppression with fullyunknown parameters but also the external disturbances aretaken into account in PMSM chaos model Newly adaptiveupdating laws of unknown parameters are designed to totallyestimate unknown parameters of PMSM chaotic model andadaptive robust backstepping controllers on the basis ofadaptive estimations and external disturbances are developedto drive PMSM to escape out of chaotic state quickly inhibitthe external disturbances and accomplish the given signals
tracking rapidlyThemethod proposed in this paper expandsthe applied range of backstepping control theory in PMSMchaotic systemMoreover the study of chaos control problemwith totally unknown parameters and external disturbancesis more general and practical and the results and conclusionsobtained are more applicable
2 PMSM Chaotic Model with FullyUnknown Parameters
For a PMSM its mathematical model in 119889119902 axis coordinatesystem can be described as follows [16]
=
3119901120601119898
2119869119902minus119861
119869 minus
120591119897
119869
119902= minus
119877
119871119902
119902minus 119901 sdot
119889minus119901120601119898
119871119902
sdot +1
119871119902
119902
119889= minus
119877
119871119889
119889+ 119901 sdot
119902+
1
119871119889
119889
(1)
where is the mechanical angular velocity of the rotatingrotor
119889and 119902are 119889 axis and 119902 axis currents of stator winding
respectively 119889and
119902are 119889 axis and 119902 axis voltages of stator
winding 119901 is the number of rotor pole pairs 120601119898is the flux
generated by permanent magnets 119869 is the moment of inertia119861 is the viscous damping coefficient 120591
119897is the load torque 119877 is
the phase resistance of the stator windings and 119871119889and 119871
119902are
119889 axis and 119902 axis inductances of stator winding respectivelyFor a PMSM with uniform air gap 119871
119889= 119871119902 Hence we use 119871
to substitute 119871119889and 119871
119902in the following paper
Selecting the affine transformation [
119902
119889
] =
[1119901120591 0 0
0 119896 0
0 0 119896
] [
120596
119894119902
119894119889
] and time scale transformation = 120591119905 thePMSMmathematicalmodel described in (1) can be convertedinto dimensionless form as follows
=1
120575(119894119902minus 120596) minus 120591
119897
119894119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902
119894119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889
(2)
where 120591 = 119871119877 119896 = 211986131199012
120591120601119898 119906119889= (1119896119877)
119889 120574 = minus120601
119898119896119871
119906119902= (1119896119877)
119902 120575 = 119869119861120591 and 120591
119897= 1199011205912
120591119897119869
As presented in (2) the dynamic performance of PMSMdepends on three parameters 120575 120574 and 120591
119897 Considering the
most general case let 120575 = 02 120574 = 50 120591119897= 32 119906
119889= minus06
and 119906119902
= 08 If the initial state is selected as (120596 119894119902 119894119889) =
(0 0 0) PMSM system will run on a chaotic state and displaythe chaotic behavior A typical chaotic attractor of PMSM ismanifested in Figure 1
In reality the three parameters 120575 120574 and 120591119897in (2) tend to
be unknownor to have uncertainties resulting fromoperatingconditions In other words when all the parameters 120575 120574and 120591119897cannot be determined (2) actually represents PMSM
chaotic system model with fully unknown parameters
Mathematical Problems in Engineering 3
minus20 minus15 minus10 minus5 0 5 10 15
minus50minus40minus30minus20minus100102030
minus100
102030405060708090
i d
iq120596
Figure 1 Chaotic attractor of PMSM system
3 Design of Adaptive Robust Controller withBackstepping Approach
Taking a more general situation into account the PMSMchaotic model described in (2) is immersed by external dis-turbances The model can be rewritten as follows
=1
120575(119894119902minus 120596) minus 120591
119897
119894119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1(x 119905)
119894119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2(x 119905)
(3)
where Δ1(x 119905) and Δ
2(x 119905) represent the external distur-
bances x indicates the system states and x = (1199091 1199092 1199093) =
(120596 119894119902 119894119889)
31 Control Objective and Assumptions Control problem inthe paper can be described as follows for PMSM chaoticsystem (3) with fully unknown parameters 120575 120574 and 120591
119897and
external disturbances Δ1and Δ
2 adaptive laws of unknown
parameters 120575 120574 and 120591119897are designed and adaptive robust
controllers 119906119889and 119906119902are constructed to ensure PMSMbreaks
away from chaos rapidly and runs into an expected orbitSimultaneously the fully unknownparameters 120575 120574 and 120591
119897can
be estimated accurately and the external disturbances can beinhibited effectively
For convenience of controller design the control systemis supposed to hold some reasonable assumptions as follows
Assumption 1 The state variables for PMSM chaotic system(120596 119894119902 119894119889) are observable
Assumption 2 The external disturbances Δ119894(x 119905) satisfy the
condition |Δ119894(x 119905)| le 119889
119894(x)119891119894(119905) 119894 = 1 2 where 119889
119894(x) is
a known function 119891119894(119905) is an unknown but bounded time-
varying function and |119891119894(119905)| le 119891
119894max where 119891119894max is a
constant
Assumption 3 Thedesired speed and 119889 axis current referencesignals 120596
lowast and 119894lowast
119889and their derivatives are known and
bounded
The estimated values of unknown system parameters aredescribed as and
119897 then the estimation errors and
119897can be expressed as follows
= minus 120575
= minus 120574
119897= 119897minus 120591119897
(4)
32 Controller Design The essence of adaptive robust back-stepping controller is to design controller through combina-tion of backstepping method and adaptive approach thena reasonably stable function is built in accordance withLyapunov stability theory to guarantee error variables to beeffectively stabilized and meanwhile ensure the output ofclosed loop system tracks reference signals quickly On thebasis of this the adaptive robust backstepping controller isdesigned as follows
Step 1 For the speed reference signal 120596lowast define the trackingerror 119890
120596as follows
119890120596= 120596 minus 120596
lowast
(5)
Taking PMSM chaotic system model (3) into account thederivative of (5) can be written as
119890120596= minus
lowast
=1
120575(119894119902minus 120596) minus 120591
119897minus lowast
(6)
Define the tracking error 119890119902of 119902 axis stator current 119894
119902as
follows119890119902= 119894119902minus 119894lowast
119902 (7)
where 119894lowast119902is the expected output value of 119894
119902
For the119889 axis current reference signal 119894lowast119889 its tracking error
119890119889is defined as follows
119890119889= 119894119889minus 119894lowast
119889 (8)
By substitution of (7) into (6) we can obtain
119890120596=
1
120575(119890119902+ 119894lowast
119902minus 120596) minus 120591
119897minus lowast
=1
120575(119894lowast
119902minus 120596) minus
119897+ 119897minus lowast
(9)
Let
119894lowast
119902= 120596 + (
119897+ lowast
minus 1198961119890120596) (10)
119894lowast
119889= 0 (11)
where 1198961represents the positive control gain
Through substitution of (10) into (9) (12) can be obtained
119890120596=
1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 119897minus lowast
=1
120575119890119902+120575 +
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 119897minus lowast
=1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596
(12)
4 Mathematical Problems in Engineering
Lyapunov function 1198811is selected as follows
1198811=
1
21198902
120596 (13)
Then the derivative of 1198811can be described as
1= 119890120596119890120596
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
(14)
Step 2 To stabilize the output 119902 axis current of PMSM thederivative of 119890
119902is conducted as follows
119890119902= 119894119902minus 119894lowast
119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897
+ lowast
minus 1198961sdot 119890120596) minus minus ( 120591
119897+ lowast
minus 1198961sdot 119890120596) = minus119894
119902
minus 120596 sdot 119894119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897+ lowast
minus 1198961
sdot 119890120596) minus 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(15)
By substitution of (12) into (15) we can get119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(16)
Combined with the mathematical model of PMSM chaoticsystem (16) can be further calculated as follows
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 119896119903sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961sdot
119894119902minus 120596 minus 120591
119897
120575minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 120591119897minus
sdot 1198961sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1
sdot (119897minus 119897) minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897+
sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
(17)
By substitution of = + 120575 into (17) the following equationcan be obtained
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) +120575 +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897
+ sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
(18)
Lyapunov function 1198812is chosen as follows
1198812= 1198811+1
21198902
119902 (19)
Mathematical Problems in Engineering 5
Then the derivative of 1198812can be described as
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902
+ Δ1minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
)
(20)
The first control variable is selected as
119906119902= 119906119902119904+ 119906119902119903 (21)
where 119906119902119904
and 119906119902119903
are the model compensation and robustcontrol inputs respectively
Then 119906119902119904and 119906
119902119903can be respectively chosen as
119906119902119904
= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894
119889minus sdot 120596
+
120575 (119897+ lowast
minus 1198961119890120596) + lowast
+ ( 120591119897+ lowast
) +
sdot 1198961sdot 119897+ sdot 119896
119903sdot lowast
minus 1198961(119894119902minus 120596)
(22)
119906119902119903
= minus1198892
1(x)
41205761
119890119902 (23)
where 1198962is another positive control gain and 120576
1is a positive
number chosen arbitrarilyBy substitution of (21) and (22) into (20) we can acquire
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903
+ Δ1minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) = 119890
120596(1
120575119890119902
+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596) + 119890119902(minus1198962119890119902
minus1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 minus
1198892
1(x)
41205761
119890119902+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1))
(24)
Additionally
119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) = minus
1198892
1(x)
41205761
1198902
119902+ Δ1119890119902
le minus1198892
1(x)
41205761
1198902
119902+ 1198891(x) 1198911max
10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
= minus(
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
(25)
Step 3 Differentiating the tracking error 119890119889of 119889 axis current
119894119889 we can get
119890119889= 119894119889minus 119894lowast
119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889 (26)
Lyapunov function 1198813is chosen as
1198813= 1198812+1
21198902
119889 (27)
Then the derivative of 1198813can be represented as
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889)
(28)
In terms of (28) 119889 axis output stator voltage 119906119889can be cal-
culated
119906119889= 119906119889119904+ 119906119889119903 (29)
where
119906119889119904
= minus1198963sdot 119890119889+ 119894119889minus 120596 sdot 119894
119902+ 119894lowast
119889 (30)
119906119889119903
= minus1198892
2(x)
41205762
119890119889 (31)
where 1198963is the positive control gain and 120576
2is a positive num-
ber chosen arbitrarilyBy substitution of (30) and (31) into (26) and (28) respec-
tively the following equations can be acquired
119890119889= minus1198963sdot 119890119889+ 119906119889119903
+ Δ2= minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2 (32)
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2)
(33)
6 Mathematical Problems in Engineering
119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2) = minus
1198892
2(x)
41205762
1198902
119889+ Δ2119890119889
le minus1198892
2(x)
41205762
1198902
119889+ 1198892(x) 1198912max
10038161003816100381610038161198901198891003816100381610038161003816
= minus(1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
(34)
Step 4 Lyapunov function 119881 of PMSM chaotic systemwith fully unknown parameters and external disturbances isselected as follows
119881 = 1198813+
1
21205791
2
119897+
1
21205792
2
+1
2120575 sdot 1205793
2
(35)
where 1205791 1205792 and 120579
3represent positive adaptive gains
Combined with equations 120575 =
120575 120574 = 120574 and 120591
119897= 120591119897
derivative of selected Lyapunov function 119881 can be calculatedas follows
= 3+
119897
1205791
120591119897+
1205792
120574 +
120575 sdot 1205793
120575 =
2+ 119890119889(minus1198963sdot 119890119889
+ 119906119889119903
+ Δ2) = 1+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus
sdot 120596 + 119906119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2)
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2) =
1
120575
sdot 119890119902119890120596+
120575(119897+ lowast
minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902
120596
minus 11989631198902
119889minus 11989621198902
119902minus 120596119890
119902minus
1
1205751198902
119902+ 119890119902(119906119902119903+ Δ1)
+ 119890119889(119906119889119903
+ Δ2) minus
120575((119897+ lowast
minus 1198961119890120596)
minus 1198961(119894119902minus 120596)) sdot 119890
119902+ 119897( sdot 1198961minus 1) 119890
119902+ 1198961119890120596119890119902
+119897
1205791
120591119897+
1205792
120574 +
1205751205793
120575 =
1
120575119890119902119890120596+ 1198961119890120596119890119902
+
120575[(119897+ lowast
minus 1198961119890120596) 119890120596minus (119897+ lowast
minus 1198961119890120596) 119890119902
+ 1198961(119894119902minus 120596) 119890
119902+
120575
1205793
] minus 11989611198902
120596minus 11989621198902
119889minus 11989631198902
119902
+ 119897[119890120596minus 119890119902+ 1198961119890119902+
120591119897
1205791
] + [minus120596119890119902+
120574
1205792
]
minus1
1205751198902
119902+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889
+ Δ2)
(36)
In terms of (36) the adaptive laws of unknown parameters 120575120574 and 120591
119897can be selected respectively as follows
120575 = minus120579
3[(119897+ lowast
minus 1198961119890120596) sdot (119890120596minus 119890119902)
+ 1198961(119894119902minus 120596) 119890
119902]
(37)
120574 = 1205792120596119890119902 (38)
120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)
By substitution of (37) (38) and (39) into (36) (36) can besimplified as follows
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(40)
33 Stability Analysis
Theorem 4 For PMSM chaotic system model (3) with fullyunknown parameters and external disturbances design ofadaptive control laws (37) (38) and (40) and selection ofsuitable controller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive
gains 1205791 1205792 and 120579
3 the proposed adaptive robust backstepping
controllers (21) and (29) can ensure the tracking error signals(5) (7) and (8) of PMSM chaotic systems are asymptoticallystableThat is to say PMSMchaotic system can run out of chaosquickly through the proposed controllers (21) and (29) and trackthe given reference signals
Through stability analysis we want to verify the correct-ness of the theorem
Mathematical Problems in Engineering 7
According to (40) new expression can be obtained as fol-lows through some mathematical computations
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
+1
21205751198902
120596+
1
21205751198902
119902minus1198961
2(119890120596minus 119890119902)2
+1198961
21198902
120596+1198961
21198902
119902minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(41)
Appropriate controller gains 1198961and 1198962are selected as follows
(1
2120575+1198961
2minus 1198961) lt 0
(1
2120575+1198961
2minus 1198962minus
1
120575) lt 0
(42)
Equation (42) can be replaced by the following
1198961gt
1
120575
1198961minus 21198962lt
1
120575
(43)
Then by substitution of (43) into (41) we can obtain
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(44)
where 1198964= minus(12120575 + 119896
12 minus 119896
1) ge 0 119896
5= minus(12120575 + 119896
12 minus
1198962minus 1120575) ge 0 and 120576
0= 12057611198912
1max + 12057621198912
2maxLet
119882(119894 (119905)) = minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
minus 11989641198902
120596minus 11989651198902
119902
minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(45)
where 119894(119905) = (119890120596 119890119902 119890119889)
By integration of (45) we can get
int
119905
1199050
119882(119894 (119905)) 119889119905 = minusint
119905
1199050
(119894 (119905)) 119889119905 997904rArr
int
119905
1199050
119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)
(46)
Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-
ing hence
lim119905rarrinfin
int
119905
1199050
119882(119894 (119905)) 119889119905 lt infin (47)
Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained
lim119905rarrinfin
119882(119894 (119905)) = 0 (48)
Apparently through selection of suitable controller gains 1198961
1198962 1198963 1205761 and 120576
2 can be ensured to be negative definite
The above derivation has proved that the selected suitablecontroller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive gains 120579
1 1205792
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
dependent on the fuzzification of Takagi-Sugeno and 119867infin
control is inclined to ignore the operating states under specialconditions [16] In essence PMSM chaotic system is highlysensitive to initial states and parameters and PMSM modelparameters are susceptible to the temperature and humidityof the surrounding environment Therefore PMSM chaoticrepression with unknownmodel parameters has applicabilityto a broader field and ismore in linewith reality [17] Actuallythe adaptive control (AC) provides a natural routine forPMSM chaotic control with unknown parameters which hasbeen presented in literatures [12 13 18]
Backstepping control (BC) is one of the most popu-lar nonlinear control methods newly proposed to addressparameter uncertainty specifically the uncertainty not sat-isfying matching condition which has been successfullyapplied to many engineering fields such as motor drivetemperature control of boiler main steam and rocket loca-tion tracking The core idea of BC is that complex high-dimensional nonlinear systems are decomposed into manysimple low-dimensional subsystems and virtual control vari-ables are introduced to backstepping process to design con-crete controllers In addition BC has been successfullyapplied to suppress Liu chaotic system [19] and Chen chaoticsystem [20] Therefore the idea of combining BC with ACprovides a useful and feasible train of thought to controlPMSM chaotic systemwith unknown parameters Literatures[21 22] have exactly practiced this idea
However the conventional backstepping approach is con-fronted with two major problems of solving complicatedldquoregression matrixrdquo [23] and encountering ldquoexplosion oftermsrdquo [24] In [25] the complexity of regression matrix issufficiently manifested which almost occupies one full pageNevertheless explosion of terms is an inherent shortcomingand is induced by repeated differentiations of virtual vari-ables particularly in design of adaptive backstepping con-troller [26] Additionally integration of BC with AC is fre-quently facedwith the singularity arising from any estimationterm emerging as a denominator of any control input Theoverparameterization caused by the number of estimationslarger than actual system parameters hinders the conven-tional adaptive backstepping control
In addition to the above problems to the extent of ourknowledgemostly existing literatures on PMSMchaotic con-trol only concentrate on the cases of single unknown param-eter and partial unknown parameters [21 22] and there isno way to address the issue of fully unknown parametersFurthermore the existing researches mainly aim at the sit-uation of sudden power failure during PMSM operation [16]the existing conclusions lack the generality Hence throughcombination of BC and AC not only does this paper studythe control issue of PMSM chaos suppression with fullyunknown parameters but also the external disturbances aretaken into account in PMSM chaos model Newly adaptiveupdating laws of unknown parameters are designed to totallyestimate unknown parameters of PMSM chaotic model andadaptive robust backstepping controllers on the basis ofadaptive estimations and external disturbances are developedto drive PMSM to escape out of chaotic state quickly inhibitthe external disturbances and accomplish the given signals
tracking rapidlyThemethod proposed in this paper expandsthe applied range of backstepping control theory in PMSMchaotic systemMoreover the study of chaos control problemwith totally unknown parameters and external disturbancesis more general and practical and the results and conclusionsobtained are more applicable
2 PMSM Chaotic Model with FullyUnknown Parameters
For a PMSM its mathematical model in 119889119902 axis coordinatesystem can be described as follows [16]
=
3119901120601119898
2119869119902minus119861
119869 minus
120591119897
119869
119902= minus
119877
119871119902
119902minus 119901 sdot
119889minus119901120601119898
119871119902
sdot +1
119871119902
119902
119889= minus
119877
119871119889
119889+ 119901 sdot
119902+
1
119871119889
119889
(1)
where is the mechanical angular velocity of the rotatingrotor
119889and 119902are 119889 axis and 119902 axis currents of stator winding
respectively 119889and
119902are 119889 axis and 119902 axis voltages of stator
winding 119901 is the number of rotor pole pairs 120601119898is the flux
generated by permanent magnets 119869 is the moment of inertia119861 is the viscous damping coefficient 120591
119897is the load torque 119877 is
the phase resistance of the stator windings and 119871119889and 119871
119902are
119889 axis and 119902 axis inductances of stator winding respectivelyFor a PMSM with uniform air gap 119871
119889= 119871119902 Hence we use 119871
to substitute 119871119889and 119871
119902in the following paper
Selecting the affine transformation [
119902
119889
] =
[1119901120591 0 0
0 119896 0
0 0 119896
] [
120596
119894119902
119894119889
] and time scale transformation = 120591119905 thePMSMmathematicalmodel described in (1) can be convertedinto dimensionless form as follows
=1
120575(119894119902minus 120596) minus 120591
119897
119894119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902
119894119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889
(2)
where 120591 = 119871119877 119896 = 211986131199012
120591120601119898 119906119889= (1119896119877)
119889 120574 = minus120601
119898119896119871
119906119902= (1119896119877)
119902 120575 = 119869119861120591 and 120591
119897= 1199011205912
120591119897119869
As presented in (2) the dynamic performance of PMSMdepends on three parameters 120575 120574 and 120591
119897 Considering the
most general case let 120575 = 02 120574 = 50 120591119897= 32 119906
119889= minus06
and 119906119902
= 08 If the initial state is selected as (120596 119894119902 119894119889) =
(0 0 0) PMSM system will run on a chaotic state and displaythe chaotic behavior A typical chaotic attractor of PMSM ismanifested in Figure 1
In reality the three parameters 120575 120574 and 120591119897in (2) tend to
be unknownor to have uncertainties resulting fromoperatingconditions In other words when all the parameters 120575 120574and 120591119897cannot be determined (2) actually represents PMSM
chaotic system model with fully unknown parameters
Mathematical Problems in Engineering 3
minus20 minus15 minus10 minus5 0 5 10 15
minus50minus40minus30minus20minus100102030
minus100
102030405060708090
i d
iq120596
Figure 1 Chaotic attractor of PMSM system
3 Design of Adaptive Robust Controller withBackstepping Approach
Taking a more general situation into account the PMSMchaotic model described in (2) is immersed by external dis-turbances The model can be rewritten as follows
=1
120575(119894119902minus 120596) minus 120591
119897
119894119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1(x 119905)
119894119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2(x 119905)
(3)
where Δ1(x 119905) and Δ
2(x 119905) represent the external distur-
bances x indicates the system states and x = (1199091 1199092 1199093) =
(120596 119894119902 119894119889)
31 Control Objective and Assumptions Control problem inthe paper can be described as follows for PMSM chaoticsystem (3) with fully unknown parameters 120575 120574 and 120591
119897and
external disturbances Δ1and Δ
2 adaptive laws of unknown
parameters 120575 120574 and 120591119897are designed and adaptive robust
controllers 119906119889and 119906119902are constructed to ensure PMSMbreaks
away from chaos rapidly and runs into an expected orbitSimultaneously the fully unknownparameters 120575 120574 and 120591
119897can
be estimated accurately and the external disturbances can beinhibited effectively
For convenience of controller design the control systemis supposed to hold some reasonable assumptions as follows
Assumption 1 The state variables for PMSM chaotic system(120596 119894119902 119894119889) are observable
Assumption 2 The external disturbances Δ119894(x 119905) satisfy the
condition |Δ119894(x 119905)| le 119889
119894(x)119891119894(119905) 119894 = 1 2 where 119889
119894(x) is
a known function 119891119894(119905) is an unknown but bounded time-
varying function and |119891119894(119905)| le 119891
119894max where 119891119894max is a
constant
Assumption 3 Thedesired speed and 119889 axis current referencesignals 120596
lowast and 119894lowast
119889and their derivatives are known and
bounded
The estimated values of unknown system parameters aredescribed as and
119897 then the estimation errors and
119897can be expressed as follows
= minus 120575
= minus 120574
119897= 119897minus 120591119897
(4)
32 Controller Design The essence of adaptive robust back-stepping controller is to design controller through combina-tion of backstepping method and adaptive approach thena reasonably stable function is built in accordance withLyapunov stability theory to guarantee error variables to beeffectively stabilized and meanwhile ensure the output ofclosed loop system tracks reference signals quickly On thebasis of this the adaptive robust backstepping controller isdesigned as follows
Step 1 For the speed reference signal 120596lowast define the trackingerror 119890
120596as follows
119890120596= 120596 minus 120596
lowast
(5)
Taking PMSM chaotic system model (3) into account thederivative of (5) can be written as
119890120596= minus
lowast
=1
120575(119894119902minus 120596) minus 120591
119897minus lowast
(6)
Define the tracking error 119890119902of 119902 axis stator current 119894
119902as
follows119890119902= 119894119902minus 119894lowast
119902 (7)
where 119894lowast119902is the expected output value of 119894
119902
For the119889 axis current reference signal 119894lowast119889 its tracking error
119890119889is defined as follows
119890119889= 119894119889minus 119894lowast
119889 (8)
By substitution of (7) into (6) we can obtain
119890120596=
1
120575(119890119902+ 119894lowast
119902minus 120596) minus 120591
119897minus lowast
=1
120575(119894lowast
119902minus 120596) minus
119897+ 119897minus lowast
(9)
Let
119894lowast
119902= 120596 + (
119897+ lowast
minus 1198961119890120596) (10)
119894lowast
119889= 0 (11)
where 1198961represents the positive control gain
Through substitution of (10) into (9) (12) can be obtained
119890120596=
1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 119897minus lowast
=1
120575119890119902+120575 +
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 119897minus lowast
=1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596
(12)
4 Mathematical Problems in Engineering
Lyapunov function 1198811is selected as follows
1198811=
1
21198902
120596 (13)
Then the derivative of 1198811can be described as
1= 119890120596119890120596
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
(14)
Step 2 To stabilize the output 119902 axis current of PMSM thederivative of 119890
119902is conducted as follows
119890119902= 119894119902minus 119894lowast
119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897
+ lowast
minus 1198961sdot 119890120596) minus minus ( 120591
119897+ lowast
minus 1198961sdot 119890120596) = minus119894
119902
minus 120596 sdot 119894119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897+ lowast
minus 1198961
sdot 119890120596) minus 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(15)
By substitution of (12) into (15) we can get119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(16)
Combined with the mathematical model of PMSM chaoticsystem (16) can be further calculated as follows
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 119896119903sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961sdot
119894119902minus 120596 minus 120591
119897
120575minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 120591119897minus
sdot 1198961sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1
sdot (119897minus 119897) minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897+
sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
(17)
By substitution of = + 120575 into (17) the following equationcan be obtained
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) +120575 +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897
+ sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
(18)
Lyapunov function 1198812is chosen as follows
1198812= 1198811+1
21198902
119902 (19)
Mathematical Problems in Engineering 5
Then the derivative of 1198812can be described as
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902
+ Δ1minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
)
(20)
The first control variable is selected as
119906119902= 119906119902119904+ 119906119902119903 (21)
where 119906119902119904
and 119906119902119903
are the model compensation and robustcontrol inputs respectively
Then 119906119902119904and 119906
119902119903can be respectively chosen as
119906119902119904
= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894
119889minus sdot 120596
+
120575 (119897+ lowast
minus 1198961119890120596) + lowast
+ ( 120591119897+ lowast
) +
sdot 1198961sdot 119897+ sdot 119896
119903sdot lowast
minus 1198961(119894119902minus 120596)
(22)
119906119902119903
= minus1198892
1(x)
41205761
119890119902 (23)
where 1198962is another positive control gain and 120576
1is a positive
number chosen arbitrarilyBy substitution of (21) and (22) into (20) we can acquire
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903
+ Δ1minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) = 119890
120596(1
120575119890119902
+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596) + 119890119902(minus1198962119890119902
minus1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 minus
1198892
1(x)
41205761
119890119902+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1))
(24)
Additionally
119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) = minus
1198892
1(x)
41205761
1198902
119902+ Δ1119890119902
le minus1198892
1(x)
41205761
1198902
119902+ 1198891(x) 1198911max
10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
= minus(
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
(25)
Step 3 Differentiating the tracking error 119890119889of 119889 axis current
119894119889 we can get
119890119889= 119894119889minus 119894lowast
119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889 (26)
Lyapunov function 1198813is chosen as
1198813= 1198812+1
21198902
119889 (27)
Then the derivative of 1198813can be represented as
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889)
(28)
In terms of (28) 119889 axis output stator voltage 119906119889can be cal-
culated
119906119889= 119906119889119904+ 119906119889119903 (29)
where
119906119889119904
= minus1198963sdot 119890119889+ 119894119889minus 120596 sdot 119894
119902+ 119894lowast
119889 (30)
119906119889119903
= minus1198892
2(x)
41205762
119890119889 (31)
where 1198963is the positive control gain and 120576
2is a positive num-
ber chosen arbitrarilyBy substitution of (30) and (31) into (26) and (28) respec-
tively the following equations can be acquired
119890119889= minus1198963sdot 119890119889+ 119906119889119903
+ Δ2= minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2 (32)
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2)
(33)
6 Mathematical Problems in Engineering
119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2) = minus
1198892
2(x)
41205762
1198902
119889+ Δ2119890119889
le minus1198892
2(x)
41205762
1198902
119889+ 1198892(x) 1198912max
10038161003816100381610038161198901198891003816100381610038161003816
= minus(1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
(34)
Step 4 Lyapunov function 119881 of PMSM chaotic systemwith fully unknown parameters and external disturbances isselected as follows
119881 = 1198813+
1
21205791
2
119897+
1
21205792
2
+1
2120575 sdot 1205793
2
(35)
where 1205791 1205792 and 120579
3represent positive adaptive gains
Combined with equations 120575 =
120575 120574 = 120574 and 120591
119897= 120591119897
derivative of selected Lyapunov function 119881 can be calculatedas follows
= 3+
119897
1205791
120591119897+
1205792
120574 +
120575 sdot 1205793
120575 =
2+ 119890119889(minus1198963sdot 119890119889
+ 119906119889119903
+ Δ2) = 1+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus
sdot 120596 + 119906119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2)
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2) =
1
120575
sdot 119890119902119890120596+
120575(119897+ lowast
minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902
120596
minus 11989631198902
119889minus 11989621198902
119902minus 120596119890
119902minus
1
1205751198902
119902+ 119890119902(119906119902119903+ Δ1)
+ 119890119889(119906119889119903
+ Δ2) minus
120575((119897+ lowast
minus 1198961119890120596)
minus 1198961(119894119902minus 120596)) sdot 119890
119902+ 119897( sdot 1198961minus 1) 119890
119902+ 1198961119890120596119890119902
+119897
1205791
120591119897+
1205792
120574 +
1205751205793
120575 =
1
120575119890119902119890120596+ 1198961119890120596119890119902
+
120575[(119897+ lowast
minus 1198961119890120596) 119890120596minus (119897+ lowast
minus 1198961119890120596) 119890119902
+ 1198961(119894119902minus 120596) 119890
119902+
120575
1205793
] minus 11989611198902
120596minus 11989621198902
119889minus 11989631198902
119902
+ 119897[119890120596minus 119890119902+ 1198961119890119902+
120591119897
1205791
] + [minus120596119890119902+
120574
1205792
]
minus1
1205751198902
119902+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889
+ Δ2)
(36)
In terms of (36) the adaptive laws of unknown parameters 120575120574 and 120591
119897can be selected respectively as follows
120575 = minus120579
3[(119897+ lowast
minus 1198961119890120596) sdot (119890120596minus 119890119902)
+ 1198961(119894119902minus 120596) 119890
119902]
(37)
120574 = 1205792120596119890119902 (38)
120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)
By substitution of (37) (38) and (39) into (36) (36) can besimplified as follows
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(40)
33 Stability Analysis
Theorem 4 For PMSM chaotic system model (3) with fullyunknown parameters and external disturbances design ofadaptive control laws (37) (38) and (40) and selection ofsuitable controller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive
gains 1205791 1205792 and 120579
3 the proposed adaptive robust backstepping
controllers (21) and (29) can ensure the tracking error signals(5) (7) and (8) of PMSM chaotic systems are asymptoticallystableThat is to say PMSMchaotic system can run out of chaosquickly through the proposed controllers (21) and (29) and trackthe given reference signals
Through stability analysis we want to verify the correct-ness of the theorem
Mathematical Problems in Engineering 7
According to (40) new expression can be obtained as fol-lows through some mathematical computations
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
+1
21205751198902
120596+
1
21205751198902
119902minus1198961
2(119890120596minus 119890119902)2
+1198961
21198902
120596+1198961
21198902
119902minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(41)
Appropriate controller gains 1198961and 1198962are selected as follows
(1
2120575+1198961
2minus 1198961) lt 0
(1
2120575+1198961
2minus 1198962minus
1
120575) lt 0
(42)
Equation (42) can be replaced by the following
1198961gt
1
120575
1198961minus 21198962lt
1
120575
(43)
Then by substitution of (43) into (41) we can obtain
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(44)
where 1198964= minus(12120575 + 119896
12 minus 119896
1) ge 0 119896
5= minus(12120575 + 119896
12 minus
1198962minus 1120575) ge 0 and 120576
0= 12057611198912
1max + 12057621198912
2maxLet
119882(119894 (119905)) = minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
minus 11989641198902
120596minus 11989651198902
119902
minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(45)
where 119894(119905) = (119890120596 119890119902 119890119889)
By integration of (45) we can get
int
119905
1199050
119882(119894 (119905)) 119889119905 = minusint
119905
1199050
(119894 (119905)) 119889119905 997904rArr
int
119905
1199050
119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)
(46)
Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-
ing hence
lim119905rarrinfin
int
119905
1199050
119882(119894 (119905)) 119889119905 lt infin (47)
Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained
lim119905rarrinfin
119882(119894 (119905)) = 0 (48)
Apparently through selection of suitable controller gains 1198961
1198962 1198963 1205761 and 120576
2 can be ensured to be negative definite
The above derivation has proved that the selected suitablecontroller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive gains 120579
1 1205792
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
minus20 minus15 minus10 minus5 0 5 10 15
minus50minus40minus30minus20minus100102030
minus100
102030405060708090
i d
iq120596
Figure 1 Chaotic attractor of PMSM system
3 Design of Adaptive Robust Controller withBackstepping Approach
Taking a more general situation into account the PMSMchaotic model described in (2) is immersed by external dis-turbances The model can be rewritten as follows
=1
120575(119894119902minus 120596) minus 120591
119897
119894119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1(x 119905)
119894119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2(x 119905)
(3)
where Δ1(x 119905) and Δ
2(x 119905) represent the external distur-
bances x indicates the system states and x = (1199091 1199092 1199093) =
(120596 119894119902 119894119889)
31 Control Objective and Assumptions Control problem inthe paper can be described as follows for PMSM chaoticsystem (3) with fully unknown parameters 120575 120574 and 120591
119897and
external disturbances Δ1and Δ
2 adaptive laws of unknown
parameters 120575 120574 and 120591119897are designed and adaptive robust
controllers 119906119889and 119906119902are constructed to ensure PMSMbreaks
away from chaos rapidly and runs into an expected orbitSimultaneously the fully unknownparameters 120575 120574 and 120591
119897can
be estimated accurately and the external disturbances can beinhibited effectively
For convenience of controller design the control systemis supposed to hold some reasonable assumptions as follows
Assumption 1 The state variables for PMSM chaotic system(120596 119894119902 119894119889) are observable
Assumption 2 The external disturbances Δ119894(x 119905) satisfy the
condition |Δ119894(x 119905)| le 119889
119894(x)119891119894(119905) 119894 = 1 2 where 119889
119894(x) is
a known function 119891119894(119905) is an unknown but bounded time-
varying function and |119891119894(119905)| le 119891
119894max where 119891119894max is a
constant
Assumption 3 Thedesired speed and 119889 axis current referencesignals 120596
lowast and 119894lowast
119889and their derivatives are known and
bounded
The estimated values of unknown system parameters aredescribed as and
119897 then the estimation errors and
119897can be expressed as follows
= minus 120575
= minus 120574
119897= 119897minus 120591119897
(4)
32 Controller Design The essence of adaptive robust back-stepping controller is to design controller through combina-tion of backstepping method and adaptive approach thena reasonably stable function is built in accordance withLyapunov stability theory to guarantee error variables to beeffectively stabilized and meanwhile ensure the output ofclosed loop system tracks reference signals quickly On thebasis of this the adaptive robust backstepping controller isdesigned as follows
Step 1 For the speed reference signal 120596lowast define the trackingerror 119890
120596as follows
119890120596= 120596 minus 120596
lowast
(5)
Taking PMSM chaotic system model (3) into account thederivative of (5) can be written as
119890120596= minus
lowast
=1
120575(119894119902minus 120596) minus 120591
119897minus lowast
(6)
Define the tracking error 119890119902of 119902 axis stator current 119894
119902as
follows119890119902= 119894119902minus 119894lowast
119902 (7)
where 119894lowast119902is the expected output value of 119894
119902
For the119889 axis current reference signal 119894lowast119889 its tracking error
119890119889is defined as follows
119890119889= 119894119889minus 119894lowast
119889 (8)
By substitution of (7) into (6) we can obtain
119890120596=
1
120575(119890119902+ 119894lowast
119902minus 120596) minus 120591
119897minus lowast
=1
120575(119894lowast
119902minus 120596) minus
119897+ 119897minus lowast
(9)
Let
119894lowast
119902= 120596 + (
119897+ lowast
minus 1198961119890120596) (10)
119894lowast
119889= 0 (11)
where 1198961represents the positive control gain
Through substitution of (10) into (9) (12) can be obtained
119890120596=
1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 119897minus lowast
=1
120575119890119902+120575 +
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 119897minus lowast
=1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596
(12)
4 Mathematical Problems in Engineering
Lyapunov function 1198811is selected as follows
1198811=
1
21198902
120596 (13)
Then the derivative of 1198811can be described as
1= 119890120596119890120596
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
(14)
Step 2 To stabilize the output 119902 axis current of PMSM thederivative of 119890
119902is conducted as follows
119890119902= 119894119902minus 119894lowast
119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897
+ lowast
minus 1198961sdot 119890120596) minus minus ( 120591
119897+ lowast
minus 1198961sdot 119890120596) = minus119894
119902
minus 120596 sdot 119894119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897+ lowast
minus 1198961
sdot 119890120596) minus 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(15)
By substitution of (12) into (15) we can get119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(16)
Combined with the mathematical model of PMSM chaoticsystem (16) can be further calculated as follows
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 119896119903sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961sdot
119894119902minus 120596 minus 120591
119897
120575minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 120591119897minus
sdot 1198961sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1
sdot (119897minus 119897) minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897+
sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
(17)
By substitution of = + 120575 into (17) the following equationcan be obtained
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) +120575 +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897
+ sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
(18)
Lyapunov function 1198812is chosen as follows
1198812= 1198811+1
21198902
119902 (19)
Mathematical Problems in Engineering 5
Then the derivative of 1198812can be described as
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902
+ Δ1minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
)
(20)
The first control variable is selected as
119906119902= 119906119902119904+ 119906119902119903 (21)
where 119906119902119904
and 119906119902119903
are the model compensation and robustcontrol inputs respectively
Then 119906119902119904and 119906
119902119903can be respectively chosen as
119906119902119904
= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894
119889minus sdot 120596
+
120575 (119897+ lowast
minus 1198961119890120596) + lowast
+ ( 120591119897+ lowast
) +
sdot 1198961sdot 119897+ sdot 119896
119903sdot lowast
minus 1198961(119894119902minus 120596)
(22)
119906119902119903
= minus1198892
1(x)
41205761
119890119902 (23)
where 1198962is another positive control gain and 120576
1is a positive
number chosen arbitrarilyBy substitution of (21) and (22) into (20) we can acquire
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903
+ Δ1minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) = 119890
120596(1
120575119890119902
+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596) + 119890119902(minus1198962119890119902
minus1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 minus
1198892
1(x)
41205761
119890119902+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1))
(24)
Additionally
119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) = minus
1198892
1(x)
41205761
1198902
119902+ Δ1119890119902
le minus1198892
1(x)
41205761
1198902
119902+ 1198891(x) 1198911max
10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
= minus(
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
(25)
Step 3 Differentiating the tracking error 119890119889of 119889 axis current
119894119889 we can get
119890119889= 119894119889minus 119894lowast
119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889 (26)
Lyapunov function 1198813is chosen as
1198813= 1198812+1
21198902
119889 (27)
Then the derivative of 1198813can be represented as
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889)
(28)
In terms of (28) 119889 axis output stator voltage 119906119889can be cal-
culated
119906119889= 119906119889119904+ 119906119889119903 (29)
where
119906119889119904
= minus1198963sdot 119890119889+ 119894119889minus 120596 sdot 119894
119902+ 119894lowast
119889 (30)
119906119889119903
= minus1198892
2(x)
41205762
119890119889 (31)
where 1198963is the positive control gain and 120576
2is a positive num-
ber chosen arbitrarilyBy substitution of (30) and (31) into (26) and (28) respec-
tively the following equations can be acquired
119890119889= minus1198963sdot 119890119889+ 119906119889119903
+ Δ2= minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2 (32)
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2)
(33)
6 Mathematical Problems in Engineering
119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2) = minus
1198892
2(x)
41205762
1198902
119889+ Δ2119890119889
le minus1198892
2(x)
41205762
1198902
119889+ 1198892(x) 1198912max
10038161003816100381610038161198901198891003816100381610038161003816
= minus(1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
(34)
Step 4 Lyapunov function 119881 of PMSM chaotic systemwith fully unknown parameters and external disturbances isselected as follows
119881 = 1198813+
1
21205791
2
119897+
1
21205792
2
+1
2120575 sdot 1205793
2
(35)
where 1205791 1205792 and 120579
3represent positive adaptive gains
Combined with equations 120575 =
120575 120574 = 120574 and 120591
119897= 120591119897
derivative of selected Lyapunov function 119881 can be calculatedas follows
= 3+
119897
1205791
120591119897+
1205792
120574 +
120575 sdot 1205793
120575 =
2+ 119890119889(minus1198963sdot 119890119889
+ 119906119889119903
+ Δ2) = 1+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus
sdot 120596 + 119906119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2)
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2) =
1
120575
sdot 119890119902119890120596+
120575(119897+ lowast
minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902
120596
minus 11989631198902
119889minus 11989621198902
119902minus 120596119890
119902minus
1
1205751198902
119902+ 119890119902(119906119902119903+ Δ1)
+ 119890119889(119906119889119903
+ Δ2) minus
120575((119897+ lowast
minus 1198961119890120596)
minus 1198961(119894119902minus 120596)) sdot 119890
119902+ 119897( sdot 1198961minus 1) 119890
119902+ 1198961119890120596119890119902
+119897
1205791
120591119897+
1205792
120574 +
1205751205793
120575 =
1
120575119890119902119890120596+ 1198961119890120596119890119902
+
120575[(119897+ lowast
minus 1198961119890120596) 119890120596minus (119897+ lowast
minus 1198961119890120596) 119890119902
+ 1198961(119894119902minus 120596) 119890
119902+
120575
1205793
] minus 11989611198902
120596minus 11989621198902
119889minus 11989631198902
119902
+ 119897[119890120596minus 119890119902+ 1198961119890119902+
120591119897
1205791
] + [minus120596119890119902+
120574
1205792
]
minus1
1205751198902
119902+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889
+ Δ2)
(36)
In terms of (36) the adaptive laws of unknown parameters 120575120574 and 120591
119897can be selected respectively as follows
120575 = minus120579
3[(119897+ lowast
minus 1198961119890120596) sdot (119890120596minus 119890119902)
+ 1198961(119894119902minus 120596) 119890
119902]
(37)
120574 = 1205792120596119890119902 (38)
120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)
By substitution of (37) (38) and (39) into (36) (36) can besimplified as follows
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(40)
33 Stability Analysis
Theorem 4 For PMSM chaotic system model (3) with fullyunknown parameters and external disturbances design ofadaptive control laws (37) (38) and (40) and selection ofsuitable controller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive
gains 1205791 1205792 and 120579
3 the proposed adaptive robust backstepping
controllers (21) and (29) can ensure the tracking error signals(5) (7) and (8) of PMSM chaotic systems are asymptoticallystableThat is to say PMSMchaotic system can run out of chaosquickly through the proposed controllers (21) and (29) and trackthe given reference signals
Through stability analysis we want to verify the correct-ness of the theorem
Mathematical Problems in Engineering 7
According to (40) new expression can be obtained as fol-lows through some mathematical computations
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
+1
21205751198902
120596+
1
21205751198902
119902minus1198961
2(119890120596minus 119890119902)2
+1198961
21198902
120596+1198961
21198902
119902minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(41)
Appropriate controller gains 1198961and 1198962are selected as follows
(1
2120575+1198961
2minus 1198961) lt 0
(1
2120575+1198961
2minus 1198962minus
1
120575) lt 0
(42)
Equation (42) can be replaced by the following
1198961gt
1
120575
1198961minus 21198962lt
1
120575
(43)
Then by substitution of (43) into (41) we can obtain
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(44)
where 1198964= minus(12120575 + 119896
12 minus 119896
1) ge 0 119896
5= minus(12120575 + 119896
12 minus
1198962minus 1120575) ge 0 and 120576
0= 12057611198912
1max + 12057621198912
2maxLet
119882(119894 (119905)) = minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
minus 11989641198902
120596minus 11989651198902
119902
minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(45)
where 119894(119905) = (119890120596 119890119902 119890119889)
By integration of (45) we can get
int
119905
1199050
119882(119894 (119905)) 119889119905 = minusint
119905
1199050
(119894 (119905)) 119889119905 997904rArr
int
119905
1199050
119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)
(46)
Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-
ing hence
lim119905rarrinfin
int
119905
1199050
119882(119894 (119905)) 119889119905 lt infin (47)
Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained
lim119905rarrinfin
119882(119894 (119905)) = 0 (48)
Apparently through selection of suitable controller gains 1198961
1198962 1198963 1205761 and 120576
2 can be ensured to be negative definite
The above derivation has proved that the selected suitablecontroller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive gains 120579
1 1205792
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
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4 Mathematical Problems in Engineering
Lyapunov function 1198811is selected as follows
1198811=
1
21198902
120596 (13)
Then the derivative of 1198811can be described as
1= 119890120596119890120596
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
(14)
Step 2 To stabilize the output 119902 axis current of PMSM thederivative of 119890
119902is conducted as follows
119890119902= 119894119902minus 119894lowast
119902= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ 120574 sdot 120596 + 119906
119902+ Δ1minus [
+
120575 (119897+ lowast
minus 1198961sdot 119890120596) + ( 120591
119897+ lowast
minus 1198961sdot 119890120596)]
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897
+ lowast
minus 1198961sdot 119890120596) minus minus ( 120591
119897+ lowast
minus 1198961sdot 119890120596) = minus119894
119902
minus 120596 sdot 119894119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1minus
120575 (119897+ lowast
minus 1198961
sdot 119890120596) minus 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(15)
By substitution of (12) into (15) we can get119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961(120596 minus
lowast
)
(16)
Combined with the mathematical model of PMSM chaoticsystem (16) can be further calculated as follows
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 119896119903sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) + sdot 1198961sdot
119894119902minus 120596 minus 120591
119897
120575minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 120591119897minus
sdot 1198961sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1
sdot (119897minus 119897) minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus120575 (119897+ lowast
minus 1198961sdot 119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast
minus ( 120591119897+ lowast
) +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897+
sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
(17)
By substitution of = + 120575 into (17) the following equationcan be obtained
119890119902= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) +120575 +
120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896
1sdot 119897
+ sdot 1198961sdot 119897minus sdot 119896
1sdot lowast
= minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902+ Δ1
minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
(18)
Lyapunov function 1198812is chosen as follows
1198812= 1198811+1
21198902
119902 (19)
Mathematical Problems in Engineering 5
Then the derivative of 1198812can be described as
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902
+ Δ1minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
)
(20)
The first control variable is selected as
119906119902= 119906119902119904+ 119906119902119903 (21)
where 119906119902119904
and 119906119902119903
are the model compensation and robustcontrol inputs respectively
Then 119906119902119904and 119906
119902119903can be respectively chosen as
119906119902119904
= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894
119889minus sdot 120596
+
120575 (119897+ lowast
minus 1198961119890120596) + lowast
+ ( 120591119897+ lowast
) +
sdot 1198961sdot 119897+ sdot 119896
119903sdot lowast
minus 1198961(119894119902minus 120596)
(22)
119906119902119903
= minus1198892
1(x)
41205761
119890119902 (23)
where 1198962is another positive control gain and 120576
1is a positive
number chosen arbitrarilyBy substitution of (21) and (22) into (20) we can acquire
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903
+ Δ1minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) = 119890
120596(1
120575119890119902
+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596) + 119890119902(minus1198962119890119902
minus1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 minus
1198892
1(x)
41205761
119890119902+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1))
(24)
Additionally
119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) = minus
1198892
1(x)
41205761
1198902
119902+ Δ1119890119902
le minus1198892
1(x)
41205761
1198902
119902+ 1198891(x) 1198911max
10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
= minus(
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
(25)
Step 3 Differentiating the tracking error 119890119889of 119889 axis current
119894119889 we can get
119890119889= 119894119889minus 119894lowast
119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889 (26)
Lyapunov function 1198813is chosen as
1198813= 1198812+1
21198902
119889 (27)
Then the derivative of 1198813can be represented as
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889)
(28)
In terms of (28) 119889 axis output stator voltage 119906119889can be cal-
culated
119906119889= 119906119889119904+ 119906119889119903 (29)
where
119906119889119904
= minus1198963sdot 119890119889+ 119894119889minus 120596 sdot 119894
119902+ 119894lowast
119889 (30)
119906119889119903
= minus1198892
2(x)
41205762
119890119889 (31)
where 1198963is the positive control gain and 120576
2is a positive num-
ber chosen arbitrarilyBy substitution of (30) and (31) into (26) and (28) respec-
tively the following equations can be acquired
119890119889= minus1198963sdot 119890119889+ 119906119889119903
+ Δ2= minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2 (32)
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2)
(33)
6 Mathematical Problems in Engineering
119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2) = minus
1198892
2(x)
41205762
1198902
119889+ Δ2119890119889
le minus1198892
2(x)
41205762
1198902
119889+ 1198892(x) 1198912max
10038161003816100381610038161198901198891003816100381610038161003816
= minus(1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
(34)
Step 4 Lyapunov function 119881 of PMSM chaotic systemwith fully unknown parameters and external disturbances isselected as follows
119881 = 1198813+
1
21205791
2
119897+
1
21205792
2
+1
2120575 sdot 1205793
2
(35)
where 1205791 1205792 and 120579
3represent positive adaptive gains
Combined with equations 120575 =
120575 120574 = 120574 and 120591
119897= 120591119897
derivative of selected Lyapunov function 119881 can be calculatedas follows
= 3+
119897
1205791
120591119897+
1205792
120574 +
120575 sdot 1205793
120575 =
2+ 119890119889(minus1198963sdot 119890119889
+ 119906119889119903
+ Δ2) = 1+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus
sdot 120596 + 119906119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2)
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2) =
1
120575
sdot 119890119902119890120596+
120575(119897+ lowast
minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902
120596
minus 11989631198902
119889minus 11989621198902
119902minus 120596119890
119902minus
1
1205751198902
119902+ 119890119902(119906119902119903+ Δ1)
+ 119890119889(119906119889119903
+ Δ2) minus
120575((119897+ lowast
minus 1198961119890120596)
minus 1198961(119894119902minus 120596)) sdot 119890
119902+ 119897( sdot 1198961minus 1) 119890
119902+ 1198961119890120596119890119902
+119897
1205791
120591119897+
1205792
120574 +
1205751205793
120575 =
1
120575119890119902119890120596+ 1198961119890120596119890119902
+
120575[(119897+ lowast
minus 1198961119890120596) 119890120596minus (119897+ lowast
minus 1198961119890120596) 119890119902
+ 1198961(119894119902minus 120596) 119890
119902+
120575
1205793
] minus 11989611198902
120596minus 11989621198902
119889minus 11989631198902
119902
+ 119897[119890120596minus 119890119902+ 1198961119890119902+
120591119897
1205791
] + [minus120596119890119902+
120574
1205792
]
minus1
1205751198902
119902+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889
+ Δ2)
(36)
In terms of (36) the adaptive laws of unknown parameters 120575120574 and 120591
119897can be selected respectively as follows
120575 = minus120579
3[(119897+ lowast
minus 1198961119890120596) sdot (119890120596minus 119890119902)
+ 1198961(119894119902minus 120596) 119890
119902]
(37)
120574 = 1205792120596119890119902 (38)
120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)
By substitution of (37) (38) and (39) into (36) (36) can besimplified as follows
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(40)
33 Stability Analysis
Theorem 4 For PMSM chaotic system model (3) with fullyunknown parameters and external disturbances design ofadaptive control laws (37) (38) and (40) and selection ofsuitable controller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive
gains 1205791 1205792 and 120579
3 the proposed adaptive robust backstepping
controllers (21) and (29) can ensure the tracking error signals(5) (7) and (8) of PMSM chaotic systems are asymptoticallystableThat is to say PMSMchaotic system can run out of chaosquickly through the proposed controllers (21) and (29) and trackthe given reference signals
Through stability analysis we want to verify the correct-ness of the theorem
Mathematical Problems in Engineering 7
According to (40) new expression can be obtained as fol-lows through some mathematical computations
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
+1
21205751198902
120596+
1
21205751198902
119902minus1198961
2(119890120596minus 119890119902)2
+1198961
21198902
120596+1198961
21198902
119902minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(41)
Appropriate controller gains 1198961and 1198962are selected as follows
(1
2120575+1198961
2minus 1198961) lt 0
(1
2120575+1198961
2minus 1198962minus
1
120575) lt 0
(42)
Equation (42) can be replaced by the following
1198961gt
1
120575
1198961minus 21198962lt
1
120575
(43)
Then by substitution of (43) into (41) we can obtain
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(44)
where 1198964= minus(12120575 + 119896
12 minus 119896
1) ge 0 119896
5= minus(12120575 + 119896
12 minus
1198962minus 1120575) ge 0 and 120576
0= 12057611198912
1max + 12057621198912
2maxLet
119882(119894 (119905)) = minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
minus 11989641198902
120596minus 11989651198902
119902
minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(45)
where 119894(119905) = (119890120596 119890119902 119890119889)
By integration of (45) we can get
int
119905
1199050
119882(119894 (119905)) 119889119905 = minusint
119905
1199050
(119894 (119905)) 119889119905 997904rArr
int
119905
1199050
119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)
(46)
Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-
ing hence
lim119905rarrinfin
int
119905
1199050
119882(119894 (119905)) 119889119905 lt infin (47)
Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained
lim119905rarrinfin
119882(119894 (119905)) = 0 (48)
Apparently through selection of suitable controller gains 1198961
1198962 1198963 1205761 and 120576
2 can be ensured to be negative definite
The above derivation has proved that the selected suitablecontroller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive gains 120579
1 1205792
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Then the derivative of 1198812can be described as
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus119894119902minus 120596 sdot 119894
119889+ sdot 120596 minus sdot 120596 + 119906
119902
+ Δ1minus
120575 (119897+ lowast
minus 1198961119890120596) minus
1
120575119890119902
minus
120575(119897+ lowast
minus 1198961119890120596) minus 119897+ 1198961119890120596minus lowast
minus ( 120591119897+ lowast
) + 1198961sdot (119894119902minus 120596) +
120575sdot 1198961sdot (119894119902minus 120596)
minus sdot 1198961sdot 119897+ sdot 119896
1sdot 119897minus sdot 119896
1sdot lowast
)
(20)
The first control variable is selected as
119906119902= 119906119902119904+ 119906119902119903 (21)
where 119906119902119904
and 119906119902119903
are the model compensation and robustcontrol inputs respectively
Then 119906119902119904and 119906
119902119903can be respectively chosen as
119906119902119904
= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894
119889minus sdot 120596
+
120575 (119897+ lowast
minus 1198961119890120596) + lowast
+ ( 120591119897+ lowast
) +
sdot 1198961sdot 119897+ sdot 119896
119903sdot lowast
minus 1198961(119894119902minus 120596)
(22)
119906119902119903
= minus1198892
1(x)
41205761
119890119902 (23)
where 1198962is another positive control gain and 120576
1is a positive
number chosen arbitrarilyBy substitution of (21) and (22) into (20) we can acquire
2= 1+ 119890119902119890119902= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897
minus 1198961119890120596) + 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903
+ Δ1minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) = 119890
120596(1
120575119890119902
+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596) + 119890119902(minus1198962119890119902
minus1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 minus
1198892
1(x)
41205761
119890119902+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1))
(24)
Additionally
119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) = minus
1198892
1(x)
41205761
1198902
119902+ Δ1119890119902
le minus1198892
1(x)
41205761
1198902
119902+ 1198891(x) 1198911max
10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
= minus(
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
(25)
Step 3 Differentiating the tracking error 119890119889of 119889 axis current
119894119889 we can get
119890119889= 119894119889minus 119894lowast
119889= minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889 (26)
Lyapunov function 1198813is chosen as
1198813= 1198812+1
21198902
119889 (27)
Then the derivative of 1198813can be represented as
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus119894119889+ 120596 sdot 119894
119902+ 119906119889+ Δ2minus 119894lowast
119889)
(28)
In terms of (28) 119889 axis output stator voltage 119906119889can be cal-
culated
119906119889= 119906119889119904+ 119906119889119903 (29)
where
119906119889119904
= minus1198963sdot 119890119889+ 119894119889minus 120596 sdot 119894
119902+ 119894lowast
119889 (30)
119906119889119903
= minus1198892
2(x)
41205762
119890119889 (31)
where 1198963is the positive control gain and 120576
2is a positive num-
ber chosen arbitrarilyBy substitution of (30) and (31) into (26) and (28) respec-
tively the following equations can be acquired
119890119889= minus1198963sdot 119890119889+ 119906119889119903
+ Δ2= minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2 (32)
3= 2+ 119890119889119890119889
= 2+ 119890119889(minus1198963sdot 119890119889minus1198892
2(x)
41205762
119890119889+ Δ2)
(33)
6 Mathematical Problems in Engineering
119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2) = minus
1198892
2(x)
41205762
1198902
119889+ Δ2119890119889
le minus1198892
2(x)
41205762
1198902
119889+ 1198892(x) 1198912max
10038161003816100381610038161198901198891003816100381610038161003816
= minus(1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
(34)
Step 4 Lyapunov function 119881 of PMSM chaotic systemwith fully unknown parameters and external disturbances isselected as follows
119881 = 1198813+
1
21205791
2
119897+
1
21205792
2
+1
2120575 sdot 1205793
2
(35)
where 1205791 1205792 and 120579
3represent positive adaptive gains
Combined with equations 120575 =
120575 120574 = 120574 and 120591
119897= 120591119897
derivative of selected Lyapunov function 119881 can be calculatedas follows
= 3+
119897
1205791
120591119897+
1205792
120574 +
120575 sdot 1205793
120575 =
2+ 119890119889(minus1198963sdot 119890119889
+ 119906119889119903
+ Δ2) = 1+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus
sdot 120596 + 119906119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2)
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2) =
1
120575
sdot 119890119902119890120596+
120575(119897+ lowast
minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902
120596
minus 11989631198902
119889minus 11989621198902
119902minus 120596119890
119902minus
1
1205751198902
119902+ 119890119902(119906119902119903+ Δ1)
+ 119890119889(119906119889119903
+ Δ2) minus
120575((119897+ lowast
minus 1198961119890120596)
minus 1198961(119894119902minus 120596)) sdot 119890
119902+ 119897( sdot 1198961minus 1) 119890
119902+ 1198961119890120596119890119902
+119897
1205791
120591119897+
1205792
120574 +
1205751205793
120575 =
1
120575119890119902119890120596+ 1198961119890120596119890119902
+
120575[(119897+ lowast
minus 1198961119890120596) 119890120596minus (119897+ lowast
minus 1198961119890120596) 119890119902
+ 1198961(119894119902minus 120596) 119890
119902+
120575
1205793
] minus 11989611198902
120596minus 11989621198902
119889minus 11989631198902
119902
+ 119897[119890120596minus 119890119902+ 1198961119890119902+
120591119897
1205791
] + [minus120596119890119902+
120574
1205792
]
minus1
1205751198902
119902+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889
+ Δ2)
(36)
In terms of (36) the adaptive laws of unknown parameters 120575120574 and 120591
119897can be selected respectively as follows
120575 = minus120579
3[(119897+ lowast
minus 1198961119890120596) sdot (119890120596minus 119890119902)
+ 1198961(119894119902minus 120596) 119890
119902]
(37)
120574 = 1205792120596119890119902 (38)
120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)
By substitution of (37) (38) and (39) into (36) (36) can besimplified as follows
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(40)
33 Stability Analysis
Theorem 4 For PMSM chaotic system model (3) with fullyunknown parameters and external disturbances design ofadaptive control laws (37) (38) and (40) and selection ofsuitable controller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive
gains 1205791 1205792 and 120579
3 the proposed adaptive robust backstepping
controllers (21) and (29) can ensure the tracking error signals(5) (7) and (8) of PMSM chaotic systems are asymptoticallystableThat is to say PMSMchaotic system can run out of chaosquickly through the proposed controllers (21) and (29) and trackthe given reference signals
Through stability analysis we want to verify the correct-ness of the theorem
Mathematical Problems in Engineering 7
According to (40) new expression can be obtained as fol-lows through some mathematical computations
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
+1
21205751198902
120596+
1
21205751198902
119902minus1198961
2(119890120596minus 119890119902)2
+1198961
21198902
120596+1198961
21198902
119902minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(41)
Appropriate controller gains 1198961and 1198962are selected as follows
(1
2120575+1198961
2minus 1198961) lt 0
(1
2120575+1198961
2minus 1198962minus
1
120575) lt 0
(42)
Equation (42) can be replaced by the following
1198961gt
1
120575
1198961minus 21198962lt
1
120575
(43)
Then by substitution of (43) into (41) we can obtain
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(44)
where 1198964= minus(12120575 + 119896
12 minus 119896
1) ge 0 119896
5= minus(12120575 + 119896
12 minus
1198962minus 1120575) ge 0 and 120576
0= 12057611198912
1max + 12057621198912
2maxLet
119882(119894 (119905)) = minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
minus 11989641198902
120596minus 11989651198902
119902
minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(45)
where 119894(119905) = (119890120596 119890119902 119890119889)
By integration of (45) we can get
int
119905
1199050
119882(119894 (119905)) 119889119905 = minusint
119905
1199050
(119894 (119905)) 119889119905 997904rArr
int
119905
1199050
119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)
(46)
Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-
ing hence
lim119905rarrinfin
int
119905
1199050
119882(119894 (119905)) 119889119905 lt infin (47)
Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained
lim119905rarrinfin
119882(119894 (119905)) = 0 (48)
Apparently through selection of suitable controller gains 1198961
1198962 1198963 1205761 and 120576
2 can be ensured to be negative definite
The above derivation has proved that the selected suitablecontroller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive gains 120579
1 1205792
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2) = minus
1198892
2(x)
41205762
1198902
119889+ Δ2119890119889
le minus1198892
2(x)
41205762
1198902
119889+ 1198892(x) 1198912max
10038161003816100381610038161198901198891003816100381610038161003816
= minus(1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
(34)
Step 4 Lyapunov function 119881 of PMSM chaotic systemwith fully unknown parameters and external disturbances isselected as follows
119881 = 1198813+
1
21205791
2
119897+
1
21205792
2
+1
2120575 sdot 1205793
2
(35)
where 1205791 1205792 and 120579
3represent positive adaptive gains
Combined with equations 120575 =
120575 120574 = 120574 and 120591
119897= 120591119897
derivative of selected Lyapunov function 119881 can be calculatedas follows
= 3+
119897
1205791
120591119897+
1205792
120574 +
120575 sdot 1205793
120575 =
2+ 119890119889(minus1198963sdot 119890119889
+ 119906119889119903
+ Δ2) = 1+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus
sdot 120596 + 119906119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2)
= 119890120596(1
120575119890119902+
120575(119897+ lowast
minus 1198961119890120596) + 119897minus 1198961119890120596)
+ 119890119902(minus1198962119890119902minus
1
120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906
119902119903+ Δ1
minus
120575((119897+ lowast
minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))
+ 119897( sdot 1198961minus 1)) + 119890
119889(minus1198963sdot 119890119889+ 119906119889119903
+ Δ2) =
1
120575
sdot 119890119902119890120596+
120575(119897+ lowast
minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902
120596
minus 11989631198902
119889minus 11989621198902
119902minus 120596119890
119902minus
1
1205751198902
119902+ 119890119902(119906119902119903+ Δ1)
+ 119890119889(119906119889119903
+ Δ2) minus
120575((119897+ lowast
minus 1198961119890120596)
minus 1198961(119894119902minus 120596)) sdot 119890
119902+ 119897( sdot 1198961minus 1) 119890
119902+ 1198961119890120596119890119902
+119897
1205791
120591119897+
1205792
120574 +
1205751205793
120575 =
1
120575119890119902119890120596+ 1198961119890120596119890119902
+
120575[(119897+ lowast
minus 1198961119890120596) 119890120596minus (119897+ lowast
minus 1198961119890120596) 119890119902
+ 1198961(119894119902minus 120596) 119890
119902+
120575
1205793
] minus 11989611198902
120596minus 11989621198902
119889minus 11989631198902
119902
+ 119897[119890120596minus 119890119902+ 1198961119890119902+
120591119897
1205791
] + [minus120596119890119902+
120574
1205792
]
minus1
1205751198902
119902+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889
+ Δ2)
(36)
In terms of (36) the adaptive laws of unknown parameters 120575120574 and 120591
119897can be selected respectively as follows
120575 = minus120579
3[(119897+ lowast
minus 1198961119890120596) sdot (119890120596minus 119890119902)
+ 1198961(119894119902minus 120596) 119890
119902]
(37)
120574 = 1205792120596119890119902 (38)
120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)
By substitution of (37) (38) and (39) into (36) (36) can besimplified as follows
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(40)
33 Stability Analysis
Theorem 4 For PMSM chaotic system model (3) with fullyunknown parameters and external disturbances design ofadaptive control laws (37) (38) and (40) and selection ofsuitable controller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive
gains 1205791 1205792 and 120579
3 the proposed adaptive robust backstepping
controllers (21) and (29) can ensure the tracking error signals(5) (7) and (8) of PMSM chaotic systems are asymptoticallystableThat is to say PMSMchaotic system can run out of chaosquickly through the proposed controllers (21) and (29) and trackthe given reference signals
Through stability analysis we want to verify the correct-ness of the theorem
Mathematical Problems in Engineering 7
According to (40) new expression can be obtained as fol-lows through some mathematical computations
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
+1
21205751198902
120596+
1
21205751198902
119902minus1198961
2(119890120596minus 119890119902)2
+1198961
21198902
120596+1198961
21198902
119902minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(41)
Appropriate controller gains 1198961and 1198962are selected as follows
(1
2120575+1198961
2minus 1198961) lt 0
(1
2120575+1198961
2minus 1198962minus
1
120575) lt 0
(42)
Equation (42) can be replaced by the following
1198961gt
1
120575
1198961minus 21198962lt
1
120575
(43)
Then by substitution of (43) into (41) we can obtain
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(44)
where 1198964= minus(12120575 + 119896
12 minus 119896
1) ge 0 119896
5= minus(12120575 + 119896
12 minus
1198962minus 1120575) ge 0 and 120576
0= 12057611198912
1max + 12057621198912
2maxLet
119882(119894 (119905)) = minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
minus 11989641198902
120596minus 11989651198902
119902
minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(45)
where 119894(119905) = (119890120596 119890119902 119890119889)
By integration of (45) we can get
int
119905
1199050
119882(119894 (119905)) 119889119905 = minusint
119905
1199050
(119894 (119905)) 119889119905 997904rArr
int
119905
1199050
119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)
(46)
Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-
ing hence
lim119905rarrinfin
int
119905
1199050
119882(119894 (119905)) 119889119905 lt infin (47)
Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained
lim119905rarrinfin
119882(119894 (119905)) = 0 (48)
Apparently through selection of suitable controller gains 1198961
1198962 1198963 1205761 and 120576
2 can be ensured to be negative definite
The above derivation has proved that the selected suitablecontroller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive gains 120579
1 1205792
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
According to (40) new expression can be obtained as fol-lows through some mathematical computations
=1
120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
+1
21205751198902
120596+
1
21205751198902
119902minus1198961
2(119890120596minus 119890119902)2
+1198961
21198902
120596+1198961
21198902
119902minus 11989611198902
120596minus 11989631198902
119889minus 11989621198902
119902minus
1
1205751198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1) + 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
(41)
Appropriate controller gains 1198961and 1198962are selected as follows
(1
2120575+1198961
2minus 1198961) lt 0
(1
2120575+1198961
2minus 1198962minus
1
120575) lt 0
(42)
Equation (42) can be replaced by the following
1198961gt
1
120575
1198961minus 21198962lt
1
120575
(43)
Then by substitution of (43) into (41) we can obtain
= minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
+ (1
2120575+1198961
2minus 1198961) 1198902
120596+ (
1
2120575+1198961
2minus 1198962minus
1
120575) 1198902
119902
+ 119890119902(minus
1198892
1(x)
41205761
119890119902+ Δ1)
+ 119890119889(minus
1198892
2(x)
41205762
119890119889+ Δ2)
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
+ 12057611198912
1max
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 12057621198912
2max
le minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889minus 11989641198902
120596
minus 11989651198902
119902minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(44)
where 1198964= minus(12120575 + 119896
12 minus 119896
1) ge 0 119896
5= minus(12120575 + 119896
12 minus
1198962minus 1120575) ge 0 and 120576
0= 12057611198912
1max + 12057621198912
2maxLet
119882(119894 (119905)) = minus1
2120575(119890120596minus 119890119902)2
minus1198961
2(119890120596minus 119890119902)2
minus 11989631198902
119889
minus 11989641198902
120596minus 11989651198902
119902
minus (
1198891(x) 10038161003816100381610038161003816119890119902
10038161003816100381610038161003816
2radic1205761
minus radic12057611198911max)
2
minus (1198892(x) 1003816100381610038161003816119890119889
1003816100381610038161003816
2radic1205762
minus radic12057621198912max)
2
+ 1205760
(45)
where 119894(119905) = (119890120596 119890119902 119890119889)
By integration of (45) we can get
int
119905
1199050
119882(119894 (119905)) 119889119905 = minusint
119905
1199050
(119894 (119905)) 119889119905 997904rArr
int
119905
1199050
119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)
(46)
Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-
ing hence
lim119905rarrinfin
int
119905
1199050
119882(119894 (119905)) 119889119905 lt infin (47)
Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained
lim119905rarrinfin
119882(119894 (119905)) = 0 (48)
Apparently through selection of suitable controller gains 1198961
1198962 1198963 1205761 and 120576
2 can be ensured to be negative definite
The above derivation has proved that the selected suitablecontroller gains 119896
1 1198962 1198963 1205761 and 120576
2and adaptive gains 120579
1 1205792
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
minus25
minus20
minus15
minus10
120596minus5
0
5
10
15
20
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906
119889and 119906
119902
and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold
In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=
119890119889= 119897= = = 0 In summary PMSM chaotic system
is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)
4 Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894
119902 119894119889) = (119909
1 1199092 1199093) and the con-
trol parameters are selected as 1198961= 10 119896
2= 30000 119896
3= 5
and 1205761= 1205762= 001 the adaptive gains are chosen as 120579
1= 62
1205792= 100 and 120579
3= 006 The simulation time is chosen as
100 s and the designed controller is put into effect at the timeof 20 s
41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591
119897= 32 In order to be
consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591
119897) = (0 0 0) and the expected
reference signals are set as 120596lowast = 10 and 119894lowast
119889= 1 Further-
more the external disturbances Δ1(x 119905) = 20119909
3sin(5119905) and
Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-
tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore
10 20 30 40 50 60 70 80 90 1000t (s)
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 3 The 119894119902curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
10 20 30 40 50 60 70 80 90 1000t (s)
0
10
20
30
40
50
60
70
80
90
100i d
Figure 4 The 119894119889curve of PMSM chaotic system with no control
inputs 119906119889and 119906
119902
minus25
minus20
minus15
minus10
120596 minus5
0
5
10
15
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906
119889and 119906
119902
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
minus50
minus40
minus30
minus20
minus10
0
10
20
30
40
i q
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 6The 119894119902curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
0
10
20
30
40
50
60
70
80
90
100
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 7The 119894119889curve of PMSMchaotic system added the controller
inputs 119906119889and 119906
119902
minus273
minus272
minus271
minus27
minus269
minus268
minus267
60 61 62 63minus27288
minus27282
30 40 50 60 70 80 90 10020t (s)
times106
times106
ud
Figure 8 The controller input 119906119889
95
952
954
956
958
96
962
964
966
60 6005 601957958959
96961962
times104
times104
uq
30 40 50 60 70 80 90 10020t (s)
Figure 9 The controller input 119906119902
minus02
minus015
minus01
minus005
0
005Es
timat
ion
erro
r of120575
30 40 50 60 70 80 90 10020t (s)
Figure 10 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus50
minus40
minus30
minus20
minus10
0
10
30 40 50 60 70 80 90 10020t (s)
Figure 11 The curve of estimation error of 120574
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
minus35
minus3
minus25
minus2
minus15
minus1
minus05
0
05
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 12 The 119897curve of estimation error of 120591
119897
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus15
minus10
minus5
0120596
5
10
15
20
25
Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances
Figures 10ndash12 show the estimated errors and 119897
of unknown parameters 120575 120574 and 120591119897for PMSM chaotic
systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters
42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591
119897in Test-I are changed into 120575 = 01 120574 = 25 and
120591119897= 16 in Test-II respectively Simultaneously the expected
reference signals are also changed and set as 120596lowast = 20 and119894lowast
119889= 0 In a word the unknown motor parameters and
expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894
119902 and 119894
119889added the
10 20 30 40 50 60 70 80 90 1000t (s)
minus30
minus20
minus10
0
10
20
30
40
i q
Figure 14 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
0
10
20
30
40
50
10 20 30 40 50 60 70 80 90 1000t (s)
i d
minus10
Figure 15 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus51
minus5095
minus509
minus5085
minus508
minus5075
469
469
246
94
469
646
98 47
minus5095minus509minus5085minus508minus5075minus507
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 16 The controller input 119906119889
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
246
94
469
646
98 47
minus2000minus1000
01000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 17 The controller input 119906119902
Estim
atio
n er
ror o
f120575
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
006
008
01
Figure 18 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
minus25
minus20
minus15
minus10
minus5
0
5
30 40 50 60 70 80 90 10020t (s)
Figure 19 The curve of estimation error of 120574
minus2
0
2
4
6
8
10
12
14
16
18
20
30 40 50 60 70 80 90 10020t (s)
Estim
atio
n er
ror o
f120591l
Figure 20 The 119897curve of estimation error of 120591
119897
0
5
10
15
20
25
minus20
minus15
minus10
120596
minus5
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 21 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906
119889and 119906
119902
controller inputs 119906119889and 119906
119902shown in Figures 16-17 which
demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed
43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ
1(x 119905) = 40119909
3sin(5119905) and Δ
2(x 119905) =
20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894
119902 and 119894
119889 which
manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
10 20 30 40 50 60 70 80 90 1000t (s)
minus20
minus10
0
10
20
30
40
i q
Figure 22 The 119894119902curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus10
0
10
20
30
40
50
i d
10 20 30 40 50 60 70 80 90 1000t (s)
Figure 23 The 119894119889curve of PMSM chaotic system added the con-
troller inputs 119906119889and 119906
119902
minus5085
minus508
minus5075
minus507
minus5065
469
469
5 4747
05
471
minus5085minus508minus5075
times104
times104
ud
30 40 50 60 70 80 90 10020t (s)
Figure 24 The controller input 119906119889
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
469
469
5 4747
05
471
minus1000minus500
0500
1000
uq
30 40 50 60 70 80 90 10020t (s)
Figure 25 The controller input 119906119902
Estim
atio
n er
ror o
f120575
minus02
minus015
minus01
minus005
0
005
01
015
02
30 40 50 60 70 80 90 10020t (s)
Figure 26 The curve of estimation error of 120575
Estim
atio
n er
ror o
f120574
30 40 50 60 70 80 90 10020t (s)
minus25
minus20
minus15
minus10
minus5
0
5
Figure 27 The curve of estimation error of 120574
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
30 40 50 60 70 80 90 10020t (s)
minus01
minus008
minus006
minus004
minus002
0
002
004
Estim
atio
n er
ror o
f120591l
Figure 28 The 119897curve of estimation error of 120591
119897
119906119889and 119906
119902shown in Figures 24-25 Figures 21ndash23 also illus-
trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again
Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591
119897into account it
undoubtedly extends the theory of parameter estimation forPMSM chaotic system
Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect
Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical
5 Conclusions
In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following
(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory
(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively
(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation
Competing Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)
References
[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994
[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002
[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004
[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012
[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014
[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006
[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009
[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002
[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009
[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013
[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013
[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014
[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012
[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867
infincontrolrdquo Acta Physica Sinica
vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control
of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014
[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014
[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007
[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010
[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011
[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014
[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006
[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000
[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995
[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of
the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002
[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of