Research ArticleRobust Observer Based Disturbance Rejection Control forEuler-Lagrange Systems
Yanjun Zhang Lu Wang Jun Zhang and Jianbo Su
Department of Automation Shanghai Jiao Tong University Shanghai China
Correspondence should be addressed to Jianbo Su jbsusjtueducn
Received 24 January 2016 Revised 11 May 2016 Accepted 5 June 2016
Academic Editor Sergey A Suslov
Copyright copy 2016 Yanjun Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Robust disturbance rejection control methodology is proposed for Euler-Lagrange systems and parameters optimization strategyfor the observer is explored First the observer based disturbance rejection methodology is analyzed based on which thedisturbance rejection paradigm is proposed Thus a disturbance observer (DOB) with partial feedback linearization and a low-pass filter is proposed for nonlinear dynamic model under relaxed restrictions of the generalized disturbanceThen the outer-loopbackstepping controller is designed for desired tracking performance Considering that the parameters of DOB cannot be obtaineddirectly based on Lyapunov stability analysis parameter of DOB is optimized under standard119867
infincontrol framework By analyzing
the influence of outer-loop controller on the inner-loop observer parameter robust stability constraint is proposed to guaranteethe robust stability of the closed-loop system Experiment on attitude tracking of an aircraft is carried out to show the effectivenessof the proposed control strategy
1 Introduction
Euler-Lagrange systems widely exist in practice such asmanipulator mobile robot underwater vehicle surface ves-sel and aerial vehicle Consequently motion control of Euler-Lagrange systems has been widely explored in the pastdecades Motion control systems usually work at unknownenvironment and inevitably they suffer from system uncer-tainties and external disturbances which will affect thecontrol performance or even make the system unstable[1] To deal with this problem numerous approaches havebeen proposed such as sliding mode control [2ndash4] adaptivecontrol [5ndash7] robust control [8ndash10] and intelligent control[11ndash13] These control methods can more or less deal withthe system uncertainties However facing the problems is stillinevitable such as chattering of slidingmode control stabilityproblem of adaptive control conservative robust control andconvergence rate of weights in neural network and fuzzysystem
The effectiveness of disturbance observer (DOB) has beenshown in many applications such as humanoid robot control[14 15] manipulator control [16ndash18] aircraft control [19 20]optical disk control [21 22] motor control [23 24] and
vibration control [25 26] Traditional DOB methodologywhich is proposed based on linear system cannot be useddirectly in nonlinear systems [27] In [28] traditional linearDOB is applied for disturbance rejection of nonlinear systemHowever only first-order binomial coefficient typed low-pass filter is used for DOB implementationThe performanceof the closed-loop system cannot be improved effectivelyMeanwhile the optimization strategy of parameters is notinvestigated Nonlinear DOB is proposed in [29 30] whichcan be directly used for disturbance estimation in nonlinearsystems In this paper we find that estimation effect ofnonlinear DOB is the same as that of linear DOB withfirst-order low-pass filter when a constant observer gain isselected And asymptotic stability is guaranteed simply basedon the assumption that the generalized disturbances andtheir first-order derivatives are bounded and that the firstderivatives go to zero in the steady state which is not realisticin most conditions Meanwhile for a closed-loop systemthe parameters of inner-loop observer depend on not onlysystem uncertainties and measurement noise but also thestructure and parameter of outer-loop controller Howeverthe existing works rarely discuss parameters optimization of
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3839505 13 pageshttpdxdoiorg10115520163839505
2 Mathematical Problems in Engineering
the observerThe influence caused by outer-loop controller isnever explored in existing researches
From the descriptions above a robust DOB based dis-turbance rejection controller is proposed and parametersoptimization strategy is investigated Nonlinear DOB andextended state observer (ESO) are first analyzed to show theessence of the disturbance estimation problem Then underrelaxed restrictions of disturbance and system perturbationa novel disturbance observer is proposed for nonlinearsystem The observer consists of a feedback linearizationcompensator and a low-pass filterThe feedback linearizationcompensator is introduced to linearize the nonlinear dynam-ics into a linear part disturbed by the equivalent disturbancewhereas the low-pass filter is employed to estimate theequivalent disturbances Then a state feedback controller ispresented for the nominal model to acquire desired perfor-mance Stability of the overall closed-loop system is analyzedbased on Lyapunov theory At last the influence on DOBparameters optimization caused by structure and parameterof outer-loop controller is analyzed The robust stabilityconstraint condition which ensures the robust stability ofthe whole system is proposed Thus the119867
infinmethod can be
employed to optimize the parameters of the DOBThe main contributions of this paper are summarized as
follows(1) The disturbance rejection paradigm of the observer
based disturbance rejection methodology is pro-posed
(2) With the proposed disturbance rejection paradigm anovel observer whose low-pass filter of its structurecan be selected to be flexible is proposed for nonlin-ear systems
(3) The parameters optimization method is investigatedto make sure the designed control system can guar-antee the robust stability of the closed-loop system
The rest of this paper is organized as follows In Section 2a mechanical system model is established based on whichthe disturbance rejection problem is formulated In Section 3DOB based control methodology is proposed and parame-ters of DOB are optimized to guarantee the robust stabilityIn Section 4 attitude tracking task is carried out to show theeffectiveness of the proposed strategy followed by conclu-sions in Section 5
2 System Model and Problem Statement
21 System Model An Euler-Lagrange equation for themechanical system is described as
where 119902 isin R119899 and isin R119899 denote the generalized coordinatesand velocities and 119906 and 119889 are the control input and externaldisturbance respectively119872(119902) isin R119899times119899 represent the positivedefinite inertial matrix119862(119902 ) isin R119899times1 represents thematrixof Coriolis and centrifugal forces and119866(119902) isin R119899times1 representsthe gravity termThe nonlinear functions119872(sdot)119862(sdot) and119866(sdot)satisfy the following assumption
Assumption 1 The unknown nonlinear functions119872(sdot) 119862(sdot)and119866(sdot) are continuously differentiable and locally Lipschitz
By introducing the definitions
1199091= 119902
1199092=
(2)
(1) can be rewritten as
1= 1199092
2= minus119872
minus1
(1199091) (119862 (119909
1 1199092) 1199092+ 119866 (119909
1))
+ 119872minus1
(1199091) (119906 + 119889)
(3)
According to the parameters perturbation it is impossibleto establish the system model accurately By introducing thenotations
119872(119902) = 1198720(119902) + 119872
Δ(119902)
119862 (119902 ) = 1198620(119902 ) + 119862
Δ(119902 )
119866 (119902) = 1198660(119902) + 119866
Δ(119902)
(4)
where subscript 0 denotes the nominal value of the corre-sponding matrix and subscript Δ denotes the part of pertur-bation then the dynamics can be described as follows
minus 119862Δ(1199091 1199092) 1199092minus 119866Δ(1199091)]
(6)
In practical applications the consumption of the externaldisturbances is finite that is the external disturbance 119889
is bounded Nevertheless internal uncertainty 119891 usuallydepends on system state Assume that the controller 119906 isdefined as 119906 = 120592(119909
1 1199092 ) nonlinear function 120592(sdot) is
continuously differentiableThus from the definition of119891 wecan also obtain that119891 is continuously differentiable From theabove analysis the following assumptions can be obtained
Assumption 2 The external disturbance 1198891015840 = 1198891+ 1198892(119905) is
boundedwhere1198891and1198892(119905) represent the constant and time-
varying component The time-varying component satisfies1198892(119905) le 119889
Assumption 3 The internal uncertainties 119891 satisfy 119891 le
120572(1199091 1199092 119889) where 120572(sdot) is classicalK function
Mathematical Problems in Engineering 3
22 Problem Formulation For the systemmodel described in(5) the key point of the antidisturbance control methodologyis the observer configuration The control accuracy androbustness of the overall system are largely determinedby the performance of observer Here several widely usedobservers are provided for analysis Based on the disturbancerejection paradigm we propose a novel observer structureand parameter optimization strategy for nonlinear systems
221 Extended State Observer (ESO) ESO is themost impor-tant part of the active disturbance rejection control (ADRC)[31] Instead of identifying the plant dynamics off-line ESOcan estimate the combined effect of plant dynamics andexternal disturbance in real time However ESO can be onlyused for the standard chained systems Here an ESO isdesigned as
By substituting (5) into (7) and introducing the LaplaceTransformation we finally get the following equation
2=
1198922
1199042 + 1198921119904 + 1198922
(119891 + 1198891015840
) (8)
where 119904 is the Laplace operator
222 Nonlinear Disturbance Observer (NDOB) The NDOBhas beenwidely used for nonlinear systemswith uncertainties[30] It can estimate the composite disturbances and compen-sate in the feedback controller The NDOB for the dynamicsof (5) is given as
Then by introducing the Laplace Transformation we finallyget
=119871 (119909)
119904 + 119871 (119909)(119891 + 119889
1015840
) (11)
In most applications observer gain 119871(119909) is usuallyselected as a positive constant
223 Disturbance Rejection Paradigm According to theanalysis above we find that the estimation of the observercan be obtained as the real composite disturbance passingthrough a low-pass filter It can be summarized that the
estimation effect of the observers should fulfill the followingdisturbance rejection paradigm
= 119876 (119904)119863 (12)
where 119863 ≜ 119891 + 1198891015840 is the composite disturbance which
contains both external disturbances and equivalent internaldisturbances119876(119904) is a low-pass filter such that can convergeto119863 asymptotically
For most researches on observer based control thestructure of the low-pass filter 119876(119904) is usually fixed by theobserver structureMeanwhile the parameters tuning usuallyrelies on trial and error rarely do researches focus on thepoint of how to optimize the observer parameters accordingto the property of system uncertainties outer-loop controllermeasurement noise and so forth Hence in this paper anovel observer whose low-pass filter can be selected to beflexible is proposed for the nonlinear system Particularly theparameters optimization strategy is explored for nonlinearsystems
3 Controller Design andParameter Optimization
31 Controller Design The objective of controller design isthat the observer is proposed to estimate the internal uncer-tainty 119891 and external disturbance 1198891015840 and thus the estimation is compensated in the closed-loop control system Thenfeedback controller 119906 is designed to stabilize the system tothe equilibrium point (119909
1= 0 119909
2= 0) The control structure
is shown in Figure 1The inner-loop observer is designed firstly By introduc-
ing a feedback linearization
119906 = 119866minus1
(119909) (V minus 119865 (119909)) (13)
the nonlinear system can be compensated as
1= 1199092
2= V + 119863 (119909 119905)
(14)
where119863(119909 119905) = 119891 + 1198891015840 is the composite disturbanceThen the observer is designed as
= minus119876 (119904) V + 119904119876 (119904) 1199092 (15)
where 119876(119904) is a low-pass filter to be optimizedAccording to (14) and (15) it can be obtained that =
minus119876(119904)V + 119876(119904)2= 119876(119904)119863(119904) that is the observer satisfies
the disturbance rejection paradigm in (12) In practicalapplications 119876(119904) and 119904119876(119904) can be realized in state-space
Then the backstepping controller can be designed for thenominal system Introduce the following notations
1198901= 1199091minus 1199091d
1198902= 1199092minus 1205731
(16)
where 1205731is the pseudo controller to be designed 119909
1d is adifferentiable reference input
4 Mathematical Problems in Engineering
DOB
Backsteppingcontroller
x1d x1d x1d
120592
d
minus
minus
u = Gminus1(x)(120592 minus F(x))u
D(x t)
x2 = F(x) + G(x)u
Q(s) sQ(s)
x2x11
s
+
Figure 1 Control structure of the closed-loop system
From the definition of 1198901and 119890
2 derivative of 119890
2is
described as
1198901= 1199092minus 1d = 1198902 + 1205731 minus 1d (17)
The pseudo controller 1205731is hence defined as
1205731= minus11987011198901+ 1d (18)
where1198701is a positive symmetric matrix
Substituting (18) into (17) yields
1198901= minus11987011198901+ 1198902 (19)
Define a Lyapunov function 1198811= (12)119890
T11198901 its derivative is
1= minus119890
T111987011198901+ 11989011198902 Notice that the derivative of 119890
2is
1198902= 2minus 1= V + 119863 (119909 119905) minus
1 (20)
where 1= minus11987011198902+1198702
11198901+1d According to the backstepping
approach and observer output the controller is finallyobtained as
119906 = 119866minus1
(119909) (minus11987021198902minus 1198901+ 1minus minus 119865 (119909)) (21)
where ≜ 119863(119909 119905) minus is disturbance estimating error ofthe observer Assume that the estimating error of observer isthe input of the above system then the unforced system isexponentially stable at the equilibrium point
32 Stability Analysis
Theorem 4 For the given second-order mechanical systemin (5) the external disturbances and equivalent internaluncertainties satisfy Assumptions 2 and 3 By adopting theobserver in (15) and controller in (21) the control error of systemstates and estimation error of observer are locally uniformlyultimately bounded (UUB)
Proof For the outer-loop controller by substituting 1198901and 1198902
into (21) it can be obtained that
V = minus (1 + 11987011198702) 1199091minus (1198701+ 1198702) 1199092
know that119860 is Hurwitz according to its definitionThat is forany given positive definite symmetric matrix 119873 there existsa positive definite symmetric matrix 119875 such that 119875119860 +119860
T119875 =
minus119873 The equilibrium point is
1205850= minus119860minus1
1198611198891+[[
[
1 0
0 1
0 0
]]
]
[
1199091d
1d] (29)
For = 120585 minus 1205850 we have the following state equation
Consequently the control error of system states and estima-tion error of observer are locally UUB
33 Parameters Optimization Theorem 4 provides us withthe parameter range such that the closed-loop system isUUBHowever it is very hard to determine the parameters directlyIn this section a parameter optimization strategy of the low-pass filter guaranteeing the robust stability is proposed
The parameter of the low-pass filter 119876(119904) is influenced bysystem uncertainties parameters of the outer-loop controllerand measurement noise First the observer is transformed as
= minus119876 (119904) V + 119904119876 (119904) 1199092= minus119876 (119904)
By assuming that the system works in a compact set Ω119909 the
uncertainty can be linearized as
119891 (1199091 1199092) = [minus119904119872
Δ(1199091) + 119862Δ(1199091 1199092)
+ 1199092
120597119862Δ(1199091 1199092)
1205971199092
]1199092+ [minus119904119909
2
120597119872Δ(1199091)
1205971199091
+ 1199092
120597119862Δ(1199091 1199092)
1205971199091
+120597119866Δ(1199091)
1205971199091
]1199091
(37)
Since 1199041199091= 1199092 the internal uncertainty satisfies the following
linear form
119891 (1199091 1199092) = minus (119870
3119904 + 1198704+1198704
119904) 1199092 (38)
where1198703= 119872Δ(1199091)
1198704= 1199092
120597119872Δ(1199091)
1205971199091
minus 119862Δ(1199091 1199092) minus 1199092
120597119862Δ(1199091 1199092)
1205971199092
1198705= minus1199092
120597119862Δ(1199091 1199092)
1205971199091
minus120597119866Δ(1199091)
1205971199091
(39)
6 Mathematical Problems in Engineering
d
d
PΔ(s)
Q(s)
Q(s)
Pminus1Δn (s)Q(s)
x2
Δ(s)
+
+
+
minus
minus
minus
1
1 minus Q(s)
2)x1d(1 + K1 + K(1 + K1K2)x1d +
Figure 2 Equivalent system transformation
It is clear that the real plant119875Δ(119904) differs if different 119909
1and
1199092are selected Define the set of equivalent systems as
119875Δ(119904)
isin 119904
(1198703+ 1) 1199042 + (119870
1+ 1198702+ 1198704) 119904 + (1 + 119870
11198702+ 1198705)
(1199091 1199092) isin Ω119909
(40)
At this time the equivalent system can be representedas the form in Figure 2 For the set of equivalent systemsand the nominal plant define the upper bound of the systemuncertainty as
Δ (119895120596) ge119875Δ(119895120596) minus 119875
Δ119899(119895120596)
119875Δ119899(119895120596)
forall120596 (41)
where scalar 120596 denote frequency From small gain theory thesufficient condition of robust stability is
where 1198821(119904) is a stable weighting function that reflects
frequency spectrum of disturbances at low frequenciesWeighting function 119882
2(119904) satisfies 119882
2(119895120596) lt Δ(119895120596) forall120596
It can be noticed that the selection of 1198822(119904) is influenced
by system uncertainties and outer-loop controller taken intoaccount meanwhile the measurement noise should also betaken into account
By defining the transfer function of virtual loop as (119904) =119876(119904)(1minus119876(119904)) = (119904)(119904) the119876 filter design problem turnsto be a standard119867
where (119904) = (119904)(119904) and (119904) and (119904) are the virtualcontrolled objective and controller respectivelyThe standardstate-space solution in 119867
infincontrol can be applied to get the
optimal solution [32] For a given virtual controlled objective(119904) if we can acquire the optimal solution of the virtualcontroller (119904) then the optimal 119876 filter can be obtained as
119876 (119904) = (119904) (119904)
1 + (119904) (119904)
(45)
Remark 5 If the weighting function1198821(119904) contains poles on
the imaginary axis the augmented controlled objective ofequivalent119867
infincontrol problemwill correspondingly contain
uncontrollable zeros on the imaginary axis There is nooptimal solution for this 119867
infincontrol problem Thus the
weighting function1198822(119904) should be transformed as follows
In this section attitude tracking of a quadrotor aircraftis implemented to verify the effectiveness of the proposedcontrol strategyThemodified Rodrigues parameters (MRPs)are applied to represent the attitude [33]The attitude trackingerror model is described as follows
with the MRPs and angular velocity error defined as
= 120590 oplus 120590minus1
d
= 120596 minus 120596d(49)
Mathematical Problems in Engineering 7
Table 1 Parameters of the quadrotor aircraft
Parameter Definition Value Error Unit119862119879
Coefficients of thrust 0012 plusmn0003119862119876
Coefficients of torque 093 times 10minus3
plusmn02 times 10minus3
120588 Density of air 1184 Kgsdotmminus3
119860 Propellerrsquos disc area 00515 plusmn0002 m2
119903 Propellerrsquos radius 0128 plusmn0001 m119897 Rotor displacement from the center 025 plusmn001 m119869120601
Rotational inertia of roll axis 0014 plusmn0002 Kgsdotm2
119869120579
Rotational inertia of pitch axis 0014 plusmn0002 Kgsdotm2
119869120595
Rotational inertia of yaw axis 0024 plusmn0004 Kgsdotm2
120596119879
Basic rotational speed of the rotor 215 plusmn5 rads
where 120590 120596 and 119869 are MRPs angular velocity and the inertiamatrix respectively 119866(120590) is a nonsingular matrix definedin [33] 120590minus1d is known as inverse of 120590d which is extractedas 120590minus1d = minus120590d and = 119877119877
Td is known as the error
of attitude transition matrix The operator oplus denotes theproduction of MRPs The control input is defined as 119906 =
[120596120601120596120579120596120595]T Then the rotational speeds of each propeller
The related parameter descriptions are shown in Table 1 [34]
41 Control System Design and Implementation Assume thatthe nominal inertia is 119869
0and inertia error as Δ119869 = 119869 minus 119869
0
Meanwhile the nominal value of119865 is given as1198650 and its error
is defined as Δ119865 = 119865 minus 1198650 Then we can use the feedback
linearization
119906 = V + 119865minus10119871 ( + 120596d) vec (1198690)
+ 119865minus1
01198690(d minus [times] 120596d)
(52)
to reduce the system dynamics to
119865minus1
01198690
120596 = V + 119889 + 119891 (53)
where the definitions of the operators 119871(sdot) and vec(sdot) satisfy119871( + 120596d)vec(1198690) = ( + 120596d) times 119869
0( + 120596d) and
operator vec(sdot) is a vector that contains all the componentsof the symmetric square matrix The external disturbance
119889 satisfies 119889 le 119889 The internal uncertainty is definedas
119891
= minus [120575 120596 + 119871 ( + 120596d) 120575lowast
+ 120575 (d minus [times] 120596d)] (54)
where 120575 ≜ (1198651198650)minus1
(1198650Δ119869 minus Δ119865119869
0)
By substituting (53) into (54) we have
119891 = (1198683+ 120575119869minus1
01198650)minus1
[minus120575119869minus1
01198650(V + 119889)
minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(55)
Since 120596d and d are all bounded and control input Vcan be rewritten into the form of state feedback the internaluncertainty 119891 satisfies Assumption 3
According to linearized model (53) the observer can bedesigned as
= minus119876 (119904) V + 119904119876 (119904) 119865minus101198690 (56)
and the backstepping controller is designed as
119906 = minus (1 + 11989611198962) minus (119896
2+ 119865minus1
011986901198961119866 ()) minus
+ 119865minus1
0119871 ( + 120596d) 119869
lowast
0
+ 119865minus1
01198690(d minus [times] 120596d)
(57)
For the variable Ω = + 1198961 and the Lyapunov function
defined as 119881rot = 2 ln(1 + T) + (12)Ω
T(119865minus1
01198690)Ω we have
2le minus120582min (1198961)
2
minus 120582min (1198962)10038171003817100381710038171003817Ω10038171003817100381710038171003817
For the controller in (57) the parameters are selected as1198961= 15 and 119896
2= 90 the system dynamics and expression of
uncertainty are given as follows
[119865minus1
01198690119904 + (119896
2+ 119865minus1
011986901198961119866 ()) + (1 + 119896
11198962) 119866 ()
1
119904]
sdot = 119889 + 119891 minus
119891 = minus [120575 120596 minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(59)
8 Mathematical Problems in Engineering
Δminus1120601 (s)
Δminus1120595 (s) Q(s)
10minus1 100 101 102 10310minus2
Frequency (rads)
minus40
minus20
0
20
40
60
80
Am
plitu
de (d
B)
minus12 (s)W
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
the observerThe influence caused by outer-loop controller isnever explored in existing researches
From the descriptions above a robust DOB based dis-turbance rejection controller is proposed and parametersoptimization strategy is investigated Nonlinear DOB andextended state observer (ESO) are first analyzed to show theessence of the disturbance estimation problem Then underrelaxed restrictions of disturbance and system perturbationa novel disturbance observer is proposed for nonlinearsystem The observer consists of a feedback linearizationcompensator and a low-pass filterThe feedback linearizationcompensator is introduced to linearize the nonlinear dynam-ics into a linear part disturbed by the equivalent disturbancewhereas the low-pass filter is employed to estimate theequivalent disturbances Then a state feedback controller ispresented for the nominal model to acquire desired perfor-mance Stability of the overall closed-loop system is analyzedbased on Lyapunov theory At last the influence on DOBparameters optimization caused by structure and parameterof outer-loop controller is analyzed The robust stabilityconstraint condition which ensures the robust stability ofthe whole system is proposed Thus the119867
infinmethod can be
employed to optimize the parameters of the DOBThe main contributions of this paper are summarized as
follows(1) The disturbance rejection paradigm of the observer
based disturbance rejection methodology is pro-posed
(2) With the proposed disturbance rejection paradigm anovel observer whose low-pass filter of its structurecan be selected to be flexible is proposed for nonlin-ear systems
(3) The parameters optimization method is investigatedto make sure the designed control system can guar-antee the robust stability of the closed-loop system
The rest of this paper is organized as follows In Section 2a mechanical system model is established based on whichthe disturbance rejection problem is formulated In Section 3DOB based control methodology is proposed and parame-ters of DOB are optimized to guarantee the robust stabilityIn Section 4 attitude tracking task is carried out to show theeffectiveness of the proposed strategy followed by conclu-sions in Section 5
2 System Model and Problem Statement
21 System Model An Euler-Lagrange equation for themechanical system is described as
where 119902 isin R119899 and isin R119899 denote the generalized coordinatesand velocities and 119906 and 119889 are the control input and externaldisturbance respectively119872(119902) isin R119899times119899 represent the positivedefinite inertial matrix119862(119902 ) isin R119899times1 represents thematrixof Coriolis and centrifugal forces and119866(119902) isin R119899times1 representsthe gravity termThe nonlinear functions119872(sdot)119862(sdot) and119866(sdot)satisfy the following assumption
Assumption 1 The unknown nonlinear functions119872(sdot) 119862(sdot)and119866(sdot) are continuously differentiable and locally Lipschitz
By introducing the definitions
1199091= 119902
1199092=
(2)
(1) can be rewritten as
1= 1199092
2= minus119872
minus1
(1199091) (119862 (119909
1 1199092) 1199092+ 119866 (119909
1))
+ 119872minus1
(1199091) (119906 + 119889)
(3)
According to the parameters perturbation it is impossibleto establish the system model accurately By introducing thenotations
119872(119902) = 1198720(119902) + 119872
Δ(119902)
119862 (119902 ) = 1198620(119902 ) + 119862
Δ(119902 )
119866 (119902) = 1198660(119902) + 119866
Δ(119902)
(4)
where subscript 0 denotes the nominal value of the corre-sponding matrix and subscript Δ denotes the part of pertur-bation then the dynamics can be described as follows
minus 119862Δ(1199091 1199092) 1199092minus 119866Δ(1199091)]
(6)
In practical applications the consumption of the externaldisturbances is finite that is the external disturbance 119889
is bounded Nevertheless internal uncertainty 119891 usuallydepends on system state Assume that the controller 119906 isdefined as 119906 = 120592(119909
1 1199092 ) nonlinear function 120592(sdot) is
continuously differentiableThus from the definition of119891 wecan also obtain that119891 is continuously differentiable From theabove analysis the following assumptions can be obtained
Assumption 2 The external disturbance 1198891015840 = 1198891+ 1198892(119905) is
boundedwhere1198891and1198892(119905) represent the constant and time-
varying component The time-varying component satisfies1198892(119905) le 119889
Assumption 3 The internal uncertainties 119891 satisfy 119891 le
120572(1199091 1199092 119889) where 120572(sdot) is classicalK function
Mathematical Problems in Engineering 3
22 Problem Formulation For the systemmodel described in(5) the key point of the antidisturbance control methodologyis the observer configuration The control accuracy androbustness of the overall system are largely determinedby the performance of observer Here several widely usedobservers are provided for analysis Based on the disturbancerejection paradigm we propose a novel observer structureand parameter optimization strategy for nonlinear systems
221 Extended State Observer (ESO) ESO is themost impor-tant part of the active disturbance rejection control (ADRC)[31] Instead of identifying the plant dynamics off-line ESOcan estimate the combined effect of plant dynamics andexternal disturbance in real time However ESO can be onlyused for the standard chained systems Here an ESO isdesigned as
By substituting (5) into (7) and introducing the LaplaceTransformation we finally get the following equation
2=
1198922
1199042 + 1198921119904 + 1198922
(119891 + 1198891015840
) (8)
where 119904 is the Laplace operator
222 Nonlinear Disturbance Observer (NDOB) The NDOBhas beenwidely used for nonlinear systemswith uncertainties[30] It can estimate the composite disturbances and compen-sate in the feedback controller The NDOB for the dynamicsof (5) is given as
Then by introducing the Laplace Transformation we finallyget
=119871 (119909)
119904 + 119871 (119909)(119891 + 119889
1015840
) (11)
In most applications observer gain 119871(119909) is usuallyselected as a positive constant
223 Disturbance Rejection Paradigm According to theanalysis above we find that the estimation of the observercan be obtained as the real composite disturbance passingthrough a low-pass filter It can be summarized that the
estimation effect of the observers should fulfill the followingdisturbance rejection paradigm
= 119876 (119904)119863 (12)
where 119863 ≜ 119891 + 1198891015840 is the composite disturbance which
contains both external disturbances and equivalent internaldisturbances119876(119904) is a low-pass filter such that can convergeto119863 asymptotically
For most researches on observer based control thestructure of the low-pass filter 119876(119904) is usually fixed by theobserver structureMeanwhile the parameters tuning usuallyrelies on trial and error rarely do researches focus on thepoint of how to optimize the observer parameters accordingto the property of system uncertainties outer-loop controllermeasurement noise and so forth Hence in this paper anovel observer whose low-pass filter can be selected to beflexible is proposed for the nonlinear system Particularly theparameters optimization strategy is explored for nonlinearsystems
3 Controller Design andParameter Optimization
31 Controller Design The objective of controller design isthat the observer is proposed to estimate the internal uncer-tainty 119891 and external disturbance 1198891015840 and thus the estimation is compensated in the closed-loop control system Thenfeedback controller 119906 is designed to stabilize the system tothe equilibrium point (119909
1= 0 119909
2= 0) The control structure
is shown in Figure 1The inner-loop observer is designed firstly By introduc-
ing a feedback linearization
119906 = 119866minus1
(119909) (V minus 119865 (119909)) (13)
the nonlinear system can be compensated as
1= 1199092
2= V + 119863 (119909 119905)
(14)
where119863(119909 119905) = 119891 + 1198891015840 is the composite disturbanceThen the observer is designed as
= minus119876 (119904) V + 119904119876 (119904) 1199092 (15)
where 119876(119904) is a low-pass filter to be optimizedAccording to (14) and (15) it can be obtained that =
minus119876(119904)V + 119876(119904)2= 119876(119904)119863(119904) that is the observer satisfies
the disturbance rejection paradigm in (12) In practicalapplications 119876(119904) and 119904119876(119904) can be realized in state-space
Then the backstepping controller can be designed for thenominal system Introduce the following notations
1198901= 1199091minus 1199091d
1198902= 1199092minus 1205731
(16)
where 1205731is the pseudo controller to be designed 119909
1d is adifferentiable reference input
4 Mathematical Problems in Engineering
DOB
Backsteppingcontroller
x1d x1d x1d
120592
d
minus
minus
u = Gminus1(x)(120592 minus F(x))u
D(x t)
x2 = F(x) + G(x)u
Q(s) sQ(s)
x2x11
s
+
Figure 1 Control structure of the closed-loop system
From the definition of 1198901and 119890
2 derivative of 119890
2is
described as
1198901= 1199092minus 1d = 1198902 + 1205731 minus 1d (17)
The pseudo controller 1205731is hence defined as
1205731= minus11987011198901+ 1d (18)
where1198701is a positive symmetric matrix
Substituting (18) into (17) yields
1198901= minus11987011198901+ 1198902 (19)
Define a Lyapunov function 1198811= (12)119890
T11198901 its derivative is
1= minus119890
T111987011198901+ 11989011198902 Notice that the derivative of 119890
2is
1198902= 2minus 1= V + 119863 (119909 119905) minus
1 (20)
where 1= minus11987011198902+1198702
11198901+1d According to the backstepping
approach and observer output the controller is finallyobtained as
119906 = 119866minus1
(119909) (minus11987021198902minus 1198901+ 1minus minus 119865 (119909)) (21)
where ≜ 119863(119909 119905) minus is disturbance estimating error ofthe observer Assume that the estimating error of observer isthe input of the above system then the unforced system isexponentially stable at the equilibrium point
32 Stability Analysis
Theorem 4 For the given second-order mechanical systemin (5) the external disturbances and equivalent internaluncertainties satisfy Assumptions 2 and 3 By adopting theobserver in (15) and controller in (21) the control error of systemstates and estimation error of observer are locally uniformlyultimately bounded (UUB)
Proof For the outer-loop controller by substituting 1198901and 1198902
into (21) it can be obtained that
V = minus (1 + 11987011198702) 1199091minus (1198701+ 1198702) 1199092
know that119860 is Hurwitz according to its definitionThat is forany given positive definite symmetric matrix 119873 there existsa positive definite symmetric matrix 119875 such that 119875119860 +119860
T119875 =
minus119873 The equilibrium point is
1205850= minus119860minus1
1198611198891+[[
[
1 0
0 1
0 0
]]
]
[
1199091d
1d] (29)
For = 120585 minus 1205850 we have the following state equation
Consequently the control error of system states and estima-tion error of observer are locally UUB
33 Parameters Optimization Theorem 4 provides us withthe parameter range such that the closed-loop system isUUBHowever it is very hard to determine the parameters directlyIn this section a parameter optimization strategy of the low-pass filter guaranteeing the robust stability is proposed
The parameter of the low-pass filter 119876(119904) is influenced bysystem uncertainties parameters of the outer-loop controllerand measurement noise First the observer is transformed as
= minus119876 (119904) V + 119904119876 (119904) 1199092= minus119876 (119904)
By assuming that the system works in a compact set Ω119909 the
uncertainty can be linearized as
119891 (1199091 1199092) = [minus119904119872
Δ(1199091) + 119862Δ(1199091 1199092)
+ 1199092
120597119862Δ(1199091 1199092)
1205971199092
]1199092+ [minus119904119909
2
120597119872Δ(1199091)
1205971199091
+ 1199092
120597119862Δ(1199091 1199092)
1205971199091
+120597119866Δ(1199091)
1205971199091
]1199091
(37)
Since 1199041199091= 1199092 the internal uncertainty satisfies the following
linear form
119891 (1199091 1199092) = minus (119870
3119904 + 1198704+1198704
119904) 1199092 (38)
where1198703= 119872Δ(1199091)
1198704= 1199092
120597119872Δ(1199091)
1205971199091
minus 119862Δ(1199091 1199092) minus 1199092
120597119862Δ(1199091 1199092)
1205971199092
1198705= minus1199092
120597119862Δ(1199091 1199092)
1205971199091
minus120597119866Δ(1199091)
1205971199091
(39)
6 Mathematical Problems in Engineering
d
d
PΔ(s)
Q(s)
Q(s)
Pminus1Δn (s)Q(s)
x2
Δ(s)
+
+
+
minus
minus
minus
1
1 minus Q(s)
2)x1d(1 + K1 + K(1 + K1K2)x1d +
Figure 2 Equivalent system transformation
It is clear that the real plant119875Δ(119904) differs if different 119909
1and
1199092are selected Define the set of equivalent systems as
119875Δ(119904)
isin 119904
(1198703+ 1) 1199042 + (119870
1+ 1198702+ 1198704) 119904 + (1 + 119870
11198702+ 1198705)
(1199091 1199092) isin Ω119909
(40)
At this time the equivalent system can be representedas the form in Figure 2 For the set of equivalent systemsand the nominal plant define the upper bound of the systemuncertainty as
Δ (119895120596) ge119875Δ(119895120596) minus 119875
Δ119899(119895120596)
119875Δ119899(119895120596)
forall120596 (41)
where scalar 120596 denote frequency From small gain theory thesufficient condition of robust stability is
where 1198821(119904) is a stable weighting function that reflects
frequency spectrum of disturbances at low frequenciesWeighting function 119882
2(119904) satisfies 119882
2(119895120596) lt Δ(119895120596) forall120596
It can be noticed that the selection of 1198822(119904) is influenced
by system uncertainties and outer-loop controller taken intoaccount meanwhile the measurement noise should also betaken into account
By defining the transfer function of virtual loop as (119904) =119876(119904)(1minus119876(119904)) = (119904)(119904) the119876 filter design problem turnsto be a standard119867
where (119904) = (119904)(119904) and (119904) and (119904) are the virtualcontrolled objective and controller respectivelyThe standardstate-space solution in 119867
infincontrol can be applied to get the
optimal solution [32] For a given virtual controlled objective(119904) if we can acquire the optimal solution of the virtualcontroller (119904) then the optimal 119876 filter can be obtained as
119876 (119904) = (119904) (119904)
1 + (119904) (119904)
(45)
Remark 5 If the weighting function1198821(119904) contains poles on
the imaginary axis the augmented controlled objective ofequivalent119867
infincontrol problemwill correspondingly contain
uncontrollable zeros on the imaginary axis There is nooptimal solution for this 119867
infincontrol problem Thus the
weighting function1198822(119904) should be transformed as follows
In this section attitude tracking of a quadrotor aircraftis implemented to verify the effectiveness of the proposedcontrol strategyThemodified Rodrigues parameters (MRPs)are applied to represent the attitude [33]The attitude trackingerror model is described as follows
with the MRPs and angular velocity error defined as
= 120590 oplus 120590minus1
d
= 120596 minus 120596d(49)
Mathematical Problems in Engineering 7
Table 1 Parameters of the quadrotor aircraft
Parameter Definition Value Error Unit119862119879
Coefficients of thrust 0012 plusmn0003119862119876
Coefficients of torque 093 times 10minus3
plusmn02 times 10minus3
120588 Density of air 1184 Kgsdotmminus3
119860 Propellerrsquos disc area 00515 plusmn0002 m2
119903 Propellerrsquos radius 0128 plusmn0001 m119897 Rotor displacement from the center 025 plusmn001 m119869120601
Rotational inertia of roll axis 0014 plusmn0002 Kgsdotm2
119869120579
Rotational inertia of pitch axis 0014 plusmn0002 Kgsdotm2
119869120595
Rotational inertia of yaw axis 0024 plusmn0004 Kgsdotm2
120596119879
Basic rotational speed of the rotor 215 plusmn5 rads
where 120590 120596 and 119869 are MRPs angular velocity and the inertiamatrix respectively 119866(120590) is a nonsingular matrix definedin [33] 120590minus1d is known as inverse of 120590d which is extractedas 120590minus1d = minus120590d and = 119877119877
Td is known as the error
of attitude transition matrix The operator oplus denotes theproduction of MRPs The control input is defined as 119906 =
[120596120601120596120579120596120595]T Then the rotational speeds of each propeller
The related parameter descriptions are shown in Table 1 [34]
41 Control System Design and Implementation Assume thatthe nominal inertia is 119869
0and inertia error as Δ119869 = 119869 minus 119869
0
Meanwhile the nominal value of119865 is given as1198650 and its error
is defined as Δ119865 = 119865 minus 1198650 Then we can use the feedback
linearization
119906 = V + 119865minus10119871 ( + 120596d) vec (1198690)
+ 119865minus1
01198690(d minus [times] 120596d)
(52)
to reduce the system dynamics to
119865minus1
01198690
120596 = V + 119889 + 119891 (53)
where the definitions of the operators 119871(sdot) and vec(sdot) satisfy119871( + 120596d)vec(1198690) = ( + 120596d) times 119869
0( + 120596d) and
operator vec(sdot) is a vector that contains all the componentsof the symmetric square matrix The external disturbance
119889 satisfies 119889 le 119889 The internal uncertainty is definedas
119891
= minus [120575 120596 + 119871 ( + 120596d) 120575lowast
+ 120575 (d minus [times] 120596d)] (54)
where 120575 ≜ (1198651198650)minus1
(1198650Δ119869 minus Δ119865119869
0)
By substituting (53) into (54) we have
119891 = (1198683+ 120575119869minus1
01198650)minus1
[minus120575119869minus1
01198650(V + 119889)
minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(55)
Since 120596d and d are all bounded and control input Vcan be rewritten into the form of state feedback the internaluncertainty 119891 satisfies Assumption 3
According to linearized model (53) the observer can bedesigned as
= minus119876 (119904) V + 119904119876 (119904) 119865minus101198690 (56)
and the backstepping controller is designed as
119906 = minus (1 + 11989611198962) minus (119896
2+ 119865minus1
011986901198961119866 ()) minus
+ 119865minus1
0119871 ( + 120596d) 119869
lowast
0
+ 119865minus1
01198690(d minus [times] 120596d)
(57)
For the variable Ω = + 1198961 and the Lyapunov function
defined as 119881rot = 2 ln(1 + T) + (12)Ω
T(119865minus1
01198690)Ω we have
2le minus120582min (1198961)
2
minus 120582min (1198962)10038171003817100381710038171003817Ω10038171003817100381710038171003817
For the controller in (57) the parameters are selected as1198961= 15 and 119896
2= 90 the system dynamics and expression of
uncertainty are given as follows
[119865minus1
01198690119904 + (119896
2+ 119865minus1
011986901198961119866 ()) + (1 + 119896
11198962) 119866 ()
1
119904]
sdot = 119889 + 119891 minus
119891 = minus [120575 120596 minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(59)
8 Mathematical Problems in Engineering
Δminus1120601 (s)
Δminus1120595 (s) Q(s)
10minus1 100 101 102 10310minus2
Frequency (rads)
minus40
minus20
0
20
40
60
80
Am
plitu
de (d
B)
minus12 (s)W
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
22 Problem Formulation For the systemmodel described in(5) the key point of the antidisturbance control methodologyis the observer configuration The control accuracy androbustness of the overall system are largely determinedby the performance of observer Here several widely usedobservers are provided for analysis Based on the disturbancerejection paradigm we propose a novel observer structureand parameter optimization strategy for nonlinear systems
221 Extended State Observer (ESO) ESO is themost impor-tant part of the active disturbance rejection control (ADRC)[31] Instead of identifying the plant dynamics off-line ESOcan estimate the combined effect of plant dynamics andexternal disturbance in real time However ESO can be onlyused for the standard chained systems Here an ESO isdesigned as
By substituting (5) into (7) and introducing the LaplaceTransformation we finally get the following equation
2=
1198922
1199042 + 1198921119904 + 1198922
(119891 + 1198891015840
) (8)
where 119904 is the Laplace operator
222 Nonlinear Disturbance Observer (NDOB) The NDOBhas beenwidely used for nonlinear systemswith uncertainties[30] It can estimate the composite disturbances and compen-sate in the feedback controller The NDOB for the dynamicsof (5) is given as
Then by introducing the Laplace Transformation we finallyget
=119871 (119909)
119904 + 119871 (119909)(119891 + 119889
1015840
) (11)
In most applications observer gain 119871(119909) is usuallyselected as a positive constant
223 Disturbance Rejection Paradigm According to theanalysis above we find that the estimation of the observercan be obtained as the real composite disturbance passingthrough a low-pass filter It can be summarized that the
estimation effect of the observers should fulfill the followingdisturbance rejection paradigm
= 119876 (119904)119863 (12)
where 119863 ≜ 119891 + 1198891015840 is the composite disturbance which
contains both external disturbances and equivalent internaldisturbances119876(119904) is a low-pass filter such that can convergeto119863 asymptotically
For most researches on observer based control thestructure of the low-pass filter 119876(119904) is usually fixed by theobserver structureMeanwhile the parameters tuning usuallyrelies on trial and error rarely do researches focus on thepoint of how to optimize the observer parameters accordingto the property of system uncertainties outer-loop controllermeasurement noise and so forth Hence in this paper anovel observer whose low-pass filter can be selected to beflexible is proposed for the nonlinear system Particularly theparameters optimization strategy is explored for nonlinearsystems
3 Controller Design andParameter Optimization
31 Controller Design The objective of controller design isthat the observer is proposed to estimate the internal uncer-tainty 119891 and external disturbance 1198891015840 and thus the estimation is compensated in the closed-loop control system Thenfeedback controller 119906 is designed to stabilize the system tothe equilibrium point (119909
1= 0 119909
2= 0) The control structure
is shown in Figure 1The inner-loop observer is designed firstly By introduc-
ing a feedback linearization
119906 = 119866minus1
(119909) (V minus 119865 (119909)) (13)
the nonlinear system can be compensated as
1= 1199092
2= V + 119863 (119909 119905)
(14)
where119863(119909 119905) = 119891 + 1198891015840 is the composite disturbanceThen the observer is designed as
= minus119876 (119904) V + 119904119876 (119904) 1199092 (15)
where 119876(119904) is a low-pass filter to be optimizedAccording to (14) and (15) it can be obtained that =
minus119876(119904)V + 119876(119904)2= 119876(119904)119863(119904) that is the observer satisfies
the disturbance rejection paradigm in (12) In practicalapplications 119876(119904) and 119904119876(119904) can be realized in state-space
Then the backstepping controller can be designed for thenominal system Introduce the following notations
1198901= 1199091minus 1199091d
1198902= 1199092minus 1205731
(16)
where 1205731is the pseudo controller to be designed 119909
1d is adifferentiable reference input
4 Mathematical Problems in Engineering
DOB
Backsteppingcontroller
x1d x1d x1d
120592
d
minus
minus
u = Gminus1(x)(120592 minus F(x))u
D(x t)
x2 = F(x) + G(x)u
Q(s) sQ(s)
x2x11
s
+
Figure 1 Control structure of the closed-loop system
From the definition of 1198901and 119890
2 derivative of 119890
2is
described as
1198901= 1199092minus 1d = 1198902 + 1205731 minus 1d (17)
The pseudo controller 1205731is hence defined as
1205731= minus11987011198901+ 1d (18)
where1198701is a positive symmetric matrix
Substituting (18) into (17) yields
1198901= minus11987011198901+ 1198902 (19)
Define a Lyapunov function 1198811= (12)119890
T11198901 its derivative is
1= minus119890
T111987011198901+ 11989011198902 Notice that the derivative of 119890
2is
1198902= 2minus 1= V + 119863 (119909 119905) minus
1 (20)
where 1= minus11987011198902+1198702
11198901+1d According to the backstepping
approach and observer output the controller is finallyobtained as
119906 = 119866minus1
(119909) (minus11987021198902minus 1198901+ 1minus minus 119865 (119909)) (21)
where ≜ 119863(119909 119905) minus is disturbance estimating error ofthe observer Assume that the estimating error of observer isthe input of the above system then the unforced system isexponentially stable at the equilibrium point
32 Stability Analysis
Theorem 4 For the given second-order mechanical systemin (5) the external disturbances and equivalent internaluncertainties satisfy Assumptions 2 and 3 By adopting theobserver in (15) and controller in (21) the control error of systemstates and estimation error of observer are locally uniformlyultimately bounded (UUB)
Proof For the outer-loop controller by substituting 1198901and 1198902
into (21) it can be obtained that
V = minus (1 + 11987011198702) 1199091minus (1198701+ 1198702) 1199092
know that119860 is Hurwitz according to its definitionThat is forany given positive definite symmetric matrix 119873 there existsa positive definite symmetric matrix 119875 such that 119875119860 +119860
T119875 =
minus119873 The equilibrium point is
1205850= minus119860minus1
1198611198891+[[
[
1 0
0 1
0 0
]]
]
[
1199091d
1d] (29)
For = 120585 minus 1205850 we have the following state equation
Consequently the control error of system states and estima-tion error of observer are locally UUB
33 Parameters Optimization Theorem 4 provides us withthe parameter range such that the closed-loop system isUUBHowever it is very hard to determine the parameters directlyIn this section a parameter optimization strategy of the low-pass filter guaranteeing the robust stability is proposed
The parameter of the low-pass filter 119876(119904) is influenced bysystem uncertainties parameters of the outer-loop controllerand measurement noise First the observer is transformed as
= minus119876 (119904) V + 119904119876 (119904) 1199092= minus119876 (119904)
By assuming that the system works in a compact set Ω119909 the
uncertainty can be linearized as
119891 (1199091 1199092) = [minus119904119872
Δ(1199091) + 119862Δ(1199091 1199092)
+ 1199092
120597119862Δ(1199091 1199092)
1205971199092
]1199092+ [minus119904119909
2
120597119872Δ(1199091)
1205971199091
+ 1199092
120597119862Δ(1199091 1199092)
1205971199091
+120597119866Δ(1199091)
1205971199091
]1199091
(37)
Since 1199041199091= 1199092 the internal uncertainty satisfies the following
linear form
119891 (1199091 1199092) = minus (119870
3119904 + 1198704+1198704
119904) 1199092 (38)
where1198703= 119872Δ(1199091)
1198704= 1199092
120597119872Δ(1199091)
1205971199091
minus 119862Δ(1199091 1199092) minus 1199092
120597119862Δ(1199091 1199092)
1205971199092
1198705= minus1199092
120597119862Δ(1199091 1199092)
1205971199091
minus120597119866Δ(1199091)
1205971199091
(39)
6 Mathematical Problems in Engineering
d
d
PΔ(s)
Q(s)
Q(s)
Pminus1Δn (s)Q(s)
x2
Δ(s)
+
+
+
minus
minus
minus
1
1 minus Q(s)
2)x1d(1 + K1 + K(1 + K1K2)x1d +
Figure 2 Equivalent system transformation
It is clear that the real plant119875Δ(119904) differs if different 119909
1and
1199092are selected Define the set of equivalent systems as
119875Δ(119904)
isin 119904
(1198703+ 1) 1199042 + (119870
1+ 1198702+ 1198704) 119904 + (1 + 119870
11198702+ 1198705)
(1199091 1199092) isin Ω119909
(40)
At this time the equivalent system can be representedas the form in Figure 2 For the set of equivalent systemsand the nominal plant define the upper bound of the systemuncertainty as
Δ (119895120596) ge119875Δ(119895120596) minus 119875
Δ119899(119895120596)
119875Δ119899(119895120596)
forall120596 (41)
where scalar 120596 denote frequency From small gain theory thesufficient condition of robust stability is
where 1198821(119904) is a stable weighting function that reflects
frequency spectrum of disturbances at low frequenciesWeighting function 119882
2(119904) satisfies 119882
2(119895120596) lt Δ(119895120596) forall120596
It can be noticed that the selection of 1198822(119904) is influenced
by system uncertainties and outer-loop controller taken intoaccount meanwhile the measurement noise should also betaken into account
By defining the transfer function of virtual loop as (119904) =119876(119904)(1minus119876(119904)) = (119904)(119904) the119876 filter design problem turnsto be a standard119867
where (119904) = (119904)(119904) and (119904) and (119904) are the virtualcontrolled objective and controller respectivelyThe standardstate-space solution in 119867
infincontrol can be applied to get the
optimal solution [32] For a given virtual controlled objective(119904) if we can acquire the optimal solution of the virtualcontroller (119904) then the optimal 119876 filter can be obtained as
119876 (119904) = (119904) (119904)
1 + (119904) (119904)
(45)
Remark 5 If the weighting function1198821(119904) contains poles on
the imaginary axis the augmented controlled objective ofequivalent119867
infincontrol problemwill correspondingly contain
uncontrollable zeros on the imaginary axis There is nooptimal solution for this 119867
infincontrol problem Thus the
weighting function1198822(119904) should be transformed as follows
In this section attitude tracking of a quadrotor aircraftis implemented to verify the effectiveness of the proposedcontrol strategyThemodified Rodrigues parameters (MRPs)are applied to represent the attitude [33]The attitude trackingerror model is described as follows
with the MRPs and angular velocity error defined as
= 120590 oplus 120590minus1
d
= 120596 minus 120596d(49)
Mathematical Problems in Engineering 7
Table 1 Parameters of the quadrotor aircraft
Parameter Definition Value Error Unit119862119879
Coefficients of thrust 0012 plusmn0003119862119876
Coefficients of torque 093 times 10minus3
plusmn02 times 10minus3
120588 Density of air 1184 Kgsdotmminus3
119860 Propellerrsquos disc area 00515 plusmn0002 m2
119903 Propellerrsquos radius 0128 plusmn0001 m119897 Rotor displacement from the center 025 plusmn001 m119869120601
Rotational inertia of roll axis 0014 plusmn0002 Kgsdotm2
119869120579
Rotational inertia of pitch axis 0014 plusmn0002 Kgsdotm2
119869120595
Rotational inertia of yaw axis 0024 plusmn0004 Kgsdotm2
120596119879
Basic rotational speed of the rotor 215 plusmn5 rads
where 120590 120596 and 119869 are MRPs angular velocity and the inertiamatrix respectively 119866(120590) is a nonsingular matrix definedin [33] 120590minus1d is known as inverse of 120590d which is extractedas 120590minus1d = minus120590d and = 119877119877
Td is known as the error
of attitude transition matrix The operator oplus denotes theproduction of MRPs The control input is defined as 119906 =
[120596120601120596120579120596120595]T Then the rotational speeds of each propeller
The related parameter descriptions are shown in Table 1 [34]
41 Control System Design and Implementation Assume thatthe nominal inertia is 119869
0and inertia error as Δ119869 = 119869 minus 119869
0
Meanwhile the nominal value of119865 is given as1198650 and its error
is defined as Δ119865 = 119865 minus 1198650 Then we can use the feedback
linearization
119906 = V + 119865minus10119871 ( + 120596d) vec (1198690)
+ 119865minus1
01198690(d minus [times] 120596d)
(52)
to reduce the system dynamics to
119865minus1
01198690
120596 = V + 119889 + 119891 (53)
where the definitions of the operators 119871(sdot) and vec(sdot) satisfy119871( + 120596d)vec(1198690) = ( + 120596d) times 119869
0( + 120596d) and
operator vec(sdot) is a vector that contains all the componentsof the symmetric square matrix The external disturbance
119889 satisfies 119889 le 119889 The internal uncertainty is definedas
119891
= minus [120575 120596 + 119871 ( + 120596d) 120575lowast
+ 120575 (d minus [times] 120596d)] (54)
where 120575 ≜ (1198651198650)minus1
(1198650Δ119869 minus Δ119865119869
0)
By substituting (53) into (54) we have
119891 = (1198683+ 120575119869minus1
01198650)minus1
[minus120575119869minus1
01198650(V + 119889)
minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(55)
Since 120596d and d are all bounded and control input Vcan be rewritten into the form of state feedback the internaluncertainty 119891 satisfies Assumption 3
According to linearized model (53) the observer can bedesigned as
= minus119876 (119904) V + 119904119876 (119904) 119865minus101198690 (56)
and the backstepping controller is designed as
119906 = minus (1 + 11989611198962) minus (119896
2+ 119865minus1
011986901198961119866 ()) minus
+ 119865minus1
0119871 ( + 120596d) 119869
lowast
0
+ 119865minus1
01198690(d minus [times] 120596d)
(57)
For the variable Ω = + 1198961 and the Lyapunov function
defined as 119881rot = 2 ln(1 + T) + (12)Ω
T(119865minus1
01198690)Ω we have
2le minus120582min (1198961)
2
minus 120582min (1198962)10038171003817100381710038171003817Ω10038171003817100381710038171003817
For the controller in (57) the parameters are selected as1198961= 15 and 119896
2= 90 the system dynamics and expression of
uncertainty are given as follows
[119865minus1
01198690119904 + (119896
2+ 119865minus1
011986901198961119866 ()) + (1 + 119896
11198962) 119866 ()
1
119904]
sdot = 119889 + 119891 minus
119891 = minus [120575 120596 minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(59)
8 Mathematical Problems in Engineering
Δminus1120601 (s)
Δminus1120595 (s) Q(s)
10minus1 100 101 102 10310minus2
Frequency (rads)
minus40
minus20
0
20
40
60
80
Am
plitu
de (d
B)
minus12 (s)W
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
where ≜ 119863(119909 119905) minus is disturbance estimating error ofthe observer Assume that the estimating error of observer isthe input of the above system then the unforced system isexponentially stable at the equilibrium point
32 Stability Analysis
Theorem 4 For the given second-order mechanical systemin (5) the external disturbances and equivalent internaluncertainties satisfy Assumptions 2 and 3 By adopting theobserver in (15) and controller in (21) the control error of systemstates and estimation error of observer are locally uniformlyultimately bounded (UUB)
Proof For the outer-loop controller by substituting 1198901and 1198902
into (21) it can be obtained that
V = minus (1 + 11987011198702) 1199091minus (1198701+ 1198702) 1199092
know that119860 is Hurwitz according to its definitionThat is forany given positive definite symmetric matrix 119873 there existsa positive definite symmetric matrix 119875 such that 119875119860 +119860
T119875 =
minus119873 The equilibrium point is
1205850= minus119860minus1
1198611198891+[[
[
1 0
0 1
0 0
]]
]
[
1199091d
1d] (29)
For = 120585 minus 1205850 we have the following state equation
Consequently the control error of system states and estima-tion error of observer are locally UUB
33 Parameters Optimization Theorem 4 provides us withthe parameter range such that the closed-loop system isUUBHowever it is very hard to determine the parameters directlyIn this section a parameter optimization strategy of the low-pass filter guaranteeing the robust stability is proposed
The parameter of the low-pass filter 119876(119904) is influenced bysystem uncertainties parameters of the outer-loop controllerand measurement noise First the observer is transformed as
= minus119876 (119904) V + 119904119876 (119904) 1199092= minus119876 (119904)
By assuming that the system works in a compact set Ω119909 the
uncertainty can be linearized as
119891 (1199091 1199092) = [minus119904119872
Δ(1199091) + 119862Δ(1199091 1199092)
+ 1199092
120597119862Δ(1199091 1199092)
1205971199092
]1199092+ [minus119904119909
2
120597119872Δ(1199091)
1205971199091
+ 1199092
120597119862Δ(1199091 1199092)
1205971199091
+120597119866Δ(1199091)
1205971199091
]1199091
(37)
Since 1199041199091= 1199092 the internal uncertainty satisfies the following
linear form
119891 (1199091 1199092) = minus (119870
3119904 + 1198704+1198704
119904) 1199092 (38)
where1198703= 119872Δ(1199091)
1198704= 1199092
120597119872Δ(1199091)
1205971199091
minus 119862Δ(1199091 1199092) minus 1199092
120597119862Δ(1199091 1199092)
1205971199092
1198705= minus1199092
120597119862Δ(1199091 1199092)
1205971199091
minus120597119866Δ(1199091)
1205971199091
(39)
6 Mathematical Problems in Engineering
d
d
PΔ(s)
Q(s)
Q(s)
Pminus1Δn (s)Q(s)
x2
Δ(s)
+
+
+
minus
minus
minus
1
1 minus Q(s)
2)x1d(1 + K1 + K(1 + K1K2)x1d +
Figure 2 Equivalent system transformation
It is clear that the real plant119875Δ(119904) differs if different 119909
1and
1199092are selected Define the set of equivalent systems as
119875Δ(119904)
isin 119904
(1198703+ 1) 1199042 + (119870
1+ 1198702+ 1198704) 119904 + (1 + 119870
11198702+ 1198705)
(1199091 1199092) isin Ω119909
(40)
At this time the equivalent system can be representedas the form in Figure 2 For the set of equivalent systemsand the nominal plant define the upper bound of the systemuncertainty as
Δ (119895120596) ge119875Δ(119895120596) minus 119875
Δ119899(119895120596)
119875Δ119899(119895120596)
forall120596 (41)
where scalar 120596 denote frequency From small gain theory thesufficient condition of robust stability is
where 1198821(119904) is a stable weighting function that reflects
frequency spectrum of disturbances at low frequenciesWeighting function 119882
2(119904) satisfies 119882
2(119895120596) lt Δ(119895120596) forall120596
It can be noticed that the selection of 1198822(119904) is influenced
by system uncertainties and outer-loop controller taken intoaccount meanwhile the measurement noise should also betaken into account
By defining the transfer function of virtual loop as (119904) =119876(119904)(1minus119876(119904)) = (119904)(119904) the119876 filter design problem turnsto be a standard119867
where (119904) = (119904)(119904) and (119904) and (119904) are the virtualcontrolled objective and controller respectivelyThe standardstate-space solution in 119867
infincontrol can be applied to get the
optimal solution [32] For a given virtual controlled objective(119904) if we can acquire the optimal solution of the virtualcontroller (119904) then the optimal 119876 filter can be obtained as
119876 (119904) = (119904) (119904)
1 + (119904) (119904)
(45)
Remark 5 If the weighting function1198821(119904) contains poles on
the imaginary axis the augmented controlled objective ofequivalent119867
infincontrol problemwill correspondingly contain
uncontrollable zeros on the imaginary axis There is nooptimal solution for this 119867
infincontrol problem Thus the
weighting function1198822(119904) should be transformed as follows
In this section attitude tracking of a quadrotor aircraftis implemented to verify the effectiveness of the proposedcontrol strategyThemodified Rodrigues parameters (MRPs)are applied to represent the attitude [33]The attitude trackingerror model is described as follows
with the MRPs and angular velocity error defined as
= 120590 oplus 120590minus1
d
= 120596 minus 120596d(49)
Mathematical Problems in Engineering 7
Table 1 Parameters of the quadrotor aircraft
Parameter Definition Value Error Unit119862119879
Coefficients of thrust 0012 plusmn0003119862119876
Coefficients of torque 093 times 10minus3
plusmn02 times 10minus3
120588 Density of air 1184 Kgsdotmminus3
119860 Propellerrsquos disc area 00515 plusmn0002 m2
119903 Propellerrsquos radius 0128 plusmn0001 m119897 Rotor displacement from the center 025 plusmn001 m119869120601
Rotational inertia of roll axis 0014 plusmn0002 Kgsdotm2
119869120579
Rotational inertia of pitch axis 0014 plusmn0002 Kgsdotm2
119869120595
Rotational inertia of yaw axis 0024 plusmn0004 Kgsdotm2
120596119879
Basic rotational speed of the rotor 215 plusmn5 rads
where 120590 120596 and 119869 are MRPs angular velocity and the inertiamatrix respectively 119866(120590) is a nonsingular matrix definedin [33] 120590minus1d is known as inverse of 120590d which is extractedas 120590minus1d = minus120590d and = 119877119877
Td is known as the error
of attitude transition matrix The operator oplus denotes theproduction of MRPs The control input is defined as 119906 =
[120596120601120596120579120596120595]T Then the rotational speeds of each propeller
The related parameter descriptions are shown in Table 1 [34]
41 Control System Design and Implementation Assume thatthe nominal inertia is 119869
0and inertia error as Δ119869 = 119869 minus 119869
0
Meanwhile the nominal value of119865 is given as1198650 and its error
is defined as Δ119865 = 119865 minus 1198650 Then we can use the feedback
linearization
119906 = V + 119865minus10119871 ( + 120596d) vec (1198690)
+ 119865minus1
01198690(d minus [times] 120596d)
(52)
to reduce the system dynamics to
119865minus1
01198690
120596 = V + 119889 + 119891 (53)
where the definitions of the operators 119871(sdot) and vec(sdot) satisfy119871( + 120596d)vec(1198690) = ( + 120596d) times 119869
0( + 120596d) and
operator vec(sdot) is a vector that contains all the componentsof the symmetric square matrix The external disturbance
119889 satisfies 119889 le 119889 The internal uncertainty is definedas
119891
= minus [120575 120596 + 119871 ( + 120596d) 120575lowast
+ 120575 (d minus [times] 120596d)] (54)
where 120575 ≜ (1198651198650)minus1
(1198650Δ119869 minus Δ119865119869
0)
By substituting (53) into (54) we have
119891 = (1198683+ 120575119869minus1
01198650)minus1
[minus120575119869minus1
01198650(V + 119889)
minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(55)
Since 120596d and d are all bounded and control input Vcan be rewritten into the form of state feedback the internaluncertainty 119891 satisfies Assumption 3
According to linearized model (53) the observer can bedesigned as
= minus119876 (119904) V + 119904119876 (119904) 119865minus101198690 (56)
and the backstepping controller is designed as
119906 = minus (1 + 11989611198962) minus (119896
2+ 119865minus1
011986901198961119866 ()) minus
+ 119865minus1
0119871 ( + 120596d) 119869
lowast
0
+ 119865minus1
01198690(d minus [times] 120596d)
(57)
For the variable Ω = + 1198961 and the Lyapunov function
defined as 119881rot = 2 ln(1 + T) + (12)Ω
T(119865minus1
01198690)Ω we have
2le minus120582min (1198961)
2
minus 120582min (1198962)10038171003817100381710038171003817Ω10038171003817100381710038171003817
For the controller in (57) the parameters are selected as1198961= 15 and 119896
2= 90 the system dynamics and expression of
uncertainty are given as follows
[119865minus1
01198690119904 + (119896
2+ 119865minus1
011986901198961119866 ()) + (1 + 119896
11198962) 119866 ()
1
119904]
sdot = 119889 + 119891 minus
119891 = minus [120575 120596 minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(59)
8 Mathematical Problems in Engineering
Δminus1120601 (s)
Δminus1120595 (s) Q(s)
10minus1 100 101 102 10310minus2
Frequency (rads)
minus40
minus20
0
20
40
60
80
Am
plitu
de (d
B)
minus12 (s)W
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
know that119860 is Hurwitz according to its definitionThat is forany given positive definite symmetric matrix 119873 there existsa positive definite symmetric matrix 119875 such that 119875119860 +119860
T119875 =
minus119873 The equilibrium point is
1205850= minus119860minus1
1198611198891+[[
[
1 0
0 1
0 0
]]
]
[
1199091d
1d] (29)
For = 120585 minus 1205850 we have the following state equation
Consequently the control error of system states and estima-tion error of observer are locally UUB
33 Parameters Optimization Theorem 4 provides us withthe parameter range such that the closed-loop system isUUBHowever it is very hard to determine the parameters directlyIn this section a parameter optimization strategy of the low-pass filter guaranteeing the robust stability is proposed
The parameter of the low-pass filter 119876(119904) is influenced bysystem uncertainties parameters of the outer-loop controllerand measurement noise First the observer is transformed as
= minus119876 (119904) V + 119904119876 (119904) 1199092= minus119876 (119904)
By assuming that the system works in a compact set Ω119909 the
uncertainty can be linearized as
119891 (1199091 1199092) = [minus119904119872
Δ(1199091) + 119862Δ(1199091 1199092)
+ 1199092
120597119862Δ(1199091 1199092)
1205971199092
]1199092+ [minus119904119909
2
120597119872Δ(1199091)
1205971199091
+ 1199092
120597119862Δ(1199091 1199092)
1205971199091
+120597119866Δ(1199091)
1205971199091
]1199091
(37)
Since 1199041199091= 1199092 the internal uncertainty satisfies the following
linear form
119891 (1199091 1199092) = minus (119870
3119904 + 1198704+1198704
119904) 1199092 (38)
where1198703= 119872Δ(1199091)
1198704= 1199092
120597119872Δ(1199091)
1205971199091
minus 119862Δ(1199091 1199092) minus 1199092
120597119862Δ(1199091 1199092)
1205971199092
1198705= minus1199092
120597119862Δ(1199091 1199092)
1205971199091
minus120597119866Δ(1199091)
1205971199091
(39)
6 Mathematical Problems in Engineering
d
d
PΔ(s)
Q(s)
Q(s)
Pminus1Δn (s)Q(s)
x2
Δ(s)
+
+
+
minus
minus
minus
1
1 minus Q(s)
2)x1d(1 + K1 + K(1 + K1K2)x1d +
Figure 2 Equivalent system transformation
It is clear that the real plant119875Δ(119904) differs if different 119909
1and
1199092are selected Define the set of equivalent systems as
119875Δ(119904)
isin 119904
(1198703+ 1) 1199042 + (119870
1+ 1198702+ 1198704) 119904 + (1 + 119870
11198702+ 1198705)
(1199091 1199092) isin Ω119909
(40)
At this time the equivalent system can be representedas the form in Figure 2 For the set of equivalent systemsand the nominal plant define the upper bound of the systemuncertainty as
Δ (119895120596) ge119875Δ(119895120596) minus 119875
Δ119899(119895120596)
119875Δ119899(119895120596)
forall120596 (41)
where scalar 120596 denote frequency From small gain theory thesufficient condition of robust stability is
where 1198821(119904) is a stable weighting function that reflects
frequency spectrum of disturbances at low frequenciesWeighting function 119882
2(119904) satisfies 119882
2(119895120596) lt Δ(119895120596) forall120596
It can be noticed that the selection of 1198822(119904) is influenced
by system uncertainties and outer-loop controller taken intoaccount meanwhile the measurement noise should also betaken into account
By defining the transfer function of virtual loop as (119904) =119876(119904)(1minus119876(119904)) = (119904)(119904) the119876 filter design problem turnsto be a standard119867
where (119904) = (119904)(119904) and (119904) and (119904) are the virtualcontrolled objective and controller respectivelyThe standardstate-space solution in 119867
infincontrol can be applied to get the
optimal solution [32] For a given virtual controlled objective(119904) if we can acquire the optimal solution of the virtualcontroller (119904) then the optimal 119876 filter can be obtained as
119876 (119904) = (119904) (119904)
1 + (119904) (119904)
(45)
Remark 5 If the weighting function1198821(119904) contains poles on
the imaginary axis the augmented controlled objective ofequivalent119867
infincontrol problemwill correspondingly contain
uncontrollable zeros on the imaginary axis There is nooptimal solution for this 119867
infincontrol problem Thus the
weighting function1198822(119904) should be transformed as follows
In this section attitude tracking of a quadrotor aircraftis implemented to verify the effectiveness of the proposedcontrol strategyThemodified Rodrigues parameters (MRPs)are applied to represent the attitude [33]The attitude trackingerror model is described as follows
with the MRPs and angular velocity error defined as
= 120590 oplus 120590minus1
d
= 120596 minus 120596d(49)
Mathematical Problems in Engineering 7
Table 1 Parameters of the quadrotor aircraft
Parameter Definition Value Error Unit119862119879
Coefficients of thrust 0012 plusmn0003119862119876
Coefficients of torque 093 times 10minus3
plusmn02 times 10minus3
120588 Density of air 1184 Kgsdotmminus3
119860 Propellerrsquos disc area 00515 plusmn0002 m2
119903 Propellerrsquos radius 0128 plusmn0001 m119897 Rotor displacement from the center 025 plusmn001 m119869120601
Rotational inertia of roll axis 0014 plusmn0002 Kgsdotm2
119869120579
Rotational inertia of pitch axis 0014 plusmn0002 Kgsdotm2
119869120595
Rotational inertia of yaw axis 0024 plusmn0004 Kgsdotm2
120596119879
Basic rotational speed of the rotor 215 plusmn5 rads
where 120590 120596 and 119869 are MRPs angular velocity and the inertiamatrix respectively 119866(120590) is a nonsingular matrix definedin [33] 120590minus1d is known as inverse of 120590d which is extractedas 120590minus1d = minus120590d and = 119877119877
Td is known as the error
of attitude transition matrix The operator oplus denotes theproduction of MRPs The control input is defined as 119906 =
[120596120601120596120579120596120595]T Then the rotational speeds of each propeller
The related parameter descriptions are shown in Table 1 [34]
41 Control System Design and Implementation Assume thatthe nominal inertia is 119869
0and inertia error as Δ119869 = 119869 minus 119869
0
Meanwhile the nominal value of119865 is given as1198650 and its error
is defined as Δ119865 = 119865 minus 1198650 Then we can use the feedback
linearization
119906 = V + 119865minus10119871 ( + 120596d) vec (1198690)
+ 119865minus1
01198690(d minus [times] 120596d)
(52)
to reduce the system dynamics to
119865minus1
01198690
120596 = V + 119889 + 119891 (53)
where the definitions of the operators 119871(sdot) and vec(sdot) satisfy119871( + 120596d)vec(1198690) = ( + 120596d) times 119869
0( + 120596d) and
operator vec(sdot) is a vector that contains all the componentsof the symmetric square matrix The external disturbance
119889 satisfies 119889 le 119889 The internal uncertainty is definedas
119891
= minus [120575 120596 + 119871 ( + 120596d) 120575lowast
+ 120575 (d minus [times] 120596d)] (54)
where 120575 ≜ (1198651198650)minus1
(1198650Δ119869 minus Δ119865119869
0)
By substituting (53) into (54) we have
119891 = (1198683+ 120575119869minus1
01198650)minus1
[minus120575119869minus1
01198650(V + 119889)
minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(55)
Since 120596d and d are all bounded and control input Vcan be rewritten into the form of state feedback the internaluncertainty 119891 satisfies Assumption 3
According to linearized model (53) the observer can bedesigned as
= minus119876 (119904) V + 119904119876 (119904) 119865minus101198690 (56)
and the backstepping controller is designed as
119906 = minus (1 + 11989611198962) minus (119896
2+ 119865minus1
011986901198961119866 ()) minus
+ 119865minus1
0119871 ( + 120596d) 119869
lowast
0
+ 119865minus1
01198690(d minus [times] 120596d)
(57)
For the variable Ω = + 1198961 and the Lyapunov function
defined as 119881rot = 2 ln(1 + T) + (12)Ω
T(119865minus1
01198690)Ω we have
2le minus120582min (1198961)
2
minus 120582min (1198962)10038171003817100381710038171003817Ω10038171003817100381710038171003817
For the controller in (57) the parameters are selected as1198961= 15 and 119896
2= 90 the system dynamics and expression of
uncertainty are given as follows
[119865minus1
01198690119904 + (119896
2+ 119865minus1
011986901198961119866 ()) + (1 + 119896
11198962) 119866 ()
1
119904]
sdot = 119889 + 119891 minus
119891 = minus [120575 120596 minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(59)
8 Mathematical Problems in Engineering
Δminus1120601 (s)
Δminus1120595 (s) Q(s)
10minus1 100 101 102 10310minus2
Frequency (rads)
minus40
minus20
0
20
40
60
80
Am
plitu
de (d
B)
minus12 (s)W
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
It is clear that the real plant119875Δ(119904) differs if different 119909
1and
1199092are selected Define the set of equivalent systems as
119875Δ(119904)
isin 119904
(1198703+ 1) 1199042 + (119870
1+ 1198702+ 1198704) 119904 + (1 + 119870
11198702+ 1198705)
(1199091 1199092) isin Ω119909
(40)
At this time the equivalent system can be representedas the form in Figure 2 For the set of equivalent systemsand the nominal plant define the upper bound of the systemuncertainty as
Δ (119895120596) ge119875Δ(119895120596) minus 119875
Δ119899(119895120596)
119875Δ119899(119895120596)
forall120596 (41)
where scalar 120596 denote frequency From small gain theory thesufficient condition of robust stability is
where 1198821(119904) is a stable weighting function that reflects
frequency spectrum of disturbances at low frequenciesWeighting function 119882
2(119904) satisfies 119882
2(119895120596) lt Δ(119895120596) forall120596
It can be noticed that the selection of 1198822(119904) is influenced
by system uncertainties and outer-loop controller taken intoaccount meanwhile the measurement noise should also betaken into account
By defining the transfer function of virtual loop as (119904) =119876(119904)(1minus119876(119904)) = (119904)(119904) the119876 filter design problem turnsto be a standard119867
where (119904) = (119904)(119904) and (119904) and (119904) are the virtualcontrolled objective and controller respectivelyThe standardstate-space solution in 119867
infincontrol can be applied to get the
optimal solution [32] For a given virtual controlled objective(119904) if we can acquire the optimal solution of the virtualcontroller (119904) then the optimal 119876 filter can be obtained as
119876 (119904) = (119904) (119904)
1 + (119904) (119904)
(45)
Remark 5 If the weighting function1198821(119904) contains poles on
the imaginary axis the augmented controlled objective ofequivalent119867
infincontrol problemwill correspondingly contain
uncontrollable zeros on the imaginary axis There is nooptimal solution for this 119867
infincontrol problem Thus the
weighting function1198822(119904) should be transformed as follows
In this section attitude tracking of a quadrotor aircraftis implemented to verify the effectiveness of the proposedcontrol strategyThemodified Rodrigues parameters (MRPs)are applied to represent the attitude [33]The attitude trackingerror model is described as follows
with the MRPs and angular velocity error defined as
= 120590 oplus 120590minus1
d
= 120596 minus 120596d(49)
Mathematical Problems in Engineering 7
Table 1 Parameters of the quadrotor aircraft
Parameter Definition Value Error Unit119862119879
Coefficients of thrust 0012 plusmn0003119862119876
Coefficients of torque 093 times 10minus3
plusmn02 times 10minus3
120588 Density of air 1184 Kgsdotmminus3
119860 Propellerrsquos disc area 00515 plusmn0002 m2
119903 Propellerrsquos radius 0128 plusmn0001 m119897 Rotor displacement from the center 025 plusmn001 m119869120601
Rotational inertia of roll axis 0014 plusmn0002 Kgsdotm2
119869120579
Rotational inertia of pitch axis 0014 plusmn0002 Kgsdotm2
119869120595
Rotational inertia of yaw axis 0024 plusmn0004 Kgsdotm2
120596119879
Basic rotational speed of the rotor 215 plusmn5 rads
where 120590 120596 and 119869 are MRPs angular velocity and the inertiamatrix respectively 119866(120590) is a nonsingular matrix definedin [33] 120590minus1d is known as inverse of 120590d which is extractedas 120590minus1d = minus120590d and = 119877119877
Td is known as the error
of attitude transition matrix The operator oplus denotes theproduction of MRPs The control input is defined as 119906 =
[120596120601120596120579120596120595]T Then the rotational speeds of each propeller
The related parameter descriptions are shown in Table 1 [34]
41 Control System Design and Implementation Assume thatthe nominal inertia is 119869
0and inertia error as Δ119869 = 119869 minus 119869
0
Meanwhile the nominal value of119865 is given as1198650 and its error
is defined as Δ119865 = 119865 minus 1198650 Then we can use the feedback
linearization
119906 = V + 119865minus10119871 ( + 120596d) vec (1198690)
+ 119865minus1
01198690(d minus [times] 120596d)
(52)
to reduce the system dynamics to
119865minus1
01198690
120596 = V + 119889 + 119891 (53)
where the definitions of the operators 119871(sdot) and vec(sdot) satisfy119871( + 120596d)vec(1198690) = ( + 120596d) times 119869
0( + 120596d) and
operator vec(sdot) is a vector that contains all the componentsof the symmetric square matrix The external disturbance
119889 satisfies 119889 le 119889 The internal uncertainty is definedas
119891
= minus [120575 120596 + 119871 ( + 120596d) 120575lowast
+ 120575 (d minus [times] 120596d)] (54)
where 120575 ≜ (1198651198650)minus1
(1198650Δ119869 minus Δ119865119869
0)
By substituting (53) into (54) we have
119891 = (1198683+ 120575119869minus1
01198650)minus1
[minus120575119869minus1
01198650(V + 119889)
minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(55)
Since 120596d and d are all bounded and control input Vcan be rewritten into the form of state feedback the internaluncertainty 119891 satisfies Assumption 3
According to linearized model (53) the observer can bedesigned as
= minus119876 (119904) V + 119904119876 (119904) 119865minus101198690 (56)
and the backstepping controller is designed as
119906 = minus (1 + 11989611198962) minus (119896
2+ 119865minus1
011986901198961119866 ()) minus
+ 119865minus1
0119871 ( + 120596d) 119869
lowast
0
+ 119865minus1
01198690(d minus [times] 120596d)
(57)
For the variable Ω = + 1198961 and the Lyapunov function
defined as 119881rot = 2 ln(1 + T) + (12)Ω
T(119865minus1
01198690)Ω we have
2le minus120582min (1198961)
2
minus 120582min (1198962)10038171003817100381710038171003817Ω10038171003817100381710038171003817
For the controller in (57) the parameters are selected as1198961= 15 and 119896
2= 90 the system dynamics and expression of
uncertainty are given as follows
[119865minus1
01198690119904 + (119896
2+ 119865minus1
011986901198961119866 ()) + (1 + 119896
11198962) 119866 ()
1
119904]
sdot = 119889 + 119891 minus
119891 = minus [120575 120596 minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(59)
8 Mathematical Problems in Engineering
Δminus1120601 (s)
Δminus1120595 (s) Q(s)
10minus1 100 101 102 10310minus2
Frequency (rads)
minus40
minus20
0
20
40
60
80
Am
plitu
de (d
B)
minus12 (s)W
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
Coefficients of thrust 0012 plusmn0003119862119876
Coefficients of torque 093 times 10minus3
plusmn02 times 10minus3
120588 Density of air 1184 Kgsdotmminus3
119860 Propellerrsquos disc area 00515 plusmn0002 m2
119903 Propellerrsquos radius 0128 plusmn0001 m119897 Rotor displacement from the center 025 plusmn001 m119869120601
Rotational inertia of roll axis 0014 plusmn0002 Kgsdotm2
119869120579
Rotational inertia of pitch axis 0014 plusmn0002 Kgsdotm2
119869120595
Rotational inertia of yaw axis 0024 plusmn0004 Kgsdotm2
120596119879
Basic rotational speed of the rotor 215 plusmn5 rads
where 120590 120596 and 119869 are MRPs angular velocity and the inertiamatrix respectively 119866(120590) is a nonsingular matrix definedin [33] 120590minus1d is known as inverse of 120590d which is extractedas 120590minus1d = minus120590d and = 119877119877
Td is known as the error
of attitude transition matrix The operator oplus denotes theproduction of MRPs The control input is defined as 119906 =
[120596120601120596120579120596120595]T Then the rotational speeds of each propeller
The related parameter descriptions are shown in Table 1 [34]
41 Control System Design and Implementation Assume thatthe nominal inertia is 119869
0and inertia error as Δ119869 = 119869 minus 119869
0
Meanwhile the nominal value of119865 is given as1198650 and its error
is defined as Δ119865 = 119865 minus 1198650 Then we can use the feedback
linearization
119906 = V + 119865minus10119871 ( + 120596d) vec (1198690)
+ 119865minus1
01198690(d minus [times] 120596d)
(52)
to reduce the system dynamics to
119865minus1
01198690
120596 = V + 119889 + 119891 (53)
where the definitions of the operators 119871(sdot) and vec(sdot) satisfy119871( + 120596d)vec(1198690) = ( + 120596d) times 119869
0( + 120596d) and
operator vec(sdot) is a vector that contains all the componentsof the symmetric square matrix The external disturbance
119889 satisfies 119889 le 119889 The internal uncertainty is definedas
119891
= minus [120575 120596 + 119871 ( + 120596d) 120575lowast
+ 120575 (d minus [times] 120596d)] (54)
where 120575 ≜ (1198651198650)minus1
(1198650Δ119869 minus Δ119865119869
0)
By substituting (53) into (54) we have
119891 = (1198683+ 120575119869minus1
01198650)minus1
[minus120575119869minus1
01198650(V + 119889)
minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(55)
Since 120596d and d are all bounded and control input Vcan be rewritten into the form of state feedback the internaluncertainty 119891 satisfies Assumption 3
According to linearized model (53) the observer can bedesigned as
= minus119876 (119904) V + 119904119876 (119904) 119865minus101198690 (56)
and the backstepping controller is designed as
119906 = minus (1 + 11989611198962) minus (119896
2+ 119865minus1
011986901198961119866 ()) minus
+ 119865minus1
0119871 ( + 120596d) 119869
lowast
0
+ 119865minus1
01198690(d minus [times] 120596d)
(57)
For the variable Ω = + 1198961 and the Lyapunov function
defined as 119881rot = 2 ln(1 + T) + (12)Ω
T(119865minus1
01198690)Ω we have
2le minus120582min (1198961)
2
minus 120582min (1198962)10038171003817100381710038171003817Ω10038171003817100381710038171003817
For the controller in (57) the parameters are selected as1198961= 15 and 119896
2= 90 the system dynamics and expression of
uncertainty are given as follows
[119865minus1
01198690119904 + (119896
2+ 119865minus1
011986901198961119866 ()) + (1 + 119896
11198962) 119866 ()
1
119904]
sdot = 119889 + 119891 minus
119891 = minus [120575 120596 minus 119871 ( + 120596d) 120575lowast
minus 120575 (d minus [times] 120596d)]
(59)
8 Mathematical Problems in Engineering
Δminus1120601 (s)
Δminus1120595 (s) Q(s)
10minus1 100 101 102 10310minus2
Frequency (rads)
minus40
minus20
0
20
40
60
80
Am
plitu
de (d
B)
minus12 (s)W
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
Figure 3 System uncertainties and weighting function constraint
From the analysis in Section 33 we get the nominalmodel of equivalent system as
119875Δ119899=
4119904
4119865minus1
011986901199042 + (4119896
2+ 119865minus1
011986901198961) 119904 + (1 + 119896
11198962) (60)
while the equivalent system is shown as
119875Δ(119904)
=4119904
(4119865minus1
01198690+ 120575) 1199042 + (4119896
2+ 119865minus1
011986901198961+ 41198963) 119904 + (1 + 119896
11198962)
(61)
where 1198963= minus(120597119871( + 120596d)120575
lowast
+ 120575(d minus [times]120596d))120597Consider that the structure of quadrotor is axially sym-
metric the corresponding parameters of pitch and rollaxes are the same Thus weighting function 119882
2(119904) can be
determined by pitch (roll) axis and yaw axis The selectionof1198822(119904) should contain the system uncertainties with all the
parameters perturbation It is also required that the designed119876 filter has at least minus30 dB attenuation against measurementnoise of gyroscope larger than 42Hz Figure 3 shows thefrequency response of Δ(119904) according to the parameters per-turbation It is illustrated that for all the possible parametersthe weighting function satisfies 119882minus1
2(119904) le Δ
120601(119904) 119882minus1
2(119904) le
Δ120595(119904) Then the optimized119876 filter is obtained while 120574 = 81
119876 (119904) =71119904 + 11415
1199042 + 71119904 + 11415 (62)
42 Simulations Numerical simulations are presented inMATLABSimulink to illustrate the efficacy of the proposedstrategy The simulation period is 5ms the same as that inexperiments We consider the parameters and their uncer-tainties depicted in Table 1 The desired MRPs are given as
120590d1 = 01 sin(120587119905
15+120587
2)
120590d2 = 01 sin(120587119905
15minus120587
2)
120590d3 = 01 sin(120587119905
15)
(63)
hence from the kinematics of MRPs we get
120596d = 119866minus1
(120590d) d
d = 119866minus1
(120590d) [d minus 119866 (120590d d) 120596d] (64)
where 119866(120590d d) is the time-derivative of 119866(120590d)The external disturbances on the dynamics are as follows
1198891= 01 sin(120587119905
2) + 01 sin(120587119905
10) + 03
1198892= 01 sin(120587119905
2) + 01 cos(120587119905
10) + 04
1198893= 01 sin(120587119905
2) + 01 cos(120587119905
10+120587
4) + 05
(65)
which contains constant and sine components with both lowand high frequencies
The measurement noise is taken into account in thissimulation Here we add the practical noise from the sensorsto the feedback channel The initial condition is 120590(119905
0) =
[01 015 005]T 120596(119905
0) = [0 0 0]
T Note that the controllerparameters are 119896
1= 10 and 119896
2= 05 Meanwhile
a nonlinear feedback controller in (57) without and atraditional DOB with first-order 119876 filter are also carried outin this simulation to compare with the proposed strategyThe bandwidth of traditional DOB is 15 which is selectedto be as large as possible to guarantee both disturbancerejection performance and robustness against measurementnoise
Figure 4 shows the tracking effect It is illustrated that thenonlinear feedback controller without DOB cannot suppressthe influence caused by internal uncertainties and externaldisturbances The compound disturbances acting on systemdynamics will cause an obvious tracking error The approxi-mation of compound disturbances in Figure 5 illustrates thatthe proposed DOB can estimate the compound disturbancessuccessfully with noise of high frequency Hence with thecompensation of the estimating disturbances the proposedcontrol strategy can enable the quadrotor to track the desiredMRPs with better performance in Figure 4 Comparing withthe proposed DOB a traditional DOB is presented and thetracking errors of these two methods are shown in Figure 6With the high frequency measurement noise the bandwidthof traditional DOB cannot be selected to be larger than15 since the high gain will enlarge the influence caused bymeasurement noise and diverge the control system Howeverthe proposed DOB has stronger suppression ability againstdisturbances with low frequency as well as attenuationagainst noise with high frequency Consequently the trackingperformance with the proposed DOB is better than that withtraditional DOBThe control performances of these methodsare given specifically in Table 2 The control structure witha DOB has higher tracking accuracy The disturbances esti-mating error of the proposed robust DOB is less than that oftraditional DOB
Mathematical Problems in Engineering 9
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
0151205901
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205961
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
Tracking effect of 1205901 Tracking effect of 1205961
(a)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205902
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
021205962
Tracking effect of 1205902 Tracking effect of 1205962
(b)
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus01
minus005
0
005
01
015
1205903
DesiredWith DOB
Without DOB
10 20 30 40 50 600Time (s)
minus02
minus015
minus01
minus005
0
005
01
015
02
1205963
Tracking effect of 1205903 Tracking effect of 1205963
(c)
Figure 4 Tracking effects of desired attitude with and without DOB
10 Mathematical Problems in Engineering
d1d1
10 20 30 40 50 600Time (s)
0
01
02
03
04
05
Dist
urba
nce (
Nm
)
(a) Approximation effect of 1198891
d2d2
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
Dist
urba
nce (
Nm
)
(b) Approximation effect of 1198892
d3d3
10 20 30 40 50 600Time (s)
01
02
03
04
05
06
07
08
Dist
urba
nce (
Nm
)
(c) Approximation effect of 1198893
Figure 5 Approximation effect of disturbances
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
Table 2 Comparison of control performances in simulations (RMSerror)
dWithout DOB 004 005 NullTraditional DOB 185 times 10
minus2 012 rads 017NsdotmProposed DOB 124 times 10
minus2 007 rads 013Nsdotm
43 Experimental Results In the experiment the desiredattitude is expressed as follows
120590d1 = minus003 sin(120587
5119905)
120590d2 = 003 cos(120587
5119905)
(66)
and 120590d3 retains 0 120596d and d are acquired by (64)It can be illustrated in (53) and (54) that even if there are
no external disturbances the existing internal uncertaintieswill also bring the system with an equivalent disturbance119891 In Figure 7 we find that with the action of DOB the
Table 3 Comparison of control performances (RMS error)
1205901
1205902
1205903
Traditional DOB 14 times 10minus3
16 times 10minus3
22 times 10minus3
Proposed DOB 79 times 10minus4
71 times 10minus4
29 times 10minus4
tracking error caused by internal uncertainties can be sup-pressed successfully At time of 75 and 90 seconds externaldisturbances are exerted on the quadrotor The proposedDOB can estimate and eliminate the disturbance quickly andaccurately Also from the enlarged view of Figure 7 we findthe convergence speed of the DOB is less than 2 secondsTheattitude error is expressed in Figure 8 and the comparison ofthe attitude tracking performances is shown in Table 3 Theattitude errors are expressed in Figure 8 and the comparisonof the attitude tracking performances is shown in Table 3The control accuracy ismuch higher with the proposedDOBSince there exists property of coupling among the 3 axes of theattitude when we exert external disturbances on the axes ofpitch or roll it will in turn affect the other two axes Howeverthe proposed DOB can also suppress the influence causedby coupling property Experimental results validate that the
Mathematical Problems in Engineering 11
Proposed DOBNDOB
times10minus3
minus1
minus05
0
05
1
1205901
10 20 30 40 50 600Time (s)
(a) Tracking error of 1205901
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205902
(b) Tracking error of 1205902
Proposed DOBNDOB
times10minus3
10 20 30 40 50 600Time (s)
minus1
minus05
0
05
1
1205903
(c) Tracking error of 1205903
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
Figure 6 Tracking error comparison of the proposed DOB andtraditional DOB
proposed control strategy can obtain strong disturbancerejection performance against external disturbances aswell asgood tracking performanceThe robust stability of the closed-loop system can be guaranteed based on the proposed DOBoptimization strategy
120590d11205901
0001002
5 10 15 20 25 300Time (s)
minus004
minus002
0
002
004
006
1205901
26 262 264258
(a) Tracking effect of 1205901
120590d21205902
minus006
minus004
minus002
0
002
004
1205902
5 10 15 20 25 300Time (s)
216 218 22214minus002
0
002
(b) Tracking effect of 1205902
120590d31205903
minus001
minus0005
0
0005
001
0015
1205903
5 10 15 20 25 300Time (s)
(c) Tracking effect of 1205903
Figure 7 Tracking effect of MRPs with DOB
5 Conclusions
This paper proposes a disturbance rejection control strat-egy for nonlinear systems with robust DOB First a DOBwith partial feedback linearization and a low-pass filteris proposed for nonlinear dynamic model under relaxedrestrictions of the generalized disturbance Then the outer-loop backstepping controller is designed for desired tracking
12 Mathematical Problems in Engineering
times10minus3
0 5 10 15 20 25 30Time (s)
minus003
minus002
minus001
0
001
002
003
10 12 14 168minus2
0
2
120590
1
2
3
Figure 8 Tracking error of the proposed method
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
performance By analyzing the influence of outer controlleron the inner-loop observer parameter the robust stabilityconstraint condition is proposed to guarantee the robuststability of the closed-loop system Experimental results onan aircraft show that the proposed strategy can increasethe control accuracy effectively The optimized DOB caneliminate the external disturbances effectively to increasethe control accuracy Meanwhile the proposed parametersoptimization strategy can guarantee the robust stability of theclosed-loop system
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
References
[1] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014
[2] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 no 8 pp 816ndash822 2012
[3] J Yang J Su S Li and X Yu ldquoHigh-order mismatcheddisturbance compensation for motion control systems via acontinuous dynamic sliding-mode approachrdquo IEEE Transac-tions on Industrial Informatics vol 10 no 1 pp 604ndash614 2014
[4] L-B Li L-L Sun S-Z Zhang andQ-Q Yang ldquoSpeed trackingand synchronization of multiple motors using ring couplingcontrol and adaptive sliding mode controlrdquo ISA Transactionsvol 58 pp 635ndash649 2015
[5] K J Astrom and B Wittenmark Adaptive Control CourierCorporation 2013
[6] Z Li S Deng C-Y Su et al ldquoDecentralised adaptive control ofcooperating Robotic manipulators with disturbance observersrdquoIET Control Theory and Applications vol 8 no 7 pp 515ndash5212014
[7] F Motallebzadeh S Ozgoli and H R Momeni ldquoMultileveladaptive control of nonlinear interconnected systemsrdquo ISATransactions vol 54 pp 83ndash91 2015
[8] K Zhou J CDoyle andKGloverRobust andOptimal Controlvol 40 Prentice Hall Upper Saddle River NJ USA 1996
[9] L Lu and B Yao ldquoEnergy-saving adaptive robust control ofa hydraulic manipulator using five cartridge valves with anaccumulatorrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 7046ndash7054 2014
[10] J P Kolhe M Shaheed T S Chandar and S E TaloleldquoRobust control of robot manipulators based on uncertaintyand disturbance estimationrdquo International Journal of Robust andNonlinear Control vol 23 no 1 pp 104ndash122 2013
[11] R-JWai and RMuthusamy ldquoDesign of fuzzy-neural-network-inherited backstepping control for robot manipulator includingactuator dynamicsrdquo IEEETransactions on Fuzzy Systems vol 22no 4 pp 709ndash722 2014
[12] J L Meza V Santibanez R Soto andM A Llama ldquoFuzzy self-tuning PID semiglobal regulator for robot manipulatorsrdquo IEEETransactions on Industrial Electronics vol 59 no 6 pp 2709ndash2717 2012
[13] J Addeh A Ebrahimzadeh M Azarbad and V RanaeeldquoStatistical process control using optimized neural networks acase studyrdquo ISA Transactions vol 53 no 5 pp 1489ndash1499 2014
[14] D Xing J Su Y Liu and J Zhong ldquoRobust approach forhumanoid joint control based on a disturbance observerrdquo IETControlTheoryampApplications vol 5 no 14 pp 1630ndash1636 2011
[15] S Lichiardopol N van de Wouw and H Nijmeijer ldquoRobustdisturbance estimation for human-robotic comanipulationrdquoInternational Journal of Robust and Nonlinear Control vol 24no 12 pp 1772ndash1796 2014
[16] J N Yun and J-B Su ldquoDesign of a disturbance observer for atwo-linkmanipulator with flexible jointsrdquo IEEE Transactions onControl Systems Technology vol 22 no 2 pp 809ndash815 2014
[17] A Mohammadi M Tavakoli H J Marquez and F Hashem-zadeh ldquoNonlinear disturbance observer design for roboticmanipulatorsrdquo Control Engineering Practice vol 21 no 3 pp253ndash267 2013
[18] Z Ma and J Su ldquoRobust uncalibrated visual servoing controlbased on disturbance observerrdquo ISA Transactions vol 59 pp193ndash204 2015
[19] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013
[20] LWang and J Su ldquoDisturbance rejection control of amorphingUAVrdquo in Proceedings of the American Control Conference (ACCrsquo13) pp 4307ndash4312 Washington DC USA June 2013
[21] K Yang Y Choi and W K Chung ldquoOn the tracking per-formance improvement of optical disk drive servo systemsusing error-based disturbance observerrdquo IEEE Transactions onIndustrial Electronics vol 52 no 1 pp 270ndash279 2005
[22] H Kim H Shim and N H Jo ldquoAdaptive add-on output regu-lator for rejection of sinusoidal disturbances and application tooptical disc drivesrdquo IEEE Transactions on Industrial Electronicsvol 61 no 10 pp 5490ndash5499 2014
[23] C Wang X Li L Guo and Y W Li ldquoA nonlinear-disturbance-observer-based DC-Bus voltage control for a hybrid ACDCmicrogridrdquo IEEE Transactions on Power Electronics vol 29 no11 pp 6162ndash6177 2014
[24] L Wang and J Su ldquoDisturbance rejection control for non-minimum phase systems with optimal disturbance observerrdquoISA Transactions vol 57 pp 1ndash9 2015
Mathematical Problems in Engineering 13
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
[25] W Li and Y Hori ldquoVibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbanceobserverrdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 117ndash126 2007
[26] S Li J Li and Y Mo ldquoPiezoelectric multimode vibrationcontrol for stiffened plate using ADRC-based accelerationcompensationrdquo IEEE Transactions on Industrial Electronics vol61 no 12 pp 6892ndash6902 2014
[27] K Ohnishi ldquoA new servo method in mechatronicsrdquo Transac-tions of the Japan Society of Mechanical Engineers vol 107 pp83ndash86 1987
[28] Z-J Yang S Hara S Kanae and K Wada ldquoRobust outputfeedback control of a class of nonlinear systems using adisturbance observerrdquo IEEE Transactions on Control SystemsTechnology vol 19 no 2 pp 256ndash268 2011
[29] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000
[30] W-H Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004
[31] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009
[32] J C Doyle K Glover P P Khargonekar et al ldquoState-spacesolutions to standard h2 and h1 control problemsrdquo IEEE Trans-actions on Automatic Control vol 34 no 8 pp 831ndash847 1989
[33] P Tsiotras ldquoFurther passivity results for the attitude controlproblemrdquo IEEE Transactions on Automatic Control vol 43 no11 pp 1597ndash1600 1998
[34] L Wang and J Su ldquoSwitching control of attitude trackingon a quadrotor UAV for large-angle rotational maneuversrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo14) pp 2907ndash2912 Hong Kong June 2014
![Delay compensation using Smith predictor for wireless ... · is based on adaptive Smith predictors. In [12], communication disturbance observer (CDOB) and network disturbance (ND)](https://static.documents.pub/doc/80x56/601eb1ef7b8fd602336ea565/delay-compensation-using-smith-predictor-for-wireless-is-based-on-adaptive-smith.jpg)
