Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 173051 9 pageshttpdxdoiorg1011552013173051
Research ArticleProportional Derivative Control with Inverse Dead-Zone forPendulum Systems
Joseacute de Jesuacutes Rubio Zizilia Zamudio Jaime Pacheco and Dante Muacutejica Vargas
Seccion de Estudios de Posgrado e Investigacion ESIME Azcapotzalco Instituto Politecnico Nacional Avenida de las Granjas No 682Colonia Santa Catarina 02250 Mexico DF Mexico
Correspondence should be addressed to Jose de Jesus Rubio jrubioaipnmx
Received 11 July 2013 Revised 30 August 2013 Accepted 2 September 2013
Academic Editor Zidong Wang
Copyright copy 2013 Jose de Jesus Rubio et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A proportional derivative controller with inverse dead-zone is proposed for the control of pendulum systemsThe proposedmethodhas the characteristic that the inverse dead-zone is cancelled with the pendulum dead-zone Asymptotic stability of the proposedtechnique is guaranteed by the Lyapunov analysis Simulations of two pendulum systems show the effectiveness of the proposedtechnique
1 Introduction
Nonsmooth nonlinear characteristics such as dead-zonebacklash and hysteresis are common in actuators sensorssuch as mechanical connections hydraulic servovalves andelectric servomotors they also appear in biomedical systemsDead-zone is one of the most important nonsmooth non-linearities in many industrial processes which can severelylimit the system performance and its study has been drawingmuch interest in the control community for a long time[1]
There is some research about control systems In [2] thestabilization of the inverted-car pendulum is presented Thestabilization of the Furuta pendulum is introduced in [3] In[4] the dissipative control problem is investigated for a classof discrete time-varying systemsThedistributed119867
infinfiltering
problem for a class of nonlinear systems is considered in [5]The recursive finite-horizon filtering problem for a class ofnonlinear time-varying systems is addressed in [6 7] In [8]the authors present a solution to the problem of the quadraticmini-max regulator for polynomial uncertain systems Theproblemof a two-player differential game affected bymatcheduncertainties with only the output measurement availablefor each player is considered by [9] The stability analysisand 119867
infincontrol for a class of discrete time switched linear
parameter-varying systems are concerned in [10]The sliding
mode control problem for uncertain nonlinear discrete-time stochastic systems with 119867
2performance constraints is
designed in [11] In [12] the sliding mode control problem isconsidered for discrete-time systems In [13] the descriptionabout the modelling and control of wind turbine system isaddressed A model to describe the dynamics of a homoge-neous viscous fluid in an open pipe is introduced in [14]In [15] the authors consider the problems of robust stabilityand 119867
infincontrol for a class of networked control systems
with long-time delays The 119867infin
control issue for a class ofnetworked control systems with packet dropouts and time-varying delays is introduced in [16] From the above studiesin [2 3 8 9 13 14] the authors propose proportionalderivative controls however none considers systems withdead-zone inputs
There is some work about the control of systems withdead-zone inputs In [17ndash22] the authors proposed thecontrol of nonlinear systems with dead-zone inputs Never-theless they do not research about the pendulum systemsThependulumdynamicmodels have different structureswithrespect to the nonlinear systems addressed in the abovepapers thus a new design may be developed
In this paper a proportional derivative controller withinverse dead-zone is proposed for the control of pendulumsystemswith dead-zone inputsOnemain contribution of thisstudy is that the pendulum dynamic model is rewritten as
2 Mathematical Problems in Engineering
a robotic dynamic model to satisfy a property and later theproperty is applied to guarantee the stability of the proposedcontroller
The paper is organized as follows In Section 2 thedynamic model of the robotic arm with dead-zone inputs ispresented In Section 3 the dynamic model of the pendulumsystems with dead-zone inputs is presented In Section 4 theproportional derivative controller with inverse dead-zone isintroduced In Section 5 the proposed method is used forthe regulation of two pendulum systems Section 6 presentsconclusions and suggests future research directions
2 Dynamic Model of the Robotic Arms withDead-Zone Inputs
The main concern of this section is to understand someconcepts of robot dynamics The equation of motion for theconstrained robotic manipulator with 119899 degrees of freedomconsidering the contact force and the constraints is given inthe joint space as follows
119872(119902) 119902 + 119862 (119902 119902) 119902 + 119866 (119902) = 120591 (1)
where 119902 isin R119899times1 denotes the joint angles or link displacementsof the manipulator 119872(119902) isin R119899times119899 is the robot inertia matrixwhich is symmetric and positive definite 119862(119902 119902) isin R119899times119899
contains the centripetal and Coriolis terms and 119866(119902) arethe gravity terms and 120591 denotes the dead-zone output Thenonsymmetric dead-zone can be represented by
120591 = DZ (V) =
119898119903(V minus 119887
119903) V ge 119887
119903
0 119887119897lt V lt 119887
119903
119898119897(V minus 119887
119897) V le 119887
119897
(2)
where119898119903and119898
119897are the right and left constant slopes for the
dead-zone characteristic and 119887119903and 119887119897represent the right and
left breakpoints Note that V is the input of the dead-zone andthe control input of the global system
Define the following two states as follows
1199091= 119902 isin R
119899times1
1199092= 119902 isin R
119899times1
119906 = 120591 isin R119899times1
(3)
where 1199091= [11990911
11990912]119879
= [1199021
1199022]119879 1199092= [11990921
11990922]119879
=
[ 1199021
1199022]119879 for 119899 = 2 Then (1) can be rewritten as
1= 1199092
119872 (1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
(4)
where 119872(1199091) 119862(119909
1 1199092) and 119866(119909
1) are described in (1) the
dead-zone 119906 is [17 19 20 22]
119906 = DZ (V) =
119898119903(V minus 119887
119903) V ge 119887
119903
0 119887119897lt V lt 119887
119903
119898119897(V minus 119887
119897) V le 119887
119897
(5)
u(t)
(t)
m
m
0 br
bl
Figure 1 The dead-zone
the parameters119898119903119898119897 119887119903 and 119887
119897are described in (2) and V is
the control input of the system Figure 1 shows the dead-zone[17]
Property 1 The inertia matrix is symmetric and positivedefinite that is [23ndash25]
1198981|119909|2
le 119909119879
119872(1199091) 119909 le 119898
2|119909|2
(6)
where 1198981 1198982are known positive scalar constants 119909 =
[1199091 1199092]119879
Property 2 The centripetal and Coriolis matrix is skew-symmetric that is satisfies the following relationship [23ndash25]
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (7)
where 119909 = [1199091 1199092]119879
The normal proportional derivative controller is
119906 = minus1198701199011199091minus 1198701198891199092 (8)
where 1199091= 1199091minus 119909119889
1and 119909
2= 1199092minus 119909119889
2and 119870
119901and 119870
119889are
positive definite symmetric and constant matrices
3 Dynamic Model of the Pendulum Systemswith Dead-Zone Inputs
The dynamic model of the pendulum systems can be rewrit-ten as the dynamic model of the robotic arms howeverProperty 2 is not directly satisfied Pendulum dynamic mod-els are rewritten as the robotic dynamic models because inthis study if the above sentence is true Property 2 of therobotic systems can be used to guarantee the stability of thecontroller applied to the pendulum systems The followinglemmas let to modify the Property 2 for its application in thependulum systems
Mathematical Problems in Engineering 3
Lemma 1 A pendulum model can be rewritten as a roboticarm model (4) Nevertheless it cannot satisfy Property 2
Proof Consider 119899 = 2 for the pendulum systems in (4) itgives
1= 1199092
119872 (1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
(9)
where
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(10)
and 119872(1199091) 119862(119909
1 1199092) and 119866(119909
1) are selected from the
pendulum dynamic model and 1199091and 119909
2are defined in (3)
Consequently
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (11)
Lemma 2 Pendulum model (4) can be rewritten as follows
1= 1199092
119872 (1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119873 (119909
1 1199092) + 119906
(12)
where
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(13)
the input 119906 is given by (5) 11988811
= 11988811
+ 11988811 11988812
= 11988812
+ 11988812
11988821
= 11988821
+ 11988821 11988822
= 11988822
+ 11988822 1198991= 1198921+ 1198881111990921
+ 1198881211990922
1198992= 1198922+ 1198882111990921+ 1198882211990922 1199091and 119909
2are defined in (3) and the
following modified property is satisfied
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (14)
Proof Consider 119899 = 2 for the pendulum systems in (4) itgives
119872(1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
119872 (1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119866 (119909
1) + 119906
(15)
Consequently a change of variables is used as follows
119872(1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119873 (119909
1 1199092) + 119906
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0
(16)
where the elements are given in (12) Note that the elementsof 119862(119909
1 1199092) and119873(119909
1 1199092) are selected such that the property
(14) is satisfied
In the following section a stable controller for the pendu-lum systems will be designed
4 Proportional Derivative Control withInverse Dead-Zone
The regulation case is considered in this study that is thedesired velocity is 119909119889
2= 0The proportional derivative control
with inverse dead-zone V is as follows
V = DZminus1 (119906pd) =
1
119898119903
119906pd + 119887119903
119906pd gt 0
0 119906pd = 0
1
119898119897
119906pd + 119887119897
119906pd lt 0
(17)
where the parameters 119898119903 119898119897 119887119903 and 119887
119897are defined as in (2)
and the auxiliary proportional derivative control is
119906pd = minus1198701199011199091minus 1198701198891199092+ minus 119870 sign (119909
2) (18)
where 1199091= 1199091minus 119909119889
1is the tracking error 119909
1and 119909
2= 1199092are
defined in (3)1199091198891= [119909119889
11119909119889
12] isin R119899times1 is the desired position
119870119901 119870119889are positive definite isin R119899times1 is an approximation
of 119873(1199091 1199092) and 119873(119909
1 1199092) isin R119899times1 are the nonlinear terms
of (12) Figure 2 shows the inverse dead-zone [17 22] andFigure 3 shows the proposed controller denoted as PDDZ Itis considered that the approximation error = minus119873(119909
1 1199092)
is bounded as1003816100381610038161003816100381610038161003816100381610038161003816le 119873 (19)
Now the convergence of the closed-loop system is dis-cussed
Theorem 3 The error of the closed-loop system with theproportional derivative control (17) and (18) for the pendulumsystems with dead-zone inputs (12) and (5) is asymptoticallystable and the error of the velocity parameter 119909
2will converge
tolim sup119879rarrinfin
100381710038171003817100381711990921003817100381710038171003817
2
= 0 (20)
where119879 is the final time 1199092= 1199092119873 le 119870119870
119901gt 0 and119870
119889gt 0
Proof The proposed Lyapunov function is
1198811=1
2119909119879
2119872(1199091) 1199092+1
2119909119879
11198701199011199091 (21)
Substituting (17) and (18) into (12) and (5) the closed-loopsystem is as follows
119872(1199091) 2= minus 119870
1199011199091minus 1198701198891199092+
minus 119870 sign (1199092) minus 119862 (119909
1 1199092) 1199092
(22)
Using the fact 1199092= 1199092 the derivative of (21) is
1= 119909119879
2119872(1199091) 2+1
2119909119879
2 (1199091) 1199092+ 119909119879
21198701199011199091 (23)
4 Mathematical Problems in Engineering
minus8
minus6
minus4
minus2
0
2
4
6
8
10
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10
0bl
br
1ml
1mr
Figure 2 The inverse dead-zone
xd1
+
minussum
PD DZminus1 DZ Plant u
x1 x2
PDDZ
upd
x1 x2
Figure 3 The proposed controller
where 1199091= 1minus 119889
1= 1199092minus 119909119889
2= 1199092= 1199092and 119909
2= 2
Substituting (22) into (23) gives
1= minus 119909
119879
21198701198891199092+ 119909119879
2[ minus 119870 sign (119909
2)]
+1
2119909119879
2[ (119909
1) minus 2119862 (119909
1 1199092)] 119909119879
2
(24)
Using (14) (17) and |1199092|119879
= 119909119879
2sign(119909
2) it gives
1le minus119909119879
21198701198891199092+10038161003816100381610038161199092
1003816100381610038161003816
119879
119873 minus10038161003816100381610038161199092
1003816100381610038161003816
119879
119870
1le minus119909119879
21198701198891199092
(25)
where 119873 le 119870 Thus the error is asymptotically stable [26]Integrating (25) from 0 to 119879 yields
int
119879
0
119909119879
21198701198891199092119889119905 le 119881
10minus 1198811119879
le 11988110
1
119879int
119879
0
119909119879
21198701198891199092119889119905 le
1
11987911988110
lim sup119879rarrinfin
(1
119879int
119879
0
119909119879
21198701198891199092119889119905) le 119881
10[lim sup119879rarrinfin
(1
119879)] = 0
(26)
If 119879 rarr infin then 11990922
= 0 (20) is established
Remark 4 The proposed controller is used for the regulationcase that is the desired velocity is 119909119889
2= 0 The general case
when 119909119889
2= 0 is not considered in this research
y
FM
mg
lx
120579
cos(120579)
Figure 4 Inverted-car pendulum
5 Simulations
In this section the proportional derivative control withinverse dead-zone denoted as PDDZ will be compared withthe proportional derivative control with gravity compensa-tion of [23] denoted by PD for the control of two pendulumsystems with dead-zone inputs In this paper the root meansquare error (RMSE) [1 26 27] is used for the comparisonresults and it is given as
RMSE = (1
119879int
119879
0
1199092
119889119905)
12
(27)
where 1199092 = 1199092
12or 1199092 = 119906
2
1
51 Example 1 Consider the inverted-car pendulum [2] ofFigure 4
Inverted-car pendulum is written as (1) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(28)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= minus119898119897 sin(11990912)11990922 the other parameters of 119862(119909
1
1199092) are zero 119892
2= minus119898119892119897 sin(119909
12) and the other parameter of
119866(1199091) is zero sin(sdot) is the sine function cos(sdot) is the cosine
function 119868 = 05 kgm2 is the pendulum inertia 119872 =
0136729 kg is the mass of the car 119898 = 0040691 kg is thependulummass 119897 = 015m is the pendulum length 120579
12is the
angle with respect of the 119910 axis 119906 = 119865 is the motion forceof the car 119909
11= 119909 is the motion distance of the car and
119892 = 981ms2 is the constant acceleration due to gravity Itcan be proven that Property 2 of (7) is not satisfied
Mathematical Problems in Engineering 5
0 1 2 3 4 5 6 7 8 9 10minus20
minus15
minus10
minus5
0
5
10
15
20
25
PDDZPD
u1
Figure 5 Input for Example 1
Inverted-car pendulum is written as (12) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(29)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= 11988821
= minus(12)119898119897 sin(11990912)11990922
and the otherparameters of 119862 (119909
1 1199092) are zero 119899
1= minus(12)119898119897 sin(119909
12)11990922
1198992= minus119898119892119897 sin(119909
12) + (12)119898119897 sin(119909
12)11990922 therefore it can be
proven that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
and 1198991= 001 with 119870 = 001 Conditions given in (20)
119873 le 119870 119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the
error of the closed-loop dynamics of the PDDZ applied forpendulum systems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198921
Comparison results for the control functions are shown inFigure 5 position states are shown in Figure 6 and compar-ison results for the controller errors are shown in Figure 7Comparison of the square norm of the velocity errors11990922 of (20) for the controllers is presented in Figure 8
From the theorem of (20) 11990922 will converge to zero for the
PDDZ Table 1 shows the RMSE results using (27)The most important variable to control is the pendulum
angle 11990912
= 120579 and this variable may reach zero even if itstarts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results From Figures 5 6 and 7 it can
0 1 2 3 4 5 6 7 8 9 10Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 6 Position states for Example 1
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus004
minus006
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 7 Position state error for Example 1
be seen that the PDDZ improves the PD because the signalof the plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in thesecond From Figure 8 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 1 it
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
Time
0
200
400
600
800
||x21||2
PDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
0
005
01
015
02
||x22||2
PDDZPD
(b)
Figure 8 Velocity errors for Example 1
Table 1 Error results for Example 1
Methods RMSE for 119909 RMSE for 119906PD 42487 times 10
minus4 155044PDDZ 21995 times 10
minus4 77534
can be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
52 Example 2 Consider the Furuta pendulum [3 27] of theFigure 9
Furuta pendulum is written as (1) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(30)
where 11989811
= 1198680+ 1198981(1198712
0+ 1198972
1sin2(119909
12)) 11989812
= 11989821
= 119898111989711198710
cos(11990912) 11989833
= 1198691+ 11989811198972
1 11988811
= 11989811198972
1sin(2119909
12)11990922 11988812
=
minus119898111989711198710sin(11990912)11990922 11988821
= minus11989811198972
1sin(11990912) cos(119909
12)11990921 1198922=
minus11989811198921198971sin(11990912) and the other parameter of 119866(119909
1) is zero
sin(sdot) is the sine function cos(sdot) is the cosine function 1198680=
05Kgm2 is the arm inertia 1198691= 05 kgm2 is the pendulum
inertia 1198982= 034 kg is the arm mass 119898
1= 024 kg is the
z
x
y
m2g
1205790 m1g
J1
l1
120591
1205791F
FI0L0
Figure 9 Furuta pendulum
pendulum mass 1198710= 0293m is the arm length 119897
1= 028m
is the pendulum length 11990911
= 1205790
is the arm angle 11990912
=
1205791is the pendulum angle 119906 = 119865 is the motion torque of the
arm and 119892 = 981ms2 is the constant acceleration due togravity It can be proven that Property 2 of (7) is not satisfied
Furuta pendulum is written as (12) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(31)
where11989811
= 119872+11989811989812
= 11989821
= 119898119897 cos(11990912)11989833
= 119868+1198981198972
11988811
= 11989811198972
1sin(11990912) cos(119909
12)11990922 11988812
= minus(12)119898111989711198710sin(11990912)
11990922 11988821
= minus(12)11989811198972
1sin(11990912) cos(119909
12)11990922 and the other
parameter of 119862 (1199091 1199092) is zero 119899
1= 11989811198972
1sin(11990912) cos(119909
12)
11990922minus (12)119898
111989711198710sin(11990912)11990922 1198992= minus119898
11198921198971sin(11990912) minus (12)
11989811198972
1sin(11990912) cos(119909
12)(11990921minus 11990922) therefore it can be proven
that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
1198991= 001 with 119870 = 001 Conditions given in (20)119873 le 119870
119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the error of
the closed-loop dynamics of the PDDZ applied for pendulumsystems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198922
Comparison results for the control functions are shownin Figure 10 position states are shown in Figure 11 andcomparison results for the controller errors are shown inFigure 12 Comparison of the square norm of the velocityerrors 119909
22 of (20) for the controllers is presented in
Figure 13 From the theorem of (20) 11990922 will converge to
zero for the PDDZ Table 2 shows the RMSE results using(27)
The most important variable to control is the pendulumangle 119909
12= 1205791 and this variable may reach zero even if it
Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 10minus15
minus10
minus5
0
5
10
15
20
25
u1
PDDZPD
Figure 10 Input for Example 2
0 1 2 3 4 5 6 7 8 9 10
Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 11 Position states for Example 2
starts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results FromFigures 10 11 and 12 it canbe seen that the PDDZ improves the PD because the signal ofthe plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in the
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus006
minus004
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 12 Position state error for Example 2
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
||x21||2
020406080100
(a)
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
000200400600801
||x22||2
(b)
Figure 13 Velocity errors for Example 2
second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 2 itcan be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
a robotic dynamic model to satisfy a property and later theproperty is applied to guarantee the stability of the proposedcontroller
The paper is organized as follows In Section 2 thedynamic model of the robotic arm with dead-zone inputs ispresented In Section 3 the dynamic model of the pendulumsystems with dead-zone inputs is presented In Section 4 theproportional derivative controller with inverse dead-zone isintroduced In Section 5 the proposed method is used forthe regulation of two pendulum systems Section 6 presentsconclusions and suggests future research directions
2 Dynamic Model of the Robotic Arms withDead-Zone Inputs
The main concern of this section is to understand someconcepts of robot dynamics The equation of motion for theconstrained robotic manipulator with 119899 degrees of freedomconsidering the contact force and the constraints is given inthe joint space as follows
119872(119902) 119902 + 119862 (119902 119902) 119902 + 119866 (119902) = 120591 (1)
where 119902 isin R119899times1 denotes the joint angles or link displacementsof the manipulator 119872(119902) isin R119899times119899 is the robot inertia matrixwhich is symmetric and positive definite 119862(119902 119902) isin R119899times119899
contains the centripetal and Coriolis terms and 119866(119902) arethe gravity terms and 120591 denotes the dead-zone output Thenonsymmetric dead-zone can be represented by
120591 = DZ (V) =
119898119903(V minus 119887
119903) V ge 119887
119903
0 119887119897lt V lt 119887
119903
119898119897(V minus 119887
119897) V le 119887
119897
(2)
where119898119903and119898
119897are the right and left constant slopes for the
dead-zone characteristic and 119887119903and 119887119897represent the right and
left breakpoints Note that V is the input of the dead-zone andthe control input of the global system
Define the following two states as follows
1199091= 119902 isin R
119899times1
1199092= 119902 isin R
119899times1
119906 = 120591 isin R119899times1
(3)
where 1199091= [11990911
11990912]119879
= [1199021
1199022]119879 1199092= [11990921
11990922]119879
=
[ 1199021
1199022]119879 for 119899 = 2 Then (1) can be rewritten as
1= 1199092
119872 (1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
(4)
where 119872(1199091) 119862(119909
1 1199092) and 119866(119909
1) are described in (1) the
dead-zone 119906 is [17 19 20 22]
119906 = DZ (V) =
119898119903(V minus 119887
119903) V ge 119887
119903
0 119887119897lt V lt 119887
119903
119898119897(V minus 119887
119897) V le 119887
119897
(5)
u(t)
(t)
m
m
0 br
bl
Figure 1 The dead-zone
the parameters119898119903119898119897 119887119903 and 119887
119897are described in (2) and V is
the control input of the system Figure 1 shows the dead-zone[17]
Property 1 The inertia matrix is symmetric and positivedefinite that is [23ndash25]
1198981|119909|2
le 119909119879
119872(1199091) 119909 le 119898
2|119909|2
(6)
where 1198981 1198982are known positive scalar constants 119909 =
[1199091 1199092]119879
Property 2 The centripetal and Coriolis matrix is skew-symmetric that is satisfies the following relationship [23ndash25]
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (7)
where 119909 = [1199091 1199092]119879
The normal proportional derivative controller is
119906 = minus1198701199011199091minus 1198701198891199092 (8)
where 1199091= 1199091minus 119909119889
1and 119909
2= 1199092minus 119909119889
2and 119870
119901and 119870
119889are
positive definite symmetric and constant matrices
3 Dynamic Model of the Pendulum Systemswith Dead-Zone Inputs
The dynamic model of the pendulum systems can be rewrit-ten as the dynamic model of the robotic arms howeverProperty 2 is not directly satisfied Pendulum dynamic mod-els are rewritten as the robotic dynamic models because inthis study if the above sentence is true Property 2 of therobotic systems can be used to guarantee the stability of thecontroller applied to the pendulum systems The followinglemmas let to modify the Property 2 for its application in thependulum systems
Mathematical Problems in Engineering 3
Lemma 1 A pendulum model can be rewritten as a roboticarm model (4) Nevertheless it cannot satisfy Property 2
Proof Consider 119899 = 2 for the pendulum systems in (4) itgives
1= 1199092
119872 (1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
(9)
where
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(10)
and 119872(1199091) 119862(119909
1 1199092) and 119866(119909
1) are selected from the
pendulum dynamic model and 1199091and 119909
2are defined in (3)
Consequently
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (11)
Lemma 2 Pendulum model (4) can be rewritten as follows
1= 1199092
119872 (1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119873 (119909
1 1199092) + 119906
(12)
where
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(13)
the input 119906 is given by (5) 11988811
= 11988811
+ 11988811 11988812
= 11988812
+ 11988812
11988821
= 11988821
+ 11988821 11988822
= 11988822
+ 11988822 1198991= 1198921+ 1198881111990921
+ 1198881211990922
1198992= 1198922+ 1198882111990921+ 1198882211990922 1199091and 119909
2are defined in (3) and the
following modified property is satisfied
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (14)
Proof Consider 119899 = 2 for the pendulum systems in (4) itgives
119872(1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
119872 (1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119866 (119909
1) + 119906
(15)
Consequently a change of variables is used as follows
119872(1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119873 (119909
1 1199092) + 119906
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0
(16)
where the elements are given in (12) Note that the elementsof 119862(119909
1 1199092) and119873(119909
1 1199092) are selected such that the property
(14) is satisfied
In the following section a stable controller for the pendu-lum systems will be designed
4 Proportional Derivative Control withInverse Dead-Zone
The regulation case is considered in this study that is thedesired velocity is 119909119889
2= 0The proportional derivative control
with inverse dead-zone V is as follows
V = DZminus1 (119906pd) =
1
119898119903
119906pd + 119887119903
119906pd gt 0
0 119906pd = 0
1
119898119897
119906pd + 119887119897
119906pd lt 0
(17)
where the parameters 119898119903 119898119897 119887119903 and 119887
119897are defined as in (2)
and the auxiliary proportional derivative control is
119906pd = minus1198701199011199091minus 1198701198891199092+ minus 119870 sign (119909
2) (18)
where 1199091= 1199091minus 119909119889
1is the tracking error 119909
1and 119909
2= 1199092are
defined in (3)1199091198891= [119909119889
11119909119889
12] isin R119899times1 is the desired position
119870119901 119870119889are positive definite isin R119899times1 is an approximation
of 119873(1199091 1199092) and 119873(119909
1 1199092) isin R119899times1 are the nonlinear terms
of (12) Figure 2 shows the inverse dead-zone [17 22] andFigure 3 shows the proposed controller denoted as PDDZ Itis considered that the approximation error = minus119873(119909
1 1199092)
is bounded as1003816100381610038161003816100381610038161003816100381610038161003816le 119873 (19)
Now the convergence of the closed-loop system is dis-cussed
Theorem 3 The error of the closed-loop system with theproportional derivative control (17) and (18) for the pendulumsystems with dead-zone inputs (12) and (5) is asymptoticallystable and the error of the velocity parameter 119909
2will converge
tolim sup119879rarrinfin
100381710038171003817100381711990921003817100381710038171003817
2
= 0 (20)
where119879 is the final time 1199092= 1199092119873 le 119870119870
119901gt 0 and119870
119889gt 0
Proof The proposed Lyapunov function is
1198811=1
2119909119879
2119872(1199091) 1199092+1
2119909119879
11198701199011199091 (21)
Substituting (17) and (18) into (12) and (5) the closed-loopsystem is as follows
119872(1199091) 2= minus 119870
1199011199091minus 1198701198891199092+
minus 119870 sign (1199092) minus 119862 (119909
1 1199092) 1199092
(22)
Using the fact 1199092= 1199092 the derivative of (21) is
1= 119909119879
2119872(1199091) 2+1
2119909119879
2 (1199091) 1199092+ 119909119879
21198701199011199091 (23)
4 Mathematical Problems in Engineering
minus8
minus6
minus4
minus2
0
2
4
6
8
10
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10
0bl
br
1ml
1mr
Figure 2 The inverse dead-zone
xd1
+
minussum
PD DZminus1 DZ Plant u
x1 x2
PDDZ
upd
x1 x2
Figure 3 The proposed controller
where 1199091= 1minus 119889
1= 1199092minus 119909119889
2= 1199092= 1199092and 119909
2= 2
Substituting (22) into (23) gives
1= minus 119909
119879
21198701198891199092+ 119909119879
2[ minus 119870 sign (119909
2)]
+1
2119909119879
2[ (119909
1) minus 2119862 (119909
1 1199092)] 119909119879
2
(24)
Using (14) (17) and |1199092|119879
= 119909119879
2sign(119909
2) it gives
1le minus119909119879
21198701198891199092+10038161003816100381610038161199092
1003816100381610038161003816
119879
119873 minus10038161003816100381610038161199092
1003816100381610038161003816
119879
119870
1le minus119909119879
21198701198891199092
(25)
where 119873 le 119870 Thus the error is asymptotically stable [26]Integrating (25) from 0 to 119879 yields
int
119879
0
119909119879
21198701198891199092119889119905 le 119881
10minus 1198811119879
le 11988110
1
119879int
119879
0
119909119879
21198701198891199092119889119905 le
1
11987911988110
lim sup119879rarrinfin
(1
119879int
119879
0
119909119879
21198701198891199092119889119905) le 119881
10[lim sup119879rarrinfin
(1
119879)] = 0
(26)
If 119879 rarr infin then 11990922
= 0 (20) is established
Remark 4 The proposed controller is used for the regulationcase that is the desired velocity is 119909119889
2= 0 The general case
when 119909119889
2= 0 is not considered in this research
y
FM
mg
lx
120579
cos(120579)
Figure 4 Inverted-car pendulum
5 Simulations
In this section the proportional derivative control withinverse dead-zone denoted as PDDZ will be compared withthe proportional derivative control with gravity compensa-tion of [23] denoted by PD for the control of two pendulumsystems with dead-zone inputs In this paper the root meansquare error (RMSE) [1 26 27] is used for the comparisonresults and it is given as
RMSE = (1
119879int
119879
0
1199092
119889119905)
12
(27)
where 1199092 = 1199092
12or 1199092 = 119906
2
1
51 Example 1 Consider the inverted-car pendulum [2] ofFigure 4
Inverted-car pendulum is written as (1) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(28)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= minus119898119897 sin(11990912)11990922 the other parameters of 119862(119909
1
1199092) are zero 119892
2= minus119898119892119897 sin(119909
12) and the other parameter of
119866(1199091) is zero sin(sdot) is the sine function cos(sdot) is the cosine
function 119868 = 05 kgm2 is the pendulum inertia 119872 =
0136729 kg is the mass of the car 119898 = 0040691 kg is thependulummass 119897 = 015m is the pendulum length 120579
12is the
angle with respect of the 119910 axis 119906 = 119865 is the motion forceof the car 119909
11= 119909 is the motion distance of the car and
119892 = 981ms2 is the constant acceleration due to gravity Itcan be proven that Property 2 of (7) is not satisfied
Mathematical Problems in Engineering 5
0 1 2 3 4 5 6 7 8 9 10minus20
minus15
minus10
minus5
0
5
10
15
20
25
PDDZPD
u1
Figure 5 Input for Example 1
Inverted-car pendulum is written as (12) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(29)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= 11988821
= minus(12)119898119897 sin(11990912)11990922
and the otherparameters of 119862 (119909
1 1199092) are zero 119899
1= minus(12)119898119897 sin(119909
12)11990922
1198992= minus119898119892119897 sin(119909
12) + (12)119898119897 sin(119909
12)11990922 therefore it can be
proven that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
and 1198991= 001 with 119870 = 001 Conditions given in (20)
119873 le 119870 119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the
error of the closed-loop dynamics of the PDDZ applied forpendulum systems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198921
Comparison results for the control functions are shown inFigure 5 position states are shown in Figure 6 and compar-ison results for the controller errors are shown in Figure 7Comparison of the square norm of the velocity errors11990922 of (20) for the controllers is presented in Figure 8
From the theorem of (20) 11990922 will converge to zero for the
PDDZ Table 1 shows the RMSE results using (27)The most important variable to control is the pendulum
angle 11990912
= 120579 and this variable may reach zero even if itstarts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results From Figures 5 6 and 7 it can
0 1 2 3 4 5 6 7 8 9 10Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 6 Position states for Example 1
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus004
minus006
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 7 Position state error for Example 1
be seen that the PDDZ improves the PD because the signalof the plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in thesecond From Figure 8 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 1 it
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
Time
0
200
400
600
800
||x21||2
PDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
0
005
01
015
02
||x22||2
PDDZPD
(b)
Figure 8 Velocity errors for Example 1
Table 1 Error results for Example 1
Methods RMSE for 119909 RMSE for 119906PD 42487 times 10
minus4 155044PDDZ 21995 times 10
minus4 77534
can be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
52 Example 2 Consider the Furuta pendulum [3 27] of theFigure 9
Furuta pendulum is written as (1) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(30)
where 11989811
= 1198680+ 1198981(1198712
0+ 1198972
1sin2(119909
12)) 11989812
= 11989821
= 119898111989711198710
cos(11990912) 11989833
= 1198691+ 11989811198972
1 11988811
= 11989811198972
1sin(2119909
12)11990922 11988812
=
minus119898111989711198710sin(11990912)11990922 11988821
= minus11989811198972
1sin(11990912) cos(119909
12)11990921 1198922=
minus11989811198921198971sin(11990912) and the other parameter of 119866(119909
1) is zero
sin(sdot) is the sine function cos(sdot) is the cosine function 1198680=
05Kgm2 is the arm inertia 1198691= 05 kgm2 is the pendulum
inertia 1198982= 034 kg is the arm mass 119898
1= 024 kg is the
z
x
y
m2g
1205790 m1g
J1
l1
120591
1205791F
FI0L0
Figure 9 Furuta pendulum
pendulum mass 1198710= 0293m is the arm length 119897
1= 028m
is the pendulum length 11990911
= 1205790
is the arm angle 11990912
=
1205791is the pendulum angle 119906 = 119865 is the motion torque of the
arm and 119892 = 981ms2 is the constant acceleration due togravity It can be proven that Property 2 of (7) is not satisfied
Furuta pendulum is written as (12) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(31)
where11989811
= 119872+11989811989812
= 11989821
= 119898119897 cos(11990912)11989833
= 119868+1198981198972
11988811
= 11989811198972
1sin(11990912) cos(119909
12)11990922 11988812
= minus(12)119898111989711198710sin(11990912)
11990922 11988821
= minus(12)11989811198972
1sin(11990912) cos(119909
12)11990922 and the other
parameter of 119862 (1199091 1199092) is zero 119899
1= 11989811198972
1sin(11990912) cos(119909
12)
11990922minus (12)119898
111989711198710sin(11990912)11990922 1198992= minus119898
11198921198971sin(11990912) minus (12)
11989811198972
1sin(11990912) cos(119909
12)(11990921minus 11990922) therefore it can be proven
that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
1198991= 001 with 119870 = 001 Conditions given in (20)119873 le 119870
119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the error of
the closed-loop dynamics of the PDDZ applied for pendulumsystems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198922
Comparison results for the control functions are shownin Figure 10 position states are shown in Figure 11 andcomparison results for the controller errors are shown inFigure 12 Comparison of the square norm of the velocityerrors 119909
22 of (20) for the controllers is presented in
Figure 13 From the theorem of (20) 11990922 will converge to
zero for the PDDZ Table 2 shows the RMSE results using(27)
The most important variable to control is the pendulumangle 119909
12= 1205791 and this variable may reach zero even if it
Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 10minus15
minus10
minus5
0
5
10
15
20
25
u1
PDDZPD
Figure 10 Input for Example 2
0 1 2 3 4 5 6 7 8 9 10
Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 11 Position states for Example 2
starts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results FromFigures 10 11 and 12 it canbe seen that the PDDZ improves the PD because the signal ofthe plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in the
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus006
minus004
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 12 Position state error for Example 2
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
||x21||2
020406080100
(a)
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
000200400600801
||x22||2
(b)
Figure 13 Velocity errors for Example 2
second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 2 itcan be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Lemma 1 A pendulum model can be rewritten as a roboticarm model (4) Nevertheless it cannot satisfy Property 2
Proof Consider 119899 = 2 for the pendulum systems in (4) itgives
1= 1199092
119872 (1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
(9)
where
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(10)
and 119872(1199091) 119862(119909
1 1199092) and 119866(119909
1) are selected from the
pendulum dynamic model and 1199091and 119909
2are defined in (3)
Consequently
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (11)
Lemma 2 Pendulum model (4) can be rewritten as follows
1= 1199092
119872 (1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119873 (119909
1 1199092) + 119906
(12)
where
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(13)
the input 119906 is given by (5) 11988811
= 11988811
+ 11988811 11988812
= 11988812
+ 11988812
11988821
= 11988821
+ 11988821 11988822
= 11988822
+ 11988822 1198991= 1198921+ 1198881111990921
+ 1198881211990922
1198992= 1198922+ 1198882111990921+ 1198882211990922 1199091and 119909
2are defined in (3) and the
following modified property is satisfied
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0 (14)
Proof Consider 119899 = 2 for the pendulum systems in (4) itgives
119872(1199091) 2+ 119862 (119909
1 1199092) 1199092+ 119866 (119909
1) = 119906
119872 (1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119866 (119909
1) + 119906
(15)
Consequently a change of variables is used as follows
119872(1199091) 2= minus119862 (119909
1 1199092) 1199092minus 119873 (119909
1 1199092) + 119906
119909119879
[ (1199091) minus 2119862 (119909
1 1199092)] 119909 = 0
(16)
where the elements are given in (12) Note that the elementsof 119862(119909
1 1199092) and119873(119909
1 1199092) are selected such that the property
(14) is satisfied
In the following section a stable controller for the pendu-lum systems will be designed
4 Proportional Derivative Control withInverse Dead-Zone
The regulation case is considered in this study that is thedesired velocity is 119909119889
2= 0The proportional derivative control
with inverse dead-zone V is as follows
V = DZminus1 (119906pd) =
1
119898119903
119906pd + 119887119903
119906pd gt 0
0 119906pd = 0
1
119898119897
119906pd + 119887119897
119906pd lt 0
(17)
where the parameters 119898119903 119898119897 119887119903 and 119887
119897are defined as in (2)
and the auxiliary proportional derivative control is
119906pd = minus1198701199011199091minus 1198701198891199092+ minus 119870 sign (119909
2) (18)
where 1199091= 1199091minus 119909119889
1is the tracking error 119909
1and 119909
2= 1199092are
defined in (3)1199091198891= [119909119889
11119909119889
12] isin R119899times1 is the desired position
119870119901 119870119889are positive definite isin R119899times1 is an approximation
of 119873(1199091 1199092) and 119873(119909
1 1199092) isin R119899times1 are the nonlinear terms
of (12) Figure 2 shows the inverse dead-zone [17 22] andFigure 3 shows the proposed controller denoted as PDDZ Itis considered that the approximation error = minus119873(119909
1 1199092)
is bounded as1003816100381610038161003816100381610038161003816100381610038161003816le 119873 (19)
Now the convergence of the closed-loop system is dis-cussed
Theorem 3 The error of the closed-loop system with theproportional derivative control (17) and (18) for the pendulumsystems with dead-zone inputs (12) and (5) is asymptoticallystable and the error of the velocity parameter 119909
2will converge
tolim sup119879rarrinfin
100381710038171003817100381711990921003817100381710038171003817
2
= 0 (20)
where119879 is the final time 1199092= 1199092119873 le 119870119870
119901gt 0 and119870
119889gt 0
Proof The proposed Lyapunov function is
1198811=1
2119909119879
2119872(1199091) 1199092+1
2119909119879
11198701199011199091 (21)
Substituting (17) and (18) into (12) and (5) the closed-loopsystem is as follows
119872(1199091) 2= minus 119870
1199011199091minus 1198701198891199092+
minus 119870 sign (1199092) minus 119862 (119909
1 1199092) 1199092
(22)
Using the fact 1199092= 1199092 the derivative of (21) is
1= 119909119879
2119872(1199091) 2+1
2119909119879
2 (1199091) 1199092+ 119909119879
21198701199011199091 (23)
4 Mathematical Problems in Engineering
minus8
minus6
minus4
minus2
0
2
4
6
8
10
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10
0bl
br
1ml
1mr
Figure 2 The inverse dead-zone
xd1
+
minussum
PD DZminus1 DZ Plant u
x1 x2
PDDZ
upd
x1 x2
Figure 3 The proposed controller
where 1199091= 1minus 119889
1= 1199092minus 119909119889
2= 1199092= 1199092and 119909
2= 2
Substituting (22) into (23) gives
1= minus 119909
119879
21198701198891199092+ 119909119879
2[ minus 119870 sign (119909
2)]
+1
2119909119879
2[ (119909
1) minus 2119862 (119909
1 1199092)] 119909119879
2
(24)
Using (14) (17) and |1199092|119879
= 119909119879
2sign(119909
2) it gives
1le minus119909119879
21198701198891199092+10038161003816100381610038161199092
1003816100381610038161003816
119879
119873 minus10038161003816100381610038161199092
1003816100381610038161003816
119879
119870
1le minus119909119879
21198701198891199092
(25)
where 119873 le 119870 Thus the error is asymptotically stable [26]Integrating (25) from 0 to 119879 yields
int
119879
0
119909119879
21198701198891199092119889119905 le 119881
10minus 1198811119879
le 11988110
1
119879int
119879
0
119909119879
21198701198891199092119889119905 le
1
11987911988110
lim sup119879rarrinfin
(1
119879int
119879
0
119909119879
21198701198891199092119889119905) le 119881
10[lim sup119879rarrinfin
(1
119879)] = 0
(26)
If 119879 rarr infin then 11990922
= 0 (20) is established
Remark 4 The proposed controller is used for the regulationcase that is the desired velocity is 119909119889
2= 0 The general case
when 119909119889
2= 0 is not considered in this research
y
FM
mg
lx
120579
cos(120579)
Figure 4 Inverted-car pendulum
5 Simulations
In this section the proportional derivative control withinverse dead-zone denoted as PDDZ will be compared withthe proportional derivative control with gravity compensa-tion of [23] denoted by PD for the control of two pendulumsystems with dead-zone inputs In this paper the root meansquare error (RMSE) [1 26 27] is used for the comparisonresults and it is given as
RMSE = (1
119879int
119879
0
1199092
119889119905)
12
(27)
where 1199092 = 1199092
12or 1199092 = 119906
2
1
51 Example 1 Consider the inverted-car pendulum [2] ofFigure 4
Inverted-car pendulum is written as (1) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(28)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= minus119898119897 sin(11990912)11990922 the other parameters of 119862(119909
1
1199092) are zero 119892
2= minus119898119892119897 sin(119909
12) and the other parameter of
119866(1199091) is zero sin(sdot) is the sine function cos(sdot) is the cosine
function 119868 = 05 kgm2 is the pendulum inertia 119872 =
0136729 kg is the mass of the car 119898 = 0040691 kg is thependulummass 119897 = 015m is the pendulum length 120579
12is the
angle with respect of the 119910 axis 119906 = 119865 is the motion forceof the car 119909
11= 119909 is the motion distance of the car and
119892 = 981ms2 is the constant acceleration due to gravity Itcan be proven that Property 2 of (7) is not satisfied
Mathematical Problems in Engineering 5
0 1 2 3 4 5 6 7 8 9 10minus20
minus15
minus10
minus5
0
5
10
15
20
25
PDDZPD
u1
Figure 5 Input for Example 1
Inverted-car pendulum is written as (12) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(29)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= 11988821
= minus(12)119898119897 sin(11990912)11990922
and the otherparameters of 119862 (119909
1 1199092) are zero 119899
1= minus(12)119898119897 sin(119909
12)11990922
1198992= minus119898119892119897 sin(119909
12) + (12)119898119897 sin(119909
12)11990922 therefore it can be
proven that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
and 1198991= 001 with 119870 = 001 Conditions given in (20)
119873 le 119870 119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the
error of the closed-loop dynamics of the PDDZ applied forpendulum systems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198921
Comparison results for the control functions are shown inFigure 5 position states are shown in Figure 6 and compar-ison results for the controller errors are shown in Figure 7Comparison of the square norm of the velocity errors11990922 of (20) for the controllers is presented in Figure 8
From the theorem of (20) 11990922 will converge to zero for the
PDDZ Table 1 shows the RMSE results using (27)The most important variable to control is the pendulum
angle 11990912
= 120579 and this variable may reach zero even if itstarts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results From Figures 5 6 and 7 it can
0 1 2 3 4 5 6 7 8 9 10Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 6 Position states for Example 1
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus004
minus006
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 7 Position state error for Example 1
be seen that the PDDZ improves the PD because the signalof the plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in thesecond From Figure 8 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 1 it
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
Time
0
200
400
600
800
||x21||2
PDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
0
005
01
015
02
||x22||2
PDDZPD
(b)
Figure 8 Velocity errors for Example 1
Table 1 Error results for Example 1
Methods RMSE for 119909 RMSE for 119906PD 42487 times 10
minus4 155044PDDZ 21995 times 10
minus4 77534
can be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
52 Example 2 Consider the Furuta pendulum [3 27] of theFigure 9
Furuta pendulum is written as (1) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(30)
where 11989811
= 1198680+ 1198981(1198712
0+ 1198972
1sin2(119909
12)) 11989812
= 11989821
= 119898111989711198710
cos(11990912) 11989833
= 1198691+ 11989811198972
1 11988811
= 11989811198972
1sin(2119909
12)11990922 11988812
=
minus119898111989711198710sin(11990912)11990922 11988821
= minus11989811198972
1sin(11990912) cos(119909
12)11990921 1198922=
minus11989811198921198971sin(11990912) and the other parameter of 119866(119909
1) is zero
sin(sdot) is the sine function cos(sdot) is the cosine function 1198680=
05Kgm2 is the arm inertia 1198691= 05 kgm2 is the pendulum
inertia 1198982= 034 kg is the arm mass 119898
1= 024 kg is the
z
x
y
m2g
1205790 m1g
J1
l1
120591
1205791F
FI0L0
Figure 9 Furuta pendulum
pendulum mass 1198710= 0293m is the arm length 119897
1= 028m
is the pendulum length 11990911
= 1205790
is the arm angle 11990912
=
1205791is the pendulum angle 119906 = 119865 is the motion torque of the
arm and 119892 = 981ms2 is the constant acceleration due togravity It can be proven that Property 2 of (7) is not satisfied
Furuta pendulum is written as (12) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(31)
where11989811
= 119872+11989811989812
= 11989821
= 119898119897 cos(11990912)11989833
= 119868+1198981198972
11988811
= 11989811198972
1sin(11990912) cos(119909
12)11990922 11988812
= minus(12)119898111989711198710sin(11990912)
11990922 11988821
= minus(12)11989811198972
1sin(11990912) cos(119909
12)11990922 and the other
parameter of 119862 (1199091 1199092) is zero 119899
1= 11989811198972
1sin(11990912) cos(119909
12)
11990922minus (12)119898
111989711198710sin(11990912)11990922 1198992= minus119898
11198921198971sin(11990912) minus (12)
11989811198972
1sin(11990912) cos(119909
12)(11990921minus 11990922) therefore it can be proven
that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
1198991= 001 with 119870 = 001 Conditions given in (20)119873 le 119870
119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the error of
the closed-loop dynamics of the PDDZ applied for pendulumsystems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198922
Comparison results for the control functions are shownin Figure 10 position states are shown in Figure 11 andcomparison results for the controller errors are shown inFigure 12 Comparison of the square norm of the velocityerrors 119909
22 of (20) for the controllers is presented in
Figure 13 From the theorem of (20) 11990922 will converge to
zero for the PDDZ Table 2 shows the RMSE results using(27)
The most important variable to control is the pendulumangle 119909
12= 1205791 and this variable may reach zero even if it
Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 10minus15
minus10
minus5
0
5
10
15
20
25
u1
PDDZPD
Figure 10 Input for Example 2
0 1 2 3 4 5 6 7 8 9 10
Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 11 Position states for Example 2
starts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results FromFigures 10 11 and 12 it canbe seen that the PDDZ improves the PD because the signal ofthe plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in the
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus006
minus004
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 12 Position state error for Example 2
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
||x21||2
020406080100
(a)
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
000200400600801
||x22||2
(b)
Figure 13 Velocity errors for Example 2
second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 2 itcan be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
minus8
minus6
minus4
minus2
0
2
4
6
8
10
minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10
0bl
br
1ml
1mr
Figure 2 The inverse dead-zone
xd1
+
minussum
PD DZminus1 DZ Plant u
x1 x2
PDDZ
upd
x1 x2
Figure 3 The proposed controller
where 1199091= 1minus 119889
1= 1199092minus 119909119889
2= 1199092= 1199092and 119909
2= 2
Substituting (22) into (23) gives
1= minus 119909
119879
21198701198891199092+ 119909119879
2[ minus 119870 sign (119909
2)]
+1
2119909119879
2[ (119909
1) minus 2119862 (119909
1 1199092)] 119909119879
2
(24)
Using (14) (17) and |1199092|119879
= 119909119879
2sign(119909
2) it gives
1le minus119909119879
21198701198891199092+10038161003816100381610038161199092
1003816100381610038161003816
119879
119873 minus10038161003816100381610038161199092
1003816100381610038161003816
119879
119870
1le minus119909119879
21198701198891199092
(25)
where 119873 le 119870 Thus the error is asymptotically stable [26]Integrating (25) from 0 to 119879 yields
int
119879
0
119909119879
21198701198891199092119889119905 le 119881
10minus 1198811119879
le 11988110
1
119879int
119879
0
119909119879
21198701198891199092119889119905 le
1
11987911988110
lim sup119879rarrinfin
(1
119879int
119879
0
119909119879
21198701198891199092119889119905) le 119881
10[lim sup119879rarrinfin
(1
119879)] = 0
(26)
If 119879 rarr infin then 11990922
= 0 (20) is established
Remark 4 The proposed controller is used for the regulationcase that is the desired velocity is 119909119889
2= 0 The general case
when 119909119889
2= 0 is not considered in this research
y
FM
mg
lx
120579
cos(120579)
Figure 4 Inverted-car pendulum
5 Simulations
In this section the proportional derivative control withinverse dead-zone denoted as PDDZ will be compared withthe proportional derivative control with gravity compensa-tion of [23] denoted by PD for the control of two pendulumsystems with dead-zone inputs In this paper the root meansquare error (RMSE) [1 26 27] is used for the comparisonresults and it is given as
RMSE = (1
119879int
119879
0
1199092
119889119905)
12
(27)
where 1199092 = 1199092
12or 1199092 = 119906
2
1
51 Example 1 Consider the inverted-car pendulum [2] ofFigure 4
Inverted-car pendulum is written as (1) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(28)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= minus119898119897 sin(11990912)11990922 the other parameters of 119862(119909
1
1199092) are zero 119892
2= minus119898119892119897 sin(119909
12) and the other parameter of
119866(1199091) is zero sin(sdot) is the sine function cos(sdot) is the cosine
function 119868 = 05 kgm2 is the pendulum inertia 119872 =
0136729 kg is the mass of the car 119898 = 0040691 kg is thependulummass 119897 = 015m is the pendulum length 120579
12is the
angle with respect of the 119910 axis 119906 = 119865 is the motion forceof the car 119909
11= 119909 is the motion distance of the car and
119892 = 981ms2 is the constant acceleration due to gravity Itcan be proven that Property 2 of (7) is not satisfied
Mathematical Problems in Engineering 5
0 1 2 3 4 5 6 7 8 9 10minus20
minus15
minus10
minus5
0
5
10
15
20
25
PDDZPD
u1
Figure 5 Input for Example 1
Inverted-car pendulum is written as (12) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(29)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= 11988821
= minus(12)119898119897 sin(11990912)11990922
and the otherparameters of 119862 (119909
1 1199092) are zero 119899
1= minus(12)119898119897 sin(119909
12)11990922
1198992= minus119898119892119897 sin(119909
12) + (12)119898119897 sin(119909
12)11990922 therefore it can be
proven that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
and 1198991= 001 with 119870 = 001 Conditions given in (20)
119873 le 119870 119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the
error of the closed-loop dynamics of the PDDZ applied forpendulum systems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198921
Comparison results for the control functions are shown inFigure 5 position states are shown in Figure 6 and compar-ison results for the controller errors are shown in Figure 7Comparison of the square norm of the velocity errors11990922 of (20) for the controllers is presented in Figure 8
From the theorem of (20) 11990922 will converge to zero for the
PDDZ Table 1 shows the RMSE results using (27)The most important variable to control is the pendulum
angle 11990912
= 120579 and this variable may reach zero even if itstarts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results From Figures 5 6 and 7 it can
0 1 2 3 4 5 6 7 8 9 10Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 6 Position states for Example 1
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus004
minus006
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 7 Position state error for Example 1
be seen that the PDDZ improves the PD because the signalof the plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in thesecond From Figure 8 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 1 it
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
Time
0
200
400
600
800
||x21||2
PDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
0
005
01
015
02
||x22||2
PDDZPD
(b)
Figure 8 Velocity errors for Example 1
Table 1 Error results for Example 1
Methods RMSE for 119909 RMSE for 119906PD 42487 times 10
minus4 155044PDDZ 21995 times 10
minus4 77534
can be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
52 Example 2 Consider the Furuta pendulum [3 27] of theFigure 9
Furuta pendulum is written as (1) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(30)
where 11989811
= 1198680+ 1198981(1198712
0+ 1198972
1sin2(119909
12)) 11989812
= 11989821
= 119898111989711198710
cos(11990912) 11989833
= 1198691+ 11989811198972
1 11988811
= 11989811198972
1sin(2119909
12)11990922 11988812
=
minus119898111989711198710sin(11990912)11990922 11988821
= minus11989811198972
1sin(11990912) cos(119909
12)11990921 1198922=
minus11989811198921198971sin(11990912) and the other parameter of 119866(119909
1) is zero
sin(sdot) is the sine function cos(sdot) is the cosine function 1198680=
05Kgm2 is the arm inertia 1198691= 05 kgm2 is the pendulum
inertia 1198982= 034 kg is the arm mass 119898
1= 024 kg is the
z
x
y
m2g
1205790 m1g
J1
l1
120591
1205791F
FI0L0
Figure 9 Furuta pendulum
pendulum mass 1198710= 0293m is the arm length 119897
1= 028m
is the pendulum length 11990911
= 1205790
is the arm angle 11990912
=
1205791is the pendulum angle 119906 = 119865 is the motion torque of the
arm and 119892 = 981ms2 is the constant acceleration due togravity It can be proven that Property 2 of (7) is not satisfied
Furuta pendulum is written as (12) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(31)
where11989811
= 119872+11989811989812
= 11989821
= 119898119897 cos(11990912)11989833
= 119868+1198981198972
11988811
= 11989811198972
1sin(11990912) cos(119909
12)11990922 11988812
= minus(12)119898111989711198710sin(11990912)
11990922 11988821
= minus(12)11989811198972
1sin(11990912) cos(119909
12)11990922 and the other
parameter of 119862 (1199091 1199092) is zero 119899
1= 11989811198972
1sin(11990912) cos(119909
12)
11990922minus (12)119898
111989711198710sin(11990912)11990922 1198992= minus119898
11198921198971sin(11990912) minus (12)
11989811198972
1sin(11990912) cos(119909
12)(11990921minus 11990922) therefore it can be proven
that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
1198991= 001 with 119870 = 001 Conditions given in (20)119873 le 119870
119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the error of
the closed-loop dynamics of the PDDZ applied for pendulumsystems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198922
Comparison results for the control functions are shownin Figure 10 position states are shown in Figure 11 andcomparison results for the controller errors are shown inFigure 12 Comparison of the square norm of the velocityerrors 119909
22 of (20) for the controllers is presented in
Figure 13 From the theorem of (20) 11990922 will converge to
zero for the PDDZ Table 2 shows the RMSE results using(27)
The most important variable to control is the pendulumangle 119909
12= 1205791 and this variable may reach zero even if it
Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 10minus15
minus10
minus5
0
5
10
15
20
25
u1
PDDZPD
Figure 10 Input for Example 2
0 1 2 3 4 5 6 7 8 9 10
Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 11 Position states for Example 2
starts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results FromFigures 10 11 and 12 it canbe seen that the PDDZ improves the PD because the signal ofthe plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in the
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus006
minus004
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 12 Position state error for Example 2
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
||x21||2
020406080100
(a)
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
000200400600801
||x22||2
(b)
Figure 13 Velocity errors for Example 2
second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 2 itcan be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
0 1 2 3 4 5 6 7 8 9 10minus20
minus15
minus10
minus5
0
5
10
15
20
25
PDDZPD
u1
Figure 5 Input for Example 1
Inverted-car pendulum is written as (12) and it is detailedas follows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(29)
where 11989811
= 119872 + 119898 11989812
= 11989821
= 119898119897 cos(11990912) 11989833
=
119868 + 1198981198972 11988812
= 11988821
= minus(12)119898119897 sin(11990912)11990922
and the otherparameters of 119862 (119909
1 1199092) are zero 119899
1= minus(12)119898119897 sin(119909
12)11990922
1198992= minus119898119892119897 sin(119909
12) + (12)119898119897 sin(119909
12)11990922 therefore it can be
proven that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
and 1198991= 001 with 119870 = 001 Conditions given in (20)
119873 le 119870 119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the
error of the closed-loop dynamics of the PDDZ applied forpendulum systems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198921
Comparison results for the control functions are shown inFigure 5 position states are shown in Figure 6 and compar-ison results for the controller errors are shown in Figure 7Comparison of the square norm of the velocity errors11990922 of (20) for the controllers is presented in Figure 8
From the theorem of (20) 11990922 will converge to zero for the
PDDZ Table 1 shows the RMSE results using (27)The most important variable to control is the pendulum
angle 11990912
= 120579 and this variable may reach zero even if itstarts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results From Figures 5 6 and 7 it can
0 1 2 3 4 5 6 7 8 9 10Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 6 Position states for Example 1
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus004
minus006
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 7 Position state error for Example 1
be seen that the PDDZ improves the PD because the signalof the plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in thesecond From Figure 8 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 1 it
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
Time
0
200
400
600
800
||x21||2
PDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
0
005
01
015
02
||x22||2
PDDZPD
(b)
Figure 8 Velocity errors for Example 1
Table 1 Error results for Example 1
Methods RMSE for 119909 RMSE for 119906PD 42487 times 10
minus4 155044PDDZ 21995 times 10
minus4 77534
can be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
52 Example 2 Consider the Furuta pendulum [3 27] of theFigure 9
Furuta pendulum is written as (1) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(30)
where 11989811
= 1198680+ 1198981(1198712
0+ 1198972
1sin2(119909
12)) 11989812
= 11989821
= 119898111989711198710
cos(11990912) 11989833
= 1198691+ 11989811198972
1 11988811
= 11989811198972
1sin(2119909
12)11990922 11988812
=
minus119898111989711198710sin(11990912)11990922 11988821
= minus11989811198972
1sin(11990912) cos(119909
12)11990921 1198922=
minus11989811198921198971sin(11990912) and the other parameter of 119866(119909
1) is zero
sin(sdot) is the sine function cos(sdot) is the cosine function 1198680=
05Kgm2 is the arm inertia 1198691= 05 kgm2 is the pendulum
inertia 1198982= 034 kg is the arm mass 119898
1= 024 kg is the
z
x
y
m2g
1205790 m1g
J1
l1
120591
1205791F
FI0L0
Figure 9 Furuta pendulum
pendulum mass 1198710= 0293m is the arm length 119897
1= 028m
is the pendulum length 11990911
= 1205790
is the arm angle 11990912
=
1205791is the pendulum angle 119906 = 119865 is the motion torque of the
arm and 119892 = 981ms2 is the constant acceleration due togravity It can be proven that Property 2 of (7) is not satisfied
Furuta pendulum is written as (12) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(31)
where11989811
= 119872+11989811989812
= 11989821
= 119898119897 cos(11990912)11989833
= 119868+1198981198972
11988811
= 11989811198972
1sin(11990912) cos(119909
12)11990922 11988812
= minus(12)119898111989711198710sin(11990912)
11990922 11988821
= minus(12)11989811198972
1sin(11990912) cos(119909
12)11990922 and the other
parameter of 119862 (1199091 1199092) is zero 119899
1= 11989811198972
1sin(11990912) cos(119909
12)
11990922minus (12)119898
111989711198710sin(11990912)11990922 1198992= minus119898
11198921198971sin(11990912) minus (12)
11989811198972
1sin(11990912) cos(119909
12)(11990921minus 11990922) therefore it can be proven
that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
1198991= 001 with 119870 = 001 Conditions given in (20)119873 le 119870
119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the error of
the closed-loop dynamics of the PDDZ applied for pendulumsystems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198922
Comparison results for the control functions are shownin Figure 10 position states are shown in Figure 11 andcomparison results for the controller errors are shown inFigure 12 Comparison of the square norm of the velocityerrors 119909
22 of (20) for the controllers is presented in
Figure 13 From the theorem of (20) 11990922 will converge to
zero for the PDDZ Table 2 shows the RMSE results using(27)
The most important variable to control is the pendulumangle 119909
12= 1205791 and this variable may reach zero even if it
Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 10minus15
minus10
minus5
0
5
10
15
20
25
u1
PDDZPD
Figure 10 Input for Example 2
0 1 2 3 4 5 6 7 8 9 10
Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 11 Position states for Example 2
starts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results FromFigures 10 11 and 12 it canbe seen that the PDDZ improves the PD because the signal ofthe plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in the
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus006
minus004
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 12 Position state error for Example 2
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
||x21||2
020406080100
(a)
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
000200400600801
||x22||2
(b)
Figure 13 Velocity errors for Example 2
second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 2 itcan be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8 9 10
Time
0
200
400
600
800
||x21||2
PDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
0
005
01
015
02
||x22||2
PDDZPD
(b)
Figure 8 Velocity errors for Example 1
Table 1 Error results for Example 1
Methods RMSE for 119909 RMSE for 119906PD 42487 times 10
minus4 155044PDDZ 21995 times 10
minus4 77534
can be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
52 Example 2 Consider the Furuta pendulum [3 27] of theFigure 9
Furuta pendulum is written as (1) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119866 (1199091) = [
1198921
1198922
]
(30)
where 11989811
= 1198680+ 1198981(1198712
0+ 1198972
1sin2(119909
12)) 11989812
= 11989821
= 119898111989711198710
cos(11990912) 11989833
= 1198691+ 11989811198972
1 11988811
= 11989811198972
1sin(2119909
12)11990922 11988812
=
minus119898111989711198710sin(11990912)11990922 11988821
= minus11989811198972
1sin(11990912) cos(119909
12)11990921 1198922=
minus11989811198921198971sin(11990912) and the other parameter of 119866(119909
1) is zero
sin(sdot) is the sine function cos(sdot) is the cosine function 1198680=
05Kgm2 is the arm inertia 1198691= 05 kgm2 is the pendulum
inertia 1198982= 034 kg is the arm mass 119898
1= 024 kg is the
z
x
y
m2g
1205790 m1g
J1
l1
120591
1205791F
FI0L0
Figure 9 Furuta pendulum
pendulum mass 1198710= 0293m is the arm length 119897
1= 028m
is the pendulum length 11990911
= 1205790
is the arm angle 11990912
=
1205791is the pendulum angle 119906 = 119865 is the motion torque of the
arm and 119892 = 981ms2 is the constant acceleration due togravity It can be proven that Property 2 of (7) is not satisfied
Furuta pendulum is written as (12) and it is detailed asfollows
119872(1199091) = [
11989811
11989812
11989821
11989822
]
119862 (1199091 1199092) = [
11988811
11988812
11988821
11988822
]
119873 (1199091 1199092) = [
1198991
1198992
]
(31)
where11989811
= 119872+11989811989812
= 11989821
= 119898119897 cos(11990912)11989833
= 119868+1198981198972
11988811
= 11989811198972
1sin(11990912) cos(119909
12)11990922 11988812
= minus(12)119898111989711198710sin(11990912)
11990922 11988821
= minus(12)11989811198972
1sin(11990912) cos(119909
12)11990922 and the other
parameter of 119862 (1199091 1199092) is zero 119899
1= 11989811198972
1sin(11990912) cos(119909
12)
11990922minus (12)119898
111989711198710sin(11990912)11990922 1198992= minus119898
11198921198971sin(11990912) minus (12)
11989811198972
1sin(11990912) cos(119909
12)(11990921minus 11990922) therefore it can be proven
that the property of the lemma of (14) is satisfiedPDDZ is given by (17) and (18) as 119906pd = minus119870
11990111990912minus11987011988911990922+
1198991minus119870 sign(119909
22)with parameters119870
119901= 200119870
119889= 20 = 119899
1
1198991= 001 with 119870 = 001 Conditions given in (20)119873 le 119870
119870119901gt 0 and 119870
119889gt 0 are satisfied consequently the error of
the closed-loop dynamics of the PDDZ applied for pendulumsystems is guaranteed to be asymptotically stable
PD is given by [23] as 119906pd = minus11987011990111990912
minus 11987011988911990922
+ 1198922with
parameters 119870119901= 400 119870
119889= 20 and 119866(119909
1) = 1198922
Comparison results for the control functions are shownin Figure 10 position states are shown in Figure 11 andcomparison results for the controller errors are shown inFigure 12 Comparison of the square norm of the velocityerrors 119909
22 of (20) for the controllers is presented in
Figure 13 From the theorem of (20) 11990922 will converge to
zero for the PDDZ Table 2 shows the RMSE results using(27)
The most important variable to control is the pendulumangle 119909
12= 1205791 and this variable may reach zero even if it
Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 10minus15
minus10
minus5
0
5
10
15
20
25
u1
PDDZPD
Figure 10 Input for Example 2
0 1 2 3 4 5 6 7 8 9 10
Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 11 Position states for Example 2
starts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results FromFigures 10 11 and 12 it canbe seen that the PDDZ improves the PD because the signal ofthe plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in the
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus006
minus004
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 12 Position state error for Example 2
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
||x21||2
020406080100
(a)
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
000200400600801
||x22||2
(b)
Figure 13 Velocity errors for Example 2
second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 2 itcan be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
0 1 2 3 4 5 6 7 8 9 10minus15
minus10
minus5
0
5
10
15
20
25
u1
PDDZPD
Figure 10 Input for Example 2
0 1 2 3 4 5 6 7 8 9 10
Time
0
5
10
15
x11
DesiredPDDZPD
(a)
0 1 2 3 4 5 6 7 8 9 10
Time
minus01
minus005
000501015
x12
DesiredPDDZPD
(b)
Figure 11 Position states for Example 2
starts with other value as in this example Note that the PDtechnique requires the bigger gains than the PDDZ methodto obtain satisfactory results FromFigures 10 11 and 12 it canbe seen that the PDDZ improves the PD because the signal ofthe plant for the first follows better the desired signal thanthe second and in the first the inputs are smaller than in the
0 1 2 3 4 5 6 7 8 9 10Time
minus008
minus006
minus004
minus002
0
002
004
006
008
01
012
PDDZPD
x12
Figure 12 Position state error for Example 2
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
||x21||2
020406080100
(a)
PDDZPD
0 1 2 3 4 5 6 7 8 9 10Time
000200400600801
||x22||2
(b)
Figure 13 Velocity errors for Example 2
second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909
22 presented by the first
is smaller than that presented by the second From Table 2 itcan be shown that the PDDZ achieves better accuracy whencompared with the PD because the RMSE is smaller for thefirst than for the second
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 2 Error results for Example 2
Methods RMSE for 119909 RMSE for 119906PD 37760 times 10
minus4 136043PDDZ 20158 times 10
minus4 68328
6 Conclusion
In this research a proportional derivative controlwith inversedead-zone for pendulum systems with dead-zone inputs ispresented The simulations showed that the proposed tech-nique achieves better performance when compared with theproportional derivative control with gravity compensationfor the regulation of two pendulum systems and the resultsillustrate the viability efficiency and the potential of theapproach especially important in pendulum systems As afuture research the proposed study will be improved consid-ering that some parameters of the controller are unknown[28ndash32] or it will consider the communication delays andpacket dropout
Conflict of Interests
The authors declare no conflict of interests about all theaspects related to this paper
Acknowledgments
The authors are grateful to the editors and the reviewersfor their valuable comments and insightful suggestionswhich helped to improve this research significantly Theauthors thank the Secretarıa de Investigacion y Posgradothe Comision de Operacion y Fomento de ActividadesAcademicas del IPN and Consejo Nacional de Ciencia yTecnologıa for their help in this research
References
[1] J J Rubio and J H Perez-Cruz ldquoEvolving intelligent systemfor the modelling of nonlinear systems with dead-zone inputrdquoApplied Soft Computing In press
[2] C Aguilar-Ibanez J C Martınez-Garcıa A Soria-Lopez andJ D J Rubio ldquoOn the stabilization of the inverted-cart pen-dulum using the saturation function approachrdquo MathematicalProblems in Engineering vol 2011 Article ID 856015 14 pages2011
[3] C Aguilar-Ibanez M S Suarez-Castanon and O O Gutierres-Frias ldquoThe direct Lyapunov method for the stabilisation of thefuruta pendulumrdquo International Journal of Control vol 83 no11 pp 2285ndash2293 2010
[4] D Ding Z Wang J Hu and H Shu ldquoDissipative control forstate-saturated discrete time-varying systems with randomlyoccurring nonlinearities and missing measurementsrdquo Interna-tional Journal of Control vol 86 no 4 pp 674ndash688 2013
[5] H Dong Z Wang J Lam and H Gao ldquoDistributed filteringin sensor networks with randomly occurring saturations andsuccessive packet dropoutsrdquo International Journal of Robust andNonlinear Control 2013
[6] J Hu Z Wang B Shen and H Gao ldquoQuantised recursivefiltering for a class of nonlinear systems with multiplicativenoises and missing measurementsrdquo International Journal ofControl vol 86 no 4 pp 650ndash663 2013
[7] J Hu Z Wang B Shen and H Gao ldquoGain-constrained recur-sive filtering with stochastic nonlinearties and probabilisticsensor delaysrdquo IEEE Transactions on Signal Processing vol 61no 5 pp 1230ndash1238 2013
[8] M Jimenez-Lizarraga M Basin and P Rodriguez-RamirezldquoRobust mini-max regulator for uncertain non-linear polyno-mial systemsrdquo IET Control Theory amp Applications vol 6 no 7pp 963ndash970 2012
[9] A F de Loza M Jimenez-Lizarraga and L Fridman ldquoRobustoutput nash strategies based on sliding mode observation in atwo-player differential gamerdquo Journal of the Franklin Institutevol 349 no 4 pp 1416ndash1429 2012
[10] Q Lu L Zhang H R Karimi and Y Shi ldquo119867infin
controlfor asynchronously switched linear parameter-varying systemswith mode-dependent average dwell timerdquo IET Control Theoryamp Applications vol 7 no 5 pp 677ndash683 2013
[11] L F Ma Z D Wang and Z Guo ldquoRobust H2sliding mode
control for non-linear discrete-time stochastic systemsrdquo IETControl Theory and Applications vol 3 no 11 pp 1537ndash15462009
[12] Y Niu D W C Ho and Z Wang ldquoImproved sliding modecontrol for discrete-time systems via reaching lawrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2245ndash2251 2010
[13] L A Soriano W Yu and J J Rubio ldquoModeling and control ofwind turbinerdquoMathematical Problems in Engineering vol 2013Article ID 982597 13 pages 2013
[14] J J Rubio G Ordaz M Jimenez-Lizarraga and R I CabreraldquoGeneral solution to the Navier-Stokes equation to describe thedynamics of a homogeneous viscous fluid in an open piperdquoRevista Mexicana de Fisica vol 59 no 3 pp 217ndash223 2013
[15] Y Wang H R Karimi and Z Xiang ldquoDelay-dependent119867infin
control for networked control systems with large delaysrdquoMathematical Problems in Engineering vol 2013 Article ID643174 10 pages 2013
[16] Y Wang H R Karimi and Z Xiang ldquo119867infin
control for net-worked control systems with time delays and packet dropoutsrdquoMathematical Problems in Engineering vol 2013 Article ID635941 10 pages 2013
[17] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[18] S Abrir ldquoInvarian-manifold approach to the stabilization offeedforward nonlinear systems having uncertain dead-zoneinputsrdquo in Proceedings of the International Sysmposium ofMechatronics and Its Applications pp 1ndash5 2012
[19] JH Perez-Cruz E Ruiz-Velazquez J J Rubio andCA deAlvaPadilla ldquoRobust adaptive neurocontrol of SISO nonlinear sys-tems preceded by unknow deadzonerdquo Mathematical Problemsin Engineering vol 2012 Article ID 342739 23 pages 2012
[20] J H Perez-Cruz J J Rubio E Ruiz-Velazquez and G Sols-Perales ldquoTracking control based on recurrent neural networksfor nonlinear systems with multiple inputs and unknowndeadzonerdquo Abstract and Applied Analysis vol 2012 Article ID471281 18 pages 2012
[21] Z Wang Y Zhang and H Fang ldquoNeural adaptive control fora class of nonlinear systems with unknown deadzonerdquo NeuralComputing and Applications vol 17 no 4 pp 339ndash345 2008
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
[22] J Zhou and X Z Shen ldquoRobust adaptive control of nonlinearuncertain plants with unknown dead-zonerdquo IET ControlTheoryand Applications vol 1 no 1 pp 25ndash32 2007
[23] F L Lewis D M Dawson and C T Abdallah Control of RobotManipulators Theory and Practice CRC Press New York NYUSA 2004
[24] J E Slotine and W Li Applied Nonlinear Control MacmillanEnglewood Clifs NJ USA 1991
[25] M W Spong and M Vidyasagar Robot Dynamics and ControlJohn Wiley amp Sons 1989
[26] J J Rubio G Gutierrez J Pacheco and J H Perez-Cruz ldquoCom-parison of three proposed controls to accelerate the growthof the croprdquo International Journal of Innovative ComputingInformation and Control vol 7 no 7 pp 4097ndash4114 2011
[27] J J Rubio M Figueroa J H Perez-Cruz and J RumboldquoControl to stabilize and mitigate disturbances in a rotaryinverted pendulumrdquo Revista Mexicana de Fisica E vol 58 no2 pp 107ndash112 2012
[28] J A Iglesias P Angelov A Ledezma and A Sanchis ldquoCreatingevolving user behavior profiles automaticallyrdquo IEEE Transac-tions onKnowledge andData Engineering vol 24 no 5 pp 854ndash867 2011
[29] D Leite R Ballini P Costa and F Gomide ldquoEvolving fuzzygranular modeling from nonstationary fuzzy data streamsrdquoEvolving Systems vol 3 no 2 pp 65ndash79 2012
[30] E Lughofer ldquoSingle pass active learning with conflict andignorancerdquo Evolving Systems vol 3 pp 251ndash271 2012
[31] E Lughofer ldquoA dynamic split-and-merge approach for evolvingcluster modelsrdquo Evolving Systems vol 3 pp 135ndash151 2012
[32] L Maciel A Lemos F Gomide and R Ballini ldquoEvolving fuzzysystems for pricing fixed income optionsrdquo Evolving Systems vol3 no 1 pp 5ndash18 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of