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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)


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)


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


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


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


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


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


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


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


0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

15

20

25

u1

PDDZPD


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)


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


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)


second From Figure 13 it is shown that the PDDZ improvesthe PD because the velocity error 119909


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





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


[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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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




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



2are defined in (3)

Consequently

119909119879

[ (1199091) minus 2119862 (119909

1 1199092)] 119909 = 0 (11)


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)


= 11988811

+ 11988811 11988812

= 11988812

+ 11988812

11988821

= 11988821

+ 11988821 11988822

= 11988822

+ 11988822 1198991= 1198921+ 1198881111990921

+ 1198881211990922

1198992= 1198922+ 1198882111990921+ 1198882211990922 1199091and 119909



119909119879

[ (1199091) minus 2119862 (119909

1 1199092)] 119909 = 0 (14)


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)


119872(1199091) 2= minus119862 (119909

1 1199092) 1199092minus 119873 (119909

1 1199092) + 119906

119909119879

[ (1199091) minus 2119862 (119909

1 1199092)] 119909 = 0

(16)


1 1199092) and119873(119909


(14) is satisfied







1

119898119903

119906pd + 119887119903

119906pd gt 0

0 119906pd = 0

1

119898119897

119906pd + 119887119897

119906pd lt 0

(17)





2) (18)

where 1199091= 1199091minus 119909119889


1and 119909

2= 1199092are

defined in (3)1199091198891= [119909119889

11119909119889



of 119873(1199091 1199092) and 119873(119909



1 1199092)

is bounded as1003816100381610038161003816100381610038161003816100381610038161003816le 119873 (19)



2will converge


100381710038171003817100381711990921003817100381710038171003817

2

= 0 (20)


119901gt 0 and119870

119889gt 0


1198811=1

2119909119879

2119872(1199091) 1199092+1

2119909119879

11198701199011199091 (21)


119872(1199091) 2= minus 119870

1199011199091minus 1198701198891199092+

minus 119870 sign (1199092) minus 119862 (119909

1 1199092) 1199092

(22)


1= 119909119879

2119872(1199091) 2+1

2119909119879

2 (1199091) 1199092+ 119909119879

21198701199011199091 (23)


minus8

minus6

minus4

minus2

0

2

4

6

8

10


0bl

br

1ml

1mr


xd1

+

minussum


x1 x2

PDDZ

upd

x1 x2


where 1199091= 1minus 119889

1= 1199092minus 119909119889

2= 1199092= 1199092and 119909

2= 2


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)


int

119879

0

119909119879

21198701198891199092119889119905 le 119881

10minus 1198811119879

le 11988110

1

119879int

119879

0

119909119879

21198701198891199092119889119905 le

1

11987911988110


(1

119879int

119879

0

119909119879

21198701198891199092119889119905) le 119881


(1

119879)] = 0

(26)





when 119909119889


y

FM

mg

lx

120579

cos(120579)


5 Simulations


RMSE = (1

119879int

119879

0

1199092

119889119905)

12

(27)

where 1199092 = 1199092

12or 1199092 = 119906

2

1



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


1

1199092) are zero 119892

2= minus119898119892119897 sin(119909





12is the





0 1 2 3 4 5 6 7 8 9 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

PDDZPD

u1



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


1 1199092) are zero 119899

1= minus(12)119898119897 sin(119909

12)11990922

1198992= minus119898119892119897 sin(119909

12) + (12)119898119897 sin(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119873 le 119870 119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198921




angle 11990912


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)


0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus004

minus006

minus002

0

002

004

006

008

01

012

PDDZPD

x12






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)





minus4 77534




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=


1) is zero




1= 024 kg is the

z

x

y

m2g

1205790 m1g

J1

l1

120591

1205791F

FI0L0



1= 028m


= 1205790


=




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



1= 11989811198972

1sin(11990912) cos(119909

12)

11990922minus (12)119898

111989711198710sin(11990912)11990922 1198992= minus119898

11198921198971sin(11990912) minus (12)

11989811198972

1sin(11990912) cos(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198922








0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

15

20

25

u1

PDDZPD


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)



0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus006

minus004

minus002

0

002

004

006

008

01

012

PDDZPD

x12


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)









minus4 68328

6 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












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



2are defined in (3)

Consequently

119909119879

[ (1199091) minus 2119862 (119909

1 1199092)] 119909 = 0 (11)


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)


= 11988811

+ 11988811 11988812

= 11988812

+ 11988812

11988821

= 11988821

+ 11988821 11988822

= 11988822

+ 11988822 1198991= 1198921+ 1198881111990921

+ 1198881211990922

1198992= 1198922+ 1198882111990921+ 1198882211990922 1199091and 119909



119909119879

[ (1199091) minus 2119862 (119909

1 1199092)] 119909 = 0 (14)


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)


119872(1199091) 2= minus119862 (119909

1 1199092) 1199092minus 119873 (119909

1 1199092) + 119906

119909119879

[ (1199091) minus 2119862 (119909

1 1199092)] 119909 = 0

(16)


1 1199092) and119873(119909


(14) is satisfied







1

119898119903

119906pd + 119887119903

119906pd gt 0

0 119906pd = 0

1

119898119897

119906pd + 119887119897

119906pd lt 0

(17)





2) (18)

where 1199091= 1199091minus 119909119889


1and 119909

2= 1199092are

defined in (3)1199091198891= [119909119889

11119909119889



of 119873(1199091 1199092) and 119873(119909



1 1199092)

is bounded as1003816100381610038161003816100381610038161003816100381610038161003816le 119873 (19)



2will converge


100381710038171003817100381711990921003817100381710038171003817

2

= 0 (20)


119901gt 0 and119870

119889gt 0


1198811=1

2119909119879

2119872(1199091) 1199092+1

2119909119879

11198701199011199091 (21)


119872(1199091) 2= minus 119870

1199011199091minus 1198701198891199092+

minus 119870 sign (1199092) minus 119862 (119909

1 1199092) 1199092

(22)


1= 119909119879

2119872(1199091) 2+1

2119909119879

2 (1199091) 1199092+ 119909119879

21198701199011199091 (23)


minus8

minus6

minus4

minus2

0

2

4

6

8

10


0bl

br

1ml

1mr


xd1

+

minussum


x1 x2

PDDZ

upd

x1 x2


where 1199091= 1minus 119889

1= 1199092minus 119909119889

2= 1199092= 1199092and 119909

2= 2


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)


int

119879

0

119909119879

21198701198891199092119889119905 le 119881

10minus 1198811119879

le 11988110

1

119879int

119879

0

119909119879

21198701198891199092119889119905 le

1

11987911988110


(1

119879int

119879

0

119909119879

21198701198891199092119889119905) le 119881


(1

119879)] = 0

(26)





when 119909119889


y

FM

mg

lx

120579

cos(120579)


5 Simulations


RMSE = (1

119879int

119879

0

1199092

119889119905)

12

(27)

where 1199092 = 1199092

12or 1199092 = 119906

2

1



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


1

1199092) are zero 119892

2= minus119898119892119897 sin(119909





12is the





0 1 2 3 4 5 6 7 8 9 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

PDDZPD

u1



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


1 1199092) are zero 119899

1= minus(12)119898119897 sin(119909

12)11990922

1198992= minus119898119892119897 sin(119909

12) + (12)119898119897 sin(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119873 le 119870 119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198921




angle 11990912


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)


0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus004

minus006

minus002

0

002

004

006

008

01

012

PDDZPD

x12






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)





minus4 77534




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=


1) is zero




1= 024 kg is the

z

x

y

m2g

1205790 m1g

J1

l1

120591

1205791F

FI0L0



1= 028m


= 1205790


=




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



1= 11989811198972

1sin(11990912) cos(119909

12)

11990922minus (12)119898

111989711198710sin(11990912)11990922 1198992= minus119898

11198921198971sin(11990912) minus (12)

11989811198972

1sin(11990912) cos(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198922








0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

15

20

25

u1

PDDZPD


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)



0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus006

minus004

minus002

0

002

004

006

008

01

012

PDDZPD

x12


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)









minus4 68328

6 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus8

minus6

minus4

minus2

0

2

4

6

8

10


0bl

br

1ml

1mr


xd1

+

minussum


x1 x2

PDDZ

upd

x1 x2


where 1199091= 1minus 119889

1= 1199092minus 119909119889

2= 1199092= 1199092and 119909

2= 2


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)


int

119879

0

119909119879

21198701198891199092119889119905 le 119881

10minus 1198811119879

le 11988110

1

119879int

119879

0

119909119879

21198701198891199092119889119905 le

1

11987911988110


(1

119879int

119879

0

119909119879

21198701198891199092119889119905) le 119881


(1

119879)] = 0

(26)





when 119909119889


y

FM

mg

lx

120579

cos(120579)


5 Simulations


RMSE = (1

119879int

119879

0

1199092

119889119905)

12

(27)

where 1199092 = 1199092

12or 1199092 = 119906

2

1



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


1

1199092) are zero 119892

2= minus119898119892119897 sin(119909





12is the





0 1 2 3 4 5 6 7 8 9 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

PDDZPD

u1



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


1 1199092) are zero 119899

1= minus(12)119898119897 sin(119909

12)11990922

1198992= minus119898119892119897 sin(119909

12) + (12)119898119897 sin(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119873 le 119870 119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198921




angle 11990912


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)


0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus004

minus006

minus002

0

002

004

006

008

01

012

PDDZPD

x12






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)





minus4 77534




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=


1) is zero




1= 024 kg is the

z

x

y

m2g

1205790 m1g

J1

l1

120591

1205791F

FI0L0



1= 028m


= 1205790


=




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



1= 11989811198972

1sin(11990912) cos(119909

12)

11990922minus (12)119898

111989711198710sin(11990912)11990922 1198992= minus119898

11198921198971sin(11990912) minus (12)

11989811198972

1sin(11990912) cos(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198922








0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

15

20

25

u1

PDDZPD


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)



0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus006

minus004

minus002

0

002

004

006

008

01

012

PDDZPD

x12


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)









minus4 68328

6 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

PDDZPD

u1



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


1 1199092) are zero 119899

1= minus(12)119898119897 sin(119909

12)11990922

1198992= minus119898119892119897 sin(119909

12) + (12)119898119897 sin(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119873 le 119870 119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198921




angle 11990912


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)


0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus004

minus006

minus002

0

002

004

006

008

01

012

PDDZPD

x12






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)





minus4 77534




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=


1) is zero




1= 024 kg is the

z

x

y

m2g

1205790 m1g

J1

l1

120591

1205791F

FI0L0



1= 028m


= 1205790


=




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



1= 11989811198972

1sin(11990912) cos(119909

12)

11990922minus (12)119898

111989711198710sin(11990912)11990922 1198992= minus119898

11198921198971sin(11990912) minus (12)

11989811198972

1sin(11990912) cos(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198922








0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

15

20

25

u1

PDDZPD


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)



0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus006

minus004

minus002

0

002

004

006

008

01

012

PDDZPD

x12


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)









minus4 68328

6 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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)





minus4 77534




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=


1) is zero




1= 024 kg is the

z

x

y

m2g

1205790 m1g

J1

l1

120591

1205791F

FI0L0



1= 028m


= 1205790


=




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



1= 11989811198972

1sin(11990912) cos(119909

12)

11990922minus (12)119898

111989711198710sin(11990912)11990922 1198992= minus119898

11198921198971sin(11990912) minus (12)

11989811198972

1sin(11990912) cos(119909



11990111990912minus11987011988911990922+

1198991minus119870 sign(119909


119901= 200119870

119889= 20 = 119899

1


119870119901gt 0 and 119870




minus 11987011988911990922

+ 1198922with

parameters 119870119901= 400 119870

119889= 20 and 119866(119909

1) = 1198922








0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

15

20

25

u1

PDDZPD


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)



0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus006

minus004

minus002

0

002

004

006

008

01

012

PDDZPD

x12


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)









minus4 68328

6 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

15

20

25

u1

PDDZPD


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)



0 1 2 3 4 5 6 7 8 9 10Time

minus008

minus006

minus004

minus002

0

002

004

006

008

01

012

PDDZPD

x12


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)









minus4 68328

6 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













minus4 68328

6 Conclusion




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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