Research Article Robust Stability for Nonlinear Systems with ...Research Article Robust Stability for Nonlinear Systems with Time-Varying Delay and Uncertainties via the Quasi-Sliding

Research ArticleRobust Stability for Nonlinear Systems with Time-Varying Delayand Uncertainties via the119867

infinQuasi-Sliding Mode Control

Yi-You Hou1 and Zhang-Lin Wan2

1 Department of Electrical Engineering Far East University Tainan 744 Taiwan2 TEL JAN Precision Machine Co LTD Kaohsiung 814 Taiwan

Correspondence should be addressed to Yi-You Hou houyi youmsahinetnet

Received 6 January 2014 Revised 16 March 2014 Accepted 17 March 2014 Published 22 April 2014

Academic Editor Chung-Liang Chang

Copyright copy 2014 Y-Y Hou and Z-L Wan 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

This paper considers the problem of the robust stability for the nonlinear system with time-varying delay and parametersuncertainties Based on the119867

infintheorem Lyapunov-Krasovskii theory and linear matrix inequality (LMI) optimization technique

the119867infinquasi-sliding mode controller and switching function are developed such that the nonlinear system is asymptotically stable

in the quasi-sliding mode and satisfies the disturbance attenuation (119867infin-norm performance)The effectiveness and accuracy of the

proposed methods are shown in numerical simulations

1 Introduction

In general in all engineering systems such as communicationsystem electronics neural networks and control systemsdisturbance usually emerges in many kinds of situationand then affects the performances of systems It deservesto be mentioned that in some special cases disturbancecan reform the instant condition to improve system per-formances For examples the proper local noise power canincrease the signal-to-noise ratio of a noisy bistable systemand optimize the stochastic resonance of coupled systemssuch as small-world and scale-free neuronal networks [1ndash3] However disturbances such as uncertainty [4ndash6] impulseinterference [7] and nonautonomous effects [8] are usuallythe unstable and destructive elements to force the mainsystem to make poor performances and abrupt changesThat is why restraining disturbance is always the first andserious study in system control Recently the119867

infincontrolled

method is usually used to suppress disturbances in manyinvestigations of the robust control Depending on its effec-tive utilization the disturbance can be restrained with thedisturbance attenuation gain in the purposed controller [9ndash14] In this paper we consider a nonlinear system with thenonautonomous noise time-varying delay and perturbation

of the uncertainty Based on the 119867infin

theorem a 119867infin

quasi-sliding mode controller is designed to guarantee the stabilityof the main nonlinear system

In these two decades sliding mode control (SMC) hasbeen a useful and distinctive robust control strategy formany kinds of engineer systems Depending on the pro-posed switching surface and discontinuous controller thetrajectories of dynamic systems can be guided to the fixedsliding manifold The proposed performance on request canbe satisfied In general there are two main advantages ofSMC which are the reducing order of dynamics from thepurposed switching functions and robustness of restrainingsystem uncertainties Many studies have been conductedon SMC [15ndash20] However the chattering phenomenon is aserious problem which is needed to overcome In the realapplication the chattering phenomenonmay cause electroniccircuits to be superheated and broken In order to solve thisproblem quasi-sliding mode control method is studiedrecently [21ndash25] Based on the quasi-sliding mode controllerwhen the trajectories of dynamic systems converge to thesliding manifold the trajectories can be bounded in thesettled region along the sliding surface and the impulse andchattering phenomenon can be avoided

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 897179 7 pageshttpdxdoiorg1011552014897179

2 International Journal of Antennas and Propagation

On the other hand in the past most of papers set thecontrol gain in advance to achieve the sufficient condition ofthe stability It is not a suitable and accurate way to definethe parameters of the system although the parameters aregotten by trial and error Therefore in this paper we use thelinearmatrix inequality (LMI) theorem to optimize the quasi-sliding mode control gain By using the computer softwareMATLAB quasi-sliding mode controller gain can be foundBased on the119867

infintheorem Lyapunov-Krasovskii theory and

LMI optimization technique the 119867infin

quasi-sliding modecontrol and switching function can be designed such thatthe resulting nonlinear system is asymptotically stable in thequasi-sliding mode and satisfies the disturbance attenuation(119867infin-norm performance)Throughout this paper 119868 denotes the identity matrix of

appropriate dimensions For a real matrix 119860 we denote thetranspose by 119860

119879 and spectral norm by 119860 Considering119876 = 119876

119879

gt 0 (119876 = 119876119879

lt 0) 119876 is a symmetric positive(negative) definite matrixThe notation lowast in symmetric blockmatrices or long matrix expressions throughout the paperrepresents an ellipsis for terms that are induced by symmetryfor example [ 119861 119862

lowast 119863] = [

119861 119862

119862119879119863] For a vector 119909 119909(119905)

means the Euclidean vector norm at time 119905 while 1199092=

radicint

infin

0

119909(119905)2

119889119905 If 1199092lt infin then 119909(119905) isin 119871

2[0infin) where

1198712[0infin) stands for the space of square integral functions on

[0infin)This paper is organized as follows In Section 2 119867

infin

theorem LMI optimization techniques and the Lyapunov-Krasovskii stability theorem are used to derive the proposedcontroller switching function and corresponding parame-ters such that the nonlinear system is asymptotically stablein the quasi-sliding mode Then the central discussion ofthis paper is illustrated by a numerical example in Section 3Finally some conclusions are presented in Section 4

2 Nonlinear System Description andProposed Controller Design

In this paper we consider the stability of a nonlinearuncertain neutral systemwith time-varying delayed state anddisturbance At first we consider the system without theuncertain segmentThen a simple extension of nonlinear sys-tems with uncertain part is considered at the next narrationAsmentioned above we firstly consider the nonlinear system(1) without uncertain part as follows

(119905) = 1198601119909 (119905) + 119860

2119909 (119905 minus ℎ (119905))

+ 119861119906 (119905) + 119862119891 (119909 (119905)) + 119863119908 (119905)

119909 (119905) = 120601 (119905) 119905 isin [minusℎ119872 0]

(1)

where 0 le ℎ(119905) le ℎ119872 ℎ(119905) le ℎ

119863lt 1 119909(119905) isin 119877

119899 isthe nonlinear system state 119891(119909(119905)) isin 119877

119898 is a continuousnonlinear function vector 119908(119905) isin 119871

2[0infin) is the noise

perturbation input 119906(119905) isin 119877119898 is a control input vector and

the matrices 1198601 1198602 119861 119862 and 119863 are some known constant

matrices with appropriate dimensions The initial vector 120601 isa differentiable function from [minusℎ

119872 0] to 119877119899

Before presenting the main result we need the followinglemma and definition

Lemma 1 Let119872119873 and 119865(119905) be real matrices of appropriatedimensions with 119865(119905) le 1 and any scalar then

119872119879

119865 (119905)119873 + 119873119879

119865119879

(119905)119872 lt 120576minus1

sdot 119872119879

119872

+ 120576minus1

sdot (120576119873)119879

(120576119873) lt 0

(2)

Lemma2 (Schur complement of [26]) For a givenmatrix 119878 =[11987811 11987812

11987821 11987822

] with 11987811= 119878119879

11 11987822= 119878119879

22 the following conditions are

equivalent

(1) 119878 lt 0

(2) 11987822lt 0 119878

11minus 11987812119878minus1

2211987821lt 0

Definition 3 119867infincontrol problemaddressed in this paper can

be formulated as finding a stabilizing controller such that thefollowing conditions are held

Under the control input 119906(119905) the states of the nonlinearsystem (1) are asymptotically stable when 119908(119905) = 0

Under the zero initial conditions the performance index119869 = int

infin

0

[119909119879

(119905)119909(119905)minus1205742

sdot119908119879

(119905)119908(119905)]119889119905 le 0 (1199092le 120574119908

2) holds

for any nonzero 119908 isin 1198712and the119867

infinperformance 120574 gt 0

This paper aims at proposing the119867infinquasi-sliding mode

controller and switching function to not only asymptoticallystabilize the nonlinear system but also guarantee a prescribedperformance of noise perturbation attenuation 119903 gt 0 Thefirst step is to select an appropriate switching surface which isshown as (3) to ensure the asymptotical stability of the slidingmotion on the sliding manifold

119878 (119905) = 119866119909 (119905) + 119903 (119905)

119903 (119905) = 119866119861119903 (119905) minus 1198661198601119909 (119905)

minus 1198661198602119909 (119905 minus ℎ (119905)) minus 119866119861119870119909 (119905) minus 119866119861119870119909 (119905 minus ℎ (119905))

(3)

where 119878(119905) isin 119877119898 is the switching surface 119870 is the quasi-

sliding mode controller gain and 119866 is chosen such that 119866119861is nonsingular and 119866119863 = 0 When the system is placed onthe switching surface it satisfies the following equations

119878 (119905) = 0119878 (119905) = 0 (4)

Then the equivalent controller 119906eq in the slidingmanifoldcan be obtained by solving for 119878(119905) = 0

119878 (119905) = 1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119861119906eq (119905)

+ 119866119862119891 (119909 (119905)) + 119866119863119908 (119905) + 119866119861119903 (119905) minus 1198661198601119909 (119905)


(5)

International Journal of Antennas and Propagation 3

From 119903(119905) = minus119866119909(119905) as 119878(119905) = 0

119878 (119905) = 1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119861119906eq (119905)

+ 119866119862119891 (119909 (119905)) + 119866119863119908 (119905) minus 119866119861119866119909 (119905) minus 1198661198601119909 (119905)

minus 1198661198602119909 (119905 minus ℎ (119905)) minus 119866119861119870119909 (119905) minus 119866119861119870119909 (119905 minus ℎ (119905)) = 0

(6)

Since119866119861 is nonsingular the equivalent controller119906eq(119905) in thesliding mode is given by

119906eq (119905) = 119870119909 (119905) + 119870119909 (119905 minus ℎ (119905))

minus (119866119861)minus1

119866119862119891 (119909 (119905)) + 119866119909 (119905)

(7)

Therefore substituting 119906eq(119905) into (1) the nonlinear dynamicsin the quasi-sliding mode can be obtained

(119905) = (1198601+ 119861119870 + 119861119866) 119909 (119905)

+ (1198602+ 119861119870) 119909 (119905 minus ℎ (119905)) + 119863119908 (119905)

(8)

Then the second step is to design the proposed119867infinquasi-

sliding mode controller In the following Theorem 4 willderive that the trajectories of the nonlinear dynamics systemconverge to the quasi-sliding manifold based on the quasi-sliding mode controller 119906(119905)

Theorem 4 Consider nonlinear dynamics system (1) with theswitching function the trajectories of the nonlinear dynamicssystem converge to the quasi-sliding manifold if the controller119906(119905) is given by

119906 (119905) = 119870119909 (119905) + 119870119909 (119905 minus ℎ (119905)) + 119866119909 (119905)

minus (119866119861)minus1

(119866119862119891 (119909 (119905)) +

119878 (119905)

(119878 (119905) + 120575119876)

)

(9)

Proof Define the Lyapunov function

119881119904(119905) =

1

2

119878119879

(119905) 119878 (119905) (10)

Taking the derivative of 119881119904(119905) and introducing (3) one has

119904(119905) = 119878

119879

(119905)119878 (119905)

= 119878119879

(119905) [1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119862119891 (119909 (119905))

+ 119866119861119906 (119905) + 119866119863119908 (119905) minus 119866119861119866119909 (119905)

minus 1198661198601119909 (119905) minus 119866119860

2119909 (119905 minus ℎ (119905))

minus119866119861119870119909 (119905) minus 119866119861119870119909 (119905 minus ℎ (119905)) ]

= minus119878119879

(119905) (

119878 (119905)

(119878 (119905) + 120575119876)

)

= minus

119878 (119905)2

(119878 (119905) + 120575119876)

= minus(119878 (119905) minus

(119878 (119905)2

+ 119878 (119905) 120575119876minus 119878 (119905)

2

)

(119878 (119905) + 120575119876)

)

= minus(119878 (119905) minus

119878 (119905) 120575119876

(119878 (119905) + 120575119876)

)

lt minus (119878 (119905) minus 120575119876)

(11)

From the above proof we can obtain that 119878(119905) gt 120575119876is stable

which satisfies the reached condition of quasi-sliding modecontrol This completes the proof

In order to solve the 119867infin

quasi-sliding mode controllergain 119870 in the switching surface and optimized performanceof noise perturbation attenuation 120574 gt 0 we use the LMIformulation to look for them such that the nonlinear dynamicsystem (1) is asymptotically stable in the quasi-sliding mode

Theorem 5 Consider nonlinear dynamics system (1) if thereare a constant 120574 gt 0 positive definite symmetric matrix 119875

1and

1198752 and matrix119870 such that the following LMI condition holds

Σ2lt 0 (12)

where Σ12

= 11986021198751+ 119861119870 Σ

13= 119863 Σ

14= 1198751 Σ22

=

minus(1 minus ℎ119863)1198752 Σ33

= minus1205742

sdot 119868 Σ44

= 119868 and Σ11

= 1198751119860119879

1+

11986011198751+119870

119879

119861119879

+119861119870+1198751119866119879

119861119879

+1198611198661198751+1198752Then the nonlinear

dynamic system (1) is asymptotically stable in the quasi-slidingmode with noise perturbation attenuation 120574 gt 0 and quasi-sliding mode controller with its gain 119870 = 119870119875

1

minus1

Proof Define Lyapunov function

119881 (119905) = 119909119879

(119905) 1198751119909 (119905) + int

119905

119905minusℎ(119905)

119909119879

(120591) 1198752119909 (120591) 119889120591 (13)


(119905) = 119909119879

(119905) (119860119879

11198751+ 11987511198601+ 119870119879

119861119879

1198751

+1198751119861119870 + 119866

119879

119861119879

1198751+ 1198751119861119866) 119909 (119905)

+ 119909119879

(119905) (11987511198602+ 1198751119861119870) 119909 (119905 minus ℎ (119905))

+ 119909119879

(119905 minus ℎ (119905)) (119860119879

11198751+ 119870119879

119861119879

1198751) 119909 (119905)

+ 2119909119879

(119905) 1198751119863119908 (119905) + 119909

119879

(119905) 1198752119909 (119905)

minus (1 minusℎ (119905)) sdot 119909

119879

(119905 minus ℎ (119905)) 1198752119909 (119905 minus ℎ (119905))

(14)


Define a function 119869(119905) as follows

119869 (119905) = (119905) + 119909119879

(119905) 119909 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

le[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

119879

Σ1

[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

(15)

where Σ1= [

Σ11 Σ12 Σ13

lowast Σ22 0

lowast lowast Σ33

] Σ11= 119860119879

11198751+ 11987511198601+ 119870119879

119861119879

1198751+

1198751119861119870 + 119866

119879

119861119879

1198751+ 1198751119861119866 + 119875

2+ 119868 Σ

12= 11987511198602+ 119875119861119870 Σ

13=

1198751119863 Σ

22= minus(1 minus ℎ

119863)1198752 and Σ

33= minus1205742

sdot 119868Pre-multiplying and post-multiplying the Σ

1by Φ =

diag lfloor1198751

minus1

1198751

minus1

119868rfloor defining 1198751= 1198751

minus1 1198752= 119875111987521198751 and

119870 = 1198701198751

minus1 we can obtain

Σ2=

[

[

[

[

Σ11

Σ12

Σ13

Σ14

lowast Σ22

0 0

lowast lowast Σ33

0

lowast lowast lowast Σ44

]

]

]

]

(16)

where Σ11= 1198751119860119879

1+11986011198751+119870

119879

119861119879

+119861119870+1198751119866119879

119861119879

+1198611198661198751+1198752

Σ12= 11986021198751+119861119870 Σ

13= 119863 Σ

14= 1198751 Σ22= minus(1minusℎ

119863)1198752 Σ33=

minus1205742

sdot 119868 and Σ44= minus119868

If matrix (16) is a negative definite matrix and 119908(119905) = 0there exists a 120588 gt 0 such that

(119905)

10038161003816100381610038161003816119908(119905)=0

le minus120588 sdot 119909(119905)2 (17)

which guarantees the stability of the system (1) Thereforeby the Lyapunov-Krasovskii stability theorem the nonlinearsystem with 119908(119905) = 0 is asymptotically stable Furthermorecombining the condition Σ

2lt 0 with (15) from 0 to infin

obviously

119881 (119909infin) minus 119881 (120601) + 119909 (119905)

2

2minus 1205742

sdot 119908 (119905)2

2le 0 (18)

In zero initial condition (120601 = 0) it can be denoted by

119881 (120601) = 0 119881 (119909infin) ge 0

119909 (119905)2le 1205742

sdot 119908 (119905)2 forall119908 (119905) isin 119871

2[0infin)

(19)

Based on Definition 3 the nonlinear dynamic system isstabilized with noise perturbation attenuation 120574 gt 0 by thequasi-sliding mode control gain given by 119870 = 119870119875

1

minus1 Thiscompletes the proof

At last we consider the nonlinear system (1) with uncer-tain part as the following form

(119905) = 1198601(119905) 119909 (119905) + 119860

2(119905) 119909 (119905 minus ℎ (119905)) + 119861 (119905) 119906 (119905)

+ 119862 (119905) 119891 (119909 (119905)) + 119863 (119905) 119908 (119905)

119909 (119905) = 120601 (119905) 119905 isin [minusℎ119872 0]

(20)

where 1198601(119905) = 119860

1+ Δ119860

1(119905) 119860

2(119905) = 119860

2+ Δ119860

2(119905) 119861(119905) =

119861 + Δ119861(119905) 119862(119905) = 119862 + Δ119862(119905) and 119863(119905) = 119863 + Δ119863(119905)Δ1198601(119905) Δ119860

2(119905) Δ119861(119905) Δ119862(119905) and Δ119863(119905) are some time-

varying continuous perturbed matrices and satisfy

[Δ1198601(119905) Δ119860

2(119905) Δ119861 (119905) Δ119862 (119905) Δ119863 (119905)]

= 119872 sdot 119865 (119905) sdot [11987311198732119873311987341198735]

(21)

where 119872 and 119873119894 119894 isin 1 sim 5 are some given constant

matrices and 119865(119905) is unknown continuous function satisfying

119865119879

(119905) sdot 119865 (119905) le 119868 forall119905 ge 0 (22)

The following result is obtained fromTheorem 5Thiswillbe a simple extension of Theorem 5

Theorem 6 Consider nonlinear dynamics system with theuncertainty part (20) if there are a constant 120574 gt 0 positivedefinite symmetric matrix 119875

1and 119875

2 and matrix 119870 such that

the following LMI condition holds

Σ =

[

[

[

[

[

[

[

[

[

Σ11

Σ12

Σ13

Σ14

Σ15

Σ16

lowast Σ22

0 0 0 Σ26

lowast lowast Σ33

0 0 Σ36


0 0

lowast lowast lowast lowast Σ55

0

lowast lowast lowast lowast lowast Σ66

]

]

]

]

]

]

]

]

]

lt 0 (23)

where Σ15= 120576119872

119879 Σ16= 1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 +

1198751119866119879

119873119879

3+ 11987331198661198751 Σ26= 11987321198751+ 1198733119870 Σ36= 1198735 and Σ

55=

Σ66= minus120576 sdot 119868Then the nonlinear dynamic system with the uncertainty

part (20) is asymptotically stable in the quasi-slidingmodewithnoise perturbation attenuation 120574 gt 0 by quasi-sliding modecontroller with its gain 119870 = 119870119875

minus1

Proof From condition (16) and nonlinear dynamic systemwith the uncertainty part (20) we can obtain the followingcondition to guarantee the119867

infinquasi-sliding mode control

Σ3= Σ2+ Π119879

119865 (119905)Ω + Ω119879

119865 (119905)Π lt 0 (24)

where

Ω = [1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 + 119875

1119866119879

119873119879

3+ 1198733119866119875111987321198751+ 1198733119870 1198735]

Π = [119872 0 0]

(25)


By Lemmas 1 and 2 the condition Σ3lt 0 in (24) is equivalent

to Σ lt 0 in (23) By similar demonstration of Theorem 5 wecan complete this proof

3 Illustrative Simulations

Consider the following nonlinear system with the nonau-tonomous noise time-varying delay and parameters uncer-tainty described as follows

(119905) = ([

[

minus2 1 02

06 minus5 minus01

0 03 01

]

]

+ (119872119865 (119905)1198731))119909 (119905)

+ ([

[

minus01 0 minus01

04 minus02 0

01 05 minus01

]

]

+ (119872119865 (119905)1198732))119909 (119905 minus ℎ (119905))

+ ([

[

0

0

2

]

]

+ (119872119865 (119905)1198733))119906 (119905)

+ ([

[

1

0

0

]

]

+ (119872119865 (119905)1198734))119891 (119909 (119905))

+ ([

[

0

1

0

]

]

+ (119872119865 (119905)1198735))119908 (119905)

(26)

where

119891 (119909 (119905)) = 1199091(119905) 1199092(119905) 119866 = [1 0 minus1]

120575119876= 001 ℎ

119872= 02 119872 = 01

1198731=[

[

01 01 01

05 01 0

03 01 01

]

]

1198732=[

[

0 01 0

0 02 0

01 001 0

]

]

1198733= [02 01 01]

119879

1198734= [05 01 01]

119879

1198735= [001 002 01]

119879

(27)

119865(119905) = sin(119905) is given and119908(119905) is the Gauss white noise whichis shown in Figure 1

Based on Theorem 6 with 120576 = 01 and disturbanceattenuation 120574 = 04 the appropriate answer can be solved as

1198751=[

[

02726 minus00917 00654

minus00917 23626 minus04097

00654 minus04097 17972

]

]

1198752=[

[

01794 minus05512 minus02904

minus05512 32315 01598

minus02904 01598 11451

]

]

119870 = [01128 minus12885 01663]

(28)

0 2 4 6 8 10 12 14 16 18 20minus5minus4minus3minus2minus1

012345

Time (s)

w(t)

Figure 1 Time responses of Gauss white noise 119908(119905)

02 4 6 8 10 12 14 16 18 200

00501

01502

01015

02

0 2 4 6 8 10 12 14 16 18 200

005

0 2 4 6 8 10 12 14 16 18 20minus02

00204

Time (s)

x1(t)

x2(t)

x3(t)

Figure 2 Time responses of states of the controlled nonlinearsystem without disturbances

0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)

Figure 3 Time responses of switching surface 119878(119905) without distur-bances

Then a quasi-sliding mode control gain can be derived as

119870 = 1198701198751

minus1

= [02409 minus05430 minus00400] (29)

Based on the proposed controller the asymptotical stabilityfor the nonlinear systemwithout the disturbances on the slid-ing manifold is shown in Figure 2 and the relative switchingsliding surface is shown in Figure 3 Oppositely under effectsof disturbances the asymptotical stability for the nonlinearsystem on the sliding manifold is shown in Figure 4 and theswitching sliding surface is shown in Figure 5 They showthat disturbances can be restrained by the main controller


0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

minus010

010203

minus020

020406

minus020

020406

Time (s)

x1(t)

x2(t)

x3(t)

Figure 4 Time responses of states of the controlled nonlinearsystem under effects of disturbances

0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)

Figure 5 Time responses of switching surface 119878(119905) under effects ofdisturbances

with noise perturbation attenuation 120574 According to theabove simulation the asymptotical stability of the nonlinearsystem with the nonautonomous noise time-varying delayand parameters uncertainties on the sliding manifold isguaranteed by the 119867

infinquasi-sliding mode controller with

noise perturbation attenuation 120574

4 Conclusions

This study investigated the robust stability of the nonlinearsystem with time-varying delay and parameters uncertain-ties Based on the119867

infintheorem Lyapunov-Krasovskii theory

and LMI optimization technique the asymptotical stability ofthe nonlinear system could be guaranteed by the purposed119867infin

quasi-sliding mode controller and switching surfacein the quasi-sliding mode and the disturbance attenuationcan be also satisfied (119867

infin-norm performance) Numerical

simulations displayed the feasibility and usefulness of thecentral discussion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the Ministry of Science and TechnologyTaiwan for supporting this work underGrantsNSC 102-2221-E-269-021 and NSC 102-2622-E-269-003-CC3 The authorsalso wish to thank the anonymous reviewers for providingconstructive suggestions

References

[1] Q Wang Z Duan M Perc and G Chen ldquoSynchronizationtransitions on small-world neuronal networks effects of infor-mation transmission delay and rewiring probabilityrdquo EPL vol83 no 5 Article ID 50008 2008

[2] QWang M Perc Z Duan and G Chen ldquoDelay-induced mul-tiple stochastic resonances on scale-free neuronal networksrdquoChaos vol 19 no 2 Article ID 023112 2009

[3] Q Wang M Perc Z Duan and G Chen ldquoSynchronizationtransitions on scale-free neuronal networks due to finite infor-mation transmission delaysrdquo Physical Review E vol 80 no 2Article ID 026206 2009

[4] Z L Wan Y Y Hou J J Yan C H Lien and T L LiaoldquoNon-fragile 119867

infinstate feedback delayed control for uncertain

neutral systems with time-varying delaysrdquo International Journalof ControlTheory and Applications vol 1 no 1 pp 77ndash92 2008

[5] X-G LiuMWuRMartin andM-L Tang ldquoDelay-dependentstability analysis for uncertain neutral systems with time-varying delaysrdquoMathematics and Computers in Simulation vol75 no 1-2 pp 15ndash27 2007

[6] K Ramakrishnan and G Ray ldquoRobust stability criteria fora class of uncertain discrete-time systems with time-varyingdelayrdquo Applied Mathematical Modelling vol 37 no 3 pp 1468ndash1479 2013

[7] S-L Wu K-L Li and J-S Zhang ldquoExponential stability ofdiscrete-timeneural networkswith delay and impulsesrdquoAppliedMathematics and Computation vol 218 no 12 pp 6972ndash69862012

[8] C Liu C Li and C Li ldquoQuasi-synchronization of delayedchaotic systems with parameters mismatch and stochastic per-turbationrdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 10 pp 4108ndash4119 2011

[9] Y-Y Hou T-L Liao and J-J Yan ldquo119867infin

synchronization ofchaotic systems using output feedback control designrdquo PhysicaA vol 379 no 1 pp 81ndash89 2007

[10] J-H Kim and D-C Oh ldquoRobust and non-fragile 119867infin

controlfor descriptor systems with parameter uncertainties and timedelayrdquo International Journal of Control Automation and Sys-tems vol 5 no 1 pp 8ndash14 2007

[11] C-H Lien K-W Yu D-H Chang et al ldquo119867infincontrol for uncer-

tain Takagi-Sugeno fuzzy systems with time-varying delays andnonlinear perturbationsrdquo Chaos Solitons and Fractals vol 39no 3 pp 1426ndash1439 2009

[12] Z Wang L Huang and X Yang ldquo119867infin

performance for aclass of uncertain stochastic nonlinearMarkovian jump systemswith time-varying delay via adaptive control methodrdquo AppliedMathematical Modelling vol 35 no 4 pp 1983ndash1993 2011

[13] X Liu ldquoDelay-dependent 119867infin

control for uncertain fuzzysystems with time-varying delaysrdquo Nonlinear Analysis TheoryMethods and Applications vol 68 no 5 pp 1352ndash1361 2008


[14] S Ma C Zhang and Z Cheng ldquoDelay-dependent robust 119867infin

control for uncertain discrete-time singular systems with time-delaysrdquo Journal of Computational and AppliedMathematics vol217 no 1 pp 194ndash211 2008

[15] Z-L Wan Y-Y Hou T-L Liao and J-J Yan ldquoPartial finite-time synchronization of switched stochastic chuarsquos circuits viasliding-mode controlrdquo Mathematical Problems in Engineeringvol 2011 Article ID 162490 13 pages 2011

[16] R LingMWu YDong andY Chai ldquoHigh order sliding-modecontrol for uncertain nonlinear systems with relative degreethreerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 8 pp 3406ndash3416 2012

[17] TWangWXie andY Zhang ldquoSlidingmode fault tolerant con-trol dealing with modeling uncertainties and actuator faultsrdquoISA Transactions vol 51 no 3 pp 386ndash392 2012

[18] M P Aghababa S Khanmohammadi andG Alizadeh ldquoFinite-time synchronization of two different chaotic systems withunknown parameters via sliding mode techniquerdquo AppliedMathematical Modelling vol 35 no 6 pp 3080ndash3091 2011

[19] J M Daly and D W L Wang ldquoOutput feedback sliding modecontrol in the presence of unknown disturbancesrdquo Systems andControl Letters vol 58 no 3 pp 188ndash193 2009

[20] R N Gasimov A Karamancıoglu and A Yazıcı ldquoA nonlinearprogramming approach for the sliding mode control designrdquoApplied Mathematical Modelling vol 29 no 11 pp 1135ndash11482005

[21] W Weihong and H Zhongsheng ldquoNew adaptive quasi-slidingmode control for nonlinear discrete-time systemsrdquo Journal ofSystems Engineering and Electronics vol 19 no 1 pp 154ndash1602008

[22] X-D Li T W S Chow and J K L Ho ldquoQuasi-sliding modebased repetitive control for nonlinear continuous-time systemswith rejection of periodic disturbancesrdquoAutomatica vol 45 no1 pp 103ndash108 2009

[23] J Cho J C Principe D Erdogmus and M A Motter ldquoQuasi-slidingmode control strategy based onmultiple-linear modelsrdquoNeurocomputing vol 70 no 4ndash6 pp 960ndash974 2007

[24] C-F Huang T-L Liao C-Y Chen and J-J Yan ldquoThe designof quasi-sliding mode control for a permanent magnet syn-chronousmotor with unmatched uncertaintiesrdquoComputers andMathematics with Applications vol 64 no 5 pp 1036ndash10432012

[25] S M M Kashani H Salarieh and G Vossoughi ldquoControl ofnonlinear systems on Poincare section via quasi-sliding modemethodrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 3 pp 645ndash654 2009

[26] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Boston Mass USA 2003

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



On the other hand in the past most of papers set thecontrol gain in advance to achieve the sufficient condition ofthe stability It is not a suitable and accurate way to definethe parameters of the system although the parameters aregotten by trial and error Therefore in this paper we use thelinearmatrix inequality (LMI) theorem to optimize the quasi-sliding mode control gain By using the computer softwareMATLAB quasi-sliding mode controller gain can be foundBased on the119867

infintheorem Lyapunov-Krasovskii theory and

LMI optimization technique the 119867infin

quasi-sliding modecontrol and switching function can be designed such thatthe resulting nonlinear system is asymptotically stable in thequasi-sliding mode and satisfies the disturbance attenuation(119867infin-norm performance)Throughout this paper 119868 denotes the identity matrix of

appropriate dimensions For a real matrix 119860 we denote thetranspose by 119860

119879 and spectral norm by 119860 Considering119876 = 119876

119879

gt 0 (119876 = 119876119879

lt 0) 119876 is a symmetric positive(negative) definite matrixThe notation lowast in symmetric blockmatrices or long matrix expressions throughout the paperrepresents an ellipsis for terms that are induced by symmetryfor example [ 119861 119862

lowast 119863] = [

119861 119862

119862119879119863] For a vector 119909 119909(119905)

means the Euclidean vector norm at time 119905 while 1199092=

radicint

infin

0

119909(119905)2

119889119905 If 1199092lt infin then 119909(119905) isin 119871

2[0infin) where

1198712[0infin) stands for the space of square integral functions on

[0infin)This paper is organized as follows In Section 2 119867

infin

theorem LMI optimization techniques and the Lyapunov-Krasovskii stability theorem are used to derive the proposedcontroller switching function and corresponding parame-ters such that the nonlinear system is asymptotically stablein the quasi-sliding mode Then the central discussion ofthis paper is illustrated by a numerical example in Section 3Finally some conclusions are presented in Section 4

2 Nonlinear System Description andProposed Controller Design

In this paper we consider the stability of a nonlinearuncertain neutral systemwith time-varying delayed state anddisturbance At first we consider the system without theuncertain segmentThen a simple extension of nonlinear sys-tems with uncertain part is considered at the next narrationAsmentioned above we firstly consider the nonlinear system(1) without uncertain part as follows

(119905) = 1198601119909 (119905) + 119860

2119909 (119905 minus ℎ (119905))

+ 119861119906 (119905) + 119862119891 (119909 (119905)) + 119863119908 (119905)

119909 (119905) = 120601 (119905) 119905 isin [minusℎ119872 0]

(1)

where 0 le ℎ(119905) le ℎ119872 ℎ(119905) le ℎ

119863lt 1 119909(119905) isin 119877

119899 isthe nonlinear system state 119891(119909(119905)) isin 119877

119898 is a continuousnonlinear function vector 119908(119905) isin 119871

2[0infin) is the noise

perturbation input 119906(119905) isin 119877119898 is a control input vector and

the matrices 1198601 1198602 119861 119862 and 119863 are some known constant

matrices with appropriate dimensions The initial vector 120601 isa differentiable function from [minusℎ

119872 0] to 119877119899

Before presenting the main result we need the followinglemma and definition

Lemma 1 Let119872119873 and 119865(119905) be real matrices of appropriatedimensions with 119865(119905) le 1 and any scalar then

119872119879

119865 (119905)119873 + 119873119879

119865119879

(119905)119872 lt 120576minus1

sdot 119872119879

119872

+ 120576minus1

sdot (120576119873)119879

(120576119873) lt 0

(2)

Lemma2 (Schur complement of [26]) For a givenmatrix 119878 =[11987811 11987812

11987821 11987822

] with 11987811= 119878119879

11 11987822= 119878119879

22 the following conditions are

equivalent

(1) 119878 lt 0

(2) 11987822lt 0 119878

11minus 11987812119878minus1

2211987821lt 0

Definition 3 119867infincontrol problemaddressed in this paper can

be formulated as finding a stabilizing controller such that thefollowing conditions are held

Under the control input 119906(119905) the states of the nonlinearsystem (1) are asymptotically stable when 119908(119905) = 0

Under the zero initial conditions the performance index119869 = int

infin

0

[119909119879

(119905)119909(119905)minus1205742

sdot119908119879

(119905)119908(119905)]119889119905 le 0 (1199092le 120574119908

2) holds

for any nonzero 119908 isin 1198712and the119867

infinperformance 120574 gt 0

This paper aims at proposing the119867infinquasi-sliding mode

controller and switching function to not only asymptoticallystabilize the nonlinear system but also guarantee a prescribedperformance of noise perturbation attenuation 119903 gt 0 Thefirst step is to select an appropriate switching surface which isshown as (3) to ensure the asymptotical stability of the slidingmotion on the sliding manifold

119878 (119905) = 119866119909 (119905) + 119903 (119905)

119903 (119905) = 119866119861119903 (119905) minus 1198661198601119909 (119905)


(3)

where 119878(119905) isin 119877119898 is the switching surface 119870 is the quasi-

sliding mode controller gain and 119866 is chosen such that 119866119861is nonsingular and 119866119863 = 0 When the system is placed onthe switching surface it satisfies the following equations

119878 (119905) = 0119878 (119905) = 0 (4)

Then the equivalent controller 119906eq in the slidingmanifoldcan be obtained by solving for 119878(119905) = 0

119878 (119905) = 1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119861119906eq (119905)

+ 119866119862119891 (119909 (119905)) + 119866119863119908 (119905) + 119866119861119903 (119905) minus 1198661198601119909 (119905)


(5)


From 119903(119905) = minus119866119909(119905) as 119878(119905) = 0

119878 (119905) = 1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119861119906eq (119905)

+ 119866119862119891 (119909 (119905)) + 119866119863119908 (119905) minus 119866119861119866119909 (119905) minus 1198661198601119909 (119905)


(6)


119906eq (119905) = 119870119909 (119905) + 119870119909 (119905 minus ℎ (119905))

minus (119866119861)minus1

119866119862119891 (119909 (119905)) + 119866119909 (119905)

(7)


(119905) = (1198601+ 119861119870 + 119861119866) 119909 (119905)

+ (1198602+ 119861119870) 119909 (119905 minus ℎ (119905)) + 119863119908 (119905)

(8)




119906 (119905) = 119870119909 (119905) + 119870119909 (119905 minus ℎ (119905)) + 119866119909 (119905)

minus (119866119861)minus1

(119866119862119891 (119909 (119905)) +

119878 (119905)

(119878 (119905) + 120575119876)

)

(9)


119881119904(119905) =

1

2

119878119879

(119905) 119878 (119905) (10)


119904(119905) = 119878

119879

(119905)119878 (119905)

= 119878119879

(119905) [1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119862119891 (119909 (119905))

+ 119866119861119906 (119905) + 119866119863119908 (119905) minus 119866119861119866119909 (119905)

minus 1198661198601119909 (119905) minus 119866119860

2119909 (119905 minus ℎ (119905))


= minus119878119879

(119905) (

119878 (119905)

(119878 (119905) + 120575119876)

)

= minus

119878 (119905)2

(119878 (119905) + 120575119876)

= minus(119878 (119905) minus

(119878 (119905)2

+ 119878 (119905) 120575119876minus 119878 (119905)

2

)

(119878 (119905) + 120575119876)

)

= minus(119878 (119905) minus

119878 (119905) 120575119876

(119878 (119905) + 120575119876)

)

lt minus (119878 (119905) minus 120575119876)

(11)






1and


Σ2lt 0 (12)

where Σ12

= 11986021198751+ 119861119870 Σ

13= 119863 Σ

14= 1198751 Σ22

=

minus(1 minus ℎ119863)1198752 Σ33

= minus1205742

sdot 119868 Σ44

= 119868 and Σ11

= 1198751119860119879

1+

11986011198751+119870

119879

119861119879

+119861119870+1198751119866119879

119861119879



1

minus1


119881 (119905) = 119909119879

(119905) 1198751119909 (119905) + int

119905

119905minusℎ(119905)

119909119879

(120591) 1198752119909 (120591) 119889120591 (13)


(119905) = 119909119879

(119905) (119860119879

11198751+ 11987511198601+ 119870119879

119861119879

1198751

+1198751119861119870 + 119866

119879

119861119879

1198751+ 1198751119861119866) 119909 (119905)

+ 119909119879

(119905) (11987511198602+ 1198751119861119870) 119909 (119905 minus ℎ (119905))

+ 119909119879

(119905 minus ℎ (119905)) (119860119879

11198751+ 119870119879

119861119879

1198751) 119909 (119905)

+ 2119909119879

(119905) 1198751119863119908 (119905) + 119909

119879

(119905) 1198752119909 (119905)


119879

(119905 minus ℎ (119905)) 1198752119909 (119905 minus ℎ (119905))

(14)



119869 (119905) = (119905) + 119909119879

(119905) 119909 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

le[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

119879

Σ1

[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

(15)

where Σ1= [

Σ11 Σ12 Σ13

lowast Σ22 0

lowast lowast Σ33

] Σ11= 119860119879

11198751+ 11987511198601+ 119870119879

119861119879

1198751+

1198751119861119870 + 119866

119879

119861119879

1198751+ 1198751119861119866 + 119875

2+ 119868 Σ

12= 11987511198602+ 119875119861119870 Σ

13=

1198751119863 Σ


119863)1198752 and Σ

33= minus1205742


1by Φ =

diag lfloor1198751

minus1

1198751

minus1

119868rfloor defining 1198751= 1198751

minus1 1198752= 119875111987521198751 and

119870 = 1198701198751


Σ2=

[

[

[

[

Σ11

Σ12

Σ13

Σ14

lowast Σ22

0 0

lowast lowast Σ33

0


]

]

]

]

(16)

where Σ11= 1198751119860119879

1+11986011198751+119870

119879

119861119879

+119861119870+1198751119866119879

119861119879

+1198611198661198751+1198752

Σ12= 11986021198751+119861119870 Σ

13= 119863 Σ


119863)1198752 Σ33=

minus1205742



(119905)

10038161003816100381610038161003816119908(119905)=0

le minus120588 sdot 119909(119905)2 (17)



obviously

119881 (119909infin) minus 119881 (120601) + 119909 (119905)

2

2minus 1205742

sdot 119908 (119905)2

2le 0 (18)


119881 (120601) = 0 119881 (119909infin) ge 0

119909 (119905)2le 1205742

sdot 119908 (119905)2 forall119908 (119905) isin 119871

2[0infin)

(19)


1



(119905) = 1198601(119905) 119909 (119905) + 119860

2(119905) 119909 (119905 minus ℎ (119905)) + 119861 (119905) 119906 (119905)

+ 119862 (119905) 119891 (119909 (119905)) + 119863 (119905) 119908 (119905)

119909 (119905) = 120601 (119905) 119905 isin [minusℎ119872 0]

(20)

where 1198601(119905) = 119860

1+ Δ119860

1(119905) 119860

2(119905) = 119860

2+ Δ119860

2(119905) 119861(119905) =

119861 + Δ119861(119905) 119862(119905) = 119862 + Δ119862(119905) and 119863(119905) = 119863 + Δ119863(119905)Δ1198601(119905) Δ119860



[Δ1198601(119905) Δ119860

2(119905) Δ119861 (119905) Δ119862 (119905) Δ119863 (119905)]

= 119872 sdot 119865 (119905) sdot [11987311198732119873311987341198735]

(21)



119865119879

(119905) sdot 119865 (119905) le 119868 forall119905 ge 0 (22)



1and 119875



Σ =

[

[

[

[

[

[

[

[

[

Σ11

Σ12

Σ13

Σ14

Σ15

Σ16

lowast Σ22

0 0 0 Σ26

lowast lowast Σ33

0 0 Σ36


0 0


0


]

]

]

]

]

]

]

]

]

lt 0 (23)

where Σ15= 120576119872

119879 Σ16= 1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 +

1198751119866119879

119873119879

3+ 11987331198661198751 Σ26= 11987321198751+ 1198733119870 Σ36= 1198735 and Σ

55=



minus1



Σ3= Σ2+ Π119879

119865 (119905)Ω + Ω119879

119865 (119905)Π lt 0 (24)

where

Ω = [1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 + 119875

1119866119879

119873119879

3+ 1198733119866119875111987321198751+ 1198733119870 1198735]

Π = [119872 0 0]

(25)






(119905) = ([

[

minus2 1 02

06 minus5 minus01

0 03 01

]

]

+ (119872119865 (119905)1198731))119909 (119905)

+ ([

[

minus01 0 minus01

04 minus02 0

01 05 minus01

]

]

+ (119872119865 (119905)1198732))119909 (119905 minus ℎ (119905))

+ ([

[

0

0

2

]

]

+ (119872119865 (119905)1198733))119906 (119905)

+ ([

[

1

0

0

]

]

+ (119872119865 (119905)1198734))119891 (119909 (119905))

+ ([

[

0

1

0

]

]

+ (119872119865 (119905)1198735))119908 (119905)

(26)

where

119891 (119909 (119905)) = 1199091(119905) 1199092(119905) 119866 = [1 0 minus1]

120575119876= 001 ℎ

119872= 02 119872 = 01

1198731=[

[

01 01 01

05 01 0

03 01 01

]

]

1198732=[

[

0 01 0

0 02 0

01 001 0

]

]

1198733= [02 01 01]

119879

1198734= [05 01 01]

119879

1198735= [001 002 01]

119879

(27)



1198751=[

[

02726 minus00917 00654

minus00917 23626 minus04097

00654 minus04097 17972

]

]

1198752=[

[

01794 minus05512 minus02904

minus05512 32315 01598

minus02904 01598 11451

]

]

119870 = [01128 minus12885 01663]

(28)


012345

Time (s)

w(t)


02 4 6 8 10 12 14 16 18 200

00501

01502

01015

02

0 2 4 6 8 10 12 14 16 18 200

005

0 2 4 6 8 10 12 14 16 18 20minus02

00204

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)



119870 = 1198701198751

minus1

= [02409 minus05430 minus00400] (29)



0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

minus010

010203

minus020

020406

minus020

020406

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)





4 Conclusions









Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










From 119903(119905) = minus119866119909(119905) as 119878(119905) = 0

119878 (119905) = 1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119861119906eq (119905)

+ 119866119862119891 (119909 (119905)) + 119866119863119908 (119905) minus 119866119861119866119909 (119905) minus 1198661198601119909 (119905)


(6)


119906eq (119905) = 119870119909 (119905) + 119870119909 (119905 minus ℎ (119905))

minus (119866119861)minus1

119866119862119891 (119909 (119905)) + 119866119909 (119905)

(7)


(119905) = (1198601+ 119861119870 + 119861119866) 119909 (119905)

+ (1198602+ 119861119870) 119909 (119905 minus ℎ (119905)) + 119863119908 (119905)

(8)




119906 (119905) = 119870119909 (119905) + 119870119909 (119905 minus ℎ (119905)) + 119866119909 (119905)

minus (119866119861)minus1

(119866119862119891 (119909 (119905)) +

119878 (119905)

(119878 (119905) + 120575119876)

)

(9)


119881119904(119905) =

1

2

119878119879

(119905) 119878 (119905) (10)


119904(119905) = 119878

119879

(119905)119878 (119905)

= 119878119879

(119905) [1198661198601119909 (119905) + 119866119860

2119909 (119905 minus ℎ (119905)) + 119866119862119891 (119909 (119905))

+ 119866119861119906 (119905) + 119866119863119908 (119905) minus 119866119861119866119909 (119905)

minus 1198661198601119909 (119905) minus 119866119860

2119909 (119905 minus ℎ (119905))


= minus119878119879

(119905) (

119878 (119905)

(119878 (119905) + 120575119876)

)

= minus

119878 (119905)2

(119878 (119905) + 120575119876)

= minus(119878 (119905) minus

(119878 (119905)2

+ 119878 (119905) 120575119876minus 119878 (119905)

2

)

(119878 (119905) + 120575119876)

)

= minus(119878 (119905) minus

119878 (119905) 120575119876

(119878 (119905) + 120575119876)

)

lt minus (119878 (119905) minus 120575119876)

(11)






1and


Σ2lt 0 (12)

where Σ12

= 11986021198751+ 119861119870 Σ

13= 119863 Σ

14= 1198751 Σ22

=

minus(1 minus ℎ119863)1198752 Σ33

= minus1205742

sdot 119868 Σ44

= 119868 and Σ11

= 1198751119860119879

1+

11986011198751+119870

119879

119861119879

+119861119870+1198751119866119879

119861119879



1

minus1


119881 (119905) = 119909119879

(119905) 1198751119909 (119905) + int

119905

119905minusℎ(119905)

119909119879

(120591) 1198752119909 (120591) 119889120591 (13)


(119905) = 119909119879

(119905) (119860119879

11198751+ 11987511198601+ 119870119879

119861119879

1198751

+1198751119861119870 + 119866

119879

119861119879

1198751+ 1198751119861119866) 119909 (119905)

+ 119909119879

(119905) (11987511198602+ 1198751119861119870) 119909 (119905 minus ℎ (119905))

+ 119909119879

(119905 minus ℎ (119905)) (119860119879

11198751+ 119870119879

119861119879

1198751) 119909 (119905)

+ 2119909119879

(119905) 1198751119863119908 (119905) + 119909

119879

(119905) 1198752119909 (119905)


119879

(119905 minus ℎ (119905)) 1198752119909 (119905 minus ℎ (119905))

(14)



119869 (119905) = (119905) + 119909119879

(119905) 119909 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

le[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

119879

Σ1

[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

(15)

where Σ1= [

Σ11 Σ12 Σ13

lowast Σ22 0

lowast lowast Σ33

] Σ11= 119860119879

11198751+ 11987511198601+ 119870119879

119861119879

1198751+

1198751119861119870 + 119866

119879

119861119879

1198751+ 1198751119861119866 + 119875

2+ 119868 Σ

12= 11987511198602+ 119875119861119870 Σ

13=

1198751119863 Σ


119863)1198752 and Σ

33= minus1205742


1by Φ =

diag lfloor1198751

minus1

1198751

minus1

119868rfloor defining 1198751= 1198751

minus1 1198752= 119875111987521198751 and

119870 = 1198701198751


Σ2=

[

[

[

[

Σ11

Σ12

Σ13

Σ14

lowast Σ22

0 0

lowast lowast Σ33

0


]

]

]

]

(16)

where Σ11= 1198751119860119879

1+11986011198751+119870

119879

119861119879

+119861119870+1198751119866119879

119861119879

+1198611198661198751+1198752

Σ12= 11986021198751+119861119870 Σ

13= 119863 Σ


119863)1198752 Σ33=

minus1205742



(119905)

10038161003816100381610038161003816119908(119905)=0

le minus120588 sdot 119909(119905)2 (17)



obviously

119881 (119909infin) minus 119881 (120601) + 119909 (119905)

2

2minus 1205742

sdot 119908 (119905)2

2le 0 (18)


119881 (120601) = 0 119881 (119909infin) ge 0

119909 (119905)2le 1205742

sdot 119908 (119905)2 forall119908 (119905) isin 119871

2[0infin)

(19)


1



(119905) = 1198601(119905) 119909 (119905) + 119860

2(119905) 119909 (119905 minus ℎ (119905)) + 119861 (119905) 119906 (119905)

+ 119862 (119905) 119891 (119909 (119905)) + 119863 (119905) 119908 (119905)

119909 (119905) = 120601 (119905) 119905 isin [minusℎ119872 0]

(20)

where 1198601(119905) = 119860

1+ Δ119860

1(119905) 119860

2(119905) = 119860

2+ Δ119860

2(119905) 119861(119905) =

119861 + Δ119861(119905) 119862(119905) = 119862 + Δ119862(119905) and 119863(119905) = 119863 + Δ119863(119905)Δ1198601(119905) Δ119860



[Δ1198601(119905) Δ119860

2(119905) Δ119861 (119905) Δ119862 (119905) Δ119863 (119905)]

= 119872 sdot 119865 (119905) sdot [11987311198732119873311987341198735]

(21)



119865119879

(119905) sdot 119865 (119905) le 119868 forall119905 ge 0 (22)



1and 119875



Σ =

[

[

[

[

[

[

[

[

[

Σ11

Σ12

Σ13

Σ14

Σ15

Σ16

lowast Σ22

0 0 0 Σ26

lowast lowast Σ33

0 0 Σ36


0 0


0


]

]

]

]

]

]

]

]

]

lt 0 (23)

where Σ15= 120576119872

119879 Σ16= 1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 +

1198751119866119879

119873119879

3+ 11987331198661198751 Σ26= 11987321198751+ 1198733119870 Σ36= 1198735 and Σ

55=



minus1



Σ3= Σ2+ Π119879

119865 (119905)Ω + Ω119879

119865 (119905)Π lt 0 (24)

where

Ω = [1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 + 119875

1119866119879

119873119879

3+ 1198733119866119875111987321198751+ 1198733119870 1198735]

Π = [119872 0 0]

(25)






(119905) = ([

[

minus2 1 02

06 minus5 minus01

0 03 01

]

]

+ (119872119865 (119905)1198731))119909 (119905)

+ ([

[

minus01 0 minus01

04 minus02 0

01 05 minus01

]

]

+ (119872119865 (119905)1198732))119909 (119905 minus ℎ (119905))

+ ([

[

0

0

2

]

]

+ (119872119865 (119905)1198733))119906 (119905)

+ ([

[

1

0

0

]

]

+ (119872119865 (119905)1198734))119891 (119909 (119905))

+ ([

[

0

1

0

]

]

+ (119872119865 (119905)1198735))119908 (119905)

(26)

where

119891 (119909 (119905)) = 1199091(119905) 1199092(119905) 119866 = [1 0 minus1]

120575119876= 001 ℎ

119872= 02 119872 = 01

1198731=[

[

01 01 01

05 01 0

03 01 01

]

]

1198732=[

[

0 01 0

0 02 0

01 001 0

]

]

1198733= [02 01 01]

119879

1198734= [05 01 01]

119879

1198735= [001 002 01]

119879

(27)



1198751=[

[

02726 minus00917 00654

minus00917 23626 minus04097

00654 minus04097 17972

]

]

1198752=[

[

01794 minus05512 minus02904

minus05512 32315 01598

minus02904 01598 11451

]

]

119870 = [01128 minus12885 01663]

(28)


012345

Time (s)

w(t)


02 4 6 8 10 12 14 16 18 200

00501

01502

01015

02

0 2 4 6 8 10 12 14 16 18 200

005

0 2 4 6 8 10 12 14 16 18 20minus02

00204

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)



119870 = 1198701198751

minus1

= [02409 minus05430 minus00400] (29)



0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

minus010

010203

minus020

020406

minus020

020406

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)





4 Conclusions









Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119869 (119905) = (119905) + 119909119879

(119905) 119909 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

le[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

119879

Σ1

[

[

119909 (119905)

119909 (119905 minus ℎ (119905))

119908 (119905)

]

]

(15)

where Σ1= [

Σ11 Σ12 Σ13

lowast Σ22 0

lowast lowast Σ33

] Σ11= 119860119879

11198751+ 11987511198601+ 119870119879

119861119879

1198751+

1198751119861119870 + 119866

119879

119861119879

1198751+ 1198751119861119866 + 119875

2+ 119868 Σ

12= 11987511198602+ 119875119861119870 Σ

13=

1198751119863 Σ


119863)1198752 and Σ

33= minus1205742


1by Φ =

diag lfloor1198751

minus1

1198751

minus1

119868rfloor defining 1198751= 1198751

minus1 1198752= 119875111987521198751 and

119870 = 1198701198751


Σ2=

[

[

[

[

Σ11

Σ12

Σ13

Σ14

lowast Σ22

0 0

lowast lowast Σ33

0


]

]

]

]

(16)

where Σ11= 1198751119860119879

1+11986011198751+119870

119879

119861119879

+119861119870+1198751119866119879

119861119879

+1198611198661198751+1198752

Σ12= 11986021198751+119861119870 Σ

13= 119863 Σ


119863)1198752 Σ33=

minus1205742



(119905)

10038161003816100381610038161003816119908(119905)=0

le minus120588 sdot 119909(119905)2 (17)



obviously

119881 (119909infin) minus 119881 (120601) + 119909 (119905)

2

2minus 1205742

sdot 119908 (119905)2

2le 0 (18)


119881 (120601) = 0 119881 (119909infin) ge 0

119909 (119905)2le 1205742

sdot 119908 (119905)2 forall119908 (119905) isin 119871

2[0infin)

(19)


1



(119905) = 1198601(119905) 119909 (119905) + 119860

2(119905) 119909 (119905 minus ℎ (119905)) + 119861 (119905) 119906 (119905)

+ 119862 (119905) 119891 (119909 (119905)) + 119863 (119905) 119908 (119905)

119909 (119905) = 120601 (119905) 119905 isin [minusℎ119872 0]

(20)

where 1198601(119905) = 119860

1+ Δ119860

1(119905) 119860

2(119905) = 119860

2+ Δ119860

2(119905) 119861(119905) =

119861 + Δ119861(119905) 119862(119905) = 119862 + Δ119862(119905) and 119863(119905) = 119863 + Δ119863(119905)Δ1198601(119905) Δ119860



[Δ1198601(119905) Δ119860

2(119905) Δ119861 (119905) Δ119862 (119905) Δ119863 (119905)]

= 119872 sdot 119865 (119905) sdot [11987311198732119873311987341198735]

(21)



119865119879

(119905) sdot 119865 (119905) le 119868 forall119905 ge 0 (22)



1and 119875



Σ =

[

[

[

[

[

[

[

[

[

Σ11

Σ12

Σ13

Σ14

Σ15

Σ16

lowast Σ22

0 0 0 Σ26

lowast lowast Σ33

0 0 Σ36


0 0


0


]

]

]

]

]

]

]

]

]

lt 0 (23)

where Σ15= 120576119872

119879 Σ16= 1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 +

1198751119866119879

119873119879

3+ 11987331198661198751 Σ26= 11987321198751+ 1198733119870 Σ36= 1198735 and Σ

55=



minus1



Σ3= Σ2+ Π119879

119865 (119905)Ω + Ω119879

119865 (119905)Π lt 0 (24)

where

Ω = [1198751119873119879

1+ 11987311198751+ 119870

119879

119873119879

3+ 1198733119870 + 119875

1119866119879

119873119879

3+ 1198733119866119875111987321198751+ 1198733119870 1198735]

Π = [119872 0 0]

(25)






(119905) = ([

[

minus2 1 02

06 minus5 minus01

0 03 01

]

]

+ (119872119865 (119905)1198731))119909 (119905)

+ ([

[

minus01 0 minus01

04 minus02 0

01 05 minus01

]

]

+ (119872119865 (119905)1198732))119909 (119905 minus ℎ (119905))

+ ([

[

0

0

2

]

]

+ (119872119865 (119905)1198733))119906 (119905)

+ ([

[

1

0

0

]

]

+ (119872119865 (119905)1198734))119891 (119909 (119905))

+ ([

[

0

1

0

]

]

+ (119872119865 (119905)1198735))119908 (119905)

(26)

where

119891 (119909 (119905)) = 1199091(119905) 1199092(119905) 119866 = [1 0 minus1]

120575119876= 001 ℎ

119872= 02 119872 = 01

1198731=[

[

01 01 01

05 01 0

03 01 01

]

]

1198732=[

[

0 01 0

0 02 0

01 001 0

]

]

1198733= [02 01 01]

119879

1198734= [05 01 01]

119879

1198735= [001 002 01]

119879

(27)



1198751=[

[

02726 minus00917 00654

minus00917 23626 minus04097

00654 minus04097 17972

]

]

1198752=[

[

01794 minus05512 minus02904

minus05512 32315 01598

minus02904 01598 11451

]

]

119870 = [01128 minus12885 01663]

(28)


012345

Time (s)

w(t)


02 4 6 8 10 12 14 16 18 200

00501

01502

01015

02

0 2 4 6 8 10 12 14 16 18 200

005

0 2 4 6 8 10 12 14 16 18 20minus02

00204

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)



119870 = 1198701198751

minus1

= [02409 minus05430 minus00400] (29)



0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

minus010

010203

minus020

020406

minus020

020406

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)





4 Conclusions









Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














(119905) = ([

[

minus2 1 02

06 minus5 minus01

0 03 01

]

]

+ (119872119865 (119905)1198731))119909 (119905)

+ ([

[

minus01 0 minus01

04 minus02 0

01 05 minus01

]

]

+ (119872119865 (119905)1198732))119909 (119905 minus ℎ (119905))

+ ([

[

0

0

2

]

]

+ (119872119865 (119905)1198733))119906 (119905)

+ ([

[

1

0

0

]

]

+ (119872119865 (119905)1198734))119891 (119909 (119905))

+ ([

[

0

1

0

]

]

+ (119872119865 (119905)1198735))119908 (119905)

(26)

where

119891 (119909 (119905)) = 1199091(119905) 1199092(119905) 119866 = [1 0 minus1]

120575119876= 001 ℎ

119872= 02 119872 = 01

1198731=[

[

01 01 01

05 01 0

03 01 01

]

]

1198732=[

[

0 01 0

0 02 0

01 001 0

]

]

1198733= [02 01 01]

119879

1198734= [05 01 01]

119879

1198735= [001 002 01]

119879

(27)



1198751=[

[

02726 minus00917 00654

minus00917 23626 minus04097

00654 minus04097 17972

]

]

1198752=[

[

01794 minus05512 minus02904

minus05512 32315 01598

minus02904 01598 11451

]

]

119870 = [01128 minus12885 01663]

(28)


012345

Time (s)

w(t)


02 4 6 8 10 12 14 16 18 200

00501

01502

01015

02

0 2 4 6 8 10 12 14 16 18 200

005

0 2 4 6 8 10 12 14 16 18 20minus02

00204

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)



119870 = 1198701198751

minus1

= [02409 minus05430 minus00400] (29)



0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

minus010

010203

minus020

020406

minus020

020406

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)





4 Conclusions









Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

minus010

010203

minus020

020406

minus020

020406

Time (s)

x1(t)

x2(t)

x3(t)


0 2 4 6 8 10 12 14 16 18 20minus005

0

005

01

015

02

Time (s)

S(t)





4 Conclusions









Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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