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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 537414 6 pageshttpdxdoiorg1011552013537414

Research ArticleObserver-Based Sliding Mode Control for Stabilization ofa Dynamic System with Delayed Output Feedback

Bo Wang12 Peng Shi34 Hamid Reza Karimi5 and Cheng Chew Lim3

1 School of Applied Mathematics University Electronic Science and Technology of China Chengdu 610054 China2 School of Electrical and Information Engineering Xihua University Chengdu 610096 China3 School of Electrical and Electronic Engineering The University of Adelaide Adelaide SA 5005 Australia4College of Engineering and Science Victoria University Melbourne VIC 8001 Australia5 Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Hamid Reza Karimi hamidrkarimiuiano

Received 10 July 2013 Accepted 29 August 2013

Academic Editor Rongni Yang

Copyright copy 2013 Bo Wang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers the sliding mode control problem for a kind of dynamic delay system First by utilizing Lyapunov stabilitytheory and a linear matrix inequality technique an observer based on delayed output feedback is constructed Then an integralsliding surface is presented to realize the sliding mode control for the system with the more available stability condition Finallysome numerical simulations are implemented to demonstrate the validity of the proposed control method

1 Introduction

For system control state feedback control is a powerful toolwhen the full information of the system states is assumed tobe accessible However in real engineering not all of themcan be available Hence the research on observer-based con-trol system is a meaningful topic Up to now many relevantinvestigations [1ndash10] have been carried out For instance in[1] by imposing some restrictions on an open-loop systemtwo classes of the observer-based output feedback controllersone finite dimensional and the other one infinite dimen-sional are constructed in [2] a static gain observer for linearcontinuous plants with intrinsic pulse-modulated feedback isdesigned to asymptotically drive the state estimation error tozero in [3] by using the orthogonality-preserving numericalalgorithm a nonlinear discrete-time partial state observer isdesigned to realize the attitude determination for a kind ofspacecraft system in [4] with the two-step observation algo-rithms an observer is constructed to force an underactuatedaircraft to asymptotically track a given reference trajectory in[5] through employing an adaptive backstepping approacha modified high-gain observer is introduced to realize theglobal asymptotic tracking for a class of nonlinear systems

In real engineering time delays exist objectively whichmakes the observer-based system control issue more com-plicated Many existing control methods that have been welldeveloped based on the conventional feedback observer suchas ones in [1ndash5] are not directly applicable In addition theLMI technique is known as a powerful tool for system controlHowever it is not easy to use any more when time delaysappear in the feedback output of system Hence the cor-responding research is meaningful Sliding mode control(SMC) has been proven to be an effective robust control strat-egy and has been successfully applied to a wide range of engi-neering systems such as spacecrafts robot manipulators air-craft underwater vehicles electrical motors power systemsand automotive engines [11ndash25] In [11] a novel integral slid-ing surface is introduced to achieve the control for a kind ofsystem based on the LMI technique with the more availablestability condition compared to the conventional linear slid-ing surface However the systemwith delay is not included inits research yet Hence all of the above motivate our work

In this paper the problem on the SMC for a kind of delaydynamic system will be discussed By utilizing Lyapunovstability theory and a linear matrix inequality techniquea new observer-based on delayed output feedback will be

2 Mathematical Problems in Engineering

constructed An integral sliding surface will be presented torealize the SMCof the systemwith themore available stabilitycondition Finally some numerical examples will be includedto demonstrate the validity of the proposed control method

Notation used in this paper is fairly standard Let 119877119899 bethe 119899-dimensional Euclidean space 119877119899times119898 represents the setof 119899 times 119898 real matrix lowast denotes the elements below the maindiagonal of a symmetric block matrix diag represents thediagonal matrix the notation 119860 gt 0 means that 119860 is the realsymmetric and positive definite 119868 denotes the identitymatrixwith appropriate dimensions and sdot refers to the Euclideanvector norm

2 Problem Statement and Preliminaries

In this paper the following dynamic system is considered

(119905) = 119860119909 (119905) + 119861119909 (119905 minus 119897) + 119871119906 (119905)

119911 (119905) = 119862119909 (119905 minus 119889)

(1)

where 119909(119905) isin 119877119899 is the system state 119906(119905) isin 119877

119898 is the systeminput 119910(119905) isin 119877

119897 is the system output 119897 is discrete delay 119911(119905) isthemeasurable feedback output and119860119861 119871119862 are the systemmatrices with appropriate dimension

Then an observer of system (1) is built as follows

119909 (119905) = 119860119909 (119905)+119861119909 (119905 minus 119897)+119871119906 (119905)+119890119860119889

119870 [119911 (119905) minus 119862119909 (119905 minus 119889)]

119909 (119905 minus 119889) = 119860119909 (119905 minus 119889) + 119861119909 (119905 minus 119889 minus 119897) + 119871119906 (119905 minus 119889)

+ 119870 [119911 (119905) minus 119862119909 (119905 minus 119889)]

(2)

where 119909(119905) isin 119877119899 is the state vector 119870 is the feedback control

matrix and 119889 is the measurable feedback delayDefine the observation error

119890 (119905) = 119909 (119905) minus 119909 (119905) (3)

The following theoretical result will be used throughout thepaper

Lemma 1 For observer (2) the following equation holds

119909 (119905) = 119890119860119889

119909 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

(119871119906 (120591) + 119861119909 (120591 minus 119897)) 119889120591

(4)

Proof The derivative of (4) is as follows

119909 (119905) = 119890119860119889 119909 (119905 minus 119889) + 119860int

119905

119905minus119889

119890119860(119905minus120591)

(119871119906 (120591) + 119861119909 (120591 minus 119897)) 119889120591

+ 119871119906 (119905) minus 119890119860119889

119871119906 (119905 minus 119889) + 119861119909 (119905 minus 119897)

minus 119890119860119889

119861119909 (119905 minus 119889 minus 119897)

= 119890119860119889 119909 (119905 minus 119889) + 119860 (119909 (119905) minus 119890

119860119889

119909 (119905 minus 119889))

+ 119871119906 (119905) minus 119890119860119889

119871119906 (119905 minus 119889) + 119861119909 (119905 minus 119897)

minus 119890119860119889

119861119909 (119905 minus 119889 minus 119897)

= 119860119909 (119905) + 119871119906 (119905) + 119861119909 (119905 minus 119897)

+ 119890119860119889

( 119909 (119905 minus 119889) minus 119860119909 (119905 minus 119889) minus 119871119906 (119905 minus 119889)

minus119861119909 (119905 minus 119889 minus 119897) )

(5)

Consider (2) we get

119909 (119905) = 119860119909 (119905)+119861119909 (119905 minus 119897)+119871119906 (119905) + 119890119860119889

119870 [119911 (119905) minus 119862119909 (119905 minus 119889)]

(6)

The proof of Lemma 1 is thus completed

3 Design of Observer

Theorem 2 If there exists a matrix 119872 and positive-definitesymmetric matrices 119875 and 119876 satisfy

[119876 + 119875119860 minus119872119862 + 119860

119879

119875 minus 119862119879

119872119879

119875119861

lowast minus119876] lt 0 (7)

119870 = 119875minus1

119872 (8)

then observer (2) can realize the observation of system (1)

Proof From system (1) we have

(119905 minus 119889) = 119860119909 (119905 minus 119889) + 119861119909 (119905 minus 119897 minus 119889) + 119871119906 (119905 minus 119889) (9)

With (2) we can get

119890 (119905 minus 119889) = (119860 minus 119870119862) 119890 (119905 minus 119889) + 119861119890 (119905 minus 119897 minus 119889) (10)

Choose the Lyapunov functional candidate as

119881 (119905 minus 119889) = 119890119879

(119905 minus 119889) 119875119890 (119905 minus 119889) + int

119905minus119889

119905minus119889minus119897

119890119879

(119904) 119876119890 (119904) 119889119904

(11)

The time derivative of 119881(119905 minus 119889) is

(119905 minus 119889) = 2119890119879

(119905 minus 119889) 119875 119890 (119905 minus 119889)

+ 119890119879

(119905 minus 119889)119876119890 (119905 minus 119889) minus 119890119879

(119905 minus 119889 minus 119897)

times 119876119890 (119905 minus 119897 minus 119889)

= 2119890119879

(119905 minus 119889) 119875 ((119860 minus 119870119862) 119890 (119905 minus 119889) + 119861119890 (119905 minus 119897 minus 119889))

+ 119890119879


(119905 minus 119889 minus 119897)


(12)

Then the following equation holds

(119905 minus 119889) = 120585119879

(119905 minus 119889) Ξ120585 (119905 minus 119889) (13)

where

120585 (119905) = [119890119879

(119905) 119890 (119905 minus 119897)]119879

Ξ = [119875 (119860 minus 119870119862) + (119860 minus 119870119862)

119879

119875 + 119876 119875119861

lowast minus119876]

(14)

Mathematical Problems in Engineering 3

Let119872 = 119875119870 and with LMI (7) we can conclude

(119905 minus 119889) lt 0 (15)

Hence

lim119905rarrinfin

119890 (119905 minus 119889) 997888rarr 0 (16)

From system (1) the following equation can be derived

119909 (119905) = 119890119860119889

119909 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

(119871119906 (120591) + 119861119909 (120591 minus 119897)) 119889120591

(17)

Considering (4) we have

119890 (119905) = 119890119860119889

119890 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

119861119890 (120591 minus 119897) 119889120591 (18)

Finally combined with (16) we can get lim119905rarrinfin

119890(119905) rarr 0which means the realization of the observation The proof ofTheorem 2 is thus completed

4 Design of Sliding Motion Control Law

First we introduce the following integral sliding surface

119904 (119905) = 119867119909 (119905) minus int

119905

0

119867(119860 + 119871119873) 119909 (120585) 119889120585

minus int

119905

0

119867119861119909 (120585 minus 119897 (120585)) 119889120585

(19)

For system (1) we have

119909 (119905) = 119909 (0) + int

119905

0

(119860119909 (120585) + 119861119909 (120585 minus 119897) + 119871119906 (120585)) 119909 (120585) 119889120585

(20)

Hence the following equation holds

119904 (119905) = 119867119909 (0) + 119867119871int

119905

0

(119906 (120585) minus 119873119909 (120585)) 119889120585 (21)

By 119904(119905) = 0 we get the equivalent control law as

119906119890119902(119905) = 119873119909 (119905) (22)

When 119906(119905) = 119873119909(119905) we can obtain

(119905) = (119860 + 119871119873) 119909 (119905) + 119861119909 (119905 minus 119897) (23)

Then based on Lyapunov theory and LMI techniques the fol-lowing theoretical result is derived

Theorem 3 If there exists a matrix 119879 and positive-definitesymmetric matrices 119865 and 119878 satisfying

[119860119865 + 119871119879 + 119865

119879

119860119879

+ 119879119879

119871119879

+ 119878 119861119865


119873 = 119879minus1

119865 (25)

then dynamic system (23) is stable

Proof Choose the Lyapunov functional candidate as

1198811(119905) = 119909

119879

(119905) 1198781119909 (119905) + int

119905

119905minus119897

119909119879

(119904) 1198782119909 (119904) 119889119904 (26)

Then the time derivative of 1198811(119905) is as

1(119905) = 2119909

119879

(119905) 1198781 (119905) + 119909

119879

(119905) 1198782119909 (119905) minus 119909

119879

(119905 minus 119897) 1198782119909 (119905 minus 119897)

= 2119909119879

(119905) 1198781((119860 + 119871119873) 119909 (119905 minus 119889) + 119861119909 (119905 minus 119897))

+ 119909119879

(119905) 1198782119909 (119905) minus 119909

119879

(119905 minus 119897) 1198782119909 (119905 minus 119897)

(27)

Hence we have

1(119905) = 120585

119879

(119905) Ξ120585 (119905) (28)

where

120585 (119905) = [119909119879

(119905) 119909 (119905 minus 119897)]119879

Ξ = [1198781(119860 + 119871119873) + (119860 + 119871119873)

119879

1198781+ 11987821198781119861

lowast minus1198782

]

(29)

Let 119865 = 119878minus1

1 pre- and postmultiply Ξ by diag119865 119865 we have

Ξ = [(119860 + 119871119873)119865 + 119865(119860 + 119871119873)

119879

+ 1198651198782119865 119861119865

lowast minus1198651198782119865] (30)

Let 119878 = 1198651198782119865 119879 = 119873119865 and consider LMI (24) we can get

1(119905) lt 0 (31)

Hence dynamic system (23) is stableThe proof ofTheorem 3is thus completed

Next based on the above theoretical results a slidingmode controller will be constructed to realize the stabiliza-tion of dynamic system (1)

Theorem4 For dynamic system (1) the state trajectory will bedriven onto the sliding surface 119904(119905) in a finite time by the follow-ing SMC law

119906 (119905) = 119873119909 (119905) minus 120578 sign (119867119879119871119879119904 (119905)) (32)

where 120578 and 120588 are positive scalars


1198812(119905) =

1

2119904119879

(119905) 119904 (119905) (33)

With (21) and (32) we have

119904 (119905) = 119867119871 (119906 (119905) minus 119873119909 (119905))

= minus119867119871120578 sign (119867119879119871119879119904 (119905) + 119873119890 (119905))

(34)


0 2 4 6 8 10 12 14 16 18 20

x1

Time (s)

times108

0

minus1

minus2

minus3

(a)

0 2 4 6 8 10 12 14 16 18 20

x2

Time (s)

times108

5

0

minus5

minus10

(b)

Figure 1 The time response of 119909(119905) of system (1) without SMC

0 2 4 6 8 10 12 14 16 18 20

002

minus02

minus04

minus06

minus08

x1

Estimation of x1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

05

1

minus05

x2

Estimation of x2

(b)

Figure 2 The time response of 119909(119905) and 119909(119905) of system (1) with SMC

When the observation of system (1) is achieved the timederivative of 119881

2(119905) is as

2(119905) = 119904

119879

(119905) 119904 (119905)

= minus119904119879

(119905)119867119871 (120578 sign (119867119879119871119879119904 (119905)))

= minus12057810038171003817100381710038171003817119873119879

119867119879

119871119879

119904 (119905)10038171003817100381710038171003817le 0

(35)

Hence the system state trajectory can converge to the slidingsurface 119904(119905) and stay on the surface This concludes the proofof Theorem 4

5 Numerical Example

In this section we will verify the proposed theoretical theo-rems by giving some illustrative examples

First consider dynamic system (1) with the followingparameters

119860=[1 05

05 08] 119861=[

minus05 0

03 minus03] 119871=[

08 0

0 12]

119867 = [02 02] 119862 = [1 1]

119889 = 2119904 119897 = 1119904 120578 = 05

1199090= [minus06 06]

119879

1199090= [0 10]

119879

(36)

Corresponding simulation results are shown in Figure 1

Then the presented observer-based SMC method in thispaper is applied to system (1) Based onTheorem 3 we can getthe control parameters as follows

119873 = [minus30937 minus01861

minus01837 minus29113] 119870 = [

140436

141551] (37)

In addition to avoid the chattering phenomenon of SMCsign(119904(119905)) is recommended to be replaced by 119904(119905)(|119904(119905)| +

005) and the corresponding simulation results are shown inFigures 2 and 3

Remark 5 From Figure 1 it can be seen that the state varia-bles of the systemdiverge to zero as timepasses byThismeansthat the considered system is not stable Figure 2 shows thetime response of system state 119909(119905) and system state esti-mate 119909(119905) based on the given observer-based SMC methodFigure 3 shows the time response of sliding mode controlinput 119906(119905) It can be seen that119909(119905) converges to 119909(119905) during 40seconds and the original unstable dynamic system becomesstable for the utilization of our controller which demonstratesthe effectiveness of the presented theoretical results

6 Conclusion

In this paper the problem on the stabilization for a kindof dynamic system has been discussed Through utiliz-ing Lyapunov stability theory and linear matrix inequalitytechniques a delayed output feedback observer has beendesigned and a slidingmode controller has been constructed


0 2 4 6 8 10 12 14 16 18 20

0

1

2

Time (s)

u1

minus1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

2

Time (s)

u2

minus2

minus4

(b)

Figure 3 The time response of control input 119906(119905) of system (1)

to realize the stabilization of the system Finally some typicalnumerical examples have been included to verify the validityof the proposed SMC method

Conflict of Interests

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

Acknowledgments

This work was partially supported by the Key Projectsof Xihua University (Z1120946) the National Key BasicResearch Program China (2011CB710706 2012CB215202)the 111 Project (B12018) and the National Natural ScienceFoundation of China (61174058 61134001)

References

[1] B Zhou Z-Y Li and Z Lin ldquoObserver based output feedbackcontrol of linear systems with input and output delaysrdquo Auto-matica vol 49 no 7 pp 2039ndash2052 2013

[2] A Churilov A Medvedev and A Shepeljavyi ldquoA state observerfor continuous oscillating systems under intrinsic pulse-modu-lated feedbackrdquo Automatica vol 48 no 6 pp 1117ndash1122 2012

[3] D S Laila M Lovera and A Astolfi ldquoA discrete-time observerdesign for spacecraft attitude determination using an orthogo-nality-preserving algorithmrdquoAutomatica vol 47 no 5 pp 975ndash980 2011

[4] X Wang J Liu and K-Y Cai ldquoTracking control for a velocity-sensorless VTOL aircraft with delayed outputsrdquo Automaticavol 45 no 12 pp 2876ndash2882 2009

[5] X Zhang and Y Lin ldquoAdaptive output feedback tracking for aclass of nonlinear systemsrdquoAutomatica vol 48 no 9 pp 2372ndash2376 2012

[6] B Jiang P Shi and Z Mao ldquoSliding mode observer-based faultestimation for nonlinear networked control systemsrdquo CircuitsSystems and Signal Processing vol 30 no 1 pp 1ndash16 2011

[7] H R Karimi and P Maass ldquoA convex optimization approach torobust observer-based 119867

infincontrol design of linear parameter-

varying delayed systemsrdquo International Journal of ModellingIdentification and Control vol 4 pp 226ndash241 2008

[8] M Liu D W C Ho and Y Niu ldquoObserver-based controllerdesign for linear systems with limited communication capacityvia a descriptor augmentation methodrdquo IET Control Theory ampApplications vol 6 no 3 pp 437ndash447 2012

[9] W-J Chang C-C Ku and P-H Huang ldquoRobust fuzzy controlvia observer feedback for passive stochastic fuzzy systems withtime-delay and multiplicative noiserdquo International Journal ofInnovative Computing Information and Control vol 7 no 1 pp345ndash364 2011

[10] H R Karimi ldquoObserver-based mixed 1198672119867infin

control designof linear systems with time-varying delays an LMI approachrdquoInternational Journal of Control Automation and Systems vol 6no 1 pp 1ndash14 2008

[11] L Wu and D W C Ho ldquoSliding mode control of singular sto-chastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash7832010

[12] M Basin L Fridman and P Shi ldquoOptimal sliding mode algo-rithms for dynamic systemsrdquo Journal of the Franklin Institutevol 349 no 4 pp 1317ndash1322 2012

[13] Z Lin Y Xia P Shi and H Wu ldquoRobust sliding mode controlfor uncertain linear discrete systems independent of time-delayrdquo International Journal of Innovative Computing Informa-tion and Control vol 7 no 2 pp 869ndash880 2011

[14] S Qu Z Lei Q Zhu and H Nouri ldquoStabilization for a class ofuncertain multi-time delays system using sliding mode con-trollerrdquo International Journal of Innovative Computing Informa-tion and Control vol 7 no 7 pp 4195ndash4205 2011

[15] LWu X Su and P Shi ldquoSlidingmode control with bounded 1198712

gain performance of Markovian jump singular time-delay sys-temsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012

[16] P Shi Y Xia G P Liu and D Rees ldquoOn designing of sliding-mode control for stochastic jump systemsrdquo IEEE Transactionson Automatic Control vol 51 no 1 pp 97ndash103 2006

[17] M Liu and P Shi ldquoSensor fault estimation and tolerant controlfor Ito stochastic systems with a descriptor sliding modeapproachrdquo Automatica vol 49 no 5 pp 1242ndash1250 2013

[18] Z He J Wu G Sun and C Gao ldquoState estimation and slidingmode control of uncertain switched hybrid systemsrdquo Interna-tional Journal of Innovative Computing Information and Con-trol vol 8 pp 7143ndash7156 2012

[19] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[20] L Wu P Shi and H Gao ldquoState estimation and sliding-modecontrol of Markovian jump singular systemsrdquo IEEE Transac-tions on Automatic Control vol 55 no 5 pp 1213ndash1219 2010

[21] T-C Lin S-W Chang and C-H Hsu ldquoRobust adaptive fuzzysliding mode control for a class of uncertain discrete-time non-linear systemsrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 1 pp 347ndash359 2012


[22] M Liu P Shi L Zhang and X Zhao ldquoFault-tolerant controlfor nonlinear Markovian jump systems via proportional andderivative sliding mode observer techniquerdquo IEEE Transactionson Circuits and Systems I vol 58 no 11 pp 2755ndash2764 2011

[23] H R Karimi ldquoA slidingmode approach to119867infinsynchronization

of master-slave time-delay systems with Markovian jumpingparameters and nonlinear uncertaintiesrdquo Journal of the FranklinInstitute vol 349 no 4 pp 1480ndash1496 2012

[24] J Zhang P Shi and Y Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010

[25] M Liu and G Sun ldquoObserver-based sliding mode control forIto stochastic time-delay systemswith limited capacity channelrdquoJournal of the Franklin Institute vol 349 no 4 pp 1602ndash16162012

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


constructed An integral sliding surface will be presented torealize the SMCof the systemwith themore available stabilitycondition Finally some numerical examples will be includedto demonstrate the validity of the proposed control method

Notation used in this paper is fairly standard Let 119877119899 bethe 119899-dimensional Euclidean space 119877119899times119898 represents the setof 119899 times 119898 real matrix lowast denotes the elements below the maindiagonal of a symmetric block matrix diag represents thediagonal matrix the notation 119860 gt 0 means that 119860 is the realsymmetric and positive definite 119868 denotes the identitymatrixwith appropriate dimensions and sdot refers to the Euclideanvector norm

2 Problem Statement and Preliminaries

In this paper the following dynamic system is considered

(119905) = 119860119909 (119905) + 119861119909 (119905 minus 119897) + 119871119906 (119905)

119911 (119905) = 119862119909 (119905 minus 119889)

(1)

where 119909(119905) isin 119877119899 is the system state 119906(119905) isin 119877

119898 is the systeminput 119910(119905) isin 119877

119897 is the system output 119897 is discrete delay 119911(119905) isthemeasurable feedback output and119860119861 119871119862 are the systemmatrices with appropriate dimension

Then an observer of system (1) is built as follows

119909 (119905) = 119860119909 (119905)+119861119909 (119905 minus 119897)+119871119906 (119905)+119890119860119889

119870 [119911 (119905) minus 119862119909 (119905 minus 119889)]

119909 (119905 minus 119889) = 119860119909 (119905 minus 119889) + 119861119909 (119905 minus 119889 minus 119897) + 119871119906 (119905 minus 119889)

+ 119870 [119911 (119905) minus 119862119909 (119905 minus 119889)]

(2)

where 119909(119905) isin 119877119899 is the state vector 119870 is the feedback control

matrix and 119889 is the measurable feedback delayDefine the observation error

119890 (119905) = 119909 (119905) minus 119909 (119905) (3)

The following theoretical result will be used throughout thepaper

Lemma 1 For observer (2) the following equation holds

119909 (119905) = 119890119860119889

119909 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

(119871119906 (120591) + 119861119909 (120591 minus 119897)) 119889120591

(4)

Proof The derivative of (4) is as follows

119909 (119905) = 119890119860119889 119909 (119905 minus 119889) + 119860int

119905

119905minus119889

119890119860(119905minus120591)

(119871119906 (120591) + 119861119909 (120591 minus 119897)) 119889120591

+ 119871119906 (119905) minus 119890119860119889

119871119906 (119905 minus 119889) + 119861119909 (119905 minus 119897)

minus 119890119860119889

119861119909 (119905 minus 119889 minus 119897)

= 119890119860119889 119909 (119905 minus 119889) + 119860 (119909 (119905) minus 119890

119860119889

119909 (119905 minus 119889))

+ 119871119906 (119905) minus 119890119860119889

119871119906 (119905 minus 119889) + 119861119909 (119905 minus 119897)

minus 119890119860119889

119861119909 (119905 minus 119889 minus 119897)

= 119860119909 (119905) + 119871119906 (119905) + 119861119909 (119905 minus 119897)

+ 119890119860119889

( 119909 (119905 minus 119889) minus 119860119909 (119905 minus 119889) minus 119871119906 (119905 minus 119889)

minus119861119909 (119905 minus 119889 minus 119897) )

(5)

Consider (2) we get

119909 (119905) = 119860119909 (119905)+119861119909 (119905 minus 119897)+119871119906 (119905) + 119890119860119889

119870 [119911 (119905) minus 119862119909 (119905 minus 119889)]

(6)

The proof of Lemma 1 is thus completed

3 Design of Observer

Theorem 2 If there exists a matrix 119872 and positive-definitesymmetric matrices 119875 and 119876 satisfy

[119876 + 119875119860 minus119872119862 + 119860

119879

119875 minus 119862119879

119872119879

119875119861


119870 = 119875minus1

119872 (8)

then observer (2) can realize the observation of system (1)

Proof From system (1) we have

(119905 minus 119889) = 119860119909 (119905 minus 119889) + 119861119909 (119905 minus 119897 minus 119889) + 119871119906 (119905 minus 119889) (9)

With (2) we can get

119890 (119905 minus 119889) = (119860 minus 119870119862) 119890 (119905 minus 119889) + 119861119890 (119905 minus 119897 minus 119889) (10)

Choose the Lyapunov functional candidate as

119881 (119905 minus 119889) = 119890119879

(119905 minus 119889) 119875119890 (119905 minus 119889) + int

119905minus119889

119905minus119889minus119897

119890119879

(119904) 119876119890 (119904) 119889119904

(11)

The time derivative of 119881(119905 minus 119889) is

(119905 minus 119889) = 2119890119879

(119905 minus 119889) 119875 119890 (119905 minus 119889)

+ 119890119879


(119905 minus 119889 minus 119897)


= 2119890119879

(119905 minus 119889) 119875 ((119860 minus 119870119862) 119890 (119905 minus 119889) + 119861119890 (119905 minus 119897 minus 119889))

+ 119890119879


(119905 minus 119889 minus 119897)


(12)

Then the following equation holds

(119905 minus 119889) = 120585119879

(119905 minus 119889) Ξ120585 (119905 minus 119889) (13)

where

120585 (119905) = [119890119879

(119905) 119890 (119905 minus 119897)]119879

Ξ = [119875 (119860 minus 119870119862) + (119860 minus 119870119862)

119879

119875 + 119876 119875119861

lowast minus119876]

(14)



(119905 minus 119889) lt 0 (15)

Hence

lim119905rarrinfin

119890 (119905 minus 119889) 997888rarr 0 (16)


119909 (119905) = 119890119860119889

119909 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

(119871119906 (120591) + 119861119909 (120591 minus 119897)) 119889120591

(17)


119890 (119905) = 119890119860119889

119890 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

119861119890 (120591 minus 119897) 119889120591 (18)





119904 (119905) = 119867119909 (119905) minus int

119905

0

119867(119860 + 119871119873) 119909 (120585) 119889120585

minus int

119905

0

119867119861119909 (120585 minus 119897 (120585)) 119889120585

(19)


119909 (119905) = 119909 (0) + int

119905

0

(119860119909 (120585) + 119861119909 (120585 minus 119897) + 119871119906 (120585)) 119909 (120585) 119889120585

(20)


119904 (119905) = 119867119909 (0) + 119867119871int

119905

0

(119906 (120585) minus 119873119909 (120585)) 119889120585 (21)


119906119890119902(119905) = 119873119909 (119905) (22)


(119905) = (119860 + 119871119873) 119909 (119905) + 119861119909 (119905 minus 119897) (23)



[119860119865 + 119871119879 + 119865

119879

119860119879

+ 119879119879

119871119879

+ 119878 119861119865


119873 = 119879minus1

119865 (25)



1198811(119905) = 119909

119879

(119905) 1198781119909 (119905) + int

119905

119905minus119897

119909119879

(119904) 1198782119909 (119904) 119889119904 (26)


1(119905) = 2119909

119879

(119905) 1198781 (119905) + 119909

119879

(119905) 1198782119909 (119905) minus 119909

119879

(119905 minus 119897) 1198782119909 (119905 minus 119897)

= 2119909119879

(119905) 1198781((119860 + 119871119873) 119909 (119905 minus 119889) + 119861119909 (119905 minus 119897))

+ 119909119879

(119905) 1198782119909 (119905) minus 119909

119879

(119905 minus 119897) 1198782119909 (119905 minus 119897)

(27)

Hence we have

1(119905) = 120585

119879

(119905) Ξ120585 (119905) (28)

where

120585 (119905) = [119909119879

(119905) 119909 (119905 minus 119897)]119879

Ξ = [1198781(119860 + 119871119873) + (119860 + 119871119873)

119879

1198781+ 11987821198781119861

lowast minus1198782

]

(29)

Let 119865 = 119878minus1


Ξ = [(119860 + 119871119873)119865 + 119865(119860 + 119871119873)

119879

+ 1198651198782119865 119861119865

lowast minus1198651198782119865] (30)


1(119905) lt 0 (31)




119906 (119905) = 119873119909 (119905) minus 120578 sign (119867119879119871119879119904 (119905)) (32)



1198812(119905) =

1

2119904119879

(119905) 119904 (119905) (33)


119904 (119905) = 119867119871 (119906 (119905) minus 119873119909 (119905))

= minus119867119871120578 sign (119867119879119871119879119904 (119905) + 119873119890 (119905))

(34)


0 2 4 6 8 10 12 14 16 18 20

x1

Time (s)

times108

0

minus1

minus2

minus3

(a)

0 2 4 6 8 10 12 14 16 18 20

x2

Time (s)

times108

5

0

minus5

minus10

(b)


0 2 4 6 8 10 12 14 16 18 20

002

minus02

minus04

minus06

minus08

x1

Estimation of x1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

05

1

minus05

x2

Estimation of x2

(b)



2(119905) is as

2(119905) = 119904

119879

(119905) 119904 (119905)

= minus119904119879

(119905)119867119871 (120578 sign (119867119879119871119879119904 (119905)))

= minus12057810038171003817100381710038171003817119873119879

119867119879

119871119879

119904 (119905)10038171003817100381710038171003817le 0

(35)


5 Numerical Example



119860=[1 05

05 08] 119861=[

minus05 0

03 minus03] 119871=[

08 0

0 12]

119867 = [02 02] 119862 = [1 1]

119889 = 2119904 119897 = 1119904 120578 = 05

1199090= [minus06 06]

119879

1199090= [0 10]

119879

(36)



119873 = [minus30937 minus01861

minus01837 minus29113] 119870 = [

140436

141551] (37)




6 Conclusion



0 2 4 6 8 10 12 14 16 18 20

0

1

2

Time (s)

u1

minus1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

2

Time (s)

u2

minus2

minus4

(b)





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905 minus 119889) lt 0 (15)

Hence

lim119905rarrinfin

119890 (119905 minus 119889) 997888rarr 0 (16)


119909 (119905) = 119890119860119889

119909 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

(119871119906 (120591) + 119861119909 (120591 minus 119897)) 119889120591

(17)


119890 (119905) = 119890119860119889

119890 (119905 minus 119889) + int

119905

119905minus119889

119890119860(119905minus120591)

119861119890 (120591 minus 119897) 119889120591 (18)





119904 (119905) = 119867119909 (119905) minus int

119905

0

119867(119860 + 119871119873) 119909 (120585) 119889120585

minus int

119905

0

119867119861119909 (120585 minus 119897 (120585)) 119889120585

(19)


119909 (119905) = 119909 (0) + int

119905

0

(119860119909 (120585) + 119861119909 (120585 minus 119897) + 119871119906 (120585)) 119909 (120585) 119889120585

(20)


119904 (119905) = 119867119909 (0) + 119867119871int

119905

0

(119906 (120585) minus 119873119909 (120585)) 119889120585 (21)


119906119890119902(119905) = 119873119909 (119905) (22)


(119905) = (119860 + 119871119873) 119909 (119905) + 119861119909 (119905 minus 119897) (23)



[119860119865 + 119871119879 + 119865

119879

119860119879

+ 119879119879

119871119879

+ 119878 119861119865


119873 = 119879minus1

119865 (25)



1198811(119905) = 119909

119879

(119905) 1198781119909 (119905) + int

119905

119905minus119897

119909119879

(119904) 1198782119909 (119904) 119889119904 (26)


1(119905) = 2119909

119879

(119905) 1198781 (119905) + 119909

119879

(119905) 1198782119909 (119905) minus 119909

119879

(119905 minus 119897) 1198782119909 (119905 minus 119897)

= 2119909119879

(119905) 1198781((119860 + 119871119873) 119909 (119905 minus 119889) + 119861119909 (119905 minus 119897))

+ 119909119879

(119905) 1198782119909 (119905) minus 119909

119879

(119905 minus 119897) 1198782119909 (119905 minus 119897)

(27)

Hence we have

1(119905) = 120585

119879

(119905) Ξ120585 (119905) (28)

where

120585 (119905) = [119909119879

(119905) 119909 (119905 minus 119897)]119879

Ξ = [1198781(119860 + 119871119873) + (119860 + 119871119873)

119879

1198781+ 11987821198781119861

lowast minus1198782

]

(29)

Let 119865 = 119878minus1


Ξ = [(119860 + 119871119873)119865 + 119865(119860 + 119871119873)

119879

+ 1198651198782119865 119861119865

lowast minus1198651198782119865] (30)


1(119905) lt 0 (31)




119906 (119905) = 119873119909 (119905) minus 120578 sign (119867119879119871119879119904 (119905)) (32)



1198812(119905) =

1

2119904119879

(119905) 119904 (119905) (33)


119904 (119905) = 119867119871 (119906 (119905) minus 119873119909 (119905))

= minus119867119871120578 sign (119867119879119871119879119904 (119905) + 119873119890 (119905))

(34)


0 2 4 6 8 10 12 14 16 18 20

x1

Time (s)

times108

0

minus1

minus2

minus3

(a)

0 2 4 6 8 10 12 14 16 18 20

x2

Time (s)

times108

5

0

minus5

minus10

(b)


0 2 4 6 8 10 12 14 16 18 20

002

minus02

minus04

minus06

minus08

x1

Estimation of x1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

05

1

minus05

x2

Estimation of x2

(b)



2(119905) is as

2(119905) = 119904

119879

(119905) 119904 (119905)

= minus119904119879

(119905)119867119871 (120578 sign (119867119879119871119879119904 (119905)))

= minus12057810038171003817100381710038171003817119873119879

119867119879

119871119879

119904 (119905)10038171003817100381710038171003817le 0

(35)


5 Numerical Example



119860=[1 05

05 08] 119861=[

minus05 0

03 minus03] 119871=[

08 0

0 12]

119867 = [02 02] 119862 = [1 1]

119889 = 2119904 119897 = 1119904 120578 = 05

1199090= [minus06 06]

119879

1199090= [0 10]

119879

(36)



119873 = [minus30937 minus01861

minus01837 minus29113] 119870 = [

140436

141551] (37)




6 Conclusion



0 2 4 6 8 10 12 14 16 18 20

0

1

2

Time (s)

u1

minus1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

2

Time (s)

u2

minus2

minus4

(b)





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20

x1

Time (s)

times108

0

minus1

minus2

minus3

(a)

0 2 4 6 8 10 12 14 16 18 20

x2

Time (s)

times108

5

0

minus5

minus10

(b)


0 2 4 6 8 10 12 14 16 18 20

002

minus02

minus04

minus06

minus08

x1

Estimation of x1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

05

1

minus05

x2

Estimation of x2

(b)



2(119905) is as

2(119905) = 119904

119879

(119905) 119904 (119905)

= minus119904119879

(119905)119867119871 (120578 sign (119867119879119871119879119904 (119905)))

= minus12057810038171003817100381710038171003817119873119879

119867119879

119871119879

119904 (119905)10038171003817100381710038171003817le 0

(35)


5 Numerical Example



119860=[1 05

05 08] 119861=[

minus05 0

03 minus03] 119871=[

08 0

0 12]

119867 = [02 02] 119862 = [1 1]

119889 = 2119904 119897 = 1119904 120578 = 05

1199090= [minus06 06]

119879

1199090= [0 10]

119879

(36)



119873 = [minus30937 minus01861

minus01837 minus29113] 119870 = [

140436

141551] (37)




6 Conclusion



0 2 4 6 8 10 12 14 16 18 20

0

1

2

Time (s)

u1

minus1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

2

Time (s)

u2

minus2

minus4

(b)





Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20

0

1

2

Time (s)

u1

minus1

(a)

0 2 4 6 8 10 12 14 16 18 20

0

2

Time (s)

u2

minus2

minus4

(b)





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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