Open Access Finite-time Stabilization for a Class of Networked Systems with Delay

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1 Send Orders for Reprins o reprins@benhamscience.ae The Open Auomaion and Conrol Sysems Journal, 214, 6, Open Access Finie-ime Sabilizaion for a Class of Neworked Sysems wih Delay Hejun Yao School of Mahemaics and Saisics, Anyang Normal Universiy, Anyang, 455, China Absrac: The finie-ime sabilizaion problem for a class of neworked conrol sysems wih sae delay is considered in his paper. Wih Lyapunov heorem, a sufficien condiion for he design of a sae feedback conroller which makes he closed loop sysem finie-ime sable is provided. And, he sufficien condiion is given in erms of linear marix inequaliy. A numerical example is presened o illusrae he proposed mehodology. Keywords: Neworked conrol sysems, delay, finie-ime sabilizaion. 1. INTRODUCTION Feedback conrol sysems wherein he conrol loops are closed hrough a real-ime nework are called neworked conrol sysems NCSs. The main feaure of NCSs is ha he componens sensors, conroller and acuaors of he sysem are conneced by a nework. Compared wih radiional poin-o-poin design, he NCSs have many advanages such as low cos, wiring reducion, simple insallaion and mainenance, and high reliabiliy [1-3]. For hese reasons, NCSs have been widely applied o many complicaed conrol sysems, such as aviaion and aerospace fields [4, 5]. However, he inserion of he communicaion nework in feedback conrol loop makes he analysis and design of NCSs complicaed. The change of communicaion archiecure induces differen forms of ime delay uncerainy beween sensors, acuaors and conrollers. These ime delays come from he ime sharing of he communicaion medium as well as he compuaion ime required for physical signal coding and communicaion processing [6-8]. I is well known ha ime delays can degrade a sysems performance and even cause sysem insabiliy. Therefore, he issues of sabiliy analysis and designing conrollers for NCSs have received much consideraion for decades [7-12]. Much work has been done on he robus conrol of NCSs over he pas en years. Mos of he resuls in his field relae o sabiliy and performance crieria defined over an infinie ime inerval. However, he main concern in many pracical applicaions is he behavior of he dynamical sysems over a fixed finie ime inerval [13], for example, large values of he sae are no accepable in he presence of sauraions. Therefore, we need o check he unaccepable values ha he sysem sae does no exceed a cerain hreshold during a fixed finie-ime inerval by giving some iniial condiions. The concep of finie-ime sabiliy referring o hese ransien performances of conrol dynamics daes back o he Sixies, when i was inroduced in he conrol lieraure [14]. Then, some aemps on finie-ime sabiliy can be found by using Lyapunov funcional approach [15]. Recenly, wih he aid of LMIs echniques, more conceps of finie-ime sabiliy have been proposed for linear coninuous-ime or discree-ime conrol sysem in he lieraures; see, for insance [16-22]. Bu he above papers consider he Lyapunov sabiliy for NCSs, a few resuls on finie-ime sabiliy for NCSs has been repored. Based on his fac, some new mehods and approaches should be developed for designing conrollers for NCSs, which moivaes his paper. Inspired by he above lieraure, in his paper, he aenion is focused on he finieime sabilizaion of a class of NCSs wih sae delay and communicaion delays. The design is divided in wo seps: he synhesis condiion of he sae feedback conroller, supposing ha he sae variables are available, and hen he sufficien condiion is given in erms of an LMI opimizaion problem. Conversely, he approach proposed in his paper leads o an LMI formulaion, which gives he opporuniy of fiing he finie-ime conrol problem in he general framework of he LMI approach o he muli-objecive synhesis. This paper is organized as follows: in Secion 2 he definiion of finie-ime sabiliy is recalled and he problem we wan o solve is formally saed. In secion 3, we address he finie-ime sabiliy problems, namely sufficien condiions for he exisence of sae feedback conrollers guaraneeing finie-ime sabilizaion of he closed loop sysems are provided. In secion4, i is shown how he proposed design condiion can be expressed in erms of LMIs and, herefore, can be efficienly deal wih, and a numerical example is provided. Some conclusions are drawn in Secion PRELIMINARIES In his paper we consider he following ypical NCSs wih sae delay as shown in Fig. 1.!x = Ax + A h x! h + Bu + B 1 " / Benham Open

2 178 The Open Auomaion and Conrol Sysems Journal, 214, Volume 6 Hejun Yao ime delay NCSs 3 seing! " is said o be finie- Acuaor Plan Sensor ime sable FTS wih respec o c 1,c 2,T, R, if x T Rx! c 1 " x T Rx < c 2 # $[,T ] 6 Where, A, A h, B, B 1 ake value in R n!n, R n!n, R n!m, R n!l, respecively. x!r n is he sae vecor, u!r m is he conrol inpu, and h denoes sysem sae delay.! "R l is he exogenous disurbance and saisfies. T "! T!d # d, d $ 2 Delay Nework Medium Delay ca sc Conroller Fig. 1. A ypical neworked conrol sysem. To simplify he analysis, based on acual engineering background, a full characerizaion of his NCS is given by he following assumpion. Assumpion 1. The sensor is ime driven; he conroller and acuaor are even driven. We use! and! sc o represen he sensor-conroller and conroller-acuaor delay, re- ca specively, hen he communicaion delay is given by! =! sc +! ca. Considering he effec of communicaion delay!, he above plan model is ransformed ino an NCS model!x = Ax + A h x! h + Bu! " + B 1 # 3 Concerning NCSs 3, we design a sae feedback conroller u = Kx 4 Where, K is a sae feedback gain marix o be deermined laer. Then, he resuling closed-loop NCSs follows ha!x = Ax + A h x! h + BKx! " + B 1 # 5 The aim of his paper is o find some sufficien condiions which guaranee he exisence of a sae feedback conroller which sabilizes he sysem 1 over he finie inerval [,T ]. By selecing he appropriae Lyapunov Krasovskii funcion, he main resuls will be given in he form of LMIs. The general idea of finie-ime conrol can be formalized hrough he following definiions over a finie-ime inerval for some given iniial condiions [17-22]. Definiion 1 Finie-ime sabiliy. Given hree posiive scalars c1, c2, T, wih c 1 < c 2 and a posiive marix R, he Remark 1. Differen wih he concep of Lyapunov asympoic sabiliy, FTS is a pracical concep used o sudy he behavior of he sysem wihin a finie inerval. A sysem is said o be finie-ime sable if, once we fix a finie-ime inerval, is sae remains wihin prescribed bounds during his ime inerval. Obviously, a sysem which is FTS may be no Lyapunov asympoically sable; conversely a Lyapunov asympoically sable sysem could no be FTS if, during he ransiens sae, is sae exceeds he prescribed bounds. Definiion 2 Finie-ime boundedness. Given hree posiive scalars c 1, c 2, T wih c 1 < c 2, a posiive definie marix R and a class of signals u, he ime delay NCSs 3 is said o be FTB wih respec o c 1, c 2, u, T, R, d, if condiion m 6 holds for all u R. Definiion 3 Finie-ime sabilizaion via sae feedback. Given hree posiive scalars c1, c2, T wih c1 < c, a posiive 2 definie marix R, he ime delay NCSs 3 is FTB wih respec o c 1, c 2, T, R, d if here exiss a sae feedback conroller in he form 4 and he condiion 6 holds. Remark 2. Noice ha condiions 7 can be guaraneed by leing! T! " d see [17 19, 23 24]. In he presence of exernal inpus, FTS can lead o he concep of FTB. Thus, FTS can be recovered as a paricular case of FTB by seing d =, ha is, FTB implies finie-ime sabiliy, bu he converse is no he fac. FTB and FTS are open-loop conceps. The finie-ime conrol problem concerns he design of a sae feedback conroller which ensures he FTS or he FTB of he closed-loop ime-delay NCSs 3.! # # " Lemma 1 [3] The LMI Yx W T x W x Rx Is equivalen o, $ > Rx >,Yx!W xr!1 xw T x > Where, Yx = Y T x, Rx = R T x and W x depend on x. 3. MAIN RESULTS In his secion, we consider he finie-ime conrol synhesis for NCSs wih sae delay and communicaion delay, in erms of LMIs, we obain he sufficien condiion for he finie-ime sabilizaion via sae feedback. c Theorem 1. Given hree posiive scalars c1, c2, T, wih < c, a posiive definie marix R, he ime delay NCSs 1 2

3 Finie-ime Sabilizaion for a Class of Neworked Sysems wih Delay The Open Auomaion and Conrol Sysems Journal, 214, Volume o is finie-ime sabilizaion via sae feedback wih respec c, c, T, R, d, if here exis a scalarα, posiive definie marices n n P R, n n Q R, T R n n, l l S R, and m n marix K R such ha he following marix inequaliies hold! PA h PBK PB 1 " #Q " " #T " " " #$S And, < 8 c 1 P! + h Q! + " T! + d S1# e #$T! min P! < c 2 e #$T 9 Where,! = PA + A T P + Q + T " # P,! P = R!1/2 PR!1/2,!Q = R!1/2 QR!1/2,! T = R!1/2 TR!1/2 and i and! min i indicae he maximum and minimum eigenvalue of he augmen, respecively. Proof; For given symmeric posiive definie marix PQT,,, we consruc he following Lyapunov Krasovskii funcion: V x = x T Px + " x T!Qx!d! + " x T!Tx!d! -# -h 1 The ime derivaive of V x along he rajecories of sysem 5 is given by! V x = x T PA + A T Px + 2x T PA h x! h + 2x T PBKx! " + 2x T PB 1 # + x T Qx + x T! hqx! h + x T Tx + x T! " Tx! " =x T PA + A T P + Q + T x + 2x T PA h x! h + 2x T PBKx! " + 2x T PB 1 # + x T! hqx! h + x T! " Tx! " T $ x $ x x! h x! h = x! " x! " # # Where, $! = PA + A T P + Q + T PA h PBK PB 1 " #Q " " #T " " " From condiion 8, we have! V x <! x T Px +!" T S" <!V x +!" T S" Muliplying 11 by e!", we can obain e!"! V x! e!" "V x < "e!" # T S# Furhermore d d e!" V x < "e!" # T S# 11 By inegraing he above inequaliy from o, wih [, T], i follows ha e!" V x!v x < "e!"# $ T #S$#d# 12 Noing ha! ",! P = R!1/2 PR!1/2,! Q = R!1/2 QR!1/2, and!t = R!1/2 TR!1/2, we can obain he following relaion: x T Px! V x < e " V x + "d# max Se " e $" d < e " [x T Px + x T Qxd $ h + x T Txd + d# max S1$ e $" ] $ < e " [x T R 1/2! PR 1/2 x + x T R 1/2 QR /2 xd $ h + x T R 1/2 TR /2 xd $ + d# max S1$ e $" ] < e " [# max! Px T Rx + # max Q! x T Rxd $ h + # max T! x T Rxd - + d# max S1$ e $" ] < e "T [c 1 # max! P + h# max! Q + # max! T + d# max S1$ e $" ] On he oher hand, i yields 13 x T Px = x T R 1/2! PR 1/2 x! " min! Px T Rx 14

4 1782 The Open Auomaion and Conrol Sysems Journal, 214, Volume 6 Hejun Yao Puing ogeher 13 and 14 we have x T Rx e!t [c 1 " max! P + h" max! Q + #" max! T + d" < max S1$ e $!T ] " min P! Condiion 9 and inequaliy 15 imply, x T Rx! c 2, " #[,T ]. This complees he proof. Therefore, he proof follows. 15 Theorem 2. Given hree posiive scalars c 1,c 2,T, wih c1 < c, a posiive definie marix R, he ime delay NCSs 2 3 is finie-ime sabilizaion via sae feedback u = KX!1 x wih respec o c 1,c 2,T, R,d if, here exis scalars! ", # i >, i = 1,2,3,4., posiive definie marices X!R n"n, Q!R n"n, T!R n"n, S!R l"l, and marix K!R m"n such ha he following marix inequaliies hold:! A h X BK B 1 " #Q " " #T " " " #$S < 16 R "1 < X < R "1 17! 2 Q < X 18! 3 T < X 19 < S <! 4 I 2 d! 4 1" e "#T " c 2 e "#T c 1 h $ + " + + < 21 "! "! 3 + Where! = AX + XA T + Q + T " # X Proof. Now we prove ha he inequaliy 8 is equivalen o he inequaliy 16. Pre-and pos-muliplying he inequaliy 8 by blockdiagonal marix diag{p -1, P -1, P -1, I}, we know he inequaliy 8 is equivalen o 1 1 Σ AP h BKP B P QP 1 1 < P TP αs 22 Where, Σ= AP + P A + P QP + P TP αp 1 1 T By leing X = P -1, K = KP!1,Q = P!1 QP!1,T = P!1 TP!1, he inequaliy 22 is equivalen o inequaliy 16. On he oher hand, we denoe!x = R!1/2 XR!1/2,! Q = R!1/2 QR!1/2,! T = R!1/2 TR!1/2 Consider ha R is he posiive-definie marix and 1 X =! min P Now inequaliies 17-2 imply ha 1 <! min!p,!p < 1,! Q <! 2!P,! T <! 3! P,!max S <! 4 23 Wih he Schur Lemma1, we know he inequaliy 21 is equivalen o d! 4 1" e "#T " c 2 e "#T + c 1 + h! 2 + $! 3 < 24 Wih 23, he condiion 9 follows ha c 1! P + h! Q + "! T + d S1# e #$T! min! P < d! 4 1# e #$T + c 1 + h! 2 + "! 3 25 Insering he inequaliy 24 ino 25, he inequaliy 9 is saisfied. This complees he proof. Remark 3. If condiion 16 in Theorem 2 is saisfied wihα =, hen sysem 3 is also asympoically sable in he sense of Lyapunov. Moreover in his case he finie-ime properies are guaraneed for all T >. Remark 4. We can see ha he condiions in Theorem1 and Theorem 2 are no LMIs wih respecive oα,c, since 2!, c 2 appear in a nonlinear fashion. However, once we fixα, hey can be urned ino LMIs based feasibiliy problem which can be solved via exising sofware for example he LMI Conrol Toolbox of MATLAB. 4. NUMERICAL EXAMPLE Consider he nework conrol sysems in he form of 3, where, "!1 2 "!x = $ x + #!1 # $ " + $ " # 1 u! + # $.1.2!.1.2!.2 1 x! h

5 Finie-ime Sabilizaion for a Class of Neworked Sysems wih Delay The Open Auomaion and Conrol Sysems Journal, 214, Volume Fig. 2. Sae x 1,x 2 responses of he sysems. In his case, we choose c 1 =.25,! = 1,T = 2, R = I 2 d = 4,! =.5,h =.8. By Solving he LMIs 16-21, we can obain he conroller u KX x x 1 = = [ ] We selec " x =.5 $ #!.5 The simulaion resul are shown in he following figure. From he above Fig. 2, i is clear ha he finie-ime sabilizaion of NCSs 3 via sae feedback of desired closed-loop properies. CONCLUSION In his paper, we considered he finie-ime conrol problem via sae feedback for a class of NCSs wih sae delay and communicaion delay. Based on finie-ime sabiliy heory combined wih LMIs echniques, sufficien condiions ha ensure he closed-loop sysem finie-ime sabiliy is obained. CONFLICT OF INTEREST The auhors confirm ha his aricle conen has no conflics of ineres. ACKNOWLEDGEMENTS The auhors wish o hank he edior and he anonymous reviewers for heir consrucive commens and suggesions o improve he qualiy and he presenaion of he paper. This work is suppored by Naional Naure Science Foundaion of China under Gran ; Naure Science Foundaion of Henan Province under Gran ; The Educaion Deparmen of Henan Province Key Foundaion under Gran 13A1123, 14A523. REFERENCES [1] G.C. Walsh, O. Beldiman, L.G. Bushnell, Asympo-ic behavior of nonlinear neworked conrol sysems, IEEE Transacions on Auomaic Conrol, vol.46, pp , 21. [2] G. C. Walsh, H. Ye, L G. Bushnell, Sabiliy analysis of neworked conrol sysems, IEEE Trans. Conrol Sys Technol, vol.1, pp , 22. [3] S. Hu, Q. Zhu, Sochasic opimal conrol and analysis of sabiliy of neworked conrol sysems wih long delay, Auomaica, vol.39, pp , 23. [4] D. Yue, Q.L. Han, and J. Lam, Nework-based robus H conrol of a sysems wih uncerainy, Auomaica, vol.4, pp , 25. [5] Z. -H. Guan, J. Huang, G. R. Chen, Sabiliy Analysis of Neworked Impulsive Conrol Sysems, Proceeding of 25h Chinese Conrol Conference, pp , 26. [6] Y. Tian, Z. Yu, Mulifracal naure of nework induced ime delay in neworked conrol sysems, Physics Leer A, vol.361, pp.13-17, 27. [7] D. Dacic, D. Nesic, Observer design for wired linear neworked conrol sysems using marix inequaliies, Auomaica, vol.44, pp , 28. [8] N. Vaanski, Neworked conrol wih delay measuremen and esimaion, Conrol Engineering Pracice, vol.17, pp , 29. [9] H. C. Yan, X. H. Huang, M. Wang, H. Zhang, Delay-dependen sabiliy crieria for a class of neworked conrol sysems wih muli-inpu and muli-oupu, Chaos, Solions Fracals, vol.34, pp , 27. [1] X. Jiang, Q.-L. Han, S.R. Liu and A. Xue, A new H sabilizaion crierion for neworked conrol sysems, IEEE Trans. Auoma. Conrol, vol.53, pp , 28.

6 1784 The Open Auomaion and Conrol Sysems Journal, 214, Volume 6 Hejun Yao [11] H. Gao, T. Chen, J. Lam, A new delay sysem approach o nework-based conrol, Auomaica, vol.44, pp.39-52, 28. [12] M. W. Hong, C. L. Lin and B. M.Shiu, Sabilizing nework conrol for pneumaic sysems wih ime-delays, vol.19, pp , 29. [13] A. EI-Gohary, Opimal conrol of an angular moion of a rigid body during infinie and finie-ime inervals, Appl. Mah. Compu, vol.141, pp , 23. [14] P. Dorao, Shor ime sabiliy in linear ime-varying sysems, Proceeding of IRE inernaional convenion record, vol.4, pp.83-87, [15] L.Weiss, E. F. Infane, Finie ime sabiliy under perurbing forces and on produc spaces, IEEE Trans Auoma Conrol, vol.12, pp.54-59, [16] F. Amao, M. Ariola, P. Dorae., Finie-ime conrol of linear sysems subjec o parameric uncerainies and disurbances, Auomaica, vol.37, pp , 21. [17] A. EI-Gohary, A.S. AI-Ruzaiza, Opimal conrol of nonhomogenous prey predaor models during infinie and finie-ime inervals, Appl. Mah. Compu., vol. 146, pp , 23. [18] F. Amao, M. Ariola, Finie-ime conrol of discree-ime linear sysem, IEEE Trans Auoma Conrol, vol.5, pp , 25. [19] E. Moulay, W. Perruquei, Finie ime sabiliy and sabilizaion of a class of coninuous sysems, J Mah Anal Appl, vol.323, pp , 26. [2] F. Amao, M. Ariola, P. Dorae, Finie-ime sabilizaion via dynamic oupu feedback, Auomaica, vol.42, pp , 26. [21] E. Moulay, M. Dambrine, N, Yeganefar, W. Perruquei, Finieime sabiliy and sabilizaion of ime-delay sysems, Sysems Conrol Leers, vol.57, pp , 28. [22] F. Amao, M. Ariola, C, Cosenino, Finie-ime conrol of discree-ime linear sysems: Analysis and design condiions, Auomaica, vol.46, pp , 21. Received: Sepember 16, 214 Revised: December 23, 214 Acceped: December 31, 214 Hejun Yao; Licensee Benham Open. This is an open access aricle licensed under he erms of he Creaive Commons Aribuion Non-Commercial License hp://creaivecommons.org/licenses/by-nc/3./ which permis unresriced, non-commercial use, disribuion and reproducion in any medium, provided he work is properly cied.