Mean-square Stability Control for Networked Systems with Stochastic Time Delay

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1 JOURNAL OF SIMULAION VOL. 5 NO. May 7 Mean-square Sabiliy Conrol for Newored Sysems wih Sochasic ime Delay YAO Hejun YUAN Fushun School of Mahemaics and Saisics Anyang Normal Universiy Anyang Henan. 455 Absrac his noe is concerned wih he problem of delaydisribuion-dependen sabiliy and sabilizaion for newored conrol sysems wih sochasic newor induced delay and daa pace dropou. he sochasic delay and daa pace dropou are viewed as sochasic ime-varying delay wihou any consrains on is derivaive. Which is assumed o be saisfying a inerval Bernoulli disribuion. Due o he probabiliy of he delay aing value in differen inervals a new approach is given o model he newored conrol sysems. Based on he Lyapunov sabiliy heory wih he linear marix inequaliy approach a new sabilizaion crierion is obained. hen he conroller is given o mae he closed-loop sysems mean-square sable. A numerical example is provided o demonsrae he validiy of he proposed design approach. Index erms newored conrol sysems;sochasic delay; pace dropou I. INRODUCION I is well nown ha a feedbac conrol sysem in which all devices(sensors conrollers and acuaors) are inerconneced by communicaion newors is called newored conrol sysems(ncss).compared wih he radiional poin-o-poin archiecure conrol sysems he newored conrol sysems have he advanages of high mobiliy low cos easy mainenance and reconfiguraion. For hese advanages he newored conrol sysems receive more and more aenion and has been a very ho research opic [-4]. However he newor iself is a dynamic sysem ha exhibis characerisics such as newor-induced delay and pace losses. he inserion of he communicaion newor induces differen forms of ime delay uncerainy beween sensors acuaors and conrollers. hese 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 [5-8]. I is well nown in conrol sysems ha ime delays can degrade a sysem's performance and even cause sysem insabiliy. Anoher significan difference beween NCSs and sandard digial conrol is he possibiliy ha daa may be los while in ransi hrough he newor because of uncerainy and noise. And he woredinduced delay or pace losses are usually sochasic and described by a wo-sae Marov chain which is especially common in wireless communicaion newors such as Blueooh and WLAN. Differen meho have been developed o deal wih he above problems [9-]. he sochasic opimal conrol of newored conrol sysems whose newor-induced delay is shorer han one sampling period is sudied using he ime-samp echnique []. By viewing he ime-varying newor-induced delay as parameer uncerainy he sabilizaion of newored conrol sysems wih delay shorer han one sampling period is sudied and sufficien condiions expressed in linear marix inequaliy are presened in []. In his paper our objecive is o consider he problem of delay-disribuion-dependen sabiliy for newored conrol sysems wih sochasic delay and daa pace dropou. A new approach is given o model he NCSs. Based on he Lyapunov sabiliy heory a new sabilizaion crierion is given. II. PROBLEM FORMULAION Consider he following conrol sysem x() x( ) Ax( ) Bu( ) y( ) Cx( ) Du( ) x( ) x n R is he sae vecor u() y R r he inpu vecor () R is he conrolled oupu is an consan vecor. A B C D are nown real consan marices wih appropriae dimensions. hroughou his noe we suppose ha all he sysem's saes are available for a sae feedbac conrol. In he presence of he conrol newor which is shown in fig. daa ransfers beween he conroller and he remoe sysem e.g. sensors and acuaors in a disribued conrol sysem will induce newor delay in addiion o he conroller proceeding delay. In his noe we mae he following assumpions: Assumpion : Sensor and conroller are clocdriven. Assumpion : Acuaor is even-driven. x sc In Fig. is he communicaion delay beween c he sensor and he conroller; is he compuaional delay in he conroller; is he communicaion delay beween he conroller and he acuaor. Here we suppose ha sc c ca [ ] is sochasic neworinduced delay is a consan. d( ) [ d ] represening he sochasic daa dropou number beween he sensor and he acuaor. ca m is ()

2 4 JOURNAL OF SIMULAION VOL. 5 NO. May 7 Figure. A ypical newored conrol sysem h () We inroduce he sochasic delay o denoe he sochasic delay and daa pace dropou h( ) d( ) h ( ) [ ] ( d ). Wih he sae feedbac conroller u () K R We now ha ( ) Kx( ) is a consan marix o be obained. u( ) Kx( ) Kx( d( ) ) Kx( h( )) Insering he above conroller ino sysem() we obain he closed sysem: x( ) Ax( ) BKx( h( )) y( ) Cx( ) Du( ) [ ] () x( ) ( ) [ ] h () [ ] is sochasic on wihou any consrain on is derivaive. he iniial condiion of he sae [ ] is supplemened as x( ) ( ) on [ ] ( ) x funcion on [ ]. () x () is a coninuous [ ) I is assumed ha here exiss a consan such ha he probabiliy of h () aing values in [ ) and [ ] can be observed. In order o employ he informaion of he probabiliy disribuion of he delay in he sysem model he following ses are proposed firsly { : h( ) [ )} { : h( ) [ ]}. Obviously R and hen we define wo funcions as: h( ) h ( ) h( ) h ( ) () Where we suppose ha is a Bernoulli disribued sequence wih Pr ob{ ( ) } E{ ( )} [] is a consan. h ( ) h ( ) and By using he new funcions sochasic variable he sysems() can be equivalenly wrien as x( ) Ax( ) ( ) BKx( h( )) ( ( )) BKx( h( )) (4) y( ) Cx( ) Du( ) [ ] () x( ) ( ) [ ] III. MAIN RESULS Lemma [] For any vecors N Y Z ab and marices wih appropriae dimensions if he following marix inequaliy hol hen we have Y Y Z a Y N a a Nb inf Y Z b Y N Z b Y ( x) W ( x) Lemma [4] he LMI * Rx ( ) equivalen o Rx ( ) Y( x) W( x) R ( x) W ( x) Y( x) Y ( x) R( x) R ( x) depend on x. heorem. For he newored conrol sysems (4) if n n here exis posiive-definie marices P R K i is Yi i wih appropriae marices R dimensions such ha he following marix inequaliies hold * * * Y Y (5) (6) (7) Corresponding o h () aing values in differen inervals a sochasic variable () is defined ()

3 JOURNAL OF SIMULAION VOL. 5 NO. May 7 5 PA A P A A A A Y Y Y Y P BK A BK A BK Y Y Y ( ) PBK ( ) A BK ( ) A BK Y Y Y BK BK Y Y ( ) BK ( ) BK Y Y wih he conroller () he newor conrol sysems(4) is mean-square sable. Proof Choose a Lyapunov funcional candidae for he sysem(4) as follows V( ) V ( ) V ( ) V ( ) (8) V ( ) x ( ) Px( ) V ( ) x ( s) x( s) d V ( ) x ( s) x( s) d P posiive-definie marices wih appropriae dimensions. hen along he soluion of sysem (4) we have ( ) V ( ) ( ) x PA (9) ( ) [ x ( ) x ( h ( )) x ( h ( ))] and A [ A ( ) BK ( ( )) BK] V ( ) ( ) A A ( ) x ( s) x( s) () V ( ) ( ) A A ( ) x ( s) x( s) We now () x( ) x( h ( )) x( s) h () x( ) x( h ( )) x( s) h () Wih wo 4n n marices N M N N M M N M By he lemma we obain ( ) N[ x( ) x( h ( ))] Y N h () x s ( ) ( ) ( ) x( s) ( ) Y [ x( ) x( h ( ))] ( ) ( ) x ( s) x( s) () and ( ) Y[ x( ) x( h ( ))] ( ) ( ) () x ( s) x( s) i i i i i i i Wih he (9-)we can obain Yi Yi Y i Y i E{ V( )} ( ) ( ) i Wih he Lyapunov sabiliy heorem and he inequaliy (5) we now ha he sysem (4) is meansquare sable. heorem For he newored conrol sysems (4) if n n here exis posiive-definie marices P R marices K R i Yi i wih appropriae dimensions such ha he following linear marix inequaliies hold PA ( ) PA ( ) ( ) PA ( ) PA ( ) ( ) Y Y (4) (5) (6)

4 6 JOURNAL OF SIMULAION VOL. 5 NO. May 7 AP PA A A Y Y Y Y BK Y Y Y ( ) BK Y Y Y Y Y Y Y u ( ) KP x( ) wih he conroller he sysems (4) is mean-square sable. Proof Now we proof ha he inequaliy (5) is equivalen o he inequaliy (4). Obviously by he lemma (5) is equivalen o A ( ) A ( ) Where ( ) A ( ) A ( ) ( ) (7) PA A P Y Y Y Y PBK Y Y Y ( ) PBK Y Y Y Y Y Y Y Pre- and Pos-muliplying he inequaliy(7) by diag{ P P P I } and giving some ransformaions P P = = K KP ij P ijp Yij P Yij P i ; j We now he inequaliy (7) is equivalen o (4). IV. SIMULAIONS Consider he newored conrol sysems in he form of (4).. A B C.4... D.4 Solving he linear marix inequaliy (4-6) we can obain he gain marix K of he sabilizing conroller u () K [ ] From he heorem we now ha he sysems (4) is mean-square sable. ACKNOWLEDGMEN he auhor would lie o han he associae edior and he anonymous reviewers for heir consrucive commens and suggesions o improve he qualiy and he presenaion of he paper. his wor was suppored by Naional Naure Science Foundaion under Projec (6765) Henan Province Science and echnology Key Projec (76) he Naional College Sudens Innovaive Enrepreneurial raining Projec (647945) Anyang Normal Universiy Innovaion Foundaion Projec. REFERENCES []. C. Yang Newored conrol sysem:a brief survey IEE Proc.Conrol heory Appl. vol. 5 pp [] Y. Wang H_infy conrol of newored conrol sysems wih ime delay and pace disordering IE Conrol heory Appl.vol. pp [] M. Das R. Ghosh ec. Muli-loop Newored process conrol:a synohronized approach ISA ransacions vol.48pp.-9. [4] D. Dacic D. Nesic Observer design for wired linear newored conrol sysems using marix inequaliies Auomaica vol.44 pp [5] Y. ianz. Yu Mulifracal naure of newor induced ime delay in newored conrol sysems Physics Leer A vol.6 pp [6]. Jiang A new H_\infy sabilizaion crierion for newored conrol sysems IEEE ransacions on Auomaic conrol vol.5 pp [7] H. Gao Sabilizaion of newored conrol sysems wih a new delay characerizaion IEEE ransacions on Auomaic Conrol vol.5 pp [8] D. Huang Dynamic oupu feedbac conrol for uncerain newored conrol sysems wih random newor induced delays Inernaional Journal of conrol Auomaion and Sysmes vol. 7pp [9].. Zhu S. S. Hu Conrollabiliy and observabiliy of newored conrol sysems. Conrol and Decision vol.9 pp

5 JOURNAL OF SIMULAION VOL. 5 NO. May 7 7 [] J. Nilsson B. Bernharson B.Wienmar Sochasic analysis and conrol of real ime sysems wih random ime delays Auomaica vol.4 pp [] G. M. ie L. Wang Sabilizaion of newored conrol sysems wih ime-varying newor-induced delay Proc. of he 4rd IEEE conference on Dicision and Conrol pp