Stability and Stabilization for Discrete Systems with Time-varying Delays Based on the Average Dwell-time Method

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1 Proceedngs of the 29 IEEE Internatonal Conference on Systems, an, and Cybernetcs San Antono, TX, USA - October 29 Stablty and Stablzaton for Dscrete Systems wth Tme-varyng Delays Based on the Average Dwell-tme ethod Jan Sun, Je Chen School of Automaton Bejng Insttute of Technology Bejng 8, Chna helos225@yahoo.com.cn, chenje@bt.edu.cn G.P. Lu, D. Rees Faculty of Advanced Technology Unversty of Glamorgan Pontyprdd CF37 DL, UK gplu@glam.ac.u, drees@glam.ac.u Abstract In ths paper, the problems of the exponental stablty and stablzaton for a class of dscrete systems wth tmevaryng delays are consdered. By convertng dscrete systems wth tme-varyng delays nto swtched systems and usng the average dwell-tme method, a new stablty crteron s obtaned and presented n terms of lnear matrx nequalty. Based on the obtaned stablty condton, a desgn method for the feedbac controller to stablze the system s also proposed. Fnally, some numercal examples are gven to show the effectveness of the proposed method. Index Terms Exponental stablty, tme-varyng delay, feedbac control, swtched systems, lnear matrx nequalty, average dwell-tme method. I. INTRODUCTION Tme-delay s often encountered n many practcal systems. It s well nown that tme-delay s often one of the sources of poor control performance and nstablty. Therefore, the subject of stablty analyss and stablzaton of systems wth delays has attracted consderable attenton durng the past few years, 2. A typcal example of a dscrete system wth tme-varyng delay s the networed control system where the delay from the sensor to the controller and the delay from the controller to the actuator are both tme-varyng 3, 4, 5. So the study of stablty and stablzaton of dscrete-tme-delay systems s mportant both n theory and n practcal applcatons. However, less attenton has been gven to dscrete-tme-delay systems 6, 7, 8, 9, compared wth contnuous-tmedelay systems, 2, 3, 4, 5, 6, 7, 8, 9. Recently, a descrptor approach has been appled to the stablty analyss of dscrete-tme-delay systems wth norm-bounded or polytopc uncertantes 2, 2. By consderng some useful terms whch were gnored n prevous results, new less conservatve stablty crtera were obtaned n 22. However, n 22 the bounds on some terms were enlarged, whch may lead to some conservatsm. Usng a new method to estmate the upper bound on the dervatve of the Lyapunov functonal wthout gnorng any terms, delay-dependent stablty condtons less conservatve than those n 22 were obtaned n 23. It should be noted that the above results for stablty of dscrete-tme-delay systems are all based on the Lyapunov functonal approach. In these results, a basc assumpton s that absolute values of the egenvalues of A + A are all less than, that s, the system s stable wthout delay. In practce, there are some systems whch are stable wth nonzero delay but unstable wthout delay 24. For such systems, all the above results wll fal to obtan a feasble soluton. To the best of the authors nowledge, few results are avalable for such systems especally n dscrete-tme doman, whch motvates ths study. A specal feature of dscrete-tme-delay systems s that they can be transformed nto systems wthout delay by the augmentaton method. A drawbac of ths method s that t may result n a large dmenson of the augmented system for large tme-delays whch maes the obtaned stablty condtons dffcult to chec. However, ths drawbac has been overcome to some extent due to the fast development of the hgh speed processor. Recently, ths augmentaton technque has been extended to dscrete-tme systems wth tme-varyng delay n 25 where systems wth tme-varyng delays are converted to swtched systems and the common quadratc Lyapunov functon approach s used to solve the problems of stablty and stablzaton for such systems. It s well nown the common quadratc Lyapunov functon method s conservatve. Therefore, the result n 25 should be further mproved. In exstng results, the asymptotc stablty for dscrete-tme systems wth tme-varyng delays has been nvestgated and lttle attenton has been pad to the exponental stablty for such systems. In exstng results, the delay s often assumed to belong to a gven nterval d d() d 2. If there are some delays larger than d 2 occurrng n the delay sequence, exstng results wll fal to determne whether the system s stable or not. Bascally, f the occurrence rate for the large delay s not hgh, the system wll be stll stable. To the best of the authors nowledge, such a problem has not been well consdered and only 26 has consdered a smlar problem. However, results n 26 are based on the Lyapunov-Krasovs functonal approach, so they cannot be appled to systems whch are unstable wthout delay. In ths paper, dscrete systems wth tme-varyng delays /9/$ IEEE 493

2 are converted nto swtched systems usng the augmentaton method. Unle exstng results, the assumpton that absolute values of the egenvalues of A + A are all less than s removed. Usng the average dwell-tme method 27, we obtan a new exponental stablty crteron. A method of desgnng a feedbac controller to stablze the system s also proposed. The controller gan can be obtaned by solvng a set of lnear matrx nequaltes. Fnally, some numercal examples are gven to llustrate the effectveness of the proposed method. II. PROBLE FORULATION AND AIN RESULTS Consder the followng dscrete-tme systems wth a tmevaryng delay x( +) = Ax()+A x( d()) + Bu() x() = ϕ(), = d 2, d 2 +,, () where x() R n s the state vector, u() R m s the control nput; A and B are constant matrces wth approprate dmensons; ϕ() s the ntal condton sequence; d() s an nteger denotng the tme-varyng delay and satsfes d d() d 2 (2) where d and d 2 are nown postve ntegers. The object of ths paper s to derve a new stablty crteron and desgn a state feedbac controller u() = Kx() to stablze system (). A. Stablty analyss Frstly, consder the nomnal system of () x( +) = Ax()+A x( d()) x() = ϕ(), = d 2, d 2 +,, (3) x() x( ) Defne z() =. and system (3) can be x( d 2 +) x( d 2 ) rewrtten as the followng swtched system z( +)=Ξ σ() z() (4) where σ() I s a pecewse constant functon and denotes the actve mode. I = {d,d +,,d 2 } and j {}}{ A A Ξ j =, j I I d2n d 2n. Remar : Swtched systems have receved much attenton n the past several years. There are plentful results for swtched systems avalable n the lterature 28, 29, 3. By convertng systems wth tme-varyng delays nto a swtched system, we can use the results for swtched systems to analyze and synthesze tme-delay systems. Let r, I, denote the occurrence rate of the th subsystem of (4) durng the nterval (,). r can be obtaned by r = n / where n denotes the tmes that the th subsystem s actvated. Before movng on, the followng defntons are ntroduced. Defnton : 3 32 System (4) s sad to be globally exponentally stable wth a decay rate λ>, f the soluton x() of system (4) satsfes x() cλ x(), where c> s a constant. Defnton 2: 28, 3 For any, letn σ,) denote the number of swtchngs of σ() over the nterval, ). If N σ,) N + /T a holds for N and T a >, then T a s called the average dwell tme and N s called the chatter bound and s often chosen as N =. For the subsystem of (4) choose the followng Lyapunov functon V () =z T ()P z() (5) The followng lemma gves an estmaton of the decay or ncrease rate of the above Lyapunov functon. Lemma : For gven scalar λ >, f there exst matrces P >, and any matrces F, G, I, such that Σ Σ = λ Ξ T GT F P G G T <, I (6) where Σ = P + λ F Ξ + λ Ξ T F T, then the Lyapunov functon (5) has the followng decay or ncrease property λ V () λ 2( ) V ( ) (7) Proof: Followng a smlar lne as n 32, defne ξ() = z(), and one can obtan ξ( +)=λ Ξ ξ() (8) For subsystems of (8), choose the Lyapunov functon V () = ξ T ()P ξ(). It s easy to obtan V () = V ( +) V () = ξ T ( +)P ξ( +) ξ T ()P ξ() (9) For any matrces F and G, the followng equaton holds 2 ξ T ( +)G + ξ T ()F ξ( +)+λ Ξ ξ() = () From (9) and (), one can obtan V () = ξ T ( +)P ξ( +) ξ T ()P ξ() = +2 ξ T ( +)G + ξ T ()F ξ( +)+λ Ξ ξ() T ξ() ξ() Σ ξ( +) ξ( +) () So, f Σ < then V () < whch mples V () < V ( ). It s clear that V () = λ 2( ) V () <λ 2( ) V ( ) = λ 2( ) V ( ) 4932

3 The proof s completed. On the bass of Lemma, the followng theorem gves a suffcent exponental stablty condton for system (3). Theorem : For gven postve scalars λ >, r > and μ>, f there exst matrces P >, and any matrces F and G, I such that the followng nequaltes hold d 2 =d λ r >λ> (2) T a >T a = 2lnλ (3) Σ <, I (4) P μp j,, j I (5) then system (3) wth a tme-varyng delay satsfyng (2) s exponentally stable and the decay rate s estmated as b x() 2Ta ln λ x() a λ + where b =max λ max(p ), a =mn λ mn(p ). I I Proof: Choose the followng pecewse Lyapunov functon V σ() () =z T ()P σ() z() (6) Let < < 2 < <, denote the swtchng nstants over the nterval,). Based on (5)-(6) and Lemma, the proof can be completed followng a smlar lne to the proof of Theorem n 32, whch s omtted here. Remar 2: Clearly n Theorem, t s not needed that every λ >, whch means that some subsystems of (4) may be unstable. The relatonshp between the occurrence rate of all the possble delay, r, and the stablty s also establshed. Remar 3: By lettng μ =, we have Ta =, whch means that the swtchng sgnals can be arbtrary, that s, the tme-delay can be arbtrary varyng wthn the gven nterval. Furthermore, f settng λ =, the results obtaned by common quadratc Lyapunov functon n 25 wll be covered. So, results n 25 can be seen as a specal case of Theorem. oreover, only asymptotcal stablty problem was consdered n 25. In ths paper, we gve an estmaton of the exponental decay rate n Theorem. B. Controller desgn Based on Theorem, we wll gve a desgn method of the state feedbac controller of system (). Substtute u() = Kx() nto () and the closed-loop system can be obtaned as x( +)=(A + BK)x()+A x( d()) (7) Smlarly, we can rewrte system (7) as a swtched system z( +)= ˆΞ σ() z() (8) where σ() I s a pecewse constant functon and denotes the actve mode, and ˆΞ j = A + BK j {}}{ A I d2n d 2n = Ξ j + ˆBKE, j I where ˆBT = B T, E = I. The followng theorem presents a method to desgn the state feedbac controller. Theorem 2: For gven postve scalars λ >, r >, μ> and δ, f there exst matrces R > and any matrces beng of the form = 2 22, I and Y, such that the followng nequaltes hold d 2. =d λ r >λ> (9) T a >T a = 2lnλ Ω λ Ξ T + λ E T Y T B T δ T R T (2) <, I (2) R μr j,, j I (22) where Ω = R +δ λ Ξ T +δ λ E T Y T ˆBT +δ λ Ξ T+ δ λ ˆBY E, then system () wth a tme-varyng delay satsfyng (2) s exponentally stable, the controller gan s gven by K T =( ) Y T and the decay rate s estmated as b x() 2Ta ln λ x() a λ + where b =max λ max( R T ), a =mn λ mn( R T ). I I Proof: Substtutng ˆΞ =Ξ+ ˆBKE nto (4) yelds ˆΣ λ ˆΞT G T F P G G T < (23) where yelds ˆΣ Σ = P + λ F ˆΞ + λ ˆΞT F T. Lettng F = δ G Σ λ ˆΞT G T δ G P G G T < (24) where = P + δ λ G ˆΞ + δ λ ˆΞT G T. The feasblty of (24) guarantees that G s nonsngular. Denote = G. In order to obtan some LI condtons, we restrct that s of a specal form, that s, = Pre- and post-multply both sdes of (24) wth dag{, } and ts transpose and ntroduce new matrces R = P T, Y T = K T, and (2) wll be obtaned. Ths completes the proof. 4933

4 Remar 4: Condton (2)-(22) n Theorem 2 are all LI, so t s possble to obtan feasble solutons usng some exstng convex optmzaton toolboxes. Remar 5: It can be seen that there are some tunng parameters δ, I, n the stablty condton. Apply a numercal optmzaton algorthm, such as fmnsearch n the optmzaton toolbox of atlab, and a locally convergent feasble soluton wll be obtaned. III. NUERICAL EXAPLES Example : Consder the followng dscrete-tme system wth a tme-varyng delay: =.2. Assume the lower bound on the delay s d =, and our objectve s to determne the upper bound on the delay, d 2, whch guarantees the above system s stable for all d d() d 2. By applyng Theorem wth μ =,we can obtan that d 2 = 5. It means that the above system s stable for any tme-varyng delay d() 5. oreover, we can see that Ξ 6 s unstable snce ts maxmum egenvalue s outsde the unt crcle. Therefore, f d() =6occurs n the system s delay sequence, the above system wll be thought to be unstable accordng to the results n 22, 23. However, t s assumed that the occurrence rates of the delays are r = 3%, r 2 = 2%, r 3 = 2%, r 4 = %, r 5 = %, r 6 = %, respectvely. Choose λ =., λ 2 =.5, λ 3 =., λ 4 =.95, λ 5 =.92, λ 6 =.87, and 6 = λr =.3. By Theorem, we obtan Ta =.4. It s easy to see that the maxmum swtchng number of σ() over the nterval,) s, so the mnmum average dwell tme T a = >. Therefore, the above system s exponentally stable under the above occurrence rates of the delays. Example 2: Consder the followng dscrete-tme system wth a tme-varyng delay: =.2.4 Clearly, the maxmum absolute value of the egenvalue of A+A s.5, so results n 6, 22, 23 are all unapplcable. By applyng Theorem wth μ =, we can see that the above system s stable for any tme-varyng delay satsfyng d() 3. Assume d =and T a =.4, the results obtaned by Theorem are lsted n Table I for dfferent occurrence rates of the delay. Comparng the result for r 4 = %, r 5 =% wth the result for r 4 =%, r 5 = %, t s not dffcult to see that the decay rate for r 4 = %, r 5 =%s bgger than the decay rate for r 4 =%, r 5 = %. Ths result s reasonable snce d() =5maes the system more unstable than d() =4. Example 3: Consder system () wth: =, B =.2. It s assumed that d() 5. Choose λ =.3, λ 2 =.3, λ 3 =.2, λ 4 =.2, λ 5 =, δ =.2, δ 2 =.3, δ 3 =., TABLE I RESULTS FOR DIFFERENT r r r 2 r 3 r 4 r 5 λ μ Ta 3% 3% 3% % % % 3% 3% 5% 5% % 3% 3% % % Delay Fg.. Tme-varyng delay δ 4 =, δ 5 = and μ =.2, the controller gan can be obtaned as K = For the purpose of smulaton, the delay durng the nterval, 3) s shown n Fg.. From Fg., we can obtan r =.2, r 2 =.2, r 3 =.2, r 4 =.2, r 5 =.2. In ths case, 6 = λr =.947. Choose λ =.94 and we obtan Ta =.54. On the other hand, the swtchng number of σ() durng the nterval, 3) s 26 and thus the average dwell tme T a =.538. Therefore, all the condtons n Theorem are all satsfed. The closed-loop system s exponentally stable wth a decay rate λ = λ 2Ta ln λ =.3. The response of the closed-loop system wth the ntal condton x() = ; s shown n Fg. 2. Smulaton results have demonstrated that the above system can be stablzed by the obtaned feedbac controller. IV. CONCLUSIONS In ths paper, the problem of the exponental stablty and stablzaton for dscrete systems wth tme-varyng delays has been nvestgated. Usng the average dwell-tme method, a new stablty crteron has been obtaned. Based on the obtaned stablty condton, a desgn method of the feedbac controller to stablze the system has also been proposed. Through some numercal examples, the method proposed n ths paper has been llustrated to be effectve. By convertng a tme-delay system nto a swtched system, plentful results for swtched systems can be used to analyze the tme-delay systems. Especally, there are necessary and suffcent stablty condtons avalable for swtched system wth arbtrary swtchng 3. Usng these stablty condtons, 4934

5 States Fg. 2. State response of the closed-loop system we can obtan some necessary and suffcent stablty condtons for systems wth tme-varyng delays. However, these condtons are only theoretcally necessary and suffcent snce the test s often nconclusve. How to mae these necessary and suffcent condtons easy to chec may be an mportant topc for future study. ACKNOWLEDGENT Ths wor s sponsored by the Bejng Educaton Commttee Cooperaton Buldng Foundaton Project. REFERENCES S.-I. Nculescu, Delay Effects on Stablty: A Robust Control Approach, Lecture Notes n Control and Informaton Scences, Sprnger Verlag, London, 2. 2 K. Gu, V.-L. Khartonov, and J. Chen, Stablty of Tme-delay Systems, Brhauser, Boston, G.P. Lu, J.X. u, D Rees, and S.C. Cha Desgn and stablty analyss of netwroed control systems wth random communcaton tme delay usng the modfed PC, Internatonal Journal of Control, vol. 79, no.4, pp , G.P. Lu, Y. Xa, J. Chen, D. Rees, and W.S. Hu, Networed predctve control of systems wth random networ delays n both forward and feedbac channels, IEEE Transactons on Industral Electroncs, vol. 54, no. 3, pp , G.P. Lu, S.C. Cha, J.X. u, and D. Rees, Networed predctve control of systems wth random delay n sgnal transmsson channels, Internatonal Journal of systems Scence, vol. 3, no., pp , H. Gao, J. Lam, C. Wang, and Y. Wang, Delay-dependent output feedbac stablzaton of dscrete-tme systems wth tme-varyng state delay, IET Control Theory & Applcatons, vol. 5, pp , S.H. Song, J.K. Km, C.H. Ym, and H.C. Km, H control of dscretetme lnear systems wth tme-varyng delays n state, Automatca, vol. 35, pp , Z. Wang, B. Huang, and H. Unbehauen, Robust H observer desgn of lnear state delayed systems wth parametrc uncertantes: the dscretetme case, Automatca, vol. 35, pp. 6 67, S. Xu, J. Lam, and Y. Zou, Improved condtons for delay-dependent robust stablty and stablzaton of uncertan dscrete tme-delay systems, Asan Journal of Control. vol. 7, pp , 25. X.. Zhang, and Q.L. Han, Delay-dependent robust H flterng for uncertan dscrete-tme systems wth tme-varyng delay based on a fnte sum nequalty, IEEE Transactons on Crcuts System II: Bref Express, vol. 53, pp , 26. x x 2 C.E. de Souza, and X. L, Delay-dependent robust H control of uncertan lnear state-delayed systems, Automatca, vol. 35, pp , Q.-L. Han, A descrptor system approach to robust stablty of uncertan neutral systems wth dscrete and dstrbuted delays, Automatca, vol. 4, pp , Y. He,. Wu, J.-H. She, and G.P. Lu, Delay-dependent robust stablty crtera for uncertan neutral systems wth mxed delays, Systems & Control Letters, vol. 5, pp , Y. He, Q.-G. Wang, L. Xe, and C. Ln, Further mprovement of freeweghtng matrces technque for systems wth tme-varyng delay, IEEE Transactons on Automatc Control, vol. 52, pp , Y. He, Q.-G. Wang, L. Xe, and C. Ln, Delay-range-dependent stablty for systems wth tme-varyng delay, Automatca, vol. 43, pp , X. Jang, and Q.-L. Han, On H control for lnear systems wth nterval tme-varyng delay, Automatca, vol. 4, pp , J.H. Km, Delay and ts tme-dervatve dependent robust stablty of tme-delayed lnear systems wth uncertanty, IEEE Transactons on Automatc Control, vol. 46, pp , 2. 8 Y.-S. oon, P. Par, W.-H. Kwon, and Y.-S. Lee, Delay-dependent robust stablzaton of uncertan state-delayed systems, Internatonal Journal of Control, vol. 74, pp , 2. 9 S. Xu, and J. Lam, Improved delay-dependent stablty crtera for tmedelay systems, IEEE Transactons on Automatc Control, vol. 5, pp , E. Frdman, and U. Shaed, Stablty and guaranteed cost control of uncertan dscrete delay systems, Internatonal Journal of Control, vol. 78, pp , E. Frdman, and U. Shaed, Delay-dependent H control of uncertan dscrete delay systems, European Journal of Control, vol., pp , H. Gao, and T. Chen, New results on stablty of dscrete-tme systems wth tme-varyng state delay, IEEE Transactons on Automatc Control, vol. 52, pp , Y. He,. Wu, G.-P. Lu, and J.-H. She, Output feedbac stablzaton for a dscrete-tme system wth a tme-varyng delay, IEEE Transactons on Automatc Control, vol. 53, pp , K. Gu, Q.-L. Han.C. Luo, and S.-I. Nculescu, Dscretzed Lyapunov functonal for systems wth dstrbuted delay and pececwse constant coeffcent, Internatonal Journal of Control, vol. 74, pp , Y.-Q. Xa, G.-P. Lu, P. Sh, D. Rees, and E.J.C. Thomas, New stablty and stablzaton condtons for systems wth tme-delay, Internatonal Journal of Systems Scence. vol. 38, pp. 7 24, X.-. Sun, G.-P. Lu, D. Rees, and W. Wang, Delay-dependent stablty for dscrete systems wth large delay sequence based on swtchng technques, Automatca. vol. 44, pp , J.P. Hespanha.S. orse, Stablty of swtched systems wth average dwell-tme, In Proceedngs of 38th IEEE Conference on Decson and Control. pp , D. Lberzon, Swtchng n systems and control, Brhauser, Boston, Z. Sun, and S.S. Ge, Swtched lnear systems-control and desgn, Sprnger-Verlag, New Yor, H. Ln, P.J. Antsals, Stablty and stablzablty of swtched lnear systems: a survey of recent results, IEEE Transactons on Automatc Control, vol. 54, pp , G.-S. Zha, B. Hu, K. Yasuda, and A.N. chel, Qualtatve analyss of dscrete-tme swtched systems, In Proceedngs of the Amercan Control Conference, pp nchorage, W.-A. Zhang, and L. Yu, Output feedbac stablzaton of networed control systems wth pacet dropouts, IEEE Transactons on Automatc Control, vol. 52, pp. 75 7,