Delay Dependent Exponential Stability and. Guaranteed Cost of Time-Varying Delay. Singular Systems

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1 Ali amaical Scincs, ol. 6,, no. 4, Dlay Dnn onnial Sabiliy an Guaran Cos of im-arying Dlay Singular Sysms Norin Caibi, l Houssain issir an Ablaziz Hmam LSSI. Darmn of Pysics, Faculy of Scincs, Dar l raz B.P : 796, 3 Fs-Alas; orocco ia_nour@yaoo.fr Absrac is ar als wi roblm of lay-nn guaran cos onnial sabiliy of singular sysms wi im-varying lay. Som imrov laynn coniions ar rsn, in form of linar mari inqualiis o nsur consir sysm o b rgular, imuls fr an onnially sabl wi isnc of a guaran cos. Numrical amls ar givn o sow usfulnss of roos rsuls. Kywors: singular sysms, im-varying lay, onnial sabiliy, laynn coniions, guaran cos, linar mari inqualiy LI.. Inroucion Ovr as cas, muc anion as bn focus on guaran cos roblm. conc of guaran cos for linar sysms was firs inrouc in []. lay-nn sabiliy roblm wi guaran cos for singular sysms is muc mor comlica an a for rgular sysms bcaus i rquirs o consir no only sabiliy, bu also rgulariy an absnc of imulss for coninuous singular sysms [5, 8, 9, 9,, ] an causaliy for iscr singular sysms s. g., [4, 6, 5, 6] a sam im. main ia of guaran cos is o obain a las ur boun of a rformanc lvl. guaran cos roblm is sui in [5, 4, 9, ] for a class of linar singular sysm wi norm-boun aramric uncrainy. robus rliabl guaran cos conrol is invsiga for uncrain singular lay sysm in [3]. No a all

2 5656 N. Caibi, l H. issir an A. Hmam abov rfrncs on consir onnial sabiliy. Howvr, i is mor imoran o suy onnial sabiliy sinc ransin rocss of a sysm may b br scrib if cay ra is rmin. onnial sabiliy of singular sysms wi mulil im-varying lays is sui in [7] an [] wiou consiring any guaran cos. In is no, w invsiga roblm of guaran cos for singular sysm. Dlay-nn coniions ar sablis in rms of LIs, wic guaran singular im-varying lay sysm o b rgular, imuls fr, an onnially sabl wi isnc of a guaran cos. rsuls ar riv by using Lyaunov-Krasovskii funcional mo an making us of Finslr s lmma. In our mo, no comosiion of cofficin marics of sysm is n. is ar is organiz as follows. In scion, roblm is sa an rquir lmmas ar formula. Scion 3 als wi guaran cos onnial sabiliy an in scion 4 w rsn numrical amls o sow usfulnss of roos rsuls.. Sysm scriion an rliminaris Consir singular sysm scrib by = A A [, ] =φ, n Wr R is sa vcor. is im-varying lay of sysm. n n mari R may b singular, w sall assum a rank=r n. A an A ar known ral marics, φ is a comaibl vcor valu coninuous iniial funcion wi φ = su [ ] φ θ. θ, Assumion.: lay is assum o saisfy following consrain:, an < 3 Wr an ar givn osiiv consans. Dfiniion. [] i. air, A is sai rgular if s- A is no inically zro. ii. air, A is sai o b imuls fr if gs- A =rank. Dfiniion. [8]: singular im lay sysm is sai o b rgular an imuls fr if airs, A is rgular an imuls fr. For mor ails on or roris an isnc of soluion of sysm, w rfr rar o [8], an rfrncs rin. In gnral, rgulariy is ofn a sufficin coniion for analysis an synsis of singular sysms.

3 Dlay nn onnial sabiliy 5657 following lmmas ar vry inrsing for our vlomn in is ar. n n n Lmma. [3]: Consir a vcor χ R, a symmric mari Q R an a m n mari B R, suc a rank B < n. following samns ar quivaln: i χ Qχ <, χ suc a Bχ =, χ ii B QB < iii μ R : Q µ B B < n m iv F R : Q FB B F < wr B nos a basis for null-sac of B. Lmma. [3]: For any consan mari n vcor funcion ω [,γ] R fin, n: η n n = R, >, scalar γ η >, : suc a ingraions in following ar wll η η η ω β ω β β ω β β ω β β In is ar, w ar inrs in sablising lay-nn coniions guaraning onnial sabiliy of singular sysm an a las ur boun for cos funcion givn by: J = Z Wr Z is a givn osiiv fini mari onnial sabiliy analysis goal of is scion consiss in vloing sufficin coniions a can b us o cck onnial sabiliy of class of sysms unr suy. coniions n on ur boun of lay as givn in Assumion.. A W A L = W[ A A ] β =λ min P β = λ ma P λ ma W λ ma Q 4 β β= β

4 5658 N. Caibi, l H. issir an A. Hmam orm 3. Givn >, if r is F i, i=,,, P, Q> an W>, suc a following coniions ol: P = P 9 3 = 3 < 33 Wi = P Q W F A A F Z = W F A A F = PF A F 3 = F A F = W Q F A A F 3 33 W F F = n sysm is rgular imuls fr an onnially sabl for any saisfying 3. Furrmor, soluion of sysm saisfis, φ β φ An guaran cos of funcion 4 is givn by: J J * =φ P φ σ φ σ W φ σ σ σ s φ σ Q φ σ σ s Proof of orm 3. : 3 From, i follows a < 3 33 L S = I A. Pr-an os-mulilying by S an S, rscivly, w g, P Q W PA A P A WA Z < 3 I Now coos wo nonsingular marics an N suc a = N = r A A An no, A = A N = ; Z = N ZN ; P = N P = ; A3 A4 3 4 w w Q = N QN ; W = W =. w w3

5 Dlay nn onnial sabiliy 5659 By using 9 i can b sow a = 3. Pr-an os-mulilying 3 by N an N rscivly w g, w P Q PA A P A WA Z < 4 From 4 i can b asily sn a A4 P 4 P 4 A4 < wic imlis a A 4 is nonsingular an consqunly air, A is rgular an imuls fr. rfor, accoring o finiion., sysm is rgular an imuls fr. N, w will sow onnial sabiliy. W consir Lyaunov funcion cania: = 3 5 wr θ = θ, θ [, ], an = P = σ σw σσs s σ 3 = σ Q σ σ I is asy o vrify a ma P λ σ σ A σ [ ] s σ σ A σ σ λmaw W A A σs 4 Consqunly w av β 3 λ ma Q β In following w will comu an boun firs rivaivs of various funcions i for i=,,3. = P σ = W σ W σ σ an w obain afr bouning lay an is rivaiv: σ W σ W σ σ 6

6 566 N. Caibi, l H. issir an A. Hmam wic by lmma. givs W W W W W W 3 3 Q Q = an w obain afr bouning lay an is rivaiv: 3 3 Q Q W g Q Q W W W W P P Now, l [ ] = χ, aing an subracing Z W obain Z χ Φχ 7 Wi P Q W Z W P W Q W Φ= Now, L [ ], I A A B = = F F F F From coniion w av;

7 Dlay nn onnial sabiliy 566 =Φ FB B F < wic by lmma.. imlis a χ Φχ < an consqunly Z < 8 is imlis a φ, aking accoun of 6 w obain β, φ φ β φ n, β φ. β Now, from 6 an 8 w can wri, Z 9 Ingraing bo sis of 9 from o an using iniial coniions w obain, Z P σ σ s σ Q σ σ σ s σ σ W σ σs W σ σs P σ σ Q σ σ As sysm is asymoically sabl, wn w av P, s σ σ Hnc, w g, Z φ σ σ Q σ σ Pφ σ φ Wic conclus a roof W σ σs, σ σ Qφ σ σ s φ σ Wφ σ σs Rmark 3.: In [-, 7-8] sabiliy criria for singular sysm wi im lay av bn vlo by mloying comosiion aroac of sysm marics of sysm. In is ar, sufficin coniions for guaran cos onnial sabiliy analysis ar riv wiou n of any comosiion. comosiion mo can g aroun crain numrical roblm arising from comosiion of marics scially for sysms wi larg imnsion. Howvr, our aroac is rlaivly siml an rliabl. Rmark 3.: mo w vlo in is orm givs lay-nn sufficin coniions wic onc y ar saisfi will guaran a consir sysm is rgular, imuls fr an onnially sabl wi

8 566 N. Caibi, l H. issir an A. Hmam isnc of a guaran cos. rsuls of is orm can b asily n o anl cas of sysms wi mulil im-varying lays scrib by: = A A P = =φ,, or in a comac from as = A A Wi A = [ A A... A ] = [... ] 3, an < 4 wr an, =,,..., ar givn osiiv consans, an = ma A W A L = W [ A A ] β =λ P λ W λ Q 5 3 ma ma ma = 4 = 6 β = β β 3 7 Wy = W W L W 8 ψ = iag W, W,..., W 9 ψ = iag F F F = F Q, Q,..., Q 3 3 orm 3. Givn >, if r is F i, i=,,, P, Q > an W >, =,,, suc a coniion 9 ols an ˆ ˆ ˆ 3 ˆ ˆ ˆ = 3 < 3 ˆ 33

9 Dlay nn onnial sabiliy 5663 wi P ˆ = P Q = ˆ = = Wy F A A F ˆ = ˆ = ψ ψ F A A P F A F 3 F ˆ F A F 3 = = ˆ 33 W F F = W F A A F n sysm is rgular imuls fr an onnially sabl for any, =,..., saisfying 4. Furrmor, soluion of sysm saisfis, φ β φ, an guaran cos of funcion 4 is givn by: * σ s = P σ φ σq φ σσ = Z J J =φ P φ φ σ W φ σσs Rmark 3.3: comuaional comliy associa o rsuls of [7] is largr an a associa o orm 3., in fac numbr of variabls rquir in orm 3. an orm 4 in [7] ar 3 n n n an n n 6 n 5 rscivly. I can b sn a solving orm 3. n n rquirs lss numbr of cision variabls a is 5 n 3. Consqunly, wi our rsuls comuaional man on sarcing for soluion of sabiliy coniions can b allvia. is avanag can b rval scially for sysms wi larg imnsion n Numrical amls! aml : Consir following singular lay sysm..9.9 =, A =, A =,. wi =., ling = an alying orm 3., corrsoning ur boun is =.67. Howvr, alying orm 4 in [7], corrsoning

10 5664 N. Caibi, l H. issir an A. Hmam ur boun is =.65. numbr of variabls rquir in orm 3. an orm 4 in cas = in [7] ar an 43 rscivly. I is clar a coniions in is ar giv br rsuls an s in [7]. aml : consir following singular lay sysm.5. = φ =. wi =, ling = an alying orm 3.. W obain ur boun =.86, an guaran cos of sysm is J*=7.87 for =.86. For comarison, w ali orm of guaran cos sabiliy in [], w foun a coniions ar infasibl. 5. Conclusion is ar als wi roblm of onnial sabiliy an guaran cos for a class of coninuous-im singular linar sysms wi im-varying lay. Dlaynn sabiliy criria ar sablis wic guaran sysm o b rgular, imuls fr an onnially sabl wi isnc of a guaran cos. LIs roos av bn obain by uilizing a Lyaunov Krasovskii funcional an Finslr s lmma. rsuls vlo can b asily solv by using alab or Scilab LI oolbos. Numrical amls ar givn o illusra ffcivnss of roos mo an o sow a our criria giv lss consrvaiv rsuls an os of som ising ons. Rfrncs [] S. Cang,. Png, Aaaiv guaran cos conrol of sysms wi uncrain aramrs, I rans. Auoma., 797, [] L. Dai, Singular Conrol Sysms, Sring-rlag Nw York, NY, U.S.A [3] P. Finslr. Obr as vorkommn fini smifinir formn is scarn quaraiscr formn, Commnarii mamaici, 937, [4]. Friman, U. Sak, Sabiliy an guaran cos conrol of uncrain iscr lay sysms, Inrnaional Journal of Conrol, 785,

11 Dlay nn onnial sabiliy 5665 [5] H. Gao, S. Zu, Z. Cng an B. Xu, Dlay nn Sa Fback Guaran Cos Conrol for Uncrain Singular im-lay Sysms, Procings of 44 I Confrnc on Dcision an Conrol, an uroan Conrol Confrnc, 5, [6] G. Garcia, B. Praina, S. arbourica, F. Zng, Robus sabilizaion an guaran cos conrol for iscr-im linar sysms by saic ouu fback, Auomaica, 393, [7] A. Haiar,.K. Boukas. onnial sabiliy of singular sysms wi mulil im-varying lay, Auomaica, 459, [8] F. Jun, Z. Suqian, C. Zaolin, guaran cos conrol of linar uncrain singular im-lay sysms, Procings of 4s I confrnc on,, [9] Q. Lan, Y. Liu, H. Niu, an J. Liang, Robus rliabl guaran cos conrol for uncrain singular sysms wi im-lay, Journal of Sysms nginring an lcronics,, 7. [] Y. Li, W. Fng, an Y. Liu, Sabiliy of consan cofficin linar singular sysms wi lay, Circuis Sysms an Signal Procssing, 9, 3 5. [] Y. Li an Y. Liu, Basic ory of singular sysms of linar iffrnial iffrnc quaions. In Proc. IFAC 3 Worl Congrss, 996, [] Y. Li an Y. Liu, Sabiliy of soluion of gnraliz funcional quaions, Aca Al. a., 999, [3]. Li, B. Yang, J. Wang, an C. Zong, On sabiliy for nural iffrnial sysms wi mi im varying lay argumns, Proc. 4 I. Conf. on cision an conrol, 53, [4] H. ukaiani, A nw aroac o robus guaran cos conrol for uncrain ulimoling sysms, Auomaica, 45, [5].-C. Pai, Guaran cos conrol of uncrain linar sysms via iscrim variabl srucur conrol, Journal of arin Scinc an cnology, 68, [6] S.-L. Wo, G.-D. Si, an Y. Zou, guaran cos conrol for iscr-im singular larg-scal sysms wi aramr uncrainy, Aca Auomaica Sinica, 35, [7] X. Xi an Y. Liu, Sabiliy for comosi singular sysms of iffrnial quaions wi a lay, Circuis Sysms an Signal Procssing, 5996, [8] S. Xu, P. an Doorn, R. Sfan, an J. Lam, Robus sabiliy an sabilizaion for singular sysms wi sa lay an aramr uncrainy. I ransacions on Auomaic Conrol, 47, 8. [9] L. Yu, J.-. Xu, an Q.-L. Han, Oimal Guaran Cos Conrol of Singular Sysms wi Dlay Sa an Paramr Uncrainis, Procing of 4 Amrican Conrol Confrnc Boson, 54,

12 5666 N. Caibi, l H. issir an A. Hmam [] D. Yu, J. Lam, D. W. C. Ho. Dlay-nn robus onnial sabiliy of uncrain scrior sysms wi im-varying lays, Dynamics of Coninuous, Discr an Imulsiv Sysms, Sris B: Alicaion an Algorims, 5, [] Z. Zao, N. li an B. Cn, Guaran cos conrol for singular nwork sysms bas on ynamic ouu fback, Inrnaional ournal of informaion an sysms scincs, 7, 5-6. [] S. Zu an Z. Cng, Dlay-nn Robus Rsilin Guaran Cos Conrol for Uncrain Singular im-lay Sysms, Amrican Conrol Confrnc, 5, Rciv: Jun,