On Robust Stability of Uncertain Neutral Systems with Discrete and Distributed Delays

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1 009 American Conrol Conference Hya Regency Riverfron S. Louis MO USA June FrC09.5 On Robus Sabiliy of Uncerain Neural Sysems wih Discree and Disribued Delays Jian Sun Jie Chen G.P. Liu Senior Member IEEE David Rees Absrac This paper is concerned wih he problem of robus sabiliy of uncerain neural sysems wih discree and disribued delays. The uncerainies under consideraion are assumed o be ime-varying bu norm bounded. By inroducing a new form of Lyapunov-Krasovskii funcional which conains some novel riple-inegral erms improved discree- disribued- and neural-delay-dependen sabiliy condiions are obained and formulaed in erms of linear marix inequaliy (LMI). Numerical examples are given o show ha he proposed mehod is effecive and leads o less conservaive resuls han he exising ones. I. INTRODUCTION Time delays are ofen encounered in many dynamic sysems such as chemical or process conrol sysems and neworked conrol sysems [] []. Time delay is always one of he sources of insabiliy and poor performance. The subjec of analysis and synhesis of ime-delay sysems hus has araced considerable aenion during he pas few years. Sabiliy crieria for ime-delay sysems can be classified ino wo major caegories namely delay-independen ones [3] [4] and delay-dependen ones [5] [6] [7] [8] [9] [0] [] [] [3]. Generally speaking delay-dependen condiions are less conservaive han delay-independen ones. So many effors have been paid o obain less conservaive delaydependen condiions. An imporan index of measuring he conservaiveness of he obained condiions is he maximum upper bond on he delay. Finding some less conservaive sabiliy condiions moivaes he presen sudy. Some pracical applicaions can be modeled by neural sysems wih disribued delays [4] [5]. So i is imporan boh in heory and in applicaion o sudy he sabiliy of neural sysems wih disribued delays. However much fewer resuls have been proposed for he sabiliy analysis of neural sysems wih disribued delays compared wih he rich resuls for neural sysems wih only discree delays [6] [7] [8] [9] [0]. Applying he discreized Lyapunov This work was suppored by he Beijing Educaion Commiee Cooperaion Building Foundaion Projec and Naional Naural Science Foundaion of China under Gran Jian Sun is wih School of Auomaion Beijing Insiue of Technology Beijing 0008 China and also wih Faculy of Advanced Technology Universiy of Glamorgan Ponypridd CF37 DL UK helios5@yahoo.com.cn Jie Chen is wih School of Auomaion Beijing Insiue of Technology Beijing 0008 China chenjie@bi.edu.cn G.P. Liu is wih Faculy of Advanced Technology Universiy of Glamorgan Ponypridd CF37 DL UK and also wih he lab of Complex Sysems and Inelligence Science Insiue of Auomaion Chinese Academy of Sciences Beijing 0090 China gpliu@glam.ac.uk D. Rees is wih Faculy of Advanced Technology Universiy of Glamorgan Ponypridd CF37 DL UK drees@glam.ac.uk funcional approach sabiliy condiions for sysems wih disribued delays were obained in [] [] [3]. However his mehod is difficul o be exended o deal wih he synhesis problems. On he basis of he descripor model ransformaion [7] [9] and he decomposiion echnique of discree-delay erm marix Han [4] pu forward a sabiliy es for neural sysems wih discree and disribued delays. Using a combinaion of he inegral inequaliy echnique and he descripor model ransformaion new delay-dependen sabiliy condiions were proposed in [5]. Mos recenly resuls in [4] [5] have been furher improved in [6] a modified Lyapunov-Krasovskii funcional has been consruced and some free-weighing marices [3] [8] have been inroduced o reduce he conservaiveness. However here sill exiss room for furher improvemens. In our previous work [7] a new form of Lyapunov- Krasovskii funcional conaining a riple-inegral erm was inroduced o derive less conservaive delay-dependen sabiliy condiions for ime-delay sysems. In his paper we exend his mehod o sudy he sabiliy of neural sysems wih disribued delays. Using some inegral inequaliies improved discree- disribued- and neural-delay-dependen sabiliy condiions are obained wihou inroducing any free-weighing marices. Two numerical examples are presened o illusrae ha our resuls are less conservaive han he exising ones. Noaions: Throughou his paper he superscrips - and T sand for he inverse and ranspose of a marix respecively; R n denoes an n-dimensional Euclidean space; R m n is he se of all m n real marices; P > 0 means ha he marix P is symmeric posiive definie; I is an appropriaely dimensional ideniy marix. II. PROBLEM FORMULATION AND PRELIMINARIES Consider he following uncerain neural sysem wih discree and disribued delays: ẋ() C()ẋ( τ) = A()x() B()x( h) D() r x(s)ds > 0 x() = φ() [ρ 0] () x() R n is he sae vecor; τ > 0 h > 0 and r > 0 are consan neural discree and disribued delay respecively; ρ = max{τ h r}; The iniial condiion φ() is a coninuously differeniable vecor-valued funcion; A() B() C() and D() are uncerain marices and can be /09/$ AACC 5469

2 described by A() = A A() B() = B B() C() = C C() D() = D D() () A B C D are known consan marices; The admissible uncerainies are assumed o saisfy he following condiion: [ A() B() C() D()] = MF () [N a N b N c N d ] (3) M N a N b N c and N d are known consan marices wih appropriae dimensions and F () is an unknown imevarying marix saisfying: F T ()F () I (4) Throughou his paper i is assumed ha he marix C() is Schur sable. The objecive of his paper is o derive less conservaive delay-dependen sabiliy condiions in erms of LMI o ensure a larger maximum upper bound on he delay. Before moving on he following lemmas are inroduced which play imporan roles in he developmen of he main resuls. Lemma : For any consan marix Z = Z T > 0 and a scalar τ > 0 such ha he following inegraions are well defined hen () () ϱ T (s)zϱ(s)ds ϱ T (s)dsz ϱ(s)ds τ ϱ T (s)zϱ(s)dsd τ τ τ ϱ T (s)dsdz ϱ(s)dsd τ Lemma : [8] For given marices Q = Q T M and N wih appropriae dimensions hen Q MF ()N N T F T ()M T < 0 for all F () saisfying F T ()F () I if and only if here exiss a scalar ε > 0 such ha Q ε MM T εn T N < 0 III. MAIN RESULTS In his secion robus sabiliy of sysem () is sudied. Firsly he following nominal sysem is considered. ẋ()cẋ() = Ax()Bx(h)D r x(s)ds (5) The following heorem presens a sabiliy condiion for he nominal sysem (5). Theorem : Given scalars τ > 0 h > 0 and r > 0 he nominal neural sysem (5) is asympoically sable if here exis marices P = [P ij ] 5 5 > 0 Q = [Q ij ] > 0 X = [X ij ] > 0 R i > 0 W i > 0 S i > 0 (i = ) and Z j > 0 (j = 3) wih appropriae dimensions such ha Ξ Γ T P Γ Γ T P Γ A T c Y A c < 0 (6) [ ] Ξ Ξ Ξ = Ξ T Ξ 3 I I Γ = 0 0 I I I 0 A 0 B C D I Γ = I I I I I A c = [ A 0 B C D 0 ] Ξ τ R HB h X Ξ = τ R Q 0 W h X Ξ = HC 0 r S τ Z Ξ 8 Ξ 9 Q h XT Ξ 3 = diag{q W r S τ R τ Z r S r Z 3 h X h Z } Ξ = Q HA A T H T τr W hx rs τ R h X r S Z Z Z 3 Ξ 8 = HD r Z 3 Ξ 9 = h Z h XT Y = Q τr W hx rs τ Z h Z r Z 3 H = Q hx Proof: Choose a Lyapunov-Krasovskii funcional candidae as V (x ) = V (x ) V (x ) V 3 (x ) V 4 (x ) (7) V (x ) = ζ T ()P ζ() V (x ) = ω T (s)qω(s)ds τ τ x T (s)r x(s)dsd ẋ T (s)r ẋ(s)dsd 5470

3 V 3 (x ) = V 4 (x ) = τ h λ x T (s)w x(s)ds h ẋ T (s)w ẋ(s)ds h h r ẋ T (s)z ẋ(s)dsdλd ω T (s)xω(s)dsd λ r r ẋ T (s)z ẋ(s)dsdλd x T (s)s x(s)dsd ẋ T (s)s ẋ(s)dsd λ ẋ T (s)z 3 ẋ(s)dsdλd [ ζ T () = x T () x T ( τ) x T ( h) xt (s)ds ] r xt (s)ds ω T (s) = [ x T (s) ẋ T (s) ]. Taking he ime derivaive of V (x ) along he rajecory of sysem (5) yields V (x ) = ζ T ()P ζ() ω T ()(Q hx)ω() ω T ( τ)qω( τ) x T ()(τr W rs )x() ẋ T ()(W rs τr τ h Z r τ h Z 3)ẋ() ẋ T (s)r ẋ(s)ds r r h ẋ T (s)z ẋ(s)dsd ẋ T (s)z ẋ(s)dsd x T (s)s x(s)ds Using Lemma yields x T (s)r x(s)ds τ ẋ T (s)r ẋ(s)ds τ ω T (s)xω(s)ds h h x T (s)s x(s)ds r r r Z x T (s)r x(s)ds ω T (s)xω(s)ds ẋ T (s)s ẋ(s)ds ẋ T (s)z 3 ẋ(s)dsd (8) h r x T (s)ds R ẋ T (s)ds R x(s)ds (9) ẋ(s)ds (0) ω T (s)ds X ω(s)ds h () x T (s)ds S x(s)ds r () ẋ T (s)s ẋ(s)ds ẋ T (s)ds S ẋ(s)ds r r r (3) ẋ T (s)z ẋ(s)dsd τ τ ẋ T (s)dsdz ẋ(s)dsd (4) τ τ ẋ T (s)z ẋ(s)dsd h h ẋ T (s)dsdz h h ẋ T (s)z 3 ẋ(s)dsd r r ẋ T (s)dsdz 3 r Subsiuing (9)-(6) ino (8) yields r ẋ(s)dsd (5) ẋ(s)dsd (6) V () ξ T ()(Ξ Γ T P Γ Γ T P Γ A T c Y A c )ξ() (7) ξ T () = [ x T () x T ( τ) x T ( h) ẋ T ( τ) ẋ T ( h) x T ( r) xt (s)ds r xt (s)ds ]. Therefore if Ξ Γ T P Γ Γ T P Γ h xt (s)ds A T c Y A c < 0 hen V (x ) < 0 which guaranees sysem (5) is asympoically sable. This complees he proof. Remark : On he basis of Lyapunov-Krasovskii funcional approach a new discree- disribued- and neuraldelay-dependen sabiliy crierion is developed. I should be noed ha wo inegral inequaliies are used o derive Theorem and no addiional free-weighing marices are inroduced in he derivaion excep for Lyapunov marices. So he mehod proposed in his paper may have less decision variables han he well-known free-weighing marix mehod. Remark : The proposed augmened Lyapunov funcional is more general han hose in [8] [6]. In paricular our Lyapunov-Krasovskii funcional conains some riple τ inegral erms ha is λ ẋt (s)z ẋ(s)dsdλd h λ ẋt (s)z ẋ(s)dsdλd and r λ ẋt (s)z 3 ẋ(s)dsdλd which play key roles in he furher reducion of conservaiveness. Furhermore we choose he augmened vecor ζ() as ζ T () = [ x T () x T ( τ) x T ( h) xt (s)ds ] r xt (s)ds. Through some numerical examples we can see ha each erm in ζ() may conribue o he furher reducion of he conservaiveness which means ha if any erms are removed from ζ() a more conservaive resul will be obained. So we hink he proposed Lyapunov funcional in his paper is differen from exising ones and can lead o less conservaive resuls. Remark 3: If leing P 4 = 0 P 4 = 0 P 34 = 0 P 5 = 0 P 5 = 0 P 35 = 0 P 45 = 0 X = 0 P 44 = ɛ I P 55 = ɛ I R = ɛ 3 I W = ɛ 4 I X = ɛ 5 I Z = ɛ 6 I Z = ɛ 7 I Z 3 = ɛ 8 I wih ɛ i (i =... 8) being sufficienly small 547

4 posiive scalars in Theorem a corollary can be direcly obained which is omied here. Furhermore following he similar line as in [9] i can be proved ha his corollary is equivalen o Theorem in [6] he improvemens over [4] [5] are demonsraed. So resuls in [6] can be covered by Theorem in his paper. This also proves heoreically ha Theorem is less conservaive han he resuls in [6]. Consider he following neural sysems wih mixed delays ẋ() Cẋ( τ) = Ax() Bx( h) (8) A sabiliy condiion for such a sysem can be obained on he basis of Theorem. Corollary : Given scalars τ > 0 and h > 0 neural sysem (8) is asympoically sable if exis P = [P ij ] 4 4 > 0 Q = [Q ij ] > 0 X = [X ij ] > 0 R i > 0 W i > 0 Z i > 0 (i = ) wih appropriae dimensions such ha Θ Λ T P Λ Λ T P Λ ÂT c Ŷ Âc < 0 (9) [ ] Θ Θ Θ = Θ T Θ 3 Λ = I I I I 0 A 0 B C Λ = I I 0 0 I I Â c = [ A 0 B C ] Θ = Θ = Θ τ R HB h X τ R Q 0 W h X h Z h XT HC 0 τ Z Q h XT Θ 3 = diag{q W τ R τ Z h X h Z } Θ = Q HA A T H T τr W hx τ R h X Z Z Ŷ = Q τr W hx τ Z h Z H = Q hx Proof: Choose he Lyapunov-Krasovskii funcional candidae as V (x ) = V (x ) V (x ) V 3 (x ) V i (x ) (i = 3) are he same as hose in Theorem and follow he same line as in Theorem and he proof will be compleed. When consider he parameer uncerainies described by ()-(4) he following heorem can be easily obained on he basis of Theorem. TABLE I MAXIMUM UPPER BOUND ON r FOR τ = 0. AND DIFFERENT h h [6] Theorem TABLE II MAXIMUM UPPER BOUND ON h FOR τ = 0. AND DIFFERENT r r [6] Theorem Theorem : Given a scalar τ > 0 h > 0 and r > 0 neural sysem () is robusly asympoically sable if exiss a scalar ε > 0 and P = [P ij ] 5 5 > 0 Q = [Q ij ] > 0 X = [X ij ] > 0 R i > 0 W i > 0 S i > 0 (i = ) and Z j > 0 (j = 3) wih appropriae dimensions such ha Π AT c Y Γ T 3 M Y Y M εi Π = Ξ Γ T P Γ Γ T P Γ ευ T Υ < 0 (0) Γ 3 = [P H P P P 4 P 5 0] Υ = [N a 0 N b N c N d 0] Proof: Replacing A B C and D in Theorem wih A MF ()N a B MF ()N b C MF ()N c and D MF ()N d respecively and using Lemma complees he proof. IV. NUMERICAL EXAMPLES In his secion wo numerical examples are given o show ha he proposed resuls are improvemens over some exising ones. Example : Consider he following uncerain neural sysem: A = B = [ ] [ C = D = M = I N a = N b = N c = N d = 0.I. This sysem has been invesigaed in [6]. Assuming τ = 0. he purpose is o calculae he maximum upper bound on r or h for differen given h or r. To he bes of auhors knowledge resuls in [6] are he leas conservaive among he exising ones so we compare our resuls wih hose in [6]. Table I liss he maximum upper bounds on r for differen given h. I is seen from Table I ha much larger value of r can be obained using Theorem. In paricular when h.5 mehod in [6] is unfeasible while our resuls are and 0.4 respecively. Table II liss he maximum upper bounds on h for differen given r. I can be seen ha our resuls are less conservaive han hose in [6] ha is larger maximum upper bounds on h can be obained by our mehod. ] 547

5 TABLE III MAXIMUM UPPER BOUND ON h FOR DIFFERENT c AND r Mehod c [6] r = Theorem r = [6] r = Theorem r = [6] r = Theorem r = [6] r = Theorem r = [6] r = Theorem r = Example : Consider he following uncerain neural sysem wih A = B = c C = D = 0 c M = I N a = N b = N c = N d = 0.I. Assume τ = 0. he objecive is o calculae he upper bound on h for differen values of c and r. Table III liss he resuls ogeher wih hose obained using he mehod in [6]. Table III shows ha he sabiliy condiion proposed in his paper yields less conservaive resuls han hose in [6]. V. CONCLUSIONS In his paper he sabiliy of linear neural sysems wih discree and disribued delays has been invesigaed. New discree- disribued- and neural-delay-dependen crieria have been proposed. These crieria are derived based on he Lyapunov-Krasovskii approach and he inegral inequaliy echnique and are presened in erms of LMI. 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