STABILITY CRITERIA FOR A CLASS OF NEUTRAL SYSTEMS VIA THE LMI APPROACH

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1 Asan Jounal of Conol, Vol. 6, No., pp. 3-9, Mach 00 3 Bef Pape SABILIY CRIERIA FOR A CLASS OF NEURAL SYSEMS VIA HE LMI APPROACH Chang-Hua Len and Jen-De Chen ABSRAC In hs pape, he asypoc sably fo a class of neual syses wh ulple cee and bued e delays s nvesgaed. he Lyapunov sably heoy s appled o guaanee he sably of he syses. he lnea ax neualy (LMI appoach s used o pove ou esuls. By eans of soe aheacal analyss, he pesened condons ae poved o be less consevave and flexble han soe ecen epos. A nuecal exaple s gven o llusae he use of ou esuls. KeyWo: Neual syse, asypoc sably, LMI, cee delay, bued delay. I. INRODUCION Ove he pas few decades, he sably poble fo dynac syses wh ulple e delays has been suded. Syses wh ulple e delays ae ofen encouneed n vaous physcal phenoena, such as he wo-body poble n elecodynacs [5 cance cheoheapy [ coecal boles [ lage scale syses [9 and uban affc [9]. Moeove, e delays ae feuenly a souce of nsably and a souce of oscllaon geneaon n any syses []. In ecen yeas, any sably sudes have concenaed on syses wh cee delays [,-3,6-7]. Ohe sably sudes consdeed syses wh bued (o connuous and xed (boh bued and cee delays [-3,7-8,5]. In hs pape, we consde a neual syse wh ulple cee and bued e delays: Manuscp eceved Januay 8, 003; evsed June 9, 003; acceped Augus 6, 003. Chang-Hua Len s wh he Depaen of Eleccal Engneeng, I-Shou Unvesy, Kaohsung, awan 80, R.O.C. Je-De Chen s wh he Insue of Engneeng Scence and echnology, Naonal Kaohsung Fs Unvesy of Scence and echnology, Kaohsung, awan 8, R.O.C. and he Depaen of Eleconc Engneeng, Yung-a Insue of echnology & Coece, Pngung, awan 909, R.O.C. he eseach was suppoed by he Naonal Scence Councl of awan, R.O.C. unde gan NSC 90-3-E x ( = Ax ( [ Bx ( h Cx ( h =, (a D x( s 0 x ( = φ(, [ H,0], (b whee x R n ; x s he sae a e defned by x (s := x( s, s [H, 0 H = ax{ h, } 0, wh H s 0 s x : = sup x( s ; h and,, ae nonnegave consans, whch epesen cee and bued delays, especvely. he aces A, B, C, D R n n, and ae known, and he nal veco φ C0. Dependng on whehe hey conan he sze of he cee delays, sably cea fo neual syses wh ulple e delays can be classfed no wo caegoes, naely, cee-delay-ndependen cea [3-,8,,6] and cee-delay-dependen cea [-,7-8,6,7]. In pas eseach on ulple e delays, he esuls wee ehe all delay-dependen [-3,7-8,6,7] o all delay-ndependen [3,8,,6] (.e., all cee delays wee ehe bounded n soe posve nevals o ndependen. Hence, a ceon ha was dependen fo soe delays and ndependen fo ohe delays would be oe suable fo paccal syse sably analyss []. In hs pape, we exend he esuls n [] o a neual syse wh ulple bued e delays, and he obaned esuls ae boh flexble and less consevave han ou pas esuls based on aheacal analyss [,3,7]. A nuecal exaple s gven o show ha ou esuls ae useful n hs neual syses wh ulple cee and bued e delays.

2 Asan Jounal of Conol, Vol. 6, No., Mach 00 II. PROBLEM FORMULAION AND SABILIY CONDIIONS By eans of soe odel ansfoaons, syse (a can be odeled as: d [ x ( Cx ( h D ( s xs ( ] d [ x( s ] = ( A B D x( B ( d = L d h L L L Bx ( h D xs (,, L, L whee L, L. Whou loss of genealy, we can se L = : = {,,, }, L = : = {,,, }, and syse (a wh he odel ansfoaon ( can be wen as d [ x ( B xs ( Cx ( h d h = = D ( s x( s ] = = = = D x( s, = = ( A B D x( B x( h whee 0 and 0. n Lea. Fo any vecos x, y R and R > 0, we have x y x Rx y R y. (3 Now we wll pesen a delay-dependen ceon fo asypoc sably of syse (. heoe. Syse ( s asypocally sable povded ha hee exs,, such ha he ax  = A B D = = e e= De s Huwz, < ρ ( h B Ck = k=, and ha fo soe posve defne syec aces P, S, S k, S 3e, S, S 5l, W (,, W (, k, W 3 (, e, W (l,, W 5 (l, k, and W 6 (l, e,, k, e, \, and l \, such ha he followng LMI condon hol: Z Z Ω= < 0, Z Z 3 whee [ ] Z Z Z Z Z Z Z Z = , ( [ ] Z = dag Z, Z, Z, Z, Z, Z, Z, Z A P PA h S S S ˆ ˆ = = = = 3 ( S h W(, W(, = 3 5 = W5(, W6(, W (, [ S h W (, Z = [ h Aˆ PB h Aˆ PB Aˆ PC Aˆ PC ˆ ˆ A PD A PD PB PB PD PD Z = dag[ S,, S, S,, S, S3,, S, S,, S, S,, S 3 ( 5( 5 Z = [ F ], F = h B PB, 3 ( Z = dag[ Y ], Y = W (,, 33 ( Z = [ F ], F = B PC ( Z = dag[ Y ], Y = W (, ( Z = [ F ], F = B PD 5 3 ( 3 Z = dag[ Y ], Y = W (, 55 3 ( 3 3 Z = [ F ], F = h D PB 6 ( Z = dag[ Y ], Y = W (, 66 ( Z = [ F ], F = D PC 7 5 ( 5 Z = dag[ Y ], Y = W (, 77 5 ( 5 5 Z = [ F ], F = D PD 8 6 ( 6 Z = dag[ Y ], Y = W (, 88 6 ( 6 6 Poof. Syse ( can be ewen n he followng fo n vew of (3: d [ x ( B xs ( Cx ( h d = h = = D ( s x( s ] = Ax ˆ ( B x ( h D x ( s. = = (5

3 C.H. Len and J.D. Chen: Sably Cea fo a Class of Neual Syses va he LMI Appoach 5 By he schu copleen of [ condon ( s euvalen o = ψ = Aˆ P PAˆ h ( S Aˆ PB S B PAˆ ( S Aˆ PC S C PAˆ = = ˆ ( S ˆ 3 A PDS3 D PA ( S PB S B P = = h W B PB W B PB ( (, (, ( W(, B PC W(, C PB = ( W3(, B PDW3(, D PB = 5 5 = ( S PD S D P h W D PB W B PD = ( (, (, ( W5(, D PC W5(, C PD = ( W6(, D PD W6(, D PD = <0. (6 he funconal gven by V( x = V( x V( x V3( x V( x V5( x V ( x V ( x, whee 6 7 V ( x = G ( x PG( x, = h = = D ( s x( s, = Gx ( x ( B xs ( Cx ( h = h = (7 (8 V ( x ( s h x ( s S x ( s, (9 3( = ( ( h = ( s x ( s S3x( s, = V x x s S x s (0 = h = V ( x x ( s K x( s, = (, K S h B PB W B PB B PC W (, C PB B PD W3 D PB (,, 5 = 5 = V ( x ( s x ( s K x( s, 5 = 5 (, K S h D PB W B PD D PC W (, C PD 5 D PDW6 D PD (,, 6 = h = V ( x [ ( s h x ( s W (, x( s h ( s x ( s W3(, x( s, ( ( x ( s W (, ( x s (3 7 = h = V ( x [ ( s h x ( s W (, x( s h x ( s W (, x( s ( 5 ( s x ( s W6 (, x( s s a legae Lyapunov funconal canddae [0]. By Lea, he e devave of V( x, 7, along he aecoes of syse (5 s gven by ( (( ˆ x x A P PAˆ x( [ h x ( Aˆ PB S B PAx ˆ ( = h = = x ( s S x( s ] [ x ( Aˆ PC S C PAx ˆ ( x ( h S x( h ] ˆ [ x ( A PD ˆ S3 D PAx( ( s x ( s S x( s ] 3

4 6 Asan Jounal of Conol, Vol. 6, No., Mach 00 [ x ( h S x ( h x ( PBS B Px ( ] = [ h x ( h B PBW (, B PBx ( h = h x ( s W (, ( x s ] [ x ( h B PC W (, C PBx( h = x ( h W (, x( h ] = [ x ( h B PD W3(, D PBx( h ( s x ( s W (, x( s ] = 3 [ x ( s S x( s x ( PD S D Px( ] 5 5 [ ( (, ( = h x s D PB W B PD x s h x ( s W (, ( x s ] [ ( 5 (, ( = x s D PC W C PD x s x ( h W (, x( h ] 5 [ x ( s D PDW6 (, D PDx( s = ( s x ( s W (, x( s = h = V ( x [ h x ( S x ( x ( s S x ( s ( x x (( S S x( 3 = 3 = = x ( h S x( h = = 6 ( s x ( s S x( s, = = ( x [ x ( K x( x ( h K x( h 5 = 5 5 = V ( x [ x ( K x ( x ( s K x ( s 6 = = 3 ( x [ h x ( W (, x( x ( W (, x( x ( W3(, x(] x h = [ ( s W (, x( s x ( h W (, x( h ( s x ( s W (, x( s 7 = = x ( W5(, x( x ( W6 (, x(] h = x h W5 x h 6 V ( x [ h x ( W (, x( [ x ( s W (, x( s ( (, ( ( s x ( s W (, x( s ]. he devave of V( x s gven by ( x = ( x ( x 3( x ( x 5( x 6( x 7( x (5 x ( ψ x(. In vew of [3] and [ he ceon ρ( < h B C D = = = guaanees ha he opeao = h = = D ( s x( s = Gx ( x ( B xs ( Cx ( h s sable. hus, by heoe 9.8. n [0] along wh (6- (5, we conclude ha syses ( and (5 ae boh asypocally sable. By sply seng = 0 and = 0, n heoe, we can oban he followng cee-delay-ndependen ceon. Coollay. Syse ( s asypocally sable fo any delays h R povded ha A s Huwz, ρ( C <, = and ˆP ha fo soe posve defne syec aces, S ˆ, S ˆ, S ˆ 5, W ˆ (,, and W ˆ 5(,, and such ha he followng LMI condon hol: 3

5 C.H. Len and J.D. Chen: Sably Cea fo a Class of Neual Syses va he LMI Appoach 7 whee Zˆ Zˆ Zˆ 3 Zˆ ˆ Z ˆ Z 0 0 0, ˆ ˆ < Z3 0 Z33 0 ˆ Z ˆ 0 0 Z = ˆ ˆ ˆ 5 = ( Wˆ ˆ (, W5(, Zˆ A Pˆ PA ˆ [ S S S ˆ ˆ Z ˆ ˆ ˆ = [ A PC A PC PB PB PD ˆ PD ˆ Zˆ = dag[ Sˆ,, Sˆ, Sˆ,, Sˆ, Sˆ,, Sˆ 5 5 Zˆ = [ Fˆ ], Fˆ = B PC ˆ, 3 Zˆ = dag[ Wˆ (, ], Zˆ = [ Fˆ ], 33 5 Fˆ D PC ˆ, Zˆ dag[ Wˆ (, ]. 5 = = 5 (6 Reak. Coollay concdes wh heoe n [3]. heoe n [3] can be seen as a specal case of ou esuls. By sply seng = and = 0 n heoe, he followng cee-delay-dependen ceon fo asypoc sably of syse ( can be obaned. Coollay. Syse ( s asypocally sable povded ha A A B s Huwz, ha he condon = = = ρ[ ( h B C ] <, and ha fo soe posve defne syec aces P, S, S, S 5, W (,, and W 5(,, and such ha he followng LMI condon hol: whee Z Z Z 3 Z Z Z 0 0 0, < Z 3 0 Z 33 0 Z 0 0 Z = [ 5 = ( h W (, W 5(, Z A P PA h S S S (7 Z = [ h A PB h A PB A PC A PC PD PD Z = dag[ S,, S, S,, S, S,, S 5 5 Z = [ F ], F = h D PB, 3 Z = dag[ W (, ], Z = [ F ], 33 5 F D PC, Z dag[ W (, ]. 5 = = 5 Reak. By seng soe aces as S 5 = ( D P ε I, W (, = ( B PD ε I, W (, = ( C PD ε I, 5 whee ε > 0, we have = S A PC S C PA h S 5 PD S 5 D P h ( h ( W (, D PB W (, B PD W 5(, D PC W 5(, C PD ]} x( x ({ A P PA [ h ( S A PB S B PA ( x ( { A P PA [ h ( S A PB S B PA = S A PC S C PA h ( D P ε I h h ( B PD ε I h ( C PD ε I]} x(. Noe ha he paaee ε > 0 can be chosen so as o be suffcenly sall. By he schu copleen of [] and = h, ou Coollay s less consevave han he esul n [7]. Reak 3. Noe ha =, =, and ha heoe concdes wh he heoe n [3]. heoe n [3] can be seen as a specal case of ou esuls. By he schu copleen gven n [] wh = = =, we can fnd ha Coollay 3 s less consevave han he esul n [] unde condons sla o hose n Reak. Reak. Noe ha D = 0, = 0,, and ou heoe concdes wh heoe n []. Hence, he esul gven n [] can be seen as a specal case of ou esuls.

6 8 Asan Jounal of Conol, Vol. 6, No., Mach 00 III. ILLUSRAIVE EXAMPLE Consde he followng neual syse wh ulple cee and bued e delays: 0. x ( = ( ( x 0 x ( ( x h x x ( h xs ( xs (, ( whee h 0 s any fne consan. In vew of ( and (8, we have h = 0.55, h = h, = 0., and = 0.. By heoe wh = and = 0, we have ρ[h B C C ] = 0.79<, and LMI ( s sasfed (even n he case whou bued delays. We copae ou esuls wh soe pevous ones n able. IV. CONCLUSION In hs pape, he Lyapunov heoy and LMI appoach have been appled o guaanee he asypoc sably of a class of neual syses. Soe ceedelay-ndependen and cee-delay-dependen cea have been poposed. he obaned esuls pesened n hs pape have been shown o be less consevave han hose pesened n eale epos. he echnues used fo paaezng ansfoaon n [8] and [] wll o be suded n fuue wok. C 0 NOAIONS = Se of all connuous funcons n fo [ H,0] o R. A (esp. x = anspose of ax A (esp., veco x. x (esp. A = Eucldean (esp., Specal no of veco x (esp., ax A. P > 0(esp. P< 0 = P s a posve (esp., negave defne syec ax. ρ [ A] = Specal adus of eal ax A. ( esp. \ = {,,, } (esp.,,,, }, whee. [ A ] n = [A A n A A n A A n ]. dag[ A ] n = dag[a,, A n, A,, A n,, A,, A n ]. REFERENCES. Boyd, S., L. El Ghaou, E. Feon, and V. Balakshnan, Lnea Max Ineuales n Syse and Conol heoy, SIAM, Phladelpha (99.. Chen, J. D., C. H. Len, K. K. Fan, and J. S. Cheng, Delay-dependen Sably Ceon fo Neual e-delays Syses, Elecon. Le., Vol. 36, pp ( Chen, J. D., C. H. Len, K. K. Fan, and J. H. Chou, Cea fo Asypoc Sably of a Class of Neual ssyses va a LMI Appoach, IEE Poc. Con. heoy. Appl., Vol. 8, pp. -7 (00.. Chen, J. D., C. H. Len, and J. H. Chou, Flexble Sably Cea of a Class of Neual Syses wh Mulple e Delays va LMI Appoach, J. Chnese Ins. Eng., Vol. 5, pp ( Dve, R. D., Odnay and Delay Dffeenal Euaons, Spnge-Velag, New Yok ( Dugad, L., and E. I. Vees, Sably and Conol of e-delay Syses, Spnge-Velag, London ( Fan, K. K., C. H. Len, and J. G. Hseh, Asypoc Sably fo a Class of Neual Syses wh Mulple e Delays, J. Op. heoy Appl., Vol., pp ( Fdan, E., New Lyapunov-Kasovsk Funconals fo Sably of Lnea Readed and Neual ype Syses, Sys. Con. Le., Vol. 3, pp ( Goeck, H., S. Fuksa, P. Gabowsk, and A. Koyowsk, Analuss and Synhess of e Delay Syses, John Wley & Sons, Chchese (989. able. Copason ou esuls wh hose pesened n ecen epos. Dscee-delay-ndependen esuls (Independen of e delay h Dscee-delay-dependen esuls (Uppe boun of e delay h Cea Wh bued delays Whou bued delays Wh bued delays Whou bued delays Ou esul [3] [8] [6] [] [3] [7] [8] [8] [7] Any h 0 Whou any LMI soluons (cea canno be sasfed Algebac condons canno be sasfed

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