Advances in Intelligent Systems Research, volume 136 4th International Conference on Sensors, Mechatronics and Automation (ICSMA 2016)

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1 Advnces in Inelligen Sysems Reserc, volume 36 4 Inernionl Conference on Sensors, Mecronics nd Auomion (ICSMA 6) New Lypunov-Krsovskii sbiliy condiion for uncerin liner sysems wi inervl ime-vrying dely Weifeng Zng,, Junjun Hui,b nd Wenqi Go,c Lnzou Insiue of ecnology, Lnzou 735,Cin Milbox 5 exension, Boji, Snxi 73, Cin 3 zngwf38@6.com, b epsone@63.com, c gowenqi@63.com Keywords: L-K funcionl; dely decomposiion; Disribued dely; Liner mrix inequliy (LMI) Absrc. is pper invesiges e robus dely-dependen sbiliy problem of clss of liner uncerin sysem wi inervl ime-vrying dely. Bsed on dely-cenrl poin meod, e wole dely inervl is divided ino wo equidisn subinervls is cenrl poin nd new Lypunov-Krsovskii (L-K) funcionls wic conins some riple-inegrl erms nd ugmen erms re inroduced on ese inervls. en, by using L-K sbiliy eorem, inegrl inequliy meod nd convex combinion ecnique, new dely-dependen sbiliy crieri for e sysem is formuled in erms of liner mrix inequliies (LMIs). Unlike exising meodologies, wen bounding e cross-erms emerge from e ime derivive of e L-K funcionl, neier superfluous free weiging mrices re inroduced nor ny useful erms re negleced, only using iger inegrl inequliies nd very few free weiging mrices for express e relionsip of e correlive erms, so i cn reduce e complexiy bo in eoreicl derivion nd in compuion. Finlly, numericl exmples re given o illusre e effeciveness nd n improvemen over some exising resuls in e lierure wi e proposed resuls. Inroducion ime dely is encounered in mny dynmic sysems suc s cemicl or process conrol sysems nd neworked conrol sysems [].ime dely is lwys one of e sources of insbiliy nd poor performnce. Hence, sbiliy nlysis nd sbilizion of sysem wi ime-delys ve received considerble enion in e ps few yers[-3].very recenly, sysems wi ime-vrying dely in known inervl ve been sudied in [,6-3],werein e ime-dely my vry in rnge for wic e lower bound is no resric o being zero. In view of e sbiliy nlysis for e sysems wi inervl ime-vrying dely, e exising sbiliy crieri re usully clssified ino wo mjor cegories, nmely, dely-independen ones, nd dely-dependen ones. Generlly speking, e dely-dependen sbiliy crierion is less conservive n dely-independen sbiliy wen e ime-dely is smll. A generl frmework for sbiliy nlysis is Lypunov-Krsovskii funcionl combine wi e Liner mrix inequliy. Under is frmework, n imporn issue is o enlrge e fesible region of sbiliy crieri. As for e nlysis meod, ere were free-weiging mrix meod[],ugmened Lypunov funcionl pproc[3] nd dely priioning pproc[5,6,8] ec., e common dvnges of e bove meod is e dely informion cn be fully used, wic vil reduce e conservism, owever, i cn lso led o e increse of compuionl complexiy since mny slck vribles re inroduced. Jensen s inegrl inequliy pproc is noer imporn meod, is pproc possesses few vribles crcerisics, so cn provide simple form of sbiliy condiions. Gu[] ws e firs one o inroduced is meod o e sbiliy nlysis of ime-dely sysems. Soon ferwrds, mny resercers, suc s Zng e l. [8],So e l.[], Rmkrisnn nd Ry[9,3] furer exended e Jensen s inegrl inequliy o some new forms nd obined some less conservive resuls. Nevereless, ere sill exiss room for furer improvemens. Moived by e bove discussions, we furer discuss e sbiliy of uncerin liner sysems wi inervl ime-vrying delys. Bsed on dely-cenrl poin meod, e wole dely inervl is Copyrig 6, e Auors. Publised by Alnis Press. is is n open ccess ricle under e CC BY-NC license (p://creivecommons.org/licenses/by-nc/4./). 59

2 Advnces in Inelligen Sysems Reserc, volume 36 divided ino wo equidisn subinervls is cenrl poin nd new L-K funcionls wic conins some riple-inegrl erms nd ugmen erms re inroduced on ese inervls. en, by using L-K sbiliy eorem, inegrl inequliy meod nd convex combinion ecnique, new dely-dependen sbiliy crieri for e sysem is formuled in erms of liner mrix inequliies (LMIs). Numericl exmples re given o illusre e effeciveness nd less conservism of e proposed meod. Problem descripion nd preliminries Noions: rougou is pper, R n denoes e n-dimensionl Euclidin spce, R n m is e se of n m rel mrice, X >, for X R n m, mens e mrix X is rel symmeric posiive definie. A B For n rbirry mrix B nd wo symmeric mrices A nd C, denoes symmeric mrix, C were denoes e enries implied by symmery, ei represens block enry mrices of pproprie dimensions, i.e., if i =,,3,en, e = [ I ], e = [ I ], e3 = [ I ]. Consider e following liner uncerin sysem wi ime-vrying dely x& ( ) = ( A + A( )) x( ) + (B + B( )) x( ( )) x( ) = ϕ ( ), [,] () Were, x( ) R n is e se vecor, A, B re consn mrices wi pproprie dimensions, ( ) is ime-vrying coninuous funcion, sisfy e following wo scenrios condiions: Cse m () M, &() µ, ; () Cse m ( ) M,, A( ), Δ B( ) denoe prmeer uncerinies sisfying e following condiion: [ A( ) B( )] = DF ( ) [ E Eb ] (3) (4) were, D, E, Eb re consn mrices wi pproprie dimensions; nd F ( ) is n unknown ime-vrying mrix,wic is Lebesque mesurble in nd sisfies F ( ) F ( ) I,. e following lemm is inroduced wic s n imporn role in e derivion of e min resuls. Lemm [8]. For ny consn mrix R R n n, sclr >,nd vecor funcion x& ( ) : [, ] Rn suc e inegrion x& ( s ) Rx& ( s )ds is well defined, en e following inequliy olds: x() R R x() x& (s)rx& (s)ds (5) x( ) R R x( ) Lemm [9]. For ny consn mrix R R n n, R = R >, sclrs ( ),nd vecor funcion x& ( ) : [, ] Rn suc e inegrion x& ( )Rx& ( )d is well defined, en e following inequliy olds: ( ) x& ( )R x& ( ) d ξ ( ) Π R + ( ( ) ) Π, R + ( ( )) Π, R ξ ( ) (6) Were ξ () = x ( ) x ( ()) x ( ) nd 593

3 Advnces in Inelligen Sysems Reserc, volume 36 R R R Π R = R R, Π, R = (e e3 ) (e e3 ), R R Π, R = (e e ) (e e ). Lemm 3[5]. Suppose γ γ () γ, Were γ (.) : R+ (orz + ) R+ (orz + ).en,for ny consn mrices Ξ, Ξ nd Ω wi proper dimensions, e following mrix inequliy Ω + (γ ( ) γ )Ξ + (γ γ ( ))Ξ < olds, if nd only if Ω + (γ γ )Ξ <, Ω + (γ γ )Ξ < Lemm 4[4]. Given mrices Q = Q, H,nd E wi pproprie dimensions, en Q + HF ( ) E + E F ( ) H < for ll F ( ) sisfying F ( ) F ( ) I,if nd only if ere exiss sclr ε >,suc Q + ε HH + ε E E < Min resul Firs consider nominl sysem () x& ( ) = Ax ( ) + Bx ( ( )) (7) x ( ) = ϕ ( ), [, ] eorem For given vlues of, nd µ,sysem (5) is sympoiclly sble, if ere exis P P3 P3 > P33 R, Q j >, Z j > j =,, S > nd pproprie dimensions suc e following LMIs old + i M <, i =, M + i M <, i =, M were mrices P P P3 PB 5 W R 33 Z 44 Z = 55 W 66 7 B P 57 4Q R W W, wi R W (8) (9) 8 B P3, 58 4Q = (e3 e4 ) ( Z )(e3 e4 ), = (e4 e5 ) ( Z )(e4 e5 ), = P A + A P + P + P + R + W + S Z 4Q 4δ Q, = R + Z, 5 = P P3, 7 = A P + P + 4Q, 8 = A P3 + P3 + 4δQ, = R R Z, 33 = R Z, 44 = ( µ)s Z, 55 = W Z, 57 = P P, 58 = P P, 66 = W W,

4 Advnces in Inelligen Sysems Reserc, volume 36 = [ A B ], M = ( ) Z + δ Z + 4Q + ( ) Q, ) Z + δ Z + 4Q + ( ) Q, = ( + ), δ = ( ). Proof. Bsed on dely-cenrl poin pproc, we dividing dely inervl ino wo equl subinervls e midpoin, is [, ] nd [, ], if we cn proof eorem olds for M = ( e wo subinervls, en eorem is rue. Cse : wen ( ), Consruc L-K funcionl cndide s V ( ) = V ( ) + V ( ) + V3 ( ) + V4 ( ) () V ( ) = ς ( ) Pς ( ), V ( ) = V3 () = ξ (s)rξ (s)ds + ξ (s)wξ (s)ds + ( ) x (s)sx(s)ds, & (s)z x& (s)dsd + ( ) x& (s)z x& (s)dsd, x + + V4 ( ) = +λ x& ( s )Q x& ( s ) dsd λ d + ( ) x& ( s )Q x& ( s ) dsd λ d, +λ Were ς ( ) = x ( ) x (s)ds x (s)ds, ξ ( ) = x ( ) x ( ), ξ ( ) = x ( ) x ( ) e ime-derivive of e L-K funcionl long e rjecory of () is given by V& ( ) = V& ( ) + V& ( ) + V& ( ) + V& ( ) 3 4 V&() = ς ()Pς&() = ς ()Px& () x () x ( ) x ( ) x ( ) () V& () = ξ ()Rξ() ξ ( )Rξ( ) + ξ ()Wξ () ξ ( )Wξ( ) + x ()Sx() ( &())x ( ())Sx( ()) V&3 () = ( ) x& ()Zx&() + ( ) x& ()Z x&() x& (s)zx& (s) ( ) x& (s)z x& (s) V&4 () = x& ()[4Q + ( ) Q ]x&() + x& (s)qx&()dsd ( ) + x& (s)q x&()dsd From e condiion (), one cn obin: V& ( ) ξ ()Rξ ( ) ξ ( )Rξ ( ) + ξ ( )Wξ ( ) ξ ( )Wξ ( ) + x ( )Sx( ) ( µ) x ( ( ))Sx( ( )) Using Lemm,one cn obin: x() x() Z Z & & x (s)zx(s) x( ) Z Z x( ) By using Lemm, i follows ( ) () (3) (4) (5) (6) (7) x& (s)z x& (s) ξ ( ) ΠZ + (( ) )Π,Z + ( ( ))Π,Z ξ ( ) (8) were ξ ( ) = x ( ) x ( ( )) x ( ), Π Z Z = Z Z Z, Z 595

5 Advnces in Inelligen Sysems Reserc, volume 36 Z Z Π, Z = (e e3 ) (e e ), (e e3 ), Π, Z = (e e ) Using Jensen s inequliy, we ve + x& (s)q x& ( )dsd ( ( x( ) + + ) Q ( x&(s)dsd ) x(s)ds ) Q ( x( ) x(s)ds ) x& (s)dsd (9) + x& (s)q x&()dsd ( x&(s)dsd ) Q ( x& (s)dsd ) ( )x() x(s)ds) Q ( ( )x() ( + + () x(s)ds ) Subsiuing () () in (), e ime derivive V& ( ) cn be expressed s follows: V& () ζ ()[+ (() )( ) + ( ())( ) + M ]ζ() Were ζ () = x () x ( ) x ( ) x ( ()) x ( ) x ( ) x (s)ds x (s)ds, One cn see if () + (() )( ) + ( ())( ) + M < en, V& () < ε x() for some sclr ε >, from wic we conclude e nominl sysem (7) is sympoiclly sble ccording o Lypunov sbiliy eory. From Lemm 3 nd Scur complimen o () yield e LMI (8) cse wen ( ), consider L-K funcionl cndide s V ( ) = V ( ) + V ( ) + V3 ( ) + V4 ( ) () V ( ) = ς ( ) Pς ( ), V ( ) = V3 ( ) = ( ) ξ (s)rξ(s)ds + ξ (s)wξ (s)ds + x (s)sx(s)ds, & & x ( s ) Z x ( s ) dsd + ( ) x& (s)z x& (s)dsd, + + V4 ( ) = Were ς () = x () +λ x& ( s )Q x& ( s ) dsd λ d + ( ) x (s)ds +λ x& ( s )Q x& ( s ) dsd λ d, x (s)ds, ξ ( ) = x ( ) x ( ), ξ ( ) = x ( ) x ( ), P, R, W, S Z, Z, Q, Q re e sme mrices used in e LK funcionl (). In e similr mnner, we cn obin V& () ζ () + (() )( ) + ( ())( ) + M ζ (), Were ζ () = x () x ( ) x ( ) x ( ()) x ( ) x ( ) x (s)ds x (s)ds 596

6 Advnces in Inelligen Sysems Reserc, volume 36 (3) ) +( ())( ) +M < en, sysem (7) is sympoiclly sble. From Lemm 3 nd Scur complimen o (3) yield e LMI (9), is complees e proof. Now consider e robus sbiliy of e uncerin sysem. eorem For given vlues of, nd µ,sysem (5) is sympoiclly sble, if ere exis If +(() )( P P3 P3 > P33 R R W W, Q j >, Z j > j =,, S >,, W R wi pproprie dimensions suc e following LMIs old ˆ Γ D εγ (4) < ε I ε I ˆ Γ D εγ < (5) ε I ε I Were + i M + i M ˆ ˆ =, i =,, =, i =, M M sclr ε i >, i =, nd P P Γ = P P P3 M, Γ = [ E Eb ], Γ3 = P P P3 M. Proof replcing A, B in (8),(9) wi A + A, B + B, respecively, nd using Lemm 4 complees e proof. Numericl exmples In is secion, we use wo numericl exmples o sow e proposed resuls re improvemens over some exiing ones Exmple. Consider nominl ime-dely sysem wi e following prmeer A=,B =.9 For given µ nd unknown µ,ble provides e mximl llowble bounds of e dely for given lower bounds.from e ble,i is cler e proposed sbiliy is less conservive n ose in [,,]. Exmple Consider n uncerin sysem described by e following mrices were, δ, δ, δ 3 nd δ 4 +δ +δ3, B= A=, +δ +δ4 unknown prmeer, sisfy δ.6, δ.5, δ 3., δ 4.3 Wen ere is no resricion on e dely derivive, ble sows e obined mximum llowble dely bound for vrying.from e ble, i is cler o see for is exmple, some exiing resuls ve been improved. ble Mximum llowble dely bound for given nd µ 597

7 Advnces in Inelligen Sysems Reserc, volume 36 µ Meod [] [].5 [] eorem [] [].9 [] eorem [] [] Any µ [] eorem = = = = 3 = ble upper dely bound for given Meod [] [3](N=) [9] eorem Conclusions is pper invesiges e robus dely-dependen sbiliy problem of clss of liner uncerin sysem wi inervl ime-vrying dely. A new L-K sbiliy condiion for is sysem ws proposed. e key feures of e sbiliy condiion include e dely-cenrl poin meod for designing e L-K funcionl nd iger inegrl inequliy for bounding e cross-erms. As resul, less conservive resuls re cieved. Numericl exmples ve illusred e effeciveness of e proposed meod. References [] Gu K, Krionov V L, Cen J. Sbiliy of ime-dely sysems[m]. Bsel: Birkuser, 3: -7. [] He Y,Wng Q, Lin C, e l. Dely-rnge-dependen sbiliy for sysems wi ime-vrying dely[j]. Auomic, 7, 43(): [3] O.M. Kwon,M.J.Prk,S.M. Lee.Augmened Lypunov funcionl pproc o sbiliy of uncerin neurl sysems wi ime-vrying delys[j],applied Memics nd Compuion,9,7():-. [4] O.M. Kwon, M.J.Prk, S.M. Lee,E.J. C. Anlysis on dely-dependen sbiliy for neurl neworks wi ime-vrying delys[j],neurocompuing,3,3: 4-. [5] Yue D, in D, Zng Y. A piecewise nlysis meod o sbiliy nlysis of coninuous/discree sysems wi ime-vrying dely[j], Inernionl Journl Robus Nonliner Conrol, 9,9(3): [6] Wng C,Sen Y. Dely priioning pproc o robus sbiliy nlysis for uncerin socsic sysems wi inervl ime-vrying dely[j],ie Conrol eory nd Applicions,,6(7):

8 Advnces in Inelligen Sysems Reserc, volume 36 [7] Oriuel, L., Milln, P., Vivs, C., Rubio, F.R.Robus sbiliy of nonliner ime-dely sysems wi inervl ime-vrying dely[j],inernionl Journl of Robus nd Nonliner Conrol,,(7): [8] Zng X-M,Hn Q-L.A dely decomposiion pproc o dely-dependen sbiliy for liner sysems wi ime-vrying delys[j],inernionl Journl of Robus nd Nonliner Conrol,9,9(7):9-93. [9] K. Rmkrisnn, G. Ry.Robus sbiliy crieri for uncerin liner sysems wi inervl ime-vrying dely [J],Journl of Conrol eory nd Applicions,, 9 (4): [] So H-Y. New dely-dependen sbiliy crieri for sysems wi inervl dely [J]. Auomic, 9,45(3): [] Jing X-F, Hn Q-L. New sbiliy crieri for liner sysems wi inervl ime vrying dely[j], Auomic, 8, 44(): [] C. Peng, Y.C.in. Improved dely-dependen robus sbiliy crieri for uncerin sysems wi inervl ime-vrying dely[j]. IE Conrol eory nd Applicion, 8,(9)9:75-76, [3] K. Rmkrisnn, G. Ry. Dely-dependen robus sbiliy crieri for liner uncerin sysems wi inervl ime vrying dely[c].encon 9-9 IEEE Region Conference. Singpore: IEEE, 9: [4] Peersen I R, Hollo C V. A Ricci equion pproc o e sbilizion of uncerin liner sysems[j]. Auomic, 986,(4):