Stability of Solution for Nonlinear Singular Systems with Delay

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1 Sed Orders for Reprits to he Ope Atomatio ad Cotrol Systems Joral Stability of Soltio for Noliear Siglar Systems with Delay Ope Access Zhag Jig * ad Lig Chog Departmet of Basic Sciece North Chia Istitte of Aerospace Egieerig Lagfag Hebei PR Chia No4 Costrctio Compay Chia Petrolem Pipelie Brea g Lagfag Hebei PR Chia Abstract: I this paper we preset ew defiitios of stability of siglar systems with delay based o the works of BBajic ad MMilic ad establish theorem o stability he the sfficiet coditios for iform stability of a class of special oliear siglar systems with delay are sggested Keywords: Noliear siglar systems siglar systems with delay stability iform stability INRODUCION From a modelig poit of view it is perhaps more realistic to model a pheomeo by a siglar sch as descriptor systems semi-state space systems ad differetialalgebraic systems se of which arise freqetly i may fields sch as optimal cotrol problems electrical circits etral etwork ad some poplatio growth models Some applicable example ad basic reslts ca be fod i [ ] Siglar systems correspod to ormal systems hey have both iteral logic ad essetial differece Bt the stdy of siglar systems had mostly referred to the give ormal systems theories which had bee geeralized ad trasplated ito siglar systems so far he methods of stdy for siglar system are mostly geometric approach freqecy domai method ad state-space techiqes Ad people still have differet views related to the qestios of siglar systems ths the research achievemet of siglar systems appears extremely fragmetary he stability ad asymptomatical stability still havet have iform distict defiitios ths there exists cofsios for some cocepts as well as icosistecies betwee the cocept ad the theorem he cosistecy of the iitial coditios still has two differet views O the other had there ofte happe implses ad jmps i the soltios So the existece ad iqeess of soltios i siglar systems havet bee resolved becase the complexity of siglar systems ad their stability havet achieved mtal recogitio herefore the research for siglar systems ot oly has a widespread practical sigificace bt also its theoretical vale has broad prospects for developmet As oe of the major research sbjects i oliear siglar systems the problem of stability attracts may researchers attetio for example [3-5] discssed the stability of *Address correspodece to this athor at the Departmet of Basic Sciece North Chia Istitte of Aerospace Egieerig Lagfag Hebei PR Chia; el: ; zhag_jig_985@6com siglar systems Sice delay ofte occrs i siglar systems sch as discssed [6] therefore the research o stability of oliear siglar systems with delay is give mch importace i practice ad theory However few stdies have bee doe o the stability of oliear siglar systems with delay herefore i this paper we cocetrate o this he discssio o stability of siglar systems compared with that of osiglar systems came p with three mai ew difficlties: the first is that it ist easy to satisfy the existece ad iqeess of soltios sice the iitial coditios may ot be cosistet; the secod is that it is difficlt to calclate the derivatives of Lyapov fctios; the third is that there ofte happe implses ad jmps i the soltios I order to overcome these difficlties this paper presets ew defiitios of stability of oliear siglar systems with delay based o the previos stdies by [7 8] Frthermore the stability theorem of soltios for the followig oliear siglar systems with delay has bee established: A!x(t) = Bx(t) Cx(t! ) f (tx(t! )) x t0 = where A is a costat siglar matrix ad B C are! costat matrices τ > 0 f (t! ) C([0) C([ 0] R ) R ) f (t0) = 0 for ay t! t 0! 0 x t0 =! is the iitial coditio of () where! C([ 0] R ) PRELIMINARIES I this paper we assme that the soltio of iitial vale problems for system () exists which is called opertrbatio soltio writte as x ( ) t t0 ϕ sometimes x () t for short At first we itrodce the followig otatios: () /5 05 Betham Ope

2 608 he Ope Atomatio ad Cotrol Systems Joral 05 Volme 7 k = [0tk ) where 0 < tk! ; 0k (t0 ) = [t0 tk ) where t0!k J is a ope iterval at R ad k! J 0 D is a ope iterval at R C!(J D) is the set of all cotios J!D differetiable fctios defied as m q(t x)!c ( J R R ) S I (t)! C([ 0] R ) S I (t) is a set of all cosistecy iitial fctios of () i [ t0 τ ) tk ) throgh (t! ) at least S k (ttk )! S I (t) ad for ay! S k (ttk ) there exists a cotios soltio of () i Jig ad Chog It is difficlt to calclate the derivatives of Lyapov fctios We itrodced q(t x) to avoid the difficlty 3 I the above otatios the iitial coditio is cosistet ad the existece of soltio is satisfied i [t0 tk ) Bt it is ot reqired that the iqeess of soltio is satisfied this is differet for the osiglar systems It is however difficlt to satisfy the iqeess of soltio for siglar systems 4 If q(t x) = x tk =! ad S k (t0 tk ) = C([! 0] R ) [ttk ) throgh (t! ) at least the the above defiitios become the defiitios of Lyapov stability of traditioal sigificace B(ϕ δ ) = {ψ C ([ τ 0] R ); ψ ϕ < δ } δ > 0 5 If A i () is a symmetric matrix the we take q((t x) = x Ax otherwise we take Q(t ε ) = {x R ; q(t x) q(t x (t )) < ε } ε > 0 I this paper we sppose x (tt0! ) exists i [t0 tk )!t0 k implies! S k (t0 tk ) Defiitio If!t0 k! > 0 there always exists! (t0 ) > 0 sch that for! B( ) S k (t0 tk ) x(tt0! ) satisfies that x(tt0! ) Q(t )!t [t0 tk ) he the soltio x (tt0! ) is said to be stable i {q(t x) k } If δ is oly related to! ad has othig to do with t0 the x (tt0! ) is said to be iformly stable i {q(t x)k } q((t x) = x A H (t ) Ax where H (t ) is a matrix fctio 3 MAIN RESUL Now we propose theorem o the stability of soltios to () by sig the Lyapov fctio heorem Sppose that there exist two K -class wedge fctios Φ Φ ad cotios V fctio: V (tq(t x)) : J! R R V (tq(t x)) C ( J! R ) sch that Defiitio ) If x (t t0 ϕ ) is stable i {q(t x) k } where tk = ad for t0 k there exists a Δ(t0 ) > 0 sch that ψ B(ϕ Δ(t0 )) Sk (t0 ) lim q(t x(t t0 ψ )) q(t x (t t0 ϕ )) = 0 t the x (tt0! ) is said to be asymptomatically stable i {q(t x)k } ; ) If x (t t0 ϕ ) is iformly stable i {q(t x)k } where tk = ad there exists a Δ > 0 for!t0 k ad ψ B(ϕ Δ) Sk (t0 ) lim q(t x(t t0 ψ )) q(t x (t t0 ϕ )) = 0 t ad which is iform at (t0! ) [0) (B( ) Sk (t0 )) the x (tt0! ) is said to be iformly asymptomatically stable i {q(t x)k } Remarks tk may take! also it is possible to take a limited mber; so the applicatio scope of stable cocept expads i) Φ ( q(t x) q(t x (t t0 ϕ )) ) V (t q(t x)) Φ ( x x (t t0 ϕ) ) t [t0 tk ); ii)! S k (t0 tk ) V! (tq(t x(tt0 ))) 0 he x (tt0! ) is iformly stable i {q(t x)k } Proof For t0![0tk )! > 0 there exists! ( ) > 0 which is oly related to ε sch that! B( ) Φ ( ψ ϕ ) < Φ (δ ) Φ (ε ) Φ ( q(t x) q(t x (t t0 ϕ )) ) V (t q(t x(t t0 ψ ))) V (t0 q(t0 x (t0 t0 ϕ ))) Φ ( x(t0 t0 ψ ) x (t0 t0 ϕ ) ) = Φ ( ψ ϕ ) < Φ (ε ) hs q(t x) q(t x (t t0 ϕ )) < ε!t [t0 tk ) he we have x(tt0! ) Q(t ) t [t0 tk ) hece x (tt0! ) is iformly stable i {q(t x)k } he proof is completed

3 Stability of Soltio for Noliear Siglar Systems with Delay he Ope Atomatio ad Cotrol Systems Joral 05 Volme 7 heorem Sppose that there exist two K -class wedge fctios!! oegative ad o-decreasig fctio V! (tq(t x(tt0! ))) w( q(t x(t)) ) w( 0 ) w ad cotios V fctio: Hece there exists a m0 big eogh sch that whe V (t! ) :[0) C([r0] R ) R t > tm 0 which satisfies V (tq(t x(tt0! ))) i) Φ ( q(t x) q(t x (t t0 ϕ )) ) 0 ) = V (t0! ) m0 w( 0 ) < 0 m0 V (t0! ) w( V (t q(t x)) Φ ( x x (t t0 ϕ) ) t [t0 tk ); m= ii)! S k (t0 tk ) V! (tq(t x(tt0 ))) w( q(t x(t)) ) he x (tt0! ) is iformly stable i {q(t x)k } where tk! If w(s) > 0 whe s > 0 the x (tt0! ) is asymptomatically stable i {q(t x(t))[0!)} Proof ) By coditios 609 of theorem we get V! (tq(t x(tt0! ))) 0 he accordig to theorem x (tt0! ) is iformly stable i {q(t x)k } ) For that x (tt0! ) is iformly stable i {q(t x)k } we set tk =!! = he there exists a! 0 > 0 sch that t0![0)! (0 0 ) S k (t0 ) we have q(t x(t t0 ψ )) q(t x (t t0 ϕ )) <!t [t0 tk ) By coditios of theorem there exists a L > 0 sch that: which is i cotradictio with the defiitio of V hs lim q(t x(t t0 ψ )) q(t x (t t0 ϕ )) = 0 t ad the soltio of () is asymptomatically stable i {q(t x(t))[0!)} he proof is completed Now that we discssed the stability of soltios for oliear siglar systems with delay by sig heorem (t) = B (t) B X (t) C (t! )C X (t! ) f (t! ) 0 = B (t) X (t) C (t! ) C X (t! ) f (t! ) xt = 0 Where! R X! R = ; Bij Cij are q(t x(tt0! )) < L!t [t0 tk ) If lim q(t x(t t0 ψ )) q(t x (t t0 ϕ )) = 0 is ot satist () i! j costat matrices f i (t)!c([0tk ) R i ) i j = We ca take tk = ; τ as a positive costat { } x = ( X X ) = (x x x ) xt0 = ϕ is the iitial codi- We ca sppose that!m tm! tm! ake δ (0) tio of system () where ϕ C ([ τ 0] R ) ad t0 0 If the iitial fctio ϕ satisfies the followig cosistecy coditio () has a iqe cotios soltio i [t0 τ ) throgh (t0! ) fied the there is a ε 0 (0) ad a seqece tm lim tm = sch that m q(tm x(tm t0 ψ )) q(tm x (tm t0 ϕ)) > ε 0 sch that L! < 0 the!m!t [tm tm ] q(t x(t t0 ψ )) q(tm x(tm t0 ψ )) q(tm x (tm t0 ϕ )) q(tm x(tm t0 ψ )) q(tm x (tm t0 ϕ )) q(t x(t t0 ψ )) ε 0 Lδ hs ε0 0 = B! (0)! (0) C! ( ) C! ( ) f (t0 ) (3) hs for!t0 [0) S k (t0 tk ) = {! C([ 0] R ); satisfies (3)} Let! S k (t0 tk ) the soltio of iitial vale problems () ca be writte as: x (tt0! ) = ( (t) X (t)) he we have the followig theorem

4 60 he Ope Atomatio ad Cotrol Systems Joral 05 Volme 7 Jig ad Chog heorem 3: For iitial vale problems () if matrix U C! B C C! B C (C! B C ) 0 0 (C! B C ) 0 0 is semi-egative defiite where U = (B! B B ) B! B B he x!) is iformly stable i {(! ) (! )[0 )} Frthermore if C < the x!) is iformly stable i {x(t)! x (t)[0 )} where t k ca be! Proof Set q(tx) = (! ) (! ) V (tq(tx)) = q(tx) Let wedge fctio! (s) = s! (s) = s s![0) he Φ ( qtx ( ) qtx ( ( t)) ) = qtx ( ) = ( X X ) ( X X ) = Vtqtx ( ( )) =Φ ( X X () t) Becase of x x () t = X X () t X X () t we have Φ ( qtx ( ) qtx ( ( t)) ) =Φ ( X X ( t) ) Φ ( x x ( t) ) For! S k (t 0 ) we ca get that! V (tq(tx! ))) = ( (t) (t)) ( (t) (t)) ( (t) (t)) ( (t) (t)) = [B (t) B X (t) C (t ) C X (t ) B (t) B X (t) C (t ) C X (t )] ( (t) (t)) ( (t) (t)) [B (t) B X (t) C (t ) C X (t ) B (t) B X (t) C (t ) C X (t )] (4) (5) (6) X (t)! X (t) =!B ( (t)! (t))! C ( (t! )! (t! ))! C ( X (t! )! X (t! )) he (6) ca be writte as! V (tq(tx! ))) = D = DED where U C B C C B C ( (C B C ) 0 0 ( (C B C ) 0 0 ( U = (B! B B ) B! B B ad D = [( (t)! (t)) ( (t! )! (t! )) ( X (t! )! X (t! )) ] If E is semi-egative defiite we get! V (tq(tx! ))) 0 from heorem x!) is iformly stable i {(! ) (! )[0 )} By the secod eqatio of system () X () t X () t = B ( X ( t) X ( t)) C ( X ( t τ) X ( t τ)) C ( X ( t τ) X ( t τ)) B X () t X () t C X ( t τ) X ( t τ) C X ( t τ) X ( t τ) = B q( t x()) t C q(( t τ) x( t τ)) C τ X ( t ) X ( t τ) D (7) (8) Sice x!) is iformly stable i {q(tx) k } therefore! > 0 there exists! () > 0 We ca take! < sch that for!t 0 k! B( ) S k (t 0 ) ad!t [t 0 ) we have B q( t x()) t C q(( t τ) x( t τ)) = B q( t x()) t q( t x ()) t C q(( t τ) x( t τ)) q(( t τ) x ( t τ)) < ( B C ) ε From the secod eqatio of system () we have

5 Stability of Soltio for Noliear Siglar Systems with Delay he Ope Atomatio ad Cotrol Systems Joral 05 Volme 7 6 Where x(t) = x! ) Becase! is arbitrary ad B ij C ij are costat matrices we ca coclde: B q( t x()) t C q(( t τ) x( t τ)) < ε Ad we also have X ( t τ) X ( t τ) ψ ϕ< δ < ε t [ t t τ ) [ t t ) So from (8) ad (9) we have X () t X () t ε C ε k Sppose that for!t [t 0 (l ) t 0 l ][t 0 ) the l X () t X () t ε C ε C ε hs whe t![t 0 l t 0 (l ) ][t 0 ) ad t! [t 0 (l!) t 0 l ][t 0 ) by idctive sppositio ad (8) (9) we get that: X () t X () t ε C X ( t τ) X ( t τ) ε C ε ε l C If C < from idctio!t [t 0 ) X () t X () t l ε C ε C ε ε C hs xt () x() t X () t X () t X () t X () t = qtxt ( ()) qtx ( ()) t X () t X () t (9) (0) () from (0) () ad x!) is iformly stable i {q(tx) k } ad ε xt () x () t ε C herefore we see that if C < x ( ) t t0 ϕ is iformly stable i { x k } he proof is completed EXAMPLE Example 33 Let s discss the followig oliear siglar system with delay ) * (t) = 0 =! X (t) 0 0 X (t) X (t! ( ) 0 0 X (t! ( ) 3(t! ( ) X (t) X (t) 0 X (t! ( ) 0 X 0 (t! ( ) (t! ( ) 3 he iitial coditio of system (0) is () x t0 =!! C([ 0] R 6 ) t 0 0 (3) Where B = B = B = C = C = C = the matrix U C B C C B C ( C BC) 0 0 ( C BC ) = C = 0 is semi-egative defiite ad C = < By heorem 3 x!) is iformly stable i {x(t)! x (t)[0)} CONCLUSION It is difficlt to discss the stability of oliear siglar systems with delay becase of implse ad delay I this paper the stability is cosidered by sig ew defiitios ad Lyapov fctio Some stable criteria are

6 6 he Ope Atomatio ad Cotrol Systems Joral 05 Volme 7 Jig ad Chog proposed hese criteria are algebraic that s why they are coveiet to se Usig these criteria the stable problem for a class of oliear siglar systems with delay ca be solved easily Moreover frther stdies o stability of soltio for oliear siglar systems with delay will be smmarized i or ext stdy CONFLIC OF INERES he athors cofirm that this article cotet has o coflict of iterest ACKNOWLEDGEMENS he athors wold like to thak the reviewers for their valable sggestios ad commets [] C Che ad YQ Li Lyapov stability of liear siglar dyamical systems I: Proceedigs of the 997 IEEE Iteratioal Coferece o Itelliget Processig Systems Beijig 997 pp [3] VLY Syrmos P Misra ad R Aripirala O the discrete geeralized lyapov eqatio Atomatica vol 3 pp [4] LQ Zhag J Lam ad Q L Zhag New lyapov ad riccati eqatios for discrete time descriptor systems I: IEEE rasactios o Atomatic Cotrol vol 44 pp [5] YQ Li ad YQ Li Basic theory of siglar system of liear differetial differece eqatio I: Proceedigs of 3 th IFAC World Cogress Sa Fracisco 996 pp79-84 [6] YQ Li ad YQ Li Stability of soltio of siglar systems with delay Cotrol heory ad Applicatios vol 5 pp [7] BB Vladimir ad MM Mirko Exteded stability of motio of semi-state systems Iteratioal Joral of Cotrol vol 46 pp [8] MM Mirko ad BB Vladimir Stability aalysis of siglar systems Circits System ad Sigal Processig vol 8 pp REFERENCES [] HH Rosebrock Strctral properties of liear dyamical systems Iteratioal Joral of Cotrol vol 0 pp Received: September 6 04 Revised: December 3 04 Accepted: December 3 04 Jig ad Chog; Licesee Betham Ope his is a ope access article licesed der the terms of the Creative Commos Attribtio No-Commercial Licese ( which permits restricted o-commercial se distribtio ad reprodctio i ay medim provided the work is properly cited