Delay and Its Time-Derivative Dependent Stable Criterion for Differential-Algebraic Systems

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1 Applied Maemaics Publised Online June 6 in SciRes p://wwwscirporg/journal/am p://dxdoiorg/46/am67 Delay and Is ime-derivaive Dependen Sable Crierion for Differenial-Algebraic Sysems Hui Liu Yucai Ding Scool of Science Souwes Universiy of Science and ecnology Mianyang Cina Received April 6; acceped June 6; publised 4 June 6 Copyrig 6 by auors and Scienific Researc Publising Inc is work is licensed under e Creaive Commons Aribuion Inernaional License (CC BY) p://creaivecommonsorg/licenses/by/4/ Absrac In is paper e sable problem for differenial-algebraic sysems is invesigaed by a convex opimizaion approac Based on e Lyapunov funcional meod and e delay pariioning approac some delay and is ime-derivaive dependen sable crieria are obained and formulaed in e form of simple linear marix inequaliies (LMIs) e obained crieria are dependen on e sizes of delay and is ime-derivaive and are less conservaive an ose produced by previous approaces Keywords Differenial-Algebraic Sysems Sabiliy Analysis Lyapunov-Krasovskii Funcional Delay Pariioning Approac Linear Marix Inequaliy (LMI) Inroducion Differenial-algebraic sysems also referred o as singular sysems descripor sysems or generalized sae-space sysems arise in a variey of pracical sysems suc as cemical processes nuclear reacors biological sysems elecrical neworks and economy sysems Differenial-algebraic sysems include no only dynamic equaions bu also saic equaions [] []; e sudy of suc sysems is muc more complicaed an a for sandard saespace sysems e exisence and uniqueness of a soluion o a given differenial-algebraic sysem are no always guaraneed and e sysem can also ave undesired impulsive beavior wic can lead o e insabiliy and poor performance [] Because of e exensive applicaions in many pracical sysems a grea number of fundamenal noions and resuls in conrol and sysem eory based on sandard sae-space sysems ave been exended successfully o How o cie is paper: Liu H and Ding YC (6) Delay and Is ime-derivaive Dependen Sable Crierion for Differenial-Algebraic Sysems Applied Maemaics 7 4- p://dxdoiorg/46/am67

2 H Liu Y C Ding differenial-algebraic sysems In recen years muc aenion as been focused on sabiliy robus sabiliy and H conrol problems for differenial-algebraic sysems and some resuls ave been derived using e ime domain meod [4]-[] e exising resuls can be classified ino wo ypes: delay-dependen condiions wic include informaion on e size of delays [] [] and delay-independen condiions wic are applicable o delays of arbirary size [4] [5] Since e sabiliy of sysems depends explicily on e ime-delay e delay-independen condiions are more conservaive especially for small delays Wile e delay-dependen condiions are usually less conservaive and e conservaism will dependen on e cosen of Lyapunov funcional e inequaliy bounding ecnique and so on wo approaces were used o prove e sabiliy of e sysem in e exising lieraure e firs approac consiss of decomposing e sysem ino fas and slow subsysems and e sabiliy of e slow subsysem is proved using some Lyapunov funcional en e fas variables are expressed explicily by an ieraive equaion in erms of e slow variables [6] e second approac consiss of consrucing a Lyapunov-Krasovskii funcional a corresponds direcly o e descripor form of e sysem [7] Some oer meods ave also been provided o reduce e conservaive for example convex analysis meod [8] and delay pariioning approac [9] [] o e bes of our knowledge mos of e exising delay-dependen asympoically sable crieria only depend on e upper bound of e delay-derivaive and o e differenial-algebraic sysems e delay and is ime-derivaive dependen sabiliy crierion as no esablised wic moivaes is paper is aricle deals wi e problem of asympoic sabiliy for a class of linear differenial-algebraic sysem wi ime-varying delay e obained crieria depend no only on e upper bound bu also on e lower bound of e delay derivaive Based on e Lyapunov funcional meod and e delay pariioning approac some delay and is ime-derivaive dependen sable crieria are obained One numerical example is provided o demonsrae e effeciveness of e proposed resuls All e developed resuls are in e LMI framework wic makes em more ineresing since e soluions are easily obained using exising powerful ools like e LMI oolbox of Malab or any equivalen ool Noaion: rougou is paper A represens e ranspose of A; e symbol in marix inequaliy denoes e symmeric erm of e marix; A > ( < ) means A is a symmerical posiive (negaive) definie Sym M sands for M + M ; I is a uni marix marix; Problem Saemen Consider e following differenial-algebraic sysem: x R is e sae vecor e marix = + ( τ ) Ex Ax Ax () E R may be singular and we assume a τ is ime-varying delay were d d are consans e iniial con- were W is e space of absoluely coninuous n n n were rank ( E) = r n A and A are consan marices wi appropriae dimensions ( ) τ ( ) [ a b] a and i is assumed o saisfy d τ ( ) d diion is given by x( + θ) = φ( θ) θ [ b ] φ( θ) W n funcion φ : [ b ] R wi e square inegrable derivaive and wi e normal φ = φ( ) + φ( s) + φ( s) ds W b e following definiion lemmas and noaion are inroduced wic will be used in e proof of e main resuls Definiion ([6]) Sysem () is said o be regular if e caracerisic polynomial de ( se A) is no idenically zero ) Sysem () is said o be impulse-free if deg ( de ( se A) ) = rank ( E) Lemma ([4]) Le H F and G be real marices of appropriae dimensions en for any scalar ε > for all marices F saisfying F F I we ave: Main Resuls HFG + G F H εhh + ε G G d d P saisfying eorem Sysem () is asympoically sable for all differeniable delays τ ( ) [ a b] wi τ k if ere exis symmeric posiive-definie marices Q R ( ) S i i = S S S 5

3 H Liu Y C Ding k = ( k = ) and appropriaely dimensioned marices P j k E P P E ( ) P j Y j Y j Z j Z j j = j W and W suc a LMIs: < τ < for τ ( ) and τ ( ) and < τ > for τ ( ) and τ ( ) were τ ( ) = d d < τ < and < τ > are denoed in (8) and () respecively Proof e proof of is eorem is divided ino wo pars Firs we prove a e resuls wen τ ( ) < τ ( ) > and τ ( ) = respecively ensure a e derivaive of Lyapunov funcional is negaive Finally we prove e resuls given in e firs par ensure a sysem is regular and impulse-free Firs of all we divide e delay inerval [ a b] ino wo segmens: [ ] and [ ] were we denoe a + b = a = b and = en sysem () can be represened as Ex( ) = Ax( ) + χ[ ] Ax( ) [ ] ( ) τ + χ Ax τ were χ[ ] : R { } is e caracerisic funcion of [ ] if s [ ] χ[ ] ( s) () Consider e following Lyapunov funcional: were ( ) = ( τ ) V x x E P x V x = x s Qx s d s τ V x = x s Sx s s d = oerwise V x = V x + V x + V x + V x + V x () S S V4 ( x ) = ξ ( s) ξ( s) d s S 4 5 i ( ) = ( i+ i) ( ) i( ) V x Ex s R Ex s dd s θ 5 i= i + + θ ( ) S S = ξ s = x s x s Q > Ri > S > > S In addiion we define e coninuous funcion P( ( ) ) τ ( ) τ ( ) ( τ ) = χ[ ] ( τ ) + Wen τ as e following form: P P P τ one can obain ( ) τ ( ) τ + χ[ ] ( τ ( ) ) P + P ( P P ) ( χ)( P P ) χ P ( τ ( ) ) = τ ( ) + ( ) τ ( ) ( ) τ ( ) V τ ( x ) = x ( ) E P ( τ ( ) ) x( ) + x ( ) χ P + P τ + ( χ ) + P P Ex (4) 6

4 H Liu Y C Ding i V 5( x) = ( Ex ( ) ) ( i+ i) Ri( Ex ( ) ) ( i+ i) ( Ex ( s) ) Ri( Ex ( s) ) ds i + i= i= o obain e main resuls we consider e following ree cases: τ ( ) [ ) ( ) ( ] ( ) Case : τ ( ) [ ) j Using e fac a f ( s ) d s τ f j j j ( s ) d s d f s s = + j were f j ( s) ( Ex ( s) ) Ri( Ex ( s) ) τ = j+ j+ τ easily obain e following inequaliies by Jensen s inequaliy: i i i ( i+ i) i ( ) i ( ) f s ds Ex s dsr Ex s d s i+ i+ i+ ( ) ( ) + + j τ τ f s d s τ υ Rυ j j j j j j j j j ( ) ( + ) ( + )( + ) j+ ds υ j were υ j = ( Ex ( s) ) τ ( ) τ ( ) j derivaive of ( ) f s d s τ υ Rυ j j j j j j j j j j = V x along e soluion of () is given by τ ( ) Ex ( s) ds j j τ + + V x V x Ex R Ex τ < τ < i= ( ) = ( ) + ( ) ( i+ i) i ( ) ( τ ) ( τ ) ( τ ) x Qx x Qx + x Sx x Sx + ( ) ( ) x S S x + x S x x x x x x x E R E x x x x E R E x x τ υ υ S S ( ) S ( ) ( ) ( ) ( ) j j+ j jrj j τ υ υ ( + )( + ) R j j j j j j Leing η ( ) = x Ex x x υ υ x τ x following erms o (5) gives τ and = we i = j = en e ime- ( ) ( ) ( ) (5) and adding e = x Y + x E Y + x τ Ex + Ex τ + τ υ ( ) ( ) = x Z + x E Z + x W Ex Ex τ + τ υ ( τ ) (6) = x P + x E P Ax + Ax Ex for some scalar α > if V x η η α x (7) τ < τ < 7

5 H Liu Y C Ding were Π Π Π ZE Π5 Π6 Π7 Π YE ZE Π5 Π6 Π7 Π S E Π Π = τ < < Π55 Π57 Π Π Π Π ( ) τ τ Π = ( ) + ( ) EW ( ) E P P Sym P A τ + Sym P A + S E RE+ AP + P A Π = P + AP Π = E RE Y E ( τ ) ( τ ) Π = Y Π = Z 5 6 ( ) τ ( ) τ Π = PA+ PA+ Y E Z E+ P A 7 P P ( i+ i) Ri 5 ( τ ( ) ) Y i= Π = + Π = ( τ ) Π = Z Π = Y E Z E+ P A 6 7 Π = Q S + S E R E Π = S + S E R E 44 ( τ ) Π = S + E RE+ EW Π = R ( τ ) ( τ ) Π = Π = R ( τ ) ( τ ) Π = W Π = Q+ E+ E (8) Π = 88 S E RE Inequaliy (8) conains wo variables wic make i difficul o solve by LMI ool In order o overcome is difficuly we seek e sufficien condiions for inequaliy (8) Wen τ ( ) and τ ( ) e inequaliy (8) leads o e following LMIs: and Π Π Π ZE Z Π7 Π YE ZE Z Π7 Π S E = Π44 Π48 < ( ) R ( ) W Π77 EW Π88 (9) 8

6 H Liu Y C Ding were Πˆ Π Π ZE Y Πˆ 7 Π YE ZE Y Π7 Π S E 44 = Π Π48 < ( ) R ( ) Π77 EW Π 88 Π =Π Π =Π Π ˆ =Π Π ˆ =Π τ( ) = 7 7 τ( ) = τ( ) = 7 7 τ( ) = () Noe a we ave omied e zero row and zero column in and Leing ( ) ( ) ji η i = x Ex x x υ x τ x e laer wo LMIs imply (8) wic is because ( ) ( ) x τ τ η ( ) η ( ) + η η τ < = η η α is convex in ( ) [ ) us τ τ < τ = d i follows from (9) a Wen i e LMI () implies (9) since = < = i ( ) i τ = d () τ i d τ d + = < d d d d us τ d d Similarly we can obain a τ d d Case : τ ( ) ( ] By e definiion of e caracerisic funcion χ = we apply e above argumens and represenaions wi i = and j = In addiion we replace (6) wi e following equaions: is convex in [ ] ( τ ) ( ) ( τ ) ( τ ) υ = x Y + x E Y + x + x W Ex + Ex + ( ) ( ) is also convex in [ ] = x Z + x E Z Ex Ex τ + τ υ ( τ ) = + + x P x E P Ax Ax Ex and leing η ( ) ( ) = x Ex x x υ υ x τ x obain a for some scala α > were V x x η η α τ > τ > I is easy o 9

7 H Liu Y C Ding were Π Π E RE Y E Π 5 Π 6 Π 7 ZE Π YE Π 5 Π 6 Π 7 ZE Π Π 4 Π 5 W E Π 44 E S = < τ > Π 55 Π 57 Π 66 Π 77 S ( ) τ τ Π = ( ) + ( ) ( ) E P P P A Sym τ + Sym P A + S E RE+ AP + P A ( τ ) ( τ ) Π = P + AP Π = Y Π = Z 5 6 τ τ Π = P A P A Y E ZE P A P P ( i+ i) Ri 5 ( τ ( ) ) Y i= Π = + Π = ( τ ) Π = Z Π = Y E Z E+ P A 6 7 Π = Q S + S E R + R E Π = E RE W E+ S 4 ( τ ) Π = W Π = S + S E RE 5 44 ( τ ) ( τ ) Π = R Π = ( τ ) ( τ ) Π = R Π = Q+ E+ E () Similar o e case I we obain e resuls wen τ ( ) and ( ) < and < respecively Furer we can verify τ ( ) > is convex in ( ) ( ] are also convex in τ ( ) [ d d ] From Case and Case we ave [ ] ( ) [ ] ( ) τ wic can be marked as τ and χ τ η η + χ τ η η α τ τ < τ > () V x for some scalar α > τ = Case : { η η η η } α max τ τ τ (4) V x ( ) = ( ) < ( ) > erefore V ( x ) α x( ) wen τ ( ) [ ] τ ( ) [ d d ] a b Now e asympoic sabiliy of sysem () can no be obained ye since e exisence and uniqueness of a soluion o sysem () are no always guaraneed and e sysem may ave undesired impulsive beavior In e following we will prove a e above-menioned resuls ensure e regular and impulse-free I follows form (8) a

8 H Liu Y C Ding Pre- and pos-muliplying (5) by Π Π S < Π ( i+ i) R i i= I A and I A ( ) respecively we obain ( ) τ τ τ ( ) + Sym + Sym < E P P P A P A E RE Since ranke = r n ere mus exis wo inverible marices G and H n n suc a ˆ I E = GEH = r Similar o (7) we define i i ˆ A A ˆ i i P P A = GAH = P G P H i i i A A = = = P P4 i E P = P E i can be sown a P = Pre- and pos-muliplying (6) by k k en using H we can formulae e following inequaliy easily < Sym ( A ( P + P )) 4 4 were will be irrelevan o e following discussion Form (8) we ge A ( P P ) 4 4 (5) (6) (7) H and Sym + < and us A is nonsingular wic implies a de ( se A) is no idenically zero and deg ( de ( se A) ) = r = rank E Hence by Definiion e above-menioned resuls guaranee sysem () is regular and impulse-free is complees e proof Wen e marix E is nonsingular e resul in is case can be obained by seing E equal o I wi e appropriae ransformaions e corresponding resul is given by e following corollary: Corollary Sysem () wi E I τ a b wi k d τ ( ) d if ere exis symmeric posiive-definie marices Q Ri ( i = ) S S S S P ( k = ) and appropriaely dimensioned marices P j P j Y j Y j Z j Z j j ( j = ) W and W suc a LMIs: < τ < for τ ( ) and τ ( ) and < τ > for τ ( ) and τ ( ) were τ ( ) = d d = is asympoically sable for all differeniable delays [ ] < τ < and < τ > are denoed in (8) and () respecively k Wen d is unknown le P = P ( k = ) and τ ( ) = d we ave e following corollary Corollary Sysem () is asympoically sable for all differeniable delays τ ( ) [ a b] wi τ ( ) d if ere exis symmeric posiive-definie marices Q R i ( i = ) S S S S P saisfying E P = P E and appropriaely dimensioned marices P j P j Y j Y j Z j Z j j ( j = ) W and W suc a LMIs: < τ < for τ ( ) and τ ( ) and < τ > for τ ( ) and τ ( ) were τ ( ) = d < τ < and < τ > are denoed in (8) and () respecively Remark I sould be poined ou a if a is big enoug delay pariioning of [ a ] may improve e N > resuls In addiion e beer resuls may be obained if we divide e delay inerval [ ] ino N segmens 4 A Numerical Example In is secion a numerical example will be presened o sow e validiy of e main resuls derived above Example Consider e following linear differenial-algebraic sysem described by sysems () wi 5 5 E = A A = = a b (8)

9 H Liu Y C Ding For a = coosing d = and d = 9 and applying eorem e maximum values of b is 57 wic guaranees a e sysem is asympoically sable 5 Conclusion In is paper e asympoic sabiliy of differenial-algebraic sysem wi ime-varying delay as been invesigaed Some delay and is ime-derivaive dependen asympoically sable crieria ave been obained by decomposing ime-varying delay in a convex se e obained crieria depend no only on e upper bu also on e lower bound of e delay derivaive One numerical example as been given o illusrae e effeciveness of e proposed main resuls Acknowledgemens We ank e Edior and e referee for eir commens is work was suppored by A Projec Suppored by Scienific Researc Fund of Sicuan Provincial Educaion Deparmen (6ZA46) and e Docoral Researc Foundaion of Souwes Universiy of Science and ecnology (zx74) References [] Lin C Wang QG and Lee H (5) Robus Normalizaion and Sabilizaion of Uncerain Descripor Sysems wi Norm-Bounded Perurbaions IEEE ransacions on Auomaic Conrol p://dxdoiorg/9/ac [] Xu SY and Lam J (6) Robus Conrol and Filering of Singular Sysems Springer-Verlag Berlin [] Dai L (989) Singular Conrol Sysems Lecure Noes in Conrol and Informaion Sciences Springer-Verlag New York [4] Boukas EK (9) Delay-Dependen Robus Sabilizabiliy of Singular Linear Sysems wi Delays Socasic Analysis and Applicaions p://dxdoiorg/8/ [5] Cou JH Cen SH and Zang QL (6) Robus Conrollabiliy for Linear Uncerain Descripor Sysems Linear Algebra and is Applicaions p://dxdoiorg/6/jlaa55 [6] Haidar A and Boukas EK (9) Exponenial Sabiliy of Singular Sysems wi Muliple ime-varying Delays Auomaica p://dxdoiorg/6/jauomaica889 [7] Lu RQ Xu Y and Xue AK () H filering for Singular Sysems wi Communicaion Delays Signal Process p://dxdoiorg/6/jsigpro97 [8] Virnik E (8) Sabiliy Analysis of Posiive Descripor Sysems Linear Algebra and Is Applicaions p://dxdoiorg/6/jlaa8 [9] Wo SL Zou Y Cen QW and Xu SY (9) Non-Fragile Conroller Design for Discree Descripor Sysems Journal of e Franklin Insiue p://dxdoiorg/6/jjfranklin978 [] Long SH and Zong SM (6) H Conrol for a Class of Discree-ime Singular Sysems via Dynamic Feedback Conroller Applied Maemaics Leers 58-8 p://dxdoiorg/6/jaml64 [] Niamsup P and Pa VN (6) A New Resul on Finie-ime Conrol of Singular Linear ime-delay Sysems Applied Maemaics Leers 6-7 p://dxdoiorg/6/jaml65 [] Deng SJ Liao XF and Guo S (9) Asympoic Sabiliy Analysis of Cerain Neural Differenial Equaions: A Descripor Sysem Approac Maemaics and Compuers in Simulaion p://dxdoiorg/6/jmacom95 [] Qian W Liu J Sun YX and Fei SM () A Less Conservaive Robus Sabiliy Crieria for Uncerain Neural Sysems wi Mixed Delays Maemaics and Compuers in Simulaion p://dxdoiorg/6/jmacom97 [4] Xu SY Lam J and Yang C () Robus H Conrol for Discree Singular Sysems wi Sae Delay and Parameer Uncerainy Dynamics of Coninuous Discree and Impulsive Sysems [5] Yang CY Zang QL and Zou LN (7) Srongly Absolue Sabiliy of Lur e ype Differenial-Algebraic Sysems Journal of Maemaical Analysis and Applicaions p://dxdoiorg/6/jjmaa767 [6] Xu SY Van Dooren P Sefan R and Lam J () Robus Sabiliy and Sabilizaion for Singular Sysems wi Sae Delay and Parameer Uncerainy IEEE ransacions on Auomaic Conrol 47-8 p://dxdoiorg/9/ac865

10 H Liu Y C Ding [7] Fridman E () Sabiliy of Linear Descripor Sysems wi Delay: A Lyapunov Based Approac Journal of Maemaical Analysis and Applicaions p://dxdoiorg/6/s-47x()- [8] Park P and Ko JW (7) Sabiliy and Robus Sabiliy for Sysems wi a ime-varying Delay Auomaica p://dxdoiorg/6/jauomaica7 [9] Han QL (9) A Discree Delay Decomposiion Approac o Sabiliy of Linear Rearded and Neural Sysems Auomaica p://dxdoiorg/6/jauomaica885 [] Fridman E Saked U and Liu K (9) New Condiions for Delay-Derivaive-Dependen Sabiliy Auomaica p://dxdoiorg/6/jauomaica98 Submi your manuscrip a: p://papersubmissionscirporg/