FINITE-TIME STABILITY AND STABILIZATION OF LINEAR TIME-DELAY SYSTEMS UDC

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1 FACA UNIVERSIAIS Series: Auomaic Conrol and Roboics Vol., N o, 22, pp FINIE-IME SABILIY AND SABILIZAION OF LINEAR IME-DELAY SYSEMS UDC Sreen B. Sojanović, Draguin Lj. Debeljković 2, Dragan S. Anić 3 Universiy of Niš, Faculy of echnology, Deparmen of Mahemaical-echnical Sciences, Leskovac, Serbia 2 Universiy of Belgrade, Faculy of Mechanical Engineering, Deparmen of Conrol Engineering, Serbia 3 Universiy of Niš, Faculy of Elecronic Engineering, Deparmen of Conrol Sysems, Serbia Absrac. In his paper, finie-ime sabiliy and sabilizaion problems for a class of linear ime-delay sysems are sudied. Firsly, he conceps of finie-ime sabiliy and finie-ime sabilizaion are exended o linear ime-delay sysems. hen, based on mehodology of linear marix inequaliies and he Lyapunov-like funcions mehod, some sufficien condiions under which he linear ime-delay sysems are finie-ime sable are given. Moreover, o solve he finie-ime sabilizaion problem, sabilizing saic conroller is designed. Finally, wo examples are employed o verify he efficiency of he proposed mehods and o show ha he obained resuls are less conservaive han some exising ones in he lieraure. Key words: linear sysems, ime-delay, LMIs, finie-ime sabiliy, sabilizing conroller. INRODUCION In he las years, a big effor has been made o sudy he sabiliy and sabilizaion problem for linear sysems. he work of conrol scieniss and engineers has mainly focused on Lyapunov sabiliy defined over an infinie-ime inerval. Ofen Lyapunov asympoic sabiliy is no enough for pracical applicaions, because here are some cases where large values of he sae are no accepable, for insance in he presence of sauraions. In hese cases, we need o check ha hese unaccepable values are no aained by he sae. For his purposes, he concep of he finie-ime sabiliy (FS) could be used. A Received May 28, 22 Corresponding auhor: Sreen B. Sojanović Universiy of Niš, Faculy of echnology, Deparmen of Mahemaical-echnical Sciences, Bulevar oslobođenja 24, Leskovac, Serbia ssreen@p.rs Acknowledgmens: Pars of his research were suppored by he Minisry of Sciences and echnology of Republic of Serbia hrough Mahemaical Insiue SANU Belgrade and Faculy of echnology Universiy of Nis Gran ON47.

2 26 S. B. SOJANOVIĆ, D. LJ. DEBELJKOVIĆ, D. S. ANIĆ sysem is said o be FS if, once a ime inerval is fixed, is sae does no exceed some bounds during his ime inerval. he finie-ime conrol problem concerns he design of a linear conroller which ensures he FS of he closed loop sysem. Some early resuls on FS problems dae back o he sixies of he 2 h cenury [-3]. Since hen, however, due o he lack of operaive es condiions for FS, he researchers ineres has moved oward he classical Lyapunov sabiliy. Recenly, he concep of FS has been revisied in he ligh of linear marix inequaliy heory, which has allowed finding less conservaive condiions guaraneeing FS and finie-ime sabilizaion linear coninuous ime sysems. Many valuable resuls have been obained for his ype of sabiliy; see, for insance [2, 4-6]. An approach based on Lyapunov differenial marix equaions for finie-ime sabilizaion via sae feedback is proposed in [7], whereas he use of Lyapunov differenial inequaliies in his conex had been inroduced in [9]. he discree-ime case is deal wih in he paper [8], where sufficien condiions for finieime sabilizaion via sae feedback, expressed in erms of LMIs, are given. More recenly, oher conribuions on FS have been given in [9] and [6] for he case of impulsive sysems. ime delays ofen occur in many indusrial sysems such as chemical process, biological sysems, populaion dynamics, neural neworks, large-scale sysems, and so on. I has been shown ha he exisence of delay is he sources of insabiliy and poor performance of conrol sysems. Considering he wide applicaion of ime-delay sysems and he requiremen for ransien behaviour in engineering fields, i moivaes us o invesigae finie-ime sabiliy and finie-ime sabilizaion for a class of linear ime-delay sysems. o he bes of he auhors knowledge, a lile work has been done for he finie-ime sabiliy and sabilizaion of ime-delay sysems. Some early resuls on finie ime sabiliy of imedelay sysems can be found in [2-26]. In [23] and [26] some basic resuls from he area of finie ime sabiliy were exended o he paricular class of linear coninuous ime delay sysems using fundamenal sysem marix. However, hese resuls are no pracically applicable, since i requires deermining he fundamenal sysem marix. Marix measure approach has been applied, for he firs ime, in [2-22, 24] for he analysis of finie ime sabiliy of linear ime delayed sysems. Anoher approach, based on very well-known Bellman-Gronwall Lemma, was applied in [2, 25]. Finally, modified Bellman-Gronwall principle, has been exended o he paricular class of coninuous non-auonomous ime delayed sysems operaing over he finie ime inerval [2]. he above mehods give conservaive resuls because hey use boundedness proprieies of he sysem response, i.e. of he soluion of sysem models. Recenly, based on linear marix inequaliy heory, some resuls have been obained for FS and finie-ime boundedness (FB) for paricular classes of ime-delay sysems [27-34]. However, according o he auhors knowledge, here is no resul available ye on finie-ime sabiliy and sabilizaion for a class of linear ime-delay sysems using linear marix inequaliy. he papers [28, 32-34] consider he problem of finie-ime boundedness (FB) of he delayed neural neworks. In [29-3] finie-ime boundedness of swiched linear sysems wih ime-varying delay and exogenous disurbances are sudded. Papers [27, 3] invesigae he finie-ime conrol problem for neworked conrol sysems wih ime-varying delay. In [3] a paricular linear ransformaion is inroduced o conver he original ime-delay sysem ino a delay-free form. Finie-ime sabiliy and sabilizaion of rearded-ype nonlinear funcional differenial equaions are developed in [3].

3 Finie-ime Sabiliy and Sabilizaion of Linear ime-delay Sysems 27 Saring from he resuls presened in he paper [25], we exend furher he resuls of finie-ime sabiliy and sabilizaion of linear ime-delay sysems via Lyapunov mehod using marix inequaliy. Some new sufficien condiions are obained for finie-ime sabiliy and sabilizaion via sae feedback in he form of marix inequaliy, which can be reduced o LMI feasibiliy problem. Finally, some examples are developed o illusrae he main resul provided in his paper and o show ha he obained resuls are less conservaive han some exising ones in he lieraure. 2. PRELIMINARIES AND PROBLEM FORMULAION he following noaions will be used hroughou he paper. Superscrip "" sands for marix ransposiion. n denoes he n-dimensional Euclidean space and nm is he se of all real marices of dimension nm. X > means ha X is real symmeric and posiive definie and X > Y means ha he marix X Y is posiive definie. In symmeric block marices or long marix expressions, we use an aserisk (*)o represen a erm ha is induced by symmery. diag{...} sands for a block-diagonal marix. Marices, if heir dimensions are no explicily saed, are assumed o be compaible for algebraic operaions. Consider he following linear sysem wih ime delay: x () Ax () Ax ( ) Bu () () x () (), [,] n m nn nn where x () is he sae vecor, u () is he conrol inpu, A, A and B nm are known consan marices, is consan ime delay. he iniial condiion, (), is a coninuous and differeniable vecor-valued funcion of [,]. In his paper we are ineresed in he design of a sabilizing saic conroller of he following form: u () Kx () (2) where K is design parameer ha has o be deermined. Plugging he conroller expression (2) in () we ge he following closed-loop dynamic: x () Ax ˆ () Ax ( ) (3) x () (), [,] where  A BK (4) his paper sudies he finie-ime sabiliy and sabilizaion of he class of sysems (). Our aim is o develop a sabilizaion mehod which provides a conrol gain K as well as an upper bound M of he delay such ha he closed-loop sysem is finie-ime sable for any saisfying < < M. Before moving on, he following definiion of finie-ime sabiliy for he ime-delay sysem () is inroduced.

4 28 S. B. SOJANOVIĆ, D. LJ. DEBELJKOVIĆ, D. S. ANIĆ Definiion. Sysem () wih u() is said o be finie-ime sabiliy (FS) wih respec o (c,c 2,), where c c 2, if sup ( ) ( ) c x ( ) x( ) c, [, ] (5) [,] 2 Remark. Lyapunov asympoic sabiliy (LAS) and FS are independen conceps: a sysem which is FS may be no asympoically sable; conversely a LAS sysem could be no FS if, during he ransiens, is sae exceeds he prescribed bounds. 3. MAIN RESULS In his secion we shall develop sufficien condiions for FS sabiliy and sabilizaion, and design sabilizing sae feedback conrollers for he ime-delay sysem (). heorem. Sysem () wih u() = and ime-delay is finie-ime sable wih respec o (c,c 2,), c < c 2, if exis a nonnegaive scalar and posiive define symmeric marices P and Q such ha he following condiions hold and AP PA Q P PA (6) Q c ( P) min max max 2 ( P) ( Q) e c (7) Proof. Le us consider he following Lyapunov-like funcion V x() x () Px() x ( ) Qx( ) d (8) hen, he ime derivaive of Vx() along he soluion of () gives V x ( ) 2 x ( Px ) ( ) x ( Qx ) ( ) x ( ) Qx ( ) x () A PPA Q x() 2 x () PAx( ) x ( ) Qx( ) () () where A PPA Q PA () x() x( ), Q From (6) and (9), we have (9)

5 Finie-ime Sabiliy and Sabilizaion of Linear ime-delay Sysems 29 Muliplying () by P V x() () () () () P () () () () P () () x() Px() x () Px() x ( ) Qx( ) d V x() e, we can obain () d ( e V ) () d Inegraing () from o, wih [, ], we have hen V( x( )) e V( x()) (2) V x() x () Px() x ( ) Qx( ) d max max max max max ( Px ) () x() max ( Q) x( ) x( ) d (3) ( P) c ( Q) c c ( P) ( Q) On he oher hand, V( x()) x () Px() min ( P) x () x() (4) Combining (2), (3) and (4) leads o x () x() cmax( P) max( Q) e (5) min ( P) Condiion (7) and he above inequaliy imply x () x() c2, for all [, ] (6) he proof is compleed. Remark 2. If condiions (6) and (7) in heorem is saisfied wih =, hen sysem () is also asympoically sable in he sense of Lyapunov. In his case, he finie-ime sabiliy is guaraneed for all. Remark 3. From (7), for upper bound M of he delay such ha he closed-loop sysem is finie-ime sable for any saisfying < < M and fixed, c and c 2 we obain:

6 3 S. B. SOJANOVIĆ, D. LJ. DEBELJKOVIĆ, D. S. ANIĆ c where marices P and Q are defined by (6). ( P) e ( P) 2 min max M (7) cmax( Q) max( Q) Remark 4. I should be poined ou ha he condiion in heorem is no sandard LMIs condiion wih respecive o, P and Q. However, i is easy o check ha condiion (7) is guaraneed by imposing he condiions I P 2I Q 3I (8) ce 2 2 c 3 c 2 3 for some posiive scalars, 2 and 3. Once we fix, condiions (6) and (7) can be urned ino LMIs based feasibiliy problem. Corollary. (LMIs based feasibiliy problem) Sysem () wih u() = and ime-delay is finie-ime sable wih respec o (c, c 2, ), c < c 2, if for fixed exis a posiive scalars, 2 and 3 and posiive define symmeric marices P and Q such ha he condiions (6) and (8) hold. Based on he above resuls, he saic sae feedback conroller can be designed. heorem 2. here exiss a sae feedback conroller of he form (2) such ha he closed-loop sysem (3)-(4) is finie-ime sable if here exis a nonnegaive scalar and posiive-define symmeric marices X and Y and marix Z such ha he following condiions hold XA AX BZ Z B Y X AX (9) Y and c max( X) e c 2 min ( X) min ( XY X) he sabilizing saic conroller gain is given by K ZX. Proof. From (6) and (4) we have ( A BK ) P P( A BK) Q P PA Q Pre and pos muliplying he above inequaliy by diag{ P, P } and le Y P QP resuls in (2) (2) X P and XA AX XK B BKX XQX X AX XQX (22)

7 Finie-ime Sabiliy and Sabilizaion of Linear ime-delay Sysems 3 Le K ZX, hen he above inequaliy is equivalen o (9). From (7), for Q P Y P XY X we have (2). P X and Remark 5. If condiions (9) and (2) in heorem 2 is saisfied wih =, hen closeloop sysem (3)-(4) is also asympoically sable in he sense of Lyapunov on he infinie inerval [, ]. In his case, he finie-ime sabiliy is guaraneed for all >. Remark 6. From (2), for upper bound M of he delay such ha he closed-loop sysem (3)-(4) is finie-ime sable for any saisfying < < M we obain: c2min( XY X) min( XY X) M e (23) c ( X) ( X) max min where marices X and Y are defined by (22). Remark 7. According o our knowledge, here is no resul available ye on finie-ime sabiliy and sabilizaion in he sense of Definiion for a class of linear ime-delay sysems which use linear marix inequaliy. 4. ILLUSRAIVE NUMERICAL EXAMPLES AND SIMULAIONS Example. Consider a linear coninuous ime-delay sysem as follows: x () Ax () Ax ( ) Bu () A.3.7, A.3.3, B, One should invesigae finie-ime sabiliy wih respec o ( c, c2, ), wih paricular choice of: c.55, c, 2, ( ).7, u( ) 2 Solving LMIs (6) and (8) (Corollary ) for fixed.95, we can obain following feasible soluions (24) P , Q , b 2.286, b , b herefore, he unforced sysem (24) is finie-ime sable wih respec o (.55,, 2). Fig. shows ime hisories of sae rajecories of he sysem (24) wih he iniial condiion () = [.7 ], [,]. I is observed ha he values of sae variables x i, i =,2,3 when, which proves ha he sysem is no asympoically sable. hus, we have shown ha Lyapunov asympoic sabiliy and FS are independen conceps: a sysem which is FS may be no asympoically sable. ime dependen norm

8 32 S. B. SOJANOVIĆ, D. LJ. DEBELJKOVIĆ, D. S. ANIĆ of he sae rajecories is illusraed on Fig. 2. However, using condiions (6)-(7) (heorem ) i can be concluded ha he sysem (24) is no FS wih respec o (.55,, 2), bu i is FS wih respec o (.55, 3, 2) wih =.8. hen, he upper bound of parameer c 2 amouns hus, heorem provides a more conservaive condiion hen Corollary. he upper bound of he delay M, such ha he sysem (24) is finie-ime sable wih respec o (.55, 3, 2), has he following value: M =.2 for =.79 (from heorem ) or M =.56 for = 2.8 (from Corollary ). he dependences of he parameers c 2 () and () for c =.55 and = 2, obained by Corollary, are shown in Fig. 3 and 4, respecively. From he figures we can see ha he smalles value of c 2 for =.2 and he larges value of for c 2 = 5 can be obained by adoping = rajecories x x 2 x 3 x () x() ime (s) Fig.. he sae rajecories of he unforced sysem (24) wih iniial condiion () = [.7 ] ime (s) Fig. 2. he norm of he sae rajecories of he unforced sysems (24). he smalles value of he parameer c 2 obained for = 2. and differen ime-delay from heorem and Corollary are shown in able. For comparison, he able also lis he smalles value of he parameer c 2 obained in [25] (heorem 2 and 3). I is clear ha heorem and Corollary in our paper give much beer resuls han hose obained in [25] because [25] does no use LMI echnique. able. he smalles value of he parameer c 2 for c =.55, =2 and = heorem Corollary [25, heorem 2] [25, heorem 3]

9 Finie-ime Sabiliy and Sabilizaion of Linear ime-delay Sysems c Fig. 3. he dependence of he parameers c 2 () for c =.55, = 2 and =.2 obained by Corollary Fig. 4. he dependence of he parameers () for c =.55, = 2 and c 2 = 5 obained by Corollary Example 2. Consider he linear coninuous ime-delay sysem (24). We shall design he sae feedback FS conroller by heorem 2 which make ha he sysem (24) is FS wih respec o (c,c 2,) = (.55,5,). Solving LMIs (9) and he inequaliy (2) for =. 5, we can obain he following feasible soluions for he closed-loop sysem X , Y , K ZX Z, c ( X) e c max 2 min ( X ) min ( XY X ) herefore, here exiss he consan sae feedback u () Kx () ZX x () such ha he closed-loop sysem is finie-ime sable wih respec o (.55,5,). Fig. 5 and 6 show he sae rajecories of he closed-loop sysem, as well as is norm. I can be concluded ha he closed-loop sysem is asympoically sable ( x i,, ). o he same conclusion can be reached in he following way. By seing =, we find ha he closed-loop sysem is also asympoically sable in he sense of Lyapunov because max { = } =.7 <. Furher, from Fig. 6 we have x ()x() c =.55 < c 2, [,], so he closed-loop sysem is finie-ime sable >. he upper bound of he ime-delay M, such ha he closed-loop ime-delay sysem is FS wih respec o (.55,5,) and =. 5 amouns M =.363. However, for he values (c,c 2,) = (.55,3,2) from Example, he upper bound of he ime-delay has significanly higher value M =

10 34 S. B. SOJANOVIĆ, D. LJ. DEBELJKOVIĆ, D. S. ANIĆ rajecories x x 2 x 3 x () x() ime (s) Fig. 5. he sae rajecories of he closedloop sysem (24) wih iniial condiion () = [.7 ] ime (s) Fig. 6. he norm of he sae rajecories of he closed-loop sysem (24). 5. CONCLUSION In his paper, finie-ime sabiliy and sabilizaion problems have been invesigaed for a class of linear ime-delay sysems. As he main conribuion of his paper, sufficien condiions which can guaranee finie-ime sabiliy of linear ime-delay sysems are proposed. Saring from hese resuls, we have provided sufficien condiion for he soluion of he saic sae feedback problem. he proposed condiions are expressed in erms of marix inequaliies, and our resuls are less conservaive han exising resuls. REFERENCES. H.D. Angelo, "Linear ime-varying sysems: analysis and synhesis",allyn and Bacon, Boson, P. Dorao, "Shor ime sabiliy in linear ime-varying sysem", Proc. IRE Inerna. Conv. Rec. Par 4, New York 96, pp L. Weiss and E.F. Infane, "Finie-ime sabiliy under perurbing forces and on produc spaces", IEEE rans. Auom. Conrol, vol. 2, pp , F. Aamo and M. Ariola, "Finie-ime conrol of discree-ime linear sysem", IEEE rans. Auom. Conrol, vol. 5 (5), pp , F. Amao, M. Ariola and C. Cosenino, "Finie-ime conrol of discree-ime linear sysems: analysis and design condiions", Auomaica, vol. 46, pp , F. Amao, M. Ariola and P. Dorao, "Finie-ime sabilizaion via dynamic oupu feedback", Auomaica, vol. 42, pp , F. Amao, M. Ariola and P. Dorao, "Finie-ime conrol of linear sysems subjec o parameric uncerainies and disurbances", Auomaica, vol. 37 (9), pp , F. Amao, M. Ariola and P. Dorao, "Sae feedback sabilizaion over a finie-ime inerval of linear sysems subjec o norm bounded parameric uncerainies", Proc. of he 36h Alleron Conference, Monicello, Sep , F. Amao, M. Ariola, C. Cosenino, C.. Abdallah and P. Dorao, "Necessary and sufficien condiions for finie-ime sabiliy of linear sysems", Proc. of American Conrol Conference, Denver, Colorado, June 23, pp J. Feng, Z. Wu and J. Sun, "Finie-ime conrol of linear singular sysems wih parameric uncerainies and disurbances", Aca Auom. Sin., vol. 3 (4), pp , 25.

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12 36 S. B. SOJANOVIĆ, D. LJ. DEBELJKOVIĆ, D. S. ANIĆ SABILNOS I SABILIZACIJA NA KONAČNOM VREMENSKOM INERVALU LINEARNIH SISEMA SA VREMENSKIM KAŠNJENJEM Sreen B. Sojanović, Draguin LJ. Debeljković, Dragan S. Anić U radu su razmarani problemi sabilnosi i sabilizacije na konačnom vremenskom inervalu za klasu linearnih sisema sa vremenskim kašnjenjem. Najpre su koncepi sabilnosi i sabilizacije na konačnom vremenskom inervalu prošireni na klasu sisema sa kašnjenjem. Koriseći meodologiju linearnih maričnih nejednakosi i meodu zasnovanu na funkcionalima sličnim Ljapunovim funkcionalima, izvedeni su dovoljni uslovi koji garanuju sabilnos na konačnom vremenskom inervalu. U cilju rešavanja sabilizacije na konačnom vremenskom inervalu, projekovan je sabilizirajući saički konroler. Na kraju, daa su dva primera da bi se proverila efikasnos predloženih meoda i pokazalo da su dobijeni rezulai manje konzervaivni od rezulaa iz posojeće lieraure. Ključne reči: linearni sisemi, vremensko kašnjenje, LMI, sabilnos na konačnom vremenskom inervalu, sabilizirajući konroler