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2 1 Exponential Stability of Integral Delay Systems wit a Class of Analytic Kernels Sabine Mondié and Daniel Melcor-Aguilar Abstract Te exponential stability of a class of integral delay systems wit analytic kernels is investigated by using te Lyapunov-Krasovskii functional approac. Sufcient delaydependent stability conditions and exponential estimates for te solutions are derived. Special attention is paid to te particular cases of polynomial and exponential kernels. Index Terms Integral delay system, exponential stability, Lyapunov-Krasovskii functionals. I. INTRODUCTION Integral delay systems play an important role in several stability problems of time-delay systems. Tis class of systems is found as delay approximations of te partial differential equations for describing te propagation penomena in excitable media [18], in te stability analysis of additional dynamics introduced by some system transformations [3], [5], [6], [7], in te internal stability problem of controllers used for nite spectrum assignment of time-delay systems [9], as well as in te stability analysis of some difference operators in neutral type functional differential equations [4], [8]. Lyapunov-Krasovskii teorems for integral delay systems ave been recently introduced in [12]. It as been sown tere tat new type of Lyapunov-Krasovskii conditions are required in order to properly address te dynamics of tis class of systems. General expressions of quadratic functionals wit a given time derivative are provided. Te functionals are sown to be useful in solutions of suc problems as te estimations of robustness bounds and calculations of exponential estimates for te solutions of exponentially stable integral delay systems. However, tere are still some tecnical difculties, associated wit te positivity ceck of suc functionals, limiting teir practical application to te stability analysis of integral delay systems. Tis motivated te work [13] were it was sown tat, based on te general expressions of functionals presented in [12], various reduced type functionals can be constructed to obtain stability conditions formulated directly in terms of te coefcients of some classes of integral delay systems. In te present paper, we address te exponential stability of a special class of integral delay systems wit analytic kernels. For suc integral systems, we derive sufcient delaydependent stability conditions, expressed in terms of linear matrix inequalities, by using te Lyapunov-Krasovskii functional approac. Tis work was supported by CONACYT under Grants and S. Mondié is wit te Department of Automatic Control, CINVESTAV-IPN, A.P , 73, México, DF, México. smondie@ctrl.cinvestav.mx D. Melcor-Aguilar is wit te Division of Applied Matematics, IPICYT, 78216, San Luis Potosí, SLP, México. dmelcor@ipicyt.edu.mx Te special class of integral delay systems under consideration includes tose wit polynomial and exponential kernels, wic found important applications in te internal stability problem of controllers used for te nite spectrum assignment of input time-delay systems, a topic tat as received a sustained attention in te past few years, see for instance [2], [14], [15], [16], [21], [22], and tat was proposed as an interesting open problem in te survey paper [2]. We ere address suc a problem from a Lyapunov-Krasovskii framework and develop a linear matrix inequalities formulation for it. More explicitly, a design metod guaranteeing a numerically safe implementation of te controllers and a prescribed decay rate of te closed-loop system solutions is proposed. We believe tat suc an approac to te problem is completely new. Te paper is structured as follows In section II, preliminaries concerning te class of integral delay systems to be considered, basic facts about te solutions and Lyapunov-Krasovskii stability conditions are introduced. Te main results are given in section III. First, we consider integral delay systems wit a general class of analytic kernels. Ten, te special cases of integral delay systems wit polynomial and exponential kernels are addressed. In all cases, exponential estimates and delay-dependent conditions for te exponential stability are expressed in terms of linear matrix inequalities. In section IV, we apply te obtained results to te internal stability problem of controllers used for te nite spectrum assignment of timedelay systems. Several concluding remarks end te paper. Notation Trougout te paper we will use te Euclidean norm for vectors and te induced norm for matrices, bot denoted by kk. We denote by A T te transpose of A, I n and n stand for te identity and zero matrices of nn dimension, wile min (A) and max (A) denote te smallest and largest eigenvalues of a symmetric matrix A, respectively. II. PRELIMINARIES In tis contribution we focus on te exponential stability of te class of integral delay systems wit analytic kernels described by were and x(t) = G T B()x(t + )d; 8t ; (1) G T = G T G T 1 G T N B() T = B () T B 1 () T B N () T ;

3 2 wit G 2 R n(n+1)m and B() 2 R n(n+1)m a continuously differentiable matrix function dened for 2 [ ; ] wic satises te following assumptions Tere exists a matrix M 2 R n(n+1)n(n+1) suc tat _ B() = MB(); 8 2 [ ; ] (2) Tere exists > suc tat for all 2 [ ; ] < min B T ()B() (3) Remark 1 Classes of kernels tat satisfy te assumptions (2) and (3) include tose written in a polynomial basis as well as transcendental kernels suc as exponential ones. Remark 2 It is important to point out tat te special class of functional difference equations (1), wit analytic kernels satisfying (2) and (3), belongs to te class of retarded type delay system, see [12] for discussions and te section below concerning solutions and stability concept. For te case of functional difference equations of te neutral type, te approac developed in [12] cannot be directly applied and Lyapunov stability conditions as tose proposed in [19] are more appropriate. A. Solutions and stability concept In order to dene a particular solution of (1) an initial vector function '(); 2 [ ; ) sould be given. We assume tat ' belongs to te space of piecewise continuous bounded functions PC([ ; ); R m ), equipped wit te uniform norm k'k = sup 2[ ;) k'()k. Given any initial function ' 2 PC([ ; ); R m ); tere exists a unique solution x(t; ') of (1) wic is dened for all t 2 [ ; 1) wit x(t; ') = '(t); t 2 [ ; ) Tis solution is continuous and differentiable for all t 2 (; 1) ; and it suffers a jump discontinuity given by x(; ') = x(; ') x( ; ') = G T B()'()d '( ) Denition 3 [4] System (1) is said to be exponentially stable if tere exist > and > suc tat every solution of (1) satises te inequality kx(t; ')k k'k e t ; 8t Remark 4 Notice tat differentiation of te integral equation (1) is not an option for investigating its exponential stability since te resulting differential system wit distributed delay _x(t) = G T B()x(t) G T B( )x(t ) G T _ B()x(t + )d; is not exponentially stable as it admits any constant vector as a solution, see [12] for a detailed analysis. For any t we denote te restriction of te solution x(t; ') on te interval [t ; t) by x t (') = fx(t + ; '); 2 [ ; )g. Wen te initial function is irrelevant, we simply write x(t) and x t instead of x(t; ') and x t (') B. Lyapunov stability conditions Te observation in Remark 4 motivated te autors of te work [12] to derive new Lyapunov stability conditions for integral delay systems of te form in (1). In particular, it was tere sown tat integral type quadratic functions are te natural lower bounds for te Lyapunov functionals in contrast wit quadratic lower bounds of Lyapunov functionals for differential delay systems. We ere present a new formulation of te general result on Lyapunov type conditions for te exponential stability of integral delay systems given in [12]. Tis formulation, in te spirit of te one introduced in [17] for retarded differential delay systems, enances te role of te exponential decay rate. Following [12], by noting tat for t 2 [; ), x t (') belongs to PC([ ; ); R m ), and tat for t ; x t (') belongs to te space of continuous vector functions C([ ; ); R m ), it ten follows tat, in a Lyapunov-Krasovskii functional framework, te functionals sould be dened on te innite-dimensional space PC([ ; ); R m ) Teorem 5 [12] Let tere exists a functional v PC([ ; ); R m )! R suc tat 1 k'()k 2 d v (') 2 k'()k 2 d; (4) for some < 1 2 Given > ; if along solutions of (1) te inequality d dt v(x t (')) + 2v(x t (')) ; 8t ; (5) olds, ten were kx(t; ')k k'k e t ; 8t ; (6) = max 2[ ;] G T B() r 2 (7) 1 Based on tis fundamental result, te following sufcient stability condition is obtained in [13] by using a particular Lyapunov functional satisfying te Teorem conditions. Lemma 6 [13] System (1) is exponentially stable if max G T B() < 1 (8) 2[ ;] Tis inequality is valid for integral delay systems wit more general kernels tan tose satisfying te assumptions (2) and (3). Indeed, te only assumption needed on te matrix function B() is continuity on te interval [ ; ] However, inequality (8) may be very conservative for kernels satisfying te assumptions (2) and (3) as it as been illustrated in [13] for te case of constant kernels. It is wort mentioning tat (8) can also be obtained via frequency-domain tecniques, see [1] and [11]. III. MAIN RESULTS In tis section, sufcient delay-dependent conditions for te exponential stability of integral delay systems of te form of (1), wit kernels satisfying (2) and (3), are obtained by presenting a particular functional satisfying te conditions of Teorem 5.

4 3 Consider te functional candidate v(') = ' T ()B T ()e 2 [P + ( + ) Q] B()'()d; (9) were P; Q 2 R n(n+1)n(n+1) are positive denite matrices and is a positive constant. From (9) it follows tat v(') max (P + Q) and max (P + Q) v(') e 2 min (P ) e 2 min (P ) ' T ()B T ()B()'()d max (B T ()B()) k'()k 2 d; ' T ()B T ()B()'()d min (B T ()B()) k'()k 2 d; ence te functional (9) satises te following inequalities 1 k'()k 2 d v(') 2 k'()k 2 d; wit 1 = e 2 min (P ) min min(b T ()B()) ; (1) 2[ ;] 2 = max (P + Q) max max(b T ()B()) (11) 2[ ;] Te time derivative of functional (9) along te trajectories of system (1) is suc tat Z d T dt v(x t) + 2v(x t ) = G T B()x(t + )d B T () [P + Q] B() G T B()x(t + )d x T (t + )B T ()e 2 fq + [P + ( + ) Q] M) +M T [P + ( + ) Q] B()x(t + )d x T (t )e 2 B T ( )P B( )x(t ) By using te Jensen integral inequality we ave T G T B()x(t + )d B T () [P + Q] B() G T B()x(t + )d x T (t + )B() T GB T () [P + Q] B()G T B()x(t + ) d We terefore get te following inequality for te derivative d dt v(x t) + 2v(x t ) e 2 x T (t + )B T () ()B()x(t + )d; were () = Q + M T [P + ( + ) Q] + [P + ( + ) Q] M e 2 GB T () [P + Q] B()G T Tus, if () > ; 8 2 [ ; ] ; ten te exponential stability of (1) follows. Observing tat ( + ) ( ) + () = Q + M T [P + ( + ) Q] + [P + ( + ) Q] M ( + ) e2 + GB T () [P + Q] B()G T ; and taking into account te convexity of te exponential function ( + ) e2 + e 2 ; 8 2 [ ; ] ; one as tat () > ; 8 2 [ ; ] ; if and only if () > and ( ) > Summarizing te above, we can state te following result. Teorem 7 An integral delay system (1), wit a kernel satisfying (2) and (3), is exponentially stable wit a decay rate if tere exist positive denite matrices P; Q 2 R n(n+1)n(n+1) suc tat and Q + M T [P + Q] + [P + Q] M GB T () [P + Q] B()G T > ; (12) Q + M T P + P M e 2 GB T () [P + Q] B()G T > ; (13) Furtermore, for any initial condition ' 2 PC([ ; ); R m ); te solution x(t; ') of (1) satises te exponential upper bound (6), were ; 1 and 2 are determined by (7), (1) and (11), respectively. A. Integral delay systems wit polynomial and exponential kernels For n = m and matrix functions B() of te form B() T = I n I n I n N ; (14) we get te following result. Corollary 8 System (1) wit a matrix function B() of te form (14) is exponentially stable wit a decay rate if tere exists positive denite matrices P; Q 2 R n(n+1)n(n+1) suc tat (12) and (13) olds, were te matrix M is given by 2 3 n n n I n n n M =. n 2I.. n n n n NI n n Note tat in tis particular case of polynomial kernels te positive constants 1 and 2 ; respectively dened by (1) and (11), take te form 1 = e 2 min (P ); 2 = ( N ) max (P + Q)

5 4 TABLE I Maximum delay value for different decays Corollary Norm condition (8) 988 For m = n and kernels of te form G T B() = G T e A ; te following result is obtained. Corollary 9 System (1) wit a matrix function B() of te form (14) is exponentially stable if tere exist positive denite matrices P; Q 2 R nn suc tat (12) and (13) old wit M = A Remark 1 In te case of integral delay systems wit constant kernels G T B() = G T ; sufcient conditions for te exponential stability can be derived eiter from Corollary 8 (N = ) or Corollary 9 (A is te null matrix). In tis case, te conditions of Teorem 7 reduce to Q e 2 G [P + Q] G T > For =, tese conditions are te same as tose obtained in [13]. Now we illustrate te main results by some examples. Example 11 Consider te integral delay system were x(t) = A = e A x(t + )d; (15) For = ; by using Corollary 9, we found a feasible solution of te matrix inequalities (12)-(13) for delay values 199 For instance, for = 199 we obtain te following numerical values for matrices P and Q P =, Q = In Table I, we present te maximum delay value computed by using Corollary 9, for different Te results are less conservative tan te norm condition (8) wic, in te case of exponential kernels, provides always maximum delay values less tan one. Example 12 Consider now te system (15) wit A = 1 Because of te particular form of matrix A; te critical delay value can be computed analytically by using frequency-domain tecniques. Simple calculations based on te caracteristic function s + e s 2 1 f(s) = s sow tat (15) is exponentially stable for all delay values < 1 For tis particular matrix A; e A = I 2 + A; tis integral system can also be viewed as one aving a polynomial kernel G T B() wit G T = I 2 A and B() T = I 2 I 2 T TABLE II Maximum delay value for different decays Corollary Corollary Norm condition (8) 77 For different ; by applying Corollaries 8 and 9, we obtain te maximum delay value given in Table II. For = ; Corollaries 8 and 9 bot give max = 999 wic coincides wit te exact critical delay value and improves te maximum max = 77 obtained from te norm condition (8). We also see tat in tis case for > ; Corollary 8 provides less conservative results tan Corollary 9. In order to illustrate te computation of te -factor involved in te exponential upper bound (6), let us consider, in tis numerical example, tat = 1 and = 5 Direct calculations derived from (7), (1) and (11) yield = 87696; 1 = 129 and 2 = 24164; respectively. Ten, it follows tat every solution x(t; ') of (15), ' 2 PC([ 5; ); R 2 ); admits te exponential upper bound kx(t; ')k k'k 5 e t ; 8t IV. APPLICATION TO THE FINITE SPECTRUM ASSIGNMENT PROBLEM In tis section, we apply te obtained results in developing a design metod for tackling te numerical implementation problem of control laws wit distributed delay wic assign a nite spectrum to input delay systems. Tis problem concerns systems of te form _x(t) = Ax(t) + Bu(t ); (16) were A 2 R nn, B 2 R nm and > is te input delay. Te predictor like control law u(t) = K e A x(t) + e A Bu(t + )d (17) assigns a nite spectrum to te delay free closed-loop system (16)-(17) wic coincides wit te spectrum of te matrix A + BK, see [9]. Te practical value of suc a result is limited by te instability penomenon linked to te numerical implementation of te integral term in (17). It as been sown in [2], [15], [21] tat if te integral is approximated by a nite sum ten te resulting closed-loop delay system may become unstable if te ideal controller is not internally stable. A denite result as been demonstrated in [14] a necessary and sufcient condition for a numerically safe implementation (by using any type of numerical integration rule) of te controller (17) is bot te stability of te ideal closed-loop system (16)-(17), i.e., A+BK is Hurwitz, and te stability of te internal dynamics of te controller (17), i.e., te stability of te integral delay system z(t) = Ke A Bz(t + )d (18) Based on tis observation, by combining known results on matrix inequalities for guaranteeing tat all eigenvalues of te

6 5 matrix A + BK ave real parts less tan a certain prescribed [1] wit Teorem 7 applied to te integral delay system (18), wit G T = K; B() = e A B and M = A; we easily arrive at te following result. Proposition 13 Te input delay system (16), were witout loss of generality we can assume tat te matrix B is full column rank, admits a numerically safe implementation of te control law (17) wit a guaranteed decay rate for te closedloop system solutions, if tere exists a matrix K 2 R mn and positive denite matrices R; P; Q 2 R nn suc tat (A + BK + I) T R + R(A + BK + I) < ; (19) Q A T [P + Q] [P + Q] A K T B T [P + Q] BK > ; (2) Q A T P P A e 2 K T B T [P + Q] BK > (21) Te controller syntesis conditions of Proposition 13 involve nonlinear terms. Of course, condition (19) can be expressed as a linear matrix inequality pre- and post multiplying inequality (19) by te matrix S = R 1 and ten setting K = KS yields SA T + AS + K T B T + B K + 2S < ; (22) wic is linear in te variables S and K As a consequence, an easy way to solve te syntesis problem is rst to nd a feasible set (S; K) satisfying inequality (22). Ten set K = KS 1 and solve te linear matrix inequalities (2) and (21) for te variables P and Q However, wit additional computational effort, te syntesis conditions can be expressed as a set of linear matrix inequalities to be solved simultaneously and for wic better results can be obtained as presented next. First, by using te elimination procedure along wit Finsler's Lemma, te inequality (22) can be expressed as SA T + AS + 2S BB T < ; (23) for some scalar, see [1], and if tere exist positive denite matrix S and a scalar suc tat (23) olds, ten a feedback gain is given by K = 2 BT S 1 (24) Because of te omogeneity of (23) in S and, we can assume tat is positive and select = 2 Substituting (24) into inequalities (2) and (21) one obtains Q A T [P + Q] [P + Q] A S 1 BB T [P + Q] BB T S 1 > Q A T P P A e 2 S 1 BB T [P + Q] BB T S 1 > Ten, by introducing a new variable Y and a positive scalar suc tat Y > S 1 and (P + Q) 1 > I, respectively, te TABLE III Maximum delay value, feedback gain and ideal closed-loop eigenvalues. 5 1 max K ( 5; 267) ( 54; 14) ( 215; 29) ( 81 s 1; i 52 52i 14 13i 2 above two inequalities can be replaced by Q A T [P + Q] [P + Q] A Y BB T BB T Y I > ; (25) Q A T P P A Y BB T BB T Y e 2 > ; (26) I Y I > ; (27) I S (P + Q) < I; (28) were Scur complement as been applied to obtain te rst tree inequalities wile te last one directly follows by noting tat (P + Q) 1 > I is equivalent to (P + Q) < I Summarizing te above, if tere exist positive denite matrices P; Q; S; Y and a positive scalar suc tat te inequalities (23), (25)-(28) old, ten te feedback gain K = B T S 1 acieves a successful numerical implementation of te controller (17), wit guaranteed decay rate for te closed-loop system solutions. We use tis linear inequality formulation in te following illustrative examples. Example 14 Let us consider te input delay system (16) wit system matrices 1 A = and B = 1 Te corresponding linear system represents a double integrator wit delayed input, wic is commonly found in mecanical systems. Te maximum delay value max and te corresponding feedback gain K; computed by solving (23), (25)-(28) for different decays, and te ideal closed-loop eigenvalues s 1;2 are displayed in Table 3. We see tat for = ; a numerical safe implementation of te controller (17) can be acieved for any satisfying < 1 In tis particular case, in order to illustrate tat a feasible solution can be found for an arbitrarily large delay, te feedback gain K and te corresponding ideal closed-loop eigenvalues are computed for a delay value = 1 in Table III. Te example tat follows represents te linearization of a uid-ow model for describing te beavior of some classes of TCP/AQM networks, see [11] and te references terein for details. For suc a system, te application of te controller (17) as been recently proposed in [11]. It was sown tere tat for any given set of network parameters tere always exists a feedback gain suc tat a numerically safe implementation of te controller (17) can be acieved. Suc a result was proved by using te sufcient condition (8) wic, unlike te present approac, does not provide an estimate of te decay rate for te closed-loop system solutions.

7 6 TABLE IV Feedback gain and ideal closed-loop eigenvalues for different K (147; 1) (328; 3) (585; 9) s 1;2 6479; i i Example 15 Te linearized TCP/AQM network model of te form (16) as system matrices 2n A = 2 c c 2 n and B = 2n c 2 were n represents te number of TCP ows, c is te link capacity and denotes te round-trip time (delay). We ere apply te results of tis section in computing te feedback gain for guaranteeing a numerical implementation of te controller along wit a certain prescribed decay rate For network parameters n = 4; = 7 and c = 3, te inequalities (23), (25)-(28) are feasible for a maximum decay rate max = 848 Te computed feedback gain and te corresponding ideal closed-loop eigenvalues for different max are displayed in Table IV. V. CONCLUDING REMARKS In tis paper, delay-dependent sufcient conditions for te exponential stability of some classes of integral delay systems wit analytic kernels, wic found important applications in several stability problems of time-delay systems, are given. Te results are obtained by using te Lyapunov-Krasovskii functional approac recently developed in [12] and [13] for integral delay systems. An application to te nite spectrum assignment problem of input delay systems is also presented. As a result, a novel linear matrix inequality formulation for guaranteeing a numerically safe implementation of te controllers and a prescribed decay rate for te closed-loop system solutions is proposed. [8] V.B. Kolmanovskii, A. Myskis, Introduction to te Teory and Applications of Functional Differential Equations, Kluwer, te Neterlands, [9] A.Z. Manitus, A.W. Olbrot, Finite spectrum assignment problem for systems wit delays, IEEE Trans. Automat. Control, 24, [1] D. Melcor-Aguilar, B. Tristán-Tristán, On te implementation of control laws for nite spectrum assignment te multiple delays case, in Proc. of 4t IEEE Conf. Electr. Electron. Eng. México City, México, 27. [11] D. Melcor-Aguilar, On te stability of AQM controllers suppporting TCP Flows, Topics in Time-Delay Systems, Lect. Notes Control Inf. Sci., 388, , Springer-Verlag, Berlin, 29. [12] D. Melcor-Aguilar, V. Karitonov, R. Lozano, Stability conditions for integral delay systems, Int. J. Robust. Nonlinear Control, 2, 1-15, 21. [13] D. Melcor-Aguilar, On stability of integral delay systems, Appl. Mat. and Comput., 217, , 21. [14] W. Miciels, S. Mondié, D. Roose and M. Dambrine, Te effect of te approximating distributed delay control law on stability, in Advances of Time-Delay Systems, Lect. Notes Comput. Sci. Eng., 38, 27-22, Springer-Verlag, 24. [15] S. Mondié, M. Dambrine and O. Santos, Approximation of control law wit distributed delays A necessary condition for stability, Kybernetica, 38 (5), , 21. [16] S. Mondié and W. Miciels, Finite spectrum assignment of unstable time-delay systems wit a safe implementation, IEEE Trans. Automat. Control, 48 (12), , 23. [17] S. Mondié and V. Karitonov, Exponential Estimates for Retarded Time-Delay Systems An LMI Approac, IEEE Trans. Automat. Control, 5 (2), , 25. [18] S.-I. Niculescu, Delay Effects on Stability A Robust Control Approac, Lect. Notes Control Inf. Sci., 269, Springer-Verlag, Berlin, 21. [19] P. Pepe, Te Liapunov's second metod for continuous time difference equations, Int. J. Control, 13, , 23. [2] J.-P. Ricard, Time-delay systems an overview of some recent advances and open problems, Automatica, 39 (1), , 23. [21] V. Van Assce, M. Dambrine and J.-F. Lafay, Some problems arising in te implementation of distributed-delay control laws, in Proc. 38t IEEE Conf. Decision Control, Poenix, AZ, , [22] Q.-C. Zong, On distributed delay in linear control laws-part I discretedelay implementations, IEEE Trans. Automat. Control, 49 (11), , 24. VI. ACKNOWLEDGMENT Te autors wis to tank te anonymous Reviewers and Associate Editor P. Pepe for teir careful reading and valuable suggestions tat elped us to improve te work. REFERENCES [1] S. Boyd, L.E. Gaoui, E. Feron and V. Blakrisnan, Linear Matrix Inequalities in System and Control Teory, 15, SIAM, Piladelpia, [2] K. Engelborgs, M. Dambrine, and D. Roose, Limitations of a class of stabilization metods for delay systems, IEEE Trans. Automat. Control., 46 (2), , 21. [3] K. Gu, S.-I. Niculescu, Additional Dynamics in Transformed Time- Delay Systems, IEEE Trans. Autom. Control, 45, , 2. [4] J. Hale, S.M. Verduyn-Lunel, Introduction to Functional Differential Equations. Springer-Verlag, New York, [5] V. Karitonov, D. Melcor-Aguilar, On delay-dependent stability conditions, Syst. Control Lett., 4,71-76, 2. [6] V. Karitonov, D. Melcor-Aguilar, On delay-dependent stability conditions for time-varying systems, Syst. Control Lett., 46, [7] V. Karitonov, D. Melcor-Aguilar, Additional dynamics for general class of time-delay systems, IEEE Trans. Autom. Control, 48, , 23.