New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems

Systems & Control Letters 43 (21 39 319 www.elsevier.com/locate/sysconle New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems E. Fridman Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel Received 1 September 2; received in revised form 14 February 21 Abstract A new (descriptor model transformation and a corresponding Lyapunov Krasovskii functional are introduced for stability analysis of systems with delays. Delay-dependent=delay-independent stability criteria are derived for linear retarded and neutral type systems with discrete and distributed delays. Conditions are given in terms of linear matrix inequalities and for the rst time refer to neutral systems with discrete and distributed delays. The proposed criteria are less conservative than other existing criteria (for retarded type systems and neutral systems with discrete delays since they are based on an equivalent model transformation and since they require bounds for fewer terms. Examples are given that illustrate advantages of our approach. c 21 Elsevier Science B.V. All rights reserved. Keywords: Time-delay systems; Stability; Linear matrix inequalities; Delay-dependent=delay-independent criteria 1. Introduction The choice of an appropriate Lyapunov Krasovskii functional is the key-point for deriving of stability criteria. It is known that the general form of this functional leads to a complicated system of partial dierential equations (see e.g. [12]. That is why many authors considered special forms of Lyapunov Krasovskii functional and thus derived simpler (but more conservative sucient conditions. Among the latter there are delay-independent and delay-dependent conditions. Three main transformations of the original system have been used for delay-dependent stability analysis of retarded type systems [7]. One of these transformations (the second one in [7] the neutral type representation of system has been used also for neutral type systems [9,13]. The conservatism of approaches based on these transformations is two-fold: the transformed system is not equivalent to the original one (see [1] concerning the rst transformation, while the second transformation requires additional assumptions as mentioned in [13] and bounds should be obtained (completion to the squares for certain terms. In the present paper, we introduce a new type of Lyapunov Krasovskii functional inspired by [3] which is based on equivalent augmented model a descriptor form representation of the system. Our approach essentially reduces the conservatism of the existing methods. This work was supported by the Ministry of Absorption of Israel. Tel.: +972-364-5313; fax: +72-364-795. E-mail address: emilia@eng.tau.ac.il (E. Fridman. 167-6911/1/$ - see front matter c 21 Elsevier Science B.V. All rights reserved. PII: S167-6911(1114-1

31 E. Fridman / Systems & Control Letters 43 (21 39 319 2. Delay-dependent stability of linear systems: discrete delays Let R n be Euclidean space and C[a; b] be the space of continuous functions :[a; b] R n with the supremum norm. Denote by x t (=x(t + ( [ h; ]. 2.1. Neutral systems Given the following system: ẋ(t D i ẋ(t h i = A i x(t h i ; x(t=(t; t [ h; ]; (1 where x(t R n ;h =, h i 6 h; ;:::;m, A i and D i are constant n n-matrices, is a continuously dierentiable initial function. To guarantee that the dierence operator D : C[ h; ] R n given by D(x t =x(t m D ix(t h i is stable (i.e. dierence equation Dx t = is asymptotically stable we assume [4,5]: A1. Let D i 1; where is any matrix norm. Note that for neutral type systems there exist two types of stability results: those corresponding to continuous initial functions and continuously dierentiable initial functions. Under A1 both types of stability are equivalent [2]. We represent (1 in the equivalent descriptor form: ẋ(t=y(t; y(t= D i y(t h i + A i x(t h i : (2 The latter can be represented in the form of descriptor system with discrete and distributed delay in the fast variable y: ( m ẋ(t=y(t; = y(t+ D i y(t h i + A i x(t A i y(sds: (3 Lyapunov Krasovskii functional for the latter system has the form introduced in [3]: x(t V (t=[x T (t y T (t]ep + V 1 + V 2 ; (4 y(t where I P1 E = ; P= ; P 1 = P1 T ; (5 P 2 P 3 V 1 = y T (sq i y(sds; Q i (6 and V 2 = y T (sr i y(sdsd; R i : (7 h i t+

E. Fridman / Systems & Control Letters 43 (21 39 319 311 The rst term of (4 corresponds to the descriptor system, V 1 corresponds to the delay-independent stability with respect to the discrete delays and V 2 to delay-dependent stability with respect to the distributed delays. The functional (4 is degenerated (i.e. nonpositive-denite as it is usual for descriptor systems (see e.g. [14]. We obtain the following: Theorem 1. Under A1 (1 is stable if there exist P 1 = P1 T;P 2;P 3 ; and Q i = Qi T ;R i = R T i ;i=1;:::;m that satisfy the following linear matrix inequality (LMI: ( m AT i P2 + P T ( m 2 A i P 1 P2 T + ( m AT i P3 h 1 P2 T A 1 h m P2 T A m P2 T D 1 P T 2 D m P 1 P 2 + P T ( m 3 A i P 3 P3 T + m (Q i + h i R i h 1 P3 T A 1 h m P3 T A m P3 T D 1 P T 3 D m h 1 A T 1 P 2 h 1 A T 1 P 3 h 1 R 1 : h m A T mp 2 h m A T mp 3 h m R m D1 T P 2 D1 T P 3 Q 1 D T mp 2 D T mp 3 Q m (8 Proof. We represent (1 in the equivalent form (3. Note that x [x T y T ]EP = x T P 1 x y and, hence, d dt [xt (t y T (t]ep x(t y(t ẋ(t =2x T (tp 1 ẋ(t=2[x T (t y T (t]p T : Due to (3 the latter relations imply that d x(t dt [xt (t y T (t]ep y(t y(t =2[x T (t y T (t]p T ( y(t+ m D i y(t h i + Dierentiating (4 in t and applying (9 we obtain P T P T D dv (t 1 D m = T dt [ D1 T ]P Q 1 + i i=1 i=1 [ Dm]P T Q m A i x(t m t A i y(sds : (9 t h i y T (sr i y(sds; (1

312 E. Fridman / Systems & Control Letters 43 (21 39 319 where, col{x(t; y(t; y(t h 1 ;:::;y(t h m } and [ ] [ (, P T I m ] ( m A + i AT i P + m I I I (Q ; (11 i + h i R i t i (t, 2 [x T (t y T (t]p T y(sds: (12 A i For any n n-matrices R i x t i 6 h i [x T y T ]P T R 1 i [ A T i ]P + y T (sr i y(sds: (13 A i y Eqs. (1 and (13 yield (by Schur complements that dv (x; t=dt if the following LMI holds h 1 P T h m P T P T P T A 1 A m D 1 D m h 1 [ A T 1 ]P h 1 R 1 h m [ A T ; (14 m]p h m R m [ D1 T ]P Q 1 [ Dm]P T Q m where is given by (11. LMI (8 results from the latter LMI by multiplying the block matrices. Functional V of (4 is degenerated and it has a negative derivative. This implies asymptotic stability of (1 in the space of continuous functions and thus, under A1, in the space of continuously dierentiable functions [2]. Remark 1. Conservatism of our method is caused by bounding (13. We are bounding however fewer terms than in the other existing criteria. Thus, there are m 2 more terms of this kind when the rst transformation of [7] is used (in the case of retarded system with D i =. Remark 2. Criterion (8 is delay-independent with respect to delays in the dierence operator D. As we mentioned above A1 guarantees delay-independent stability of D. The latter is necessary for robustness of stability of (1 with respect to small delays [1,5]. Remark 3. Note that (8 yields the following inequality: P 3 P3 T + m Q i P3 T D 1 P T 3 D m D1 T P 3 Q 1 : (15 DmP T 3 Q m If there exists a solution to (8 then there exists a solution to (15. If moreover P 3 = P3 T, then P 3 and the fast system ẏ(t= y(t+ D i y(t g i (16 is asymptotically stable for all g i (see e.g. [6]. Note also that (15 implies by Schur complements that P 3 P3 T + Q i + P3 T D i Qi 1 Di T P 3 : (17

E. Fridman / Systems & Control Letters 43 (21 39 319 313 Remark 4. In the scalar case (17 implies A1 and therefore Theorem 1 holds without assumption A1. Really, since Q i + P 2 3D 2 i =Q i 2P 3 D i : We have from (17 2P 3 (1 D i 6 2P 3 + The latter yields A1 since P 3. (Q i + P3D 2 i 2 =Q i : Remark 5. In the case of a single delay in the dierence operator Dx t = x(t D 1 x(t h 1 assumption A1 should be changed to the following standard assumption for neutral systems ([4,11,13,15]: A1. Assume that all eigenvalues of D 1 are inside of unit circle. Under A1 similarly to [2] the stability in the space of continuous and in the space of continuously dierentiable functions is equivalent. As in the scalar case (17 implies A1 and therefore Theorem 1 holds without assumption A1. Really, multiplying (17 by y R n from the right and by y T from the left we have 2 y T P 3 y + Q 1=2 1 y 2 + Q 1=2 1 D1 T P 3 y 2 : Since Q 1=2 1 y 2 + Q 1=2 1 D1 T P 3 y 2 2 y T D1 T P 3 y : We obtain from the previous inequality that 2 y T P 3 y +2 y T D T 1 P 3 y : Choose y to be an eigenvector of D 1 conclude that that corresponds to the eigenvalue. From the latter inequality we y T P 3 y + y T P 3 y and thus 1. Remark 6. Comparing our approach with the neutral type representation of [7,9,13] in the form [ d ] t x(t D i x(t h i + A i x(sds = A i x(t; dt we see that there is the same number of terms in the bounding of the type (13, though these terms are dierent. Our advantage is that unlike [7,9,13] we have no additional assumption on stability of D : C[ h; ] R n given by D(x t =x(t D i x(t h i + A i x(sds; which is dicult to verify. Sucient condition for stability of D is as follows: D i + h i A i 1; but the latter may lead to conservative results (see Example 2 below.

314 E. Fridman / Systems & Control Letters 43 (21 39 319 2.2. Retarded type systems Given the system ẋ(t= A i x(t h i ; we represent it in the equivalent descriptor form: ( m ẋ(t=y(t; y(t= A i x(t A i y(sds: (19 Since the fast variable y has no discrete delays, the Lyapunov Krasovskii functional has the form (4, where V 1 =, i.e. Q ;i=1;:::;m. Theorem 1 implies the following Corollary 1. Eq. (18 is asymptotically stable if there exist P 1 = P1 T;P 2;P 3 ; and R i = R T i ;i=1;:::;m that satisfy the following LMI: ( m AT i P2 + P T ( m 2 A i P 1 P2 T + ( m AT i P3 h 1 P2 T A 1 h m P T 2 A m P 1 P 2 + P T ( m 3 A i P 3 P3 T + m h ir i h 1 P3 T A 1 h m P T 3 A m h 1 A T 1 P 2 h 1 A T 1 P 3 h 1 R 1 : (2 h m A T mp 2 h m A T mp 3 h m R m 2.3. Examples We applied our criteria to examples considered in [7 9] and [13]. We solved LMIs by using LMI Toolbox of Matlab. None of the results we obtained for these examples were more conservative than the existing results and in some examples our results were less conservative. Two of these examples (one for retarded and one for neutral type cases are given below. Example 1 (Kolmanovskii andrichard[7] (retarded type case. Consider the system ẋ(t=a x(t+a 1 x(t h 1 (18 with 1 1=2 2 2 A = ; A 1 = : 1=2 1 2 2 This is Example 2:2 of [7] with =1;= 1 2 ;= =2: In [7] it was found that the second transformation leads to the less restrictive (than the other two transformations condition: h 1 :369: Our Corollary 1 improves this result and our condition is h 1 6 :271. Thus for h 1 =:271 we obtain the following solution to LMI (2: 94:169 :1653 93:5589 :1872 P 1 = ; P 2 = ; :1653 94:469 :1872 94:6599 18:517 :93 68:2748 :349 P 3 = ; R 1 = : :93 18:488 :349 68:181 Note that applying LMI conditions of [8] and [13] we obtained h 1 6 :268 and h 1 6 :271, respectively.

E. Fridman / Systems & Control Letters 43 (21 39 319 315 Example 2 (Lien et al. [9] (neutral type case. Consider the following system: ẋ(t D 1 ẋ(t h 1 =A x 1 (t+a 1 x(t h 1 with :9 :2 1:1 :2 :2 A = ; A 1 = ; D 1 = : (21 :1 :9 :1 1:1 :2 :1 The stability condition of [9] is h 1 6 :3. We obtain h 1 6 :74 and for h 1 =:74 we have the following solution to (8: 7:623 2:6912 :7783 :259 3:811 1:3542 P 1 = ; P 2 = ; P 3 = ; 2:6912 7:463 :259 :7739 1:3542 3:6974 :6499 :1261 4:2419 2:425 Q 1 = ; R 1 = : :1261 :485 2:425 4:4669 By applying LMI of [13] we nd that LMI has a solution for h 1 6 :71 (which is still less than.74, but the stability of D should be veried. Sucient condition for the latter D 1 + h 1 A 1 1 (where similarly to [9] spectral norm is taken leads to more restrictive condition h 1 6 :61. 3. Delay-dependent/delay-independent stability: distributed delays 3.1. Main result We generalize our results to systems with distributed delays: ẋ(t D i ẋ(t g i = A i x(t h i + A i x(sds + F i x(t g i ; (22 t i where g i. Similarly to [6,7], we are looking for stability criterion which is delay-dependent with respect to one part of delays (h i and the distributed delays over [ i ; ], and delay-independent with respect to the other delays (g i. The descriptor form representation for this system has the form: ( m ẋ(t=y(t; y(t = D i y(t g i + A i x(t A i y(sds + A i x(sds + F i x(t g i : (23 t i The corresponding (degenerate Lyapunov Krasovskii functional is given by x(t V (t=[x T (t y T P1 (t]ep + V 1 + V 2 + V 3 + V 4 ; P= ; P 1 = P1 T ; (24 y(t P 2 P 3 where V 2 is dened by (7, and where V 1 = y T (sq i y(sds; V 3 = x T (su i x(sds; Q i ; U i ; t g i t g i V 4 = i t+ x T (sr i x(sds d; R i : (25

316 E. Fridman / Systems & Control Letters 43 (21 39 319 The form of (24 (similarly to (4 corresponds to discrete-delay independent=distributed-delay dependent conditions for descriptor system (23. We obtain the following result: Theorem 2. Under A1 (22 is stable for all g i ; i=1;:::;k if there exist P 1 = P1 T;P 2;P 3 ;Q i = Qi T ; U i = Ui T ;i=1;:::;k and R j = R T j ;R j = R T j ;j=1;:::;m that satisfy the following LMI: P 1 P2 T + ( m AT i P3 h 1 P2 T A 1 h m P2 T A m P 1 P 2 + P T ( m 3 A i P 3 P3 T + k Q i + m h ir i h 1 P3 T A 1 h m P3 T A m h 1 A T 1 P 2 h 1 A T 1 P 3 h 1 R 1 h m A T mp 2 h m A T mp 3 h m R m D1 T P 2 D1 T P 3 Dk T P 2 Dk T P 3 1 A T 1P 2 1 A T 1P 3 m A T mp 2 m A T mp 3 F1 T P 2 F1 T P 3 Fk T P 2 Fk T P 3 P2 T D 1 P2 T D k 1 P2 T A 1 m P2 T A m P2 T F 1 P2 T F m P3 T D 1 P3 T D k 1 P3 T A 1 m P3 T A m P3 T F 1 P3 T F m Q 1 :: : (26 Q k 1 R 1 m R m U 1 U k where ( m = ( m P 2 + P2 T A i + A T i U i + i R i :

E. Fridman / Systems & Control Letters 43 (21 39 319 317 Proof. Similar to the proof of Theorem 1. Note that in this case we have 1 P T P T P T P T D 1 D k F 1 F k [ D T 1 ]P Q 1 dv (t = T dt [ D k T ]P Q k [ F1 T ]P U 1 [ Fk T ]P U k + i + i i=1 i=1 i=1 y T (sr i y(sds i=1 t i x T (sr i x(sds; (27 where, col{x(t; y(t; y(t g 1 ;:::;y(t g k ;x(t g 1 ;:::;x(t g k }; i is given by (12 and [ ] [ ( 1, P T I m ] [ k ( m A + i AT i P + U i + m ] ir i I I I k Q i + m h ; ir i t i (t, 2 [x T (t y T (t]p T x(sds: A i 3.2. Delay-independent stability Consider the system ẋ(t D i ẋ(t g i =A x(t+ F i x(t g i : (28 Delay-independent stability conditions can be derived by applying Lyapunov Krasovskii functional of (24, where V 2 = V 4 =. Theorem 2 implies the following delay-independent stability criterion: Corollary 2. Under A1 (28 is stable for all g i ; i=1;:::;k if there exist P 1 = P T 1 ;P 2;P 3 ; and Q i = Q T i ;U i = U T i ;i=1;:::;k that satisfy the following LMI: A T P 2 + P T 2 A + k U i P 1 P T 2 + A T P 3 P T 2 F 1 P T 2 F k P T 2 D 1 P T 2 D k P 1 P 2 + P T 3 A P 3 P T 3 + k Q i P T 3 F 1 P T 3 F k P T 3 D 1 P T 3 D k F1 T P 2 F1 T P 3 U 1 Fk T P 2 Fk T P 3 U k D1 T P 2 D1 T P 3 Q 1 Dk T P 2 Dk T P 3 Q k : (29

318 E. Fridman / Systems & Control Letters 43 (21 39 319 Example 3. Consider a two-dimensional system ẋ(t D 1 ẋ(t g 1 =A x(t+f 1 x(t g 1 +A 1 x(sds (3 t 1 with a1 b1 b 2 c1 c 2 A = ; F 1 = ; A 1 = : a 2 b 2 b 1 c 2 c 1 For D 1 = this is Example 4:1 from [7]. From results of [7] it follows that for a 1 = a 2 =1:5; b 1 = b 2 = 1 and c 1 =1;c 2 =:5 the system (3 is stable for all delays g 1 and for 1 6 :3. Our Theorem 2 leads to less restrictive condition: 1 6 :7. For D 1 = and a 1 =2; a 2 =15; b 1 =1; b 2 =3; c 1 =1; c 2 =:5 (31 stability conditions of [7] do not hold (even for 1 =. For this case by Theorem 2 we nd that (3 is stable for all g 1 and 1 6 1:1. Choosing D 1 given by (21 and the other parameters given by (31 we obtain that (3 is stable for all g 1 and 1 6 1. Thus for = 1 we obtain the following solution to (26: 272:1194 47:1695 27:1438 41:3581 67:5927 4:2914 P 1 = ; P 2 = ; P 3 = ; 47:1695 254:182 41:3581 192:5597 4:2914 15:8292 295:8 498:8 276:4176 171:143 62:7484 3:8686 U 1 = ; R 1 = ; Q 1 = : 498:8 311:4 171:143 464:256 3:8686 9:791 4. Conclusions New Lyapunov Krasovskii functionals have been introduced for stability of linear retarded and neutral type systems with discrete and distributed delays. These degenerate functionals are based on equivalent descriptor form of the original system and lead to new results (for neutral systems with distributed delays and to results that are less conservative than existing results for both, retarded and neutral type systems. Delay-dependent=delay-independent conditions have been obtained in terms of LMI. The new model transformation and functionals can be applied further to H control of linear systems with delay and to analysis and synthesis of some nonlinear time-delay systems. This work is currently in progress. References [1] V. Charitonov, D. Melchor-Aguilar, On delay-dependent stability conditions, Systems Control Lett. 4 (2 71 76. [2] L. El sgol ts, S. Norkin, Introduction to the Theory and Applications of Dierential Equations with Deviating Arguments, Mathematics in Science and Engineering, Vol. 15, Academic Press, New York, 1973. [3] E. Fridman, Eects of small delays on stability of singularly perturbed systems, Automatica, submitted for publication. [4] J. Hale, S. Lunel, Introduction to Functional Dierential Equations, Springer, New York, 1993. [5] J. Hale, S. Lunel, Eects of small delays on stability and control, Rapportnr. WS-528, Vrije University, Amsterdam, 1999. [6] V. Kolmanovskii, S.-I. Niculescu, J.P. Richard, On the Liapunov Krasovskii functionals for stability analysis of linear delay systems, Internat. J. Control 72 (1999 374 384. [7] V. Kolmanovskii, J-P. Richard, Stability of some linear systems with delays, IEEE Trans. Automat. Control 44 (1999 984 989. [8] X. Li, C. de Souza, Criteria for robust stability and stabilization of uncertain linear systems with state delay, Automatica 33 (1997 1657 1662. [9] C.-H. Lien, K.-W Yu, J.-G. Hsieh, Stability conditions for a class of neutral systems with multiple time delays, J. Math. Anal. Appl. 245 (2 2 27. [1] H. Logemann, S. Townley, The eect of small delays in the feedback loop on the stability of the neutral systems, Systems Control Lett. 27 (1996 267 274. [11] M. Mahmoud, Robust Control and Filtering for Time-Delay Systems, Marcel Dekker, New York, 2.

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