On Delay-Dependent Robust H Control of Uncertain Continuous- and Discrete-Time Linear Systems with Lumped Delays

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1 On Delay-Dependent Robust H Control of Uncertain Continuous- and Discrete-Time Linear Systems with Lumped Delays R. M. Palhares, C. D. Campos, M. C. R. Leles DELT/UFMG Av. Antônio Carlos , Belo Horizonte - MG Brazil palhares@cpdee.ufmg.br P. Ya. Ekel PPGEE PUC/Minas Av. Dom José Gaspar, , Belo Horizonte - MG Brazil ekel@pucminas.br M. F. S. V. D Angelo DCC UNIMONTES Av. Rui Braga, s/n Montes Claros - MG Brazil Abstract This paper develops a Linear Matrix Inequality (LMI) approach to the robust H control problem of uncertain continuous- and discrete-time linear time-invariant systems with time-delay in the state vector and control input. The main results provide sufficient delay-dependent conditions for the control problem, where the explicit size of the time delay plays a crucial role for the closed-loop stability. The solutions that are found for the H control problem are less conservative, when compared to other approaches. I. INTRODUCTION The control design problem for systems subject to timedelay in the state vector and/or control input has received a lot of attention in the last decades. This increasing interest about this topic can be understood to the fact that time-delays appear as an important source of instability or performance debasement in a great number of important engineering problems involving: material, information or energy transportation. Besides the stability analysis of systems with timedelay itself, many others subjects has been incorporated to the control design problem: parameter uncertainties, H index performance, multiple time-delays, delay-independent conditions, delay-dependent conditions, lumped and distributed delay cases and optimization techniques involving LMI descriptions. Commonly, most of the approaches employs only the traditional delay-independent condition, in which the controller design is provided irrespective of the size of the time-delay. On the other hand, it is well known that for systems which the stability issue depends explicitly on the time-delay, a delay-independent condition does not work. In fact, in order to overcome this difficult, it is necessary to evidence the time-delay in the control design, namely to obtain a delay-dependent condition. The reader can find a great number of references about this topic in the literature, particularly the following ones systemize the main ideas, [1], [2], [3], [4], [5], [6], [7], [8], [9], [1],. In this paper the main contribution is to state LMI sufficient delay-dependent conditions for the robust statefeedback control design, which guarantees an H level of disturbance attenuation for both uncertain continuous- and discrete-time state and control-delayed systems. Moreover, only the lumped delay case is handled, but with an appropriate Lyapunov-Krasovskii functional candidate, the distributed delay case can be included by means of an extra timedelay parameter. The parameter uncertainties dealt with are of polytopic type, which allows to extend the LMI results obtained for precisely known systems for a set where the vertices are elements of LMI type. The main results for continuous- and discrete-time systems take as initial point a recent work which introduces a new upper bound for the inner product of two vectors [3]. Finally, for the continuoustime case, a numerical example is presented and the results are compared with recent approaches [2], [8], [11], [7]. Throughout the paper, L 2 denotes both the Lebesgue spaces L 2 (for continuous-time systems) and l 2 (for discretetime systems). II. GENERAL PROBLEM STATEMENT Consider the following linear time-invariant dynamic system with time-delay: δ[] = A + A d x(t τ) + Bu(t) + B d u(t τ) + E z(t) = C + Du(t) + F = φ(t), t [ τ, ) (1) where : R R n is the system state vector, u : R R m is the control input vector and w : R R p is the exogenous disturbance vector. φ(t) is a given initial vector function which is continuous on the segment [ τ, ), and τ > is the time-delay of the system. Assume perfect state measurement. δ[ ] indicates the time derivative ẋ(t), for continuous-time systems, or the shift operator x(t + 1), for discrete-time systems. The matrices of the system are assumed to be uncertain but belonging to a polytopic set described by κ vertices, namely { (A, A d, B, B d, E, C, D, F ) P (A,..., F ) (A,..., F ) = κ ξ i (A i,..., F i ) ; ξ i ; i=1 κ i=1 } ξ i = 1 (2) For control purposes it is considered the memoryless statefeedback control law: u(t) = K. Although the control with an additional memory term can also be included in

2 a straightforward way. Then the closed-loop state-delayed system is given by δ[] = à + Ãdx(t τ) + E z(t) = C + F with à A+BK, à d A d +B d K, and C C +DK. The main control problem to be addressed in this paper is stated in the sequel. (P ) τ The robust H control problem with timedelay. Given scalars τ > and γ >, determine a controller gain K such that the closed-loop state-delayed system (3) is robustly stable and ensures a prescribed H disturbance attenuation for any time-delay τ [, τ], namely under zero initial conditions and for any non-zero w L 2 z L2 γ w L2 (A, A d, B, B d, C, D, E, F ) P In this situation, the closed loop system is said to be robustly stable with γ disturbance attenuation level. The notion of robust stability above for uncertain system means that: the closed-loop system in (3) with is robustly stable if the trivial solution of δ[] = Ã+Ãdx(t τ) is globally uniformly asymptotically stable for all admissible uncertainties and time-delay τ [, τ]. III. MAIN RESULTS In the following are presented the delay-dependent LMI robust H performance analysis and control design for both continuous- and discrete-time systems subject to lumped time-delay in the state and control input. A. Continuous-Time Robust H Performance Analysis Theorem 1: Consider the closed-loop system (3) and let τ > and γ > be given scalars. If there exists symmetric matrices X, H, Q and Z and matrix V satisfying Υ i XÃdi V XE i τãt i Z C i T Q τãt di Z γ 2 I τe i T Z F i T τ Z (4) I (3) H V (5) Z where Υ i XÃi + ÃT i X + τh + V + V T + Q. Then the closed-loop system is robustly stable with disturbance attenuation γ for any time-delay τ [, τ]. Proof: Define a Lyapunov-Krasovskii functional as V (x(t τ), τ [, τ]) = x T (t)x + τ t+β ẋ T (α)zẋ(α) dα dβ + x T (α)qx(α) dα (6) Considering the following identity (Leibniz-Newton) b a ν(t) dt = ν(b) ν(a) (7) system (3) can be rewritten as ẋ(t) = (à + Ãd) Ãd ẋ(α) dα + E t τ z(t) = C + F (8) Assuming the worst case τ = τ the derivative of (6) yields V ( ) = x T (t)[2x(ã + Ãd) + Q] + 2x T (t)xe + τẋ T (t)zẋ(t) τ x T (t τ)qx(t τ) 2x T (t)xãd ẋ T (t + β)zẋ(t + β) dβ ẋ(α) dα (9) Applying the upper bound for the inner product of two vectors presented in [3] to the last term of (9) it follows that 2x T (t)xãd ẋ(α) dα τx T (t)h +2x T (t)(v XÃd) ẋ(α) dα + ẋ T (α)zẋ(α) dα (1) which is valid when matrices H, V and Z satisfies H V V T Z (11) Now, applying (1) in (9), it follows that V ( ) T x(t τ) Λ x(t τ) (12) where Π XÃd V + τãt ZÃd XE + τãt ZE Λ Q + τãt d ZÃd τãt d ZE τe T ZE Π Xà + ÃT X + τh + V + V T + Q + ÃT Zà when H, V and Z satisfies (11). Next, introduce the following H performance index J = [z T (t)z(t) γ 2 w T (t)] dt (13) Assuming (3) with zero initial conditions it follows that J = [z T (t)z(t) γ 2 w T (t) + V ( )] dt (14) since V ( ) t= = and V ( ) t (when = ). Then

3 J = x(t τ) T x(t τ) dt (15) where Ξ XÃd V + τãt ZÃd XE + τãt ZE + C T F Q + τãt d ZÃd τãt d ZE γ 2 I + τe T ZE + F T F Ξ Xà + ÃT X + τh + V + V T + Q + ÃT Zà + C T C when H, V and Z satisfies (11). Next, using Schur s complement it follows that LMIs (4) ensures that < over the entire uncertain domain P. This implies that J < for any non-zero w L 2 and over the entire uncertain domain P, i.e., the closed-loop system (3) is robustly stable with disturbance attenuation γ for any timedelay τ [, τ], which concludes the proof. B. Continuous-Time Robust H Control Synthesis Theorem 2: Consider system (1) and let τ > and γ > be given scalars. If there exists symmetric matrices Y, M, W, R, and matrices L, N, satisfying Ψ 11i Ψ 12i E i Ψ 14i Y Ci T + L T Di T W Ψ 24i γ 2 I τe i T Fi T τ R (16) I M N Y R 1 (17) Y where Ψ 11i = A i Y +Y A T i +B il+l T B T i + τm +N +N T + W, Ψ 12i = A di Y + B di L N, Ψ 14i = τ(y A T i + L T B T i ) and Ψ 24i = τ(y A T di + LT B T di ), then the problem (P τ ) is solvable for any time-delay τ [, τ], with the control gain K = LY 1. Proof: Pre- and post-multiplying (4) by diag{x 1, X 1, I, Z 1, I} and replacing à i, à di and C i with (A i + B i K), (A di + B di K) and (C i + D i K), one obtains Υ 11i Υ12i E i Υ14i Υ15i X 1 QX 1 Υ24i γ 2 I τe i T Fi T τz 1 I where Υ 11i = A i X 1 +X 1 A T i +B ikx 1 +X 1 K T B i + τx 1 HX 1 + X 1 V X 1 + X 1 V T X 1 + X 1 QX 1, Υ 12i = A di X 1 + B di KX 1 X 1 V X 1, Υ14i = τ ( X 1 A T i + X 1 KB i ), Υ15i = X 1 C T i + X 1 K T D T i and Υ 24i = τ ( X 1 A T di + X 1 K T Bdi) T. Making the following linearization change of variables: Y = X 1, M = X 1 HX 1, N = X 1 V X 1, W = X 1 QX 1, R = Z 1, L = KX 1 and applying Schur s complement one obtain (16). Finally pre- and post-multiplying (5) by diag{x 1, X 1 }, and using the change of variables one can obtain (17), which concludes the proof. In order to solve the non-convex LMI problem above one can either do the linearization R = Y, which turns (16)-(17) in LMI conditions, but is somehow conservative, or choose to proceed with the same cone complementarity linearization algorithm, proposed in [3], [12] replacing (17) with M N T1 T, 2, S T 3 S I Y I R I,, (18) T 1 T 2 T 3 In this context the following algorithm is proposed 1) For a constant time-delay τ > and a disturbance attenuation level γ > given. Set k =. Find a feasible set of matrices (S, Y, R, T 1, T 2, T 3 ) satisfying (16) and (18). 2) min Trace {f k } M,W,L,N,S,Y,R,T 1,T 2,T 3 s.t. (16) and (18) where f k S k T 1 + T 1k S + Y k T 2 + T 2k Y + R k T 3 + T 3k R. 3) If Trace {f k } 3n and condition (17) holds, then there exists an H controller which ensures the disturbance attenuation level γ for τ [, τ]. If (17) is not verified, set S k+1 = S, Y k+1 = Y, R k+1 = R, T 1k+1 = T 1, T 2k+1 = T 2, T 3k+1 = T 3, solutions of the optimization problem in step 2. Set k = k + 1, if k < k max (where k max is the number of maximum iterations) return to step 2, otherwise stop. Related to this algorithm, two paths can be implemented. The first one is concerned with finding the maximum timedelay τ. For that, an additional information must be introduced at step 3, namely, if the condition (17) holds, then the time-delay τ can be increased and algorithm return to step 2, and go on. The second one deals with the minimization of the disturbance attenuation level, γ, i.e. for τ fixed, one can find the minimum γ by implementing any line search algorithm and proceeding in the same way as indicated at steps 2 and 3. Moreover, if an initial γ is required, one can compute the minimum disturbance attenuation level, γ, by taking the (16)- (17) with the linear change of variables R = Y, which is an LMI problem for a time-delay τ > given. Thus solving this linearized problem can be seen as a starting point for the above algorithm. Nevertheless, it must be noted that this

4 linearized problem could be infeasible for increasing timedelay, say τ m > τ, but for that same τ m, the algorithm above will work. C. Discrete-Time Robust H Performance Analysis The following results are the counterpart for the discretetime analysis and synthesis case. For the sake of space the proofs are omitted. Theorem 3: Consider the closed-loop system (3) and let τ > and γ > be given scalars. If there exists symmetric matrices X, H, Q and Z and matrix V satisfying Γ V Ã T i X τ(ãi I) T Z C i T Q Ã T di X τãt di Z γ 2 I Ei T X τet i Z F i T X τ Z I (19) H V (2) Z where Γ = X + τh + V + V T + Q, then system is robustly stable with disturbance attenuation γ for any timedelay τ [, τ]. D. Discrete-Time Robust H Control Synthesis Next, the discrete-time version is presented and the same cone algorithm indicated above can be used. Theorem 4: Consider system (1) and let τ > and γ > be given scalars. If there exists symmetric matrices Y, M, W, R, and matrices L, N, satisfying Φ 11 N Φ 14i τ(φ 15i L) Φ 16i W Φ 24i τφ 25i γ 2 I Ei T τe i T Fi T Y τ R I (21) M N R 1 (22) where Φ 11 = Y + τm + N + N T + W, Φ 14i = Y A T i + L T B T i, Φ 15i = Φ 14i, Φ 16i = Y C T i + L T D T i, Φ 24i = Y A T di + LT B T di, Φ 25i = Φ 24i, then problem (P τ ) is solvable for any discrete time-delay τ [, τ], τ N, with the control gain K = LY 1. IV. EXAMPLE Consider the same example as shown in [2], namely a uncertain continuou-time system with time-delay given by ẋ(t) = A + A d x(t τ 1 ) + Bu(t) + E z(t) = C with matrices α 1 + β 1 A =, A 1 + α d =.9 + β 1 B =, E = 1 1 C = [ 1 ] α.2, β.2 (23) By Theorem 3.2 in [2], the maximum time-delay found, such that the closed-loop system remains robustly stable, was τ =.3346s. And for τ =.3s one can find γ [2] = On the other hand, taking into account the approach proposed in [8], the maximum value obtained for the timedelay was τ = 1.512s with γ [8] = and control gain K [8] = 1 9 [ ] Now, if one considers the approach developed in [7] the maximum time-delay achieved was τ = 1.496s with γ [7] = and associated gain K [7] = 1 9 [ ] In both last two approaches it is necessary to adjust a free parameter in the LMIs. The values of these parameters for the approaches in [8] and [7] were ɛ =.24 and ɛ =.23, respectively. In another scenery, as discussed, considering the linearized case as an option to estimate the size of the attenuation level for the cone algorithm, one can obtain for τ =.624s, γ = and K = [ ]. Taking this value of attenuation level, γ = , as a starting point to the algorithm, one can find after 826 iterations, τ = 1.1s with the associated control gain K 1 = [ ] This maximum time-delay is allowed for that value of γ fixed, although a posterior better solution can still be found in the sense of: (i) decreasing γ for τ = 1.1s, or (ii) increasing τ for γ > (but with performance debasement). The time-response of the state variable x 1 (t) of the system (23) in closed-loop with the control gain K 1, considering the four vertices and the nominal model, for the initial condition = [1 1] T t [ τ 1, ) and initial condition for the timedelay τ = 1.1s, is depicted in figure 1. As another test, considering the upper bound to the timedelay fixed in τ 1 =.3s, one can achieve by the approach

5 proposed here, the following guaranteed H disturbance attenuation level γ =.2 with the control gain K 2 = [ ] From Theorem 1, for analysis purposes, considering the closed-loop system with the control gain K 2, one can conclude that K 2 guarantees the H disturbance attenuation level of γ a = Which can be verified, taking a look to the singular value diagram presented in figure 2, considering the four vertices and nominal model. Moreover, computing the H norms in each vertex of the uncertain systems, one can find { , , , } which allows one to conclude that γ a = is the guaranteed H disturbance attenuation. Figure 3 presents a disturbance signal, w, of magnitude [ 1, 1] and time persistence of no more than.1s, entering the state space by means of matrix E. And figure 4 depicts the controlled output signal z 2 (t) considering the H control K 2 and the disturbance signal, w. As an extra point, computing the L 2 norm, one can achieve the following relation for the signals z 2 (t) and : z 2 (t) L2 L2 = Note that the last relation is consistent when compared to γ a = V. CONCLUSIONS The presented results allow to deal with the H control design problem of uncertain systems where the size of the time-delay is a fundamental issue with respect to stability and/or performance of the closed loop system. However, the fundamental question about the applicability of the results is concerned with numeric characteristics of the approach adopted. Other related problems as multiple time-delays, memory control, distributed delay, energy-to-peak performance or filter design could be analyzed under the proposed tools, with minor adaptations. VI. ACKNOWLEDGMENTS This research was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Brazil, under grants 5263/2-3/PQ, for R. M. Palhares, 5263/2-3(NV) for M. C. R. Leles and /9 for P. Ya. Ekel, by CAPES, Brazil, for C. D. Campos and by UNIMONTES for M. F. S. V. D Angelo. VII. REFERENCES [1] V. I. Skorodinskii, Iterational method of construction of Lyapunov-Krasovskii functionals for linear systems with delay, Automation and Remote Control, vol. 51, no. 9, pp , 199. [2] C. E. de Souza and X. Li, Delay-dependent robust H control of uncertain linear state-delayed systems, Automatica, vol. 35, no. 7, pp , [3] Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, International Journal of Control, vol. 74, no. 14, pp , 21. [4] J. H. Kim and H. B. Park, H state feedback control for generalized continuous/discrete time-delay system, Automatica, vol. 35, pp , [5] C. E. de Souza, R. M. Palhares, and P. L. D. Peres, Robust H filtering for uncertain linear systems with multiple time-varying state delays, IEEE Transactions on Signal Processing, vol. 49, no. 3, pp , 21. [6] F. Zheng and P. M. Frank, Robust control of uncertain distributed delay systems with application to the stabilization of combustion in rocket motor chambers, Automatica, vol. 38, pp , 22. [7] E. Fridman and U. Shaked, Stability and H control of systems with time-varying delays, in Proceedings of the 15th IFAC Triennial World Congress, Barcelona, Spain, July 22. [8] E. Fridman and U. Shaked, A descriptor system approach to H control of linear time-delay systems, IEEE Transactions on Automatic Control, vol. 47, no. 2, pp , 22. [9] Y. Y. Cao, Y. X. Sun, and J. Lam, Delay-dependent robust H control for uncertain systems with timevarying delays, IEE Proceedings Control Theory and Applications, vol. 145, no. 3, pp , [1] M. S. Mahmoud, Robust Control and Filtering for Time- Delay Systems, Control Engineering Series. Marcekl Dekker, Inc., New York, 2. [11] E. Fridman and U. Shaked, New bounded real lemma representations for time-delay systems and their applications, IEEE Transactions on Automatic Control, vol. 46, no. 12, pp , 21. [12] L. El Ghaoui, F. Oustry, and M. AitRami, A cone complementarity linearization algorithm for state outputfeedback and related problems, IEEE Transactions on Automatic Control, vol. 42, no. 8, pp , 1997.

6 eplacements x1(t) Time (s) Fig. 1. Time-response of the state variable x 1 (t) for the closed-loop system with K 1 p, in nominal conditions (solid line) and with the four vertices (dashed line) for the constant time-delay τ 1 = 1.1s. 4.5 x γ a = x 1 4 ag replacements Singular Values Freq. ω (rad/s) PSfrag replacements z2(t) Time (s) Fig. 4. Controlled output signal z 2 (t) of system (23) with the control gain K 2 closing the loop considering the disturbance signal. Fig. 2. Singular value plot with the control gain K 2, and considering the four vertices and the nominal model for the time-delay τ 1 =.3s. 1 5 eplacements Time (s) Fig. 3. Disturbance signal,, entering the state space for a time period of.1s.