Stability and stabilization of 2D continuous state-delayed systems

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1 20 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 2-5, 20 Stability and stabilization of 2D continuous state-delayed systems Mariem Ghamgui, Nima Yeganefar, Olivier Bachelier and Driss Mehdi Abstract In this paper, we consider the problem of stability and stabilization of 2D continuous systems with state delays. The asymptotic stability of this class of systems described by the Roesser model is addressed via Lyapunov techniques. It is shown that linear matrix inequalities (LMIs) can be used to check the asymptotic stability of 2D linear delayed systems and this is applied to the case of state feedback stabilization. A numerical example is introduced to show the efficiency of the proposed criterion for a 2D linear delayed system. Keywords: 2D Roesser model, delayed systems, stabilization, Lyapunov-Krasovskii functional, LMI. I. INTRODUCTION As usual in the development of science, the theory of multidimensional systems (n-d systems) emerged from the growing complexity of modern technology particularly in image and signal processing, coding/decoding, filtering, etc. (for an overview, see 3). Another interesting class of systems which appeared recently is the repetitive processes (or multipass processes) such as longwall coal cutting or metal rolling operations that can be modeled as n-d systems 4. It was more than enough to catch the interest of the scientific community both in engineering and mathematical areas as the growing number of recent studies shows and it is not a surprise that the theory is now advancing faster than the applications. The field is not recent though. The first studies on multidimensional systems started in the early 70s with the introduction of two well known in the multidimensional community state space models to describe nd systems; the so called Roesser and Fornasini-Marchesini models 3, 5 still used today both for discrete systems. It is easy to show that both models are equivalent; one can be transformed into another with a simple state substitution (see for instance ). More recently, 2 introduced the first hybrid system based on an extension of M. Ghamgui, N. Yeganefar, O. Bachelier and D. Mehdi are with the University of Poitiers, LAII-ENSIP, Bâtiment B25, 2 rue Pierre Brousse, B.P. 633, Poitiers CEDEX, France, Mariem.Ghamgui@etu.univ-poitiers.fr, Nima.Yeganefar@univ-poitiers.fr, Olivier.Bachelier@univ-poitiers.fr, Driss.Mehdi@univ-poitiers.fr the Roesser model adding a continuous part to the original discrete system. In this paper, our attention is focused on the continuous part of this model, generalized to the delay case (for a survey on time delay systems, the reader can refer to 7, 6, 5. Delayed multidimensional systems have been recently introduced but in the majority of the existing studies only the discrete case have been analyzed (see e.g. 2, 6, 0) except for a few recent papers that inspired our work, 9 a Lyapunov approach is applied to continuous Roesser models. The problem of stability/stabilization is addressed by the above authors but the results given in terms of LMIs (linear matrix inequalities, see 4) are all delay independent. The aim of this work is to give less conservative results to design a state-feedback controller for 2D delayed systems which stabilizes the system. The paper is organized as follows. In section II, we introduce the mathematical background we need to address the problem. In section III, we introduce our main results: a sufficient condition to check the asymptotic stability and the stabilization of a 2D delayed system. Our approach is based on Lyapunov techniques; stability and stabilization criteria are given in form of LMIs. Finally section IV presents an illustrative example. II. PROBLEM FORMULATION The class of 2-D systems with delays under consideration is represented by an extension of the Roesser model with (constant) state delays (see 3 and 2) of the form: h (t,t 2) A A v (t,t 2) = 2 x h (t,t 2 ) A 2 A 22 x v () (t,t 2 ) 2 Ad A + 2d x h (t τ,t 2 ) B x v + u(t (t,t 2 τ 2 ),t 2 ) A 2d A 22d B 2 x h (t,t 2 ) is the horizontal state in IR n h, x v (t,t 2 ) is the vertical state in IR nv, u(t,t 2 ) is the control vector in IR m, τ and τ 2 are the delays in horizontal and vertical directions respectively and A ii, A iid and B i are real constant matrices of appropriate dimensions. The initial conditions are //$ IEEE 878

2 given by x h (θ,t 2 ) = f(θ,t 2 ), t 2 and τ max < θ < 0 x v (t,θ) = g(t,θ), t and τ 2max < θ < 0 f and g are continuous functions. For such a system we denote and x x(t,t 2) h (t,t 2) x v, (t,t 2) x x(t τ,t 2 τ 2) h (t τ,t 2) x v (t,t 2 τ 2) ẋ(t,t 2) h (t,t 2 ) v (t,t 2 ) 2 A A A = 2 Ad A,A A 2 A d = 2d B,B = 22 A 2d A 22d B 2 which allows us to write () in the usual form ẋ(t,t 2) =Ax(t,t 2)+A d x(t τ,t 2 τ 2) +Bu(t,t 2) (2) Consider the state feedback control: the matrix u(t,t 2) = Kx(t,t 2) (3) K = K K 2 is the state feedback gain to be determined. The problem is then to compute a static feedback control given by (3) such that the closed-loop 2D system () is asymptotically stable. A. Stability criteria III. MAIN RESULTS In this section, we investigate stability conditions for the 2D delayed system (). Theorem : If there exist symmetric bloc diagonal matrices P > 0, Q > 0 and R > 0 such that the following LMI AT P +PA+Q R PA d +R τa T R Q R τa T dr 0 R (4) τi τ = nh 0, 0 τ 2I nv P 0 Q 0 R 0 P =,Q =,R = 0 P 2 0 Q 2 0 R 2 holds true, then the 2D linear delayed system () is asymptotically stable. Proof: Proof of theorem is given in the appendix VI-A. B. Stabilization The objective of this section is the design of a stabilizing state-feedback controller for system (). Using the state-feedback control (3), () can be rewritten as: h (t,t 2 ) = A x h (t,t 2) + A 2x v (t,t 2) + B K x h (t,t 2) +B K 2x v (t,t 2) + A d x h (t τ,t 2) + A 2d x v (t,t 2 τ 2) v (t,t 2 ) 2 = A 2x v (t,t 2) + A 22x v (t,t 2) + B 2K x v (t,t 2) +B 2K 2x v (t,t 2) + A 2d x v (t τ,t 2) + A 22d x v (t,t 2 τ 2) or equivalently, ẋ(t,t 2) = A cx(t,t 2)+A d x(t τ,t 2 τ 2) (5) : A c = (A +B K ) (A 2 +B K 2) (A 2 +B 2K ) (A 22 +B 2K 2) Ad A A d = 2d A 2d A 22d Theorem 2: If there exist symmetric bloc diagonal matrices X > 0, Q > 0 and Y > 0 such that the following LMI XAT + AX + Y T B T + BY + Q X A d X + X Q X τxa T + τy T B T τxa T d X 0 (6) with X = P, Y = KX and Q = P QP is verified, then (5) is asymptotically stable. The gains K and K 2 of the controller law (3) are given by K = Y X,K 2 = Y 2X 2 (7) Proof: Proof of theorem 2 is given in the appendix VI-B. IV. EXAMPLE In order to show the applicability of our results, consider a2d continuous system represented by () with: A =,A =, A 2 =,A =, 0.6 B =,B 0 2 = 0 0. The delayed matrices are given by: A d =,A d =, A 2d =,A d = For τ 2 = 0. and τ =.2, the solutions are given by: P =,P =, Q =,Q = The stabilizing control law (5) is given by (7): K =,K =

3 Fig.. The state evolution of the first component of x h (t,t 2 ) Fig. 2. The state evolution of the second component of x h (t,t 2 ) For τ 2 = 0. and τ = 0.2, we have: P =,P =, Q =,Q = and K = ,K = Now, in both cases, injecting the values of K and K 2 back into the system, it is possible to check that the closed loop system is asymptotically stable using theorem 2. But it is also nice to have a look at the evolution of the components of states x h (t,t 2) and x v (t,t 2) in 3D graphics (Fig., Fig. 2, Fig. 3 and Fig. 4). The curves are obtained using a simple discretization method (Euler type). We have used the sampling periods T h = 0.0s and T v = 0.0s and the following initial conditions: x h (θ,t 2) =, t 2 2 and 0.2 < θ < 0 x v 3 (t,θ) =, t and 0. < θ < 0 Fig. 3. The state evolution of the first component of x v (t,t 2 ) Note that i and j presented in these figures are the indexes of the discretized times t and t 2 respectively (t = it h, t 2 = jt v). Both vertical and horizontal states quickly converge to the origin. This can be better seen in the next set of graphs. Indeed, Fig. 5 shows the evolution of the horizontal state x h (t,t 2) for a fixed value of t 2 equal to 6s. Same goes for Fig. 6 which represents the evolution of the horizontal state x h (t,t 2) for t = 8s. V. CONCLUSION To conclude, let us highlight the general contribution of this paper. We first developed a sufficient condition of asymptotic stability for 2D continuous systems with state delays. This system is based on the generalization of the (discrete) commonly used Roesser state space model. Using Lyapunov approach, we proposed the synthesis of a state feedback controller. In order to design the controller, it is necessary to solve an LMI. It is also important to note that this is the first time a delay dependant criteria is proposed (see introduction) thus leading to less conservative results. A numerical example is provided to illustrate the results. Fig. 4. The state evolution of the second component of x v (t,t 2 ) 880

4 State vector xh(t,6) t(s) Fig. 5. The state evolution of the first component of x h (t,t 2 ) at time instant t 2 = 6s State vector xh(8,t2)) t2(s) Fig. 6. The state evolution of the first component of x h (t,t 2 ) at time instant t = 8s (what we refer to by the trajectory of () in the sequel) is given by: ζ V(x(t,t 2)) = ( V) T V V ζ(t,t 2) = ζ(t,t 2) h v = V(t,t2) h (t,t 2) + V(t,t2) v (t,t 2) h (t,t 2) v (t,t 2) 2 = V(t,t2) h (t,t 2) h (t,t 2) + V2(t,t2) v (t,t 2) v (t,t 2) 2 (0) V is the gradient of the function V. Remark : In 9, the authors introduce the notion of unidirectional derivative. In this paper we refer to this derivation as the derivative of the function V along the vector ζ(t,t 2) defined by (9). In 9, the authors also proved that using this derivation, the usual Lyapunov theorem is true (roughly speaking V < 0 implies asymptotic stability). Computing the derivative of V (t,t 2) along the trajectories of () gives: V (t,t 2 ) h (t,t 2 ) h (t,t 2 ) = ( ) h T ( ) (t,t 2 ) P x h (t,t 2) + x ht (t h (t,t 2)P,t 2 ) +x ht (t,t 2)Q x h (t,t 2) x ht (t τ,t 2)Q x h (t τ ( ),t 2) + h T ( ) (t,t 2 ) (τ 2 R) h (t,t 2 ) ( ) t h T ( (θ,t 2 ) t τ (τ h (θ,t R ) 2 ) )dθ Lemma (8): For any constant matrix W IR n n,w = W T 0, scalar γ > 0, and vector function ẋ : γ,0 IR n such that the following integration is well defined, then γ 0 γ ẋt (t + ξ)wẋ T (t + ξ)dξ x T (t) x T (t γ) W W x(t) W W x(t γ) Using lemma, we obtain ( ) t h T ( t τ (θ,t2)(τ R ) h )(θ,t 2)dθ x h T (t,t 2) R R x h (t,t 2) x h (t τ,t 2) R R x h (t τ,t 2) A. Proof of Theorem Proof: Let us define VI. APPENDIX V(x(t,t 2)) = V (t,t 2) + V 2(t,t 2) (8) as a possible LK functional candidate for the system () with: V (t,t 2) = x ht (t,t 2)P x h (t,t 2) + t t τ x ht (θ,t 2)Q x h (θ,t 2)dθ + ( ) t t τ (τ t + θ) h T ( (θ,t 2 ) (τ h (θ,t R ) 2 ) )dθ ζ(t,t 2) = h (t,t 2) v (t,t 2) 2 (9) Let J = I I, then R R = J T ( R R R )J Hence ( t h ) T ( t τ (θ,t2)(τ R ) h )(θ,t 2)dθ x h T (t,t 2) x h J T x h (t ( R,t 2) )J (t τ,t 2) x h (t τ,t 2) Now, let us introduce the augmented state ξ as: x h (t,t 2) x(t ξ(t,t 2) =,t 2) = x v (t,t 2) x(t τ,t 2 τ 2) x h (t τ,t 2) x v (t,t 2 τ 2) V 2(t,t 2) = x vt (t,t 2)P 2x v (t,t 2) + t 2 t 2 τ x vt (t 2,θ)Q 2x v (t,θ)dθ With this in mind, we can write + ( t v ) ( (t,θ) T t 2 τ (τ 2 t 2 + θ) h (t 2 (τ2r 2 2),θ) ( ) )dθ h T ( 2 (t,t 2)(τ 2 R) h )(t,t 2) = A T A T T The derivative of function V(x(t,t 2)) along the vector ξ(t,t 2) T A T 2 A T (τ2 R) A T 2 ξ(t,t 2) d A T 2d 88 A T d A T 2d To make the proof easier to follow for the reader, we will only explicit the calculations in one dimension, the horizontal one, as the expressions are similar in the vertical one.

5 Then, following the same logic with the other dimension, we can show that Θ = V(t,t 2 ) ξ T (t,t 2 )Θξ(t,t 2 ) A T P + PA + Q R + A T (τ 2 R)A which can be written as with A T d P + R + AT d (τ2 R)A PA d + A T (τ 2 R)A d + R Q R + A T d (τ2 R)A d Θ Θ 2 Θ 3 Θ 4 Θ Θ = 22 Θ 23 Θ 24 Θ 33 Θ 34 Θ 44 0 Θ = A T P + PA + Q R + AT (τ2 R)A + AT 2 (τ2 2 R2)A2 Θ 2 = P A 2A T 2 P2 + AT (τ2 R)A2+ A T 2 (τ2 2 R2)A22 Θ 3 = P A d + A (τ 2 R)A d + R + A T 2 (τ2 2 R2)A 2d Θ 4 = P A 2d + A (τ 2 R)A 2d+ Γ = A T 2 (τ2 2 R2)A 22d Θ 22 = A T 22 P2 + P2A22 + Q2 R2 + AT 2 (τ2 R)A2 + AT 22 (τ2 2 R2)A22 Θ 23 = P 2A 2d + A T 2 (τ2 R)A d + A T 22 (τ2 2 R2)A 2d Θ 24 = P 2A 22d + R 2 + A T 2 (τ2 R)A 2d + A T 22 (τ2 2 R2)A 22d Θ 33 = A T d (τ2 R)A d R Q + A T 2d (τ2 2 R2)A 2d Θ 34 = A T d (τ2 R)A 2d + A T 2d (τ2 2 R2)A 22d Θ 44 = R 2 Q 2 + A T 2d (τ2 R)A 2d+ A T 22d (τ2 2 R2)A 22d As we still have nonlinear terms in these last inequalities, we need to apply the Schur complement twice. After the first one, we have φ φ 2 φ 3 φ 4 τ A T R φ 22 φ 23 φ 24 τ A T 2 Θ = R φ 33 φ 34 τ A T d R φ 44 τ A T 2d R 0 () R with φ = Γ +A T 2(τ 2 2R 2)A 2 φ 2 = Γ 2 +A T 2(τ 2 2R 2)A 22 φ 3 = Γ 3 +A T 2(τ 2 2R 2)A 2d φ 4 = Γ 4 +A T 2(τ 2 2R 2)A 22d φ 22 = Γ 22 +A T 22(τ 2 2R 2)A 22 φ 23 = Γ 23 +A T 22(τ 2 2R 2)A 2d φ 24 = Γ 24 +A T 22(τ 2 2R 2)A 22d φ 33 = Γ 33 +A T 2d(τ 2 2R 2)A 2d φ 34 = Γ 34 +A T 2d(τ 2 2R 2)A 22d φ 44 = Γ 44 +A T 22d(τ 2 2R 2)A 22d Γ = (A T +αi nh )P +P (A +αi nh )+Q R Γ 2 = P A 2 +A T 2P 2 Γ 3 = P A d e ατ +R Γ 4 = P A 2d e ατ 2 Γ 22 = (A T 22 +αi nv )P 2 +P 2(A 22 +αi nv )+Q 2 R 2 Γ 23 = P 2A 2d e ατ Γ 24 = P 2A 22d e ατ 2 +R 2 Γ 33 = R Q Γ 34 = 0 Γ 44 = R 2 Q Applying the Schur complement a second time to eliminate the last non linear terms gives Γ Γ 2 Γ 3 Γ 4 τ A T R τ2at 2 R2 Γ 22 Γ 23 Γ 24 τ A T 2 R τ2at 22 R2 Θ = Γ 33 Γ 34 τ A T d R τ2at 2d R2 Γ 44 τ A T 0 2d R τ2at 22d R2 R 0 R 2 (2) which corresponds to the LMI (4) and thus concludes the proof. B. Proof of Theorem2 Proof: Using the same Lyapunov functional as mentioned in the section III, we have Γ = V(t,t 2) ξ T (t,t 2)Γξ(t,t 2) A T c P + PAc + Q R + AT c (τ2 R)A c A T d P + R + AT d (τ2 R)A c PA d + A T c (τ2 R)A d + R Q R + A T d (τ2 R)A d Consider the case R = P, then A T c P + PAc + Q P + AT c (τ2 P)A c A T d P + P + AT d (τ2 P)A c PA d + A T c (τ2 P)A d + P Q P + A T d (τ2 P)A d In view of Schur complement, Γ 0 if there exist real matrices P 0,Q 0 such that AT c P + PAc + Q P PA d + P τa T c P Q P τa T d P 0 (3) P Note that this last condition is bilinear matrix variables P and K and therefore it may be considered as a BMI problem. To obtain the LMI (6), it is necessary to pre- and post-multiply inequality (3) by diag { P,P,P }. REFERENCES M. Benhayoun, A. Benzaouia, F. Mesquine, and F. Tadeo. Stabilization of 2d continuous systems with multi-delays and saturated control. In 8th Mediterranean Conference on Control and Automation, MED 0 - Conference Proceedings, pages , J. Bochniak and K. Galkowski. LMI-based analysis for continuous-discrete linear shift invariant nd-systems. Journal of Circuits, Systems and Computers, 4(2): , N.K. Bose. Multidimensional systems theory and applications. Kluwer Academic Publishers, Dordrecht, The Netherlands, L. Boyd, L.El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, E. Fornasini and G. Marchesini. Doubly-indexed dynamical systems: State-space models and structural properties. Mathematical Systems Theory, 2():59 72, K. Gu, V. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Birkhauser, Boston, USA, J.K. Hale and S.M. Verduyn-Lunel. Introduction to Functional Differential Equations. Springer-Verlag, New York, Q.-L. Han. Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, 4(2):27 276, A. Hmamed, F. Mesquine, F. Tadeo, M. Benhayoun, and A. Benzaouia. Stabilization of 2-d saturated systems by state feedback control. Multidimensional Systems and Signal Process, 2(3): , 200.

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