Marcus Pantoja da Silva 1 and Celso Pascoli Bottura 2. Abstract: Nonlinear systems with time-varying uncertainties

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1 A NEW PROPOSAL FOR H NORM CHARACTERIZATION AND THE OPTIMAL H CONTROL OF NONLINEAR SSTEMS WITH TIME-VARING UNCERTAINTIES WITH KNOWN NORM BOUND AND EXOGENOUS DISTURBANCES Marcus Pantoja da Silva 1 and Celso Pascoli Bottura 2 1 UNICAMP, CAMPINAS, BRAZIL, marcuspantoja@yahoo.com.br 2 UNICAMP, CAMPINAS, BRAZIL, bottura@dmcsi.fee Abstract: Nonlinear systems with time-varying uncertainties with known norm bound and exogenous disturbances are investigated in this work. Conditions for the determination of the H norm of this class of systems are obtained in the form of a convex optimization problem in terms of LMIs. It is also proposed a H optimal control design that aims to stabilize a class of nonlinear systems with time-varying uncertainties with known norm bound and exogenous disturbances and minimize its H norm. The optimal gain matrix is obtained by solving a convex optimization problem in terms of LMIs such as its H norm is minimized. Numerical examples are proposed for both methods. Keywords: Nonlinear Systems, Optimal Control, H norm. 1. INTRODUCTION During the last decades, many authors [2, [5, paid attention to the problem of optimal control for linear systems with norm bounded time varying uncertainties, where the goal is to design a controller that stabilizes the system. However, the modeling of a real process often results in a nonlinear model with time-varying uncertainties and exogenous disturbances associated. The time-varying uncertainties addressed in this work are modeled as being norm bounded with known limits. Several problems can be included in this class of systems like power systems, design of spacecraft, vehicle control, etc [8. In this work we treat the problem of H performance of nonlinear systems with time-varying uncertainties with known norm bound and exogenous disturbances. A new characterization of H norm is proposed in terms of Linear Matrix Inequalities (LMIs) for this class of systems. We also develope a method for the H optimal control of nonlinear systems with time-varying uncertainties and norm bounded with known limits of uncertainty. For the controller design, we use the Lyapunov second method with which we characterize the H control problem and thus construct a convex optimization problem in terms of LMIs which results in a controller that stabilizes the system and at the same time minimizes its H norm. For having the optimal H in a LMI format we utilize a similarity transformation as well as the Schur complement [9. The LMI conditions obtained are used to obtain the optimal gain matrix. Since the proposed method does not involve any non-convex restrictions, we can guarantee that the method has a global optimum solution in the set of feasible solutions. The proposed method has several advantages, among which we highlight the fact that the methodology is flexible because it allows the inclusion of additional design parameters such as structure and gain matrix limiting, and in addition the controller obtained is linear, which makes its implementation simpler. The paper is organized as follows: in the next section, we develop a method for characterizing the H norm of nonlinear systems with time-varying uncertainties with known norm bound and exogenous disturbances in terms of LMIs. We also present numerical examples that illustrate its application. In section III we develop a method for H optimal control for the class of systems discussed previously in terms of LMIs. In Section IV we present numerical examples that illustrate their application. In section V final comments on the proposed method and on the results obtained are made. Notation Throughout this paper, capital letters denote matrices and small letters denote vectors. For simmetric matrices, M> ( ) indicates that M is positive definite (semipositive definite). M T represents the transpose of matrix M. I and represent identity matrices and zero matrices with appropriate dimensions, respectively. Matrices, if not explicitly stated, are assumed to have appropriate dimensions. 2. NONLINEAR CONTINUOS-TIME SSTEMS H NORM CHARACTERIZATION: A PROPOSAL One of the reasons because the H norm characterization of a system is important is the study of its robustness in rela- Serra Negra, SP - ISSN

2 tion to exogenous disturbances. The robustness with respect to exogenous disturbances can be expressed quantitatively in terms of the limits of the H norm. In this section we will study the H performance of the following continuos-time nonlinear system: ẋ = A Bw(t)+h(t, x) C Dw(t) (1) where x(t) R n is the state of the system, A is a matrix n n, B has dimension n m, C is a matrix l n, D has dimension l m, w(t) R m is the exogenous disturbance, h : R n+1 R n is the uncertain time-variant nonlinear term. Regarding h(t, x), we consider that it is norm bounded [1 as: h(t, x) T h(t, x) α 2 x T H T Hx (2) where the uncertainties limits: the scalar α and the matrix H are known. Knowing that w is an limited energy signal, then + w T (τ)w(τ)dτ < (3) the H norm may be characterized by the lower value of γ [1 such that y(t) 2 γ w(t) 2 (4) where. 2 is the Euclidian norm. The H norm of a system with transfer matrix G(s) can also be characterized by G(s) <γ y(t) T y(t) <γ 2 w T (t)w(t) (5) For a stable system the H norm can be characterized by a Lyapunov function V (x) =x T Px, P = P T > imposing V (x)+y T y γ 2 w T w< (6) We note that V (x) is expressed by Raplacing (7) in (6) we have V (x) =ẋ T Px+ x T P ẋ (7) ẋ T Px+ x T P ẋ + y T y γ 2 w T w< (8) Now substituting the equation of system (1) in (8) we have x T (A T P + PA+ C T C) w T (B T P + D T C) +x T (PB + C T D)w + w T (D T D γ 2 I)w + +h T Px+ x T Ph <, P> (9) We use the following lemma to obtain a quadratic form in the state for h T Px+ x T Ph Lemma 1. For any matrices (or vectors) X and with appropriate dimensions, we have the following inequality: X T + T X X T X + T (1) Proof. Since X and are matrices or vectors of appropriate dimensions we have It follows that ( X) T I( X) T T X X T + X T X X T + X T X T X + T (11) h T Px+ x T Ph h T h + x T PPx x T (α 2 H T H + PP)x (12) Now we have x T (A T P + PA+ C T C + PP + α 2 H T H) +w T (B T P + D T C) +x T (PB + C T D)w + w T (D T D γ 2 I)w < P> (13) that can be written as P>, [ T [ x S w B T P + D T C [ PB + C T D x D T D γ 2 < (14) I w where S = A T P + PA+ C T C + PP + α 2 H T H. Using the Schur complement formula [9, (14) can be written as P>, X PB + C T D P B T P + D T C D T D γ 2 I P I < (15) where X = A T P + PA+ C T C + α 2 H T H. Assuming that α and H are known we minimize γ solving the following LMI problem in P and γ min µ s.t. P = P T >, X B T P + D T C P PB + C T D P D T D µi < (16) I where γ = µ. If problem (16) is feasible, then its solution minimizes H norm for the system (1). Serra Negra, SP - ISSN

3 2.1. Examples To illustrate the proposed method for H norm characterization for nonlinear systems we will use the following examples Example 1. We consider the nonlinear system described by the following equations: ẋ = w(t)+h(t, x) where α =2and H = I. w(t) (17) In this paper all examples are solved using a computer with an AMD Turion 64 X2 (2. GHz) processor with 2 GB of RAM (8 MHz) with operational system Windows Vista 32-bits. The solver used is the SeDuMI [3 (Self-Dual- Minimization) interfaced by ALMIP [4 (et Another LMI Parser) both running in Matlab 7.. Using the method proposed in (16) to find the minimum H norm for the system (17) of example 1 we find that γ =4.878 and P = 1 9 Example 2. Consider the system [ ẋ = h(t, x) [ where α =1.5 and H = I. [ [ w(t) w(t) (18) Using again the method proposed in (16) to determine the minimum H norm we find that γ = and [ P = for the system (18) of example 2 3. H OPTIMAL CONTROL DESIGN In this section, we consider the optimal H control problem for the following nonlinear system with time-varying uncertainties and exogenous disturbances ẋ = A B u u + B w w(t)+h(t, x) C Dw(t) (19) where u R m is the control input, w(t) R q is the exogenous disturbance. A R n n, B u R n m, B w R n m, C R l n and D R l m are know constants matrices, h : R n+1 R n is the uncertain time-variant nonlinear term satisfying (2). Let us consider the problem of finding a linear controller for the system (19) that at the same time stabilizes the system and minimizes the H norm. Applying the linear feedback control law u = Kx (2) where K is a constant matrix, to the open-loop system (19), we obtain the closed-loop system ẋ =(A + B u K) B w w(t)+h(t, x) C Dw(t) (21) We consider that pair (A, B u ) is controllable. For a stable system the H norm can be characterized by a Lyapunov function V (x) =x T Px, P = P T > imposing (6). Now replacing (21) in (6) and using Lemma 1 and equation (2) we have x T (A T P + PA+ K T Bu T P + PB u K + PP + α 2 H T H) w T (BwP T + D T C)x +x T (PB w + C T D)w + w T (D T D γ 2 I)w < (22) that can be written as P>, [ T [ x X w BwP T + D T C [ PB w + C T D x D T D γ 2 I w < (23) where X = A T P + PA + K T B T u P + PB u K + PP + α 2 H T H. Multiplying (23) on the left and on the right by [ P 1 I (24) where P 1 =, and making the change of variable proposed in [5 L = K so (25) K = L 1 (26) Serra Negra, SP - ISSN

4 we have [ Z Bw T + D T C B w + C T D D T D γ 2 < (27) I where Z = A T + A + L T B T u + B u L + I + (C T C + α 2 H T H). Relying on the Schur complement formula [9, (27) can be written as Bw T + D T C D T D γ 2 I < (28) where W = A T +A +L T B T u +B u L+I. The minimum H norm can be computed using a convex optimization procedure Define µ = γ 2 min µ s.t. = T > Bw T + D T C D T D µi < (29) The H norm procedure stabilizes the system (21) and minimizes its H norm if the optimization problem (29) is feasible. We should note, however, that the procedure outlined in (29) is aimed only at minimizing the H norm, with no restritions on the size of the gain. To limit the gain, we apply the following modifications of the optimization problem [1: min µ + k L + k s.t. = T > Bw T + D T C D T D γ 2 I < [ kl I L T < L I [ k I I > (3) I where k and k L are constraints on the magnitudes of gain K, satisfying [11: L T L<k L I, 1 <k I (31) 4. NUMERICAL EXAMPLES In this section numerical examples of H optimal control design illustrating the proposed method are presented. Example 3. Consider the system ẋ = u w(t)+h(t, x) where α =2and H = I. w(t) (32) Using the design method proprosed in (3) for the system (32) of the example 3 such that it be stable and have the lowest possible H norm we obtain γ =.949, = and the optimal gain matrix is: K = [ Example 4. [ [ ẋ = [ w(t)+h(t, x).7621 [ where α =1.5 and H = I. [ u w(t)(33) Again solving the problem (3) for the system (33) of the example 4 such that it be stable and have the lowest possible H norm we obtain γ =1.5482, = [ Serra Negra, SP - ISSN

5 and the optimal gain matrix: 5. CONCLUSIONS K = [ In this paper we proposed a new method for characterization of the H norm for nonlinear systems with time-varying uncertainties with known norm bound and exogenous disturbances through the construction of a convex optimization problem using LMIs. In addtion we also proposed a H optimal control design method for the same type of system ensuring stability and the lowest H norm possible. Numerical examples were presented to illustrate both proposed methods. [9 Boyd, S., Ghaoui, L., Feron, E. and Balakrishnan, V., 1994, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA SIAM Studies in Applied Mathematics [1 Kemin Zhou, John C. Doyle and Keith Glover, 1995, Robust and Optimal Control, Prentice Hall. [11 Chen, G.,. L., and Chu, J., Decentralized stabilization of large-scale linear systems with time-delay. Proceedings of the 14th IFAC Congres, 1(1), May, pp ACKNOWLEDGMENTS This work has been supported by grants from "Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES - Brazil". REFERENCES [1 D. M. Stipanovic and D.D. Siljak, 2, Robust Stabilization of Nonlinear Systems: The LMI Approach. Mathematical Problems in Engineering, vol. 6, n. 1, pp [2 K. Zhou and P.P. Khargonekar, 1988, Robust Stabilization of Linear Systems with norm bounded time varying uncertainty, Systems Control Letters, vol. 1, n. 1, pp [3 Sturm, J.F., 1999, Using SeDuMi 1.2, a MAT- LAB toolbox for optimization over symmetric cones, Optim. Methods Softw., vol. 1, n. 1, pp , [4 Lofberg, J., 24, ALMIP: A toolbox for modeling and optimization in MATLAB, Proc. 24 IEEE Int. Symp. Computer Aided Control Systems Design, Taipei, Taiwan, vol. 1, pp [5 Bernussou, J., Peres, P., and Geromel, J. C., 1989, A linear programming oriented procedure for quadratic stabilization of uncertain systems, Systems and Control Letters vol. 6, n. 1, pp [6 Geromel, J. C. Bernussou, J, and Peres, P. L. D., 1994, Decentralized control through parameter space optimization, Automatica, vol. 3, n. 1, pp [7 De Oliveira, M.C., Geromel, J.C. and Bernussou, J., 22, Extend H 2 and H norm characterization and controller parametrizations for discrete-time systems. Int. J. Control, vol 75, n. 9, pp [8 D. D. Siljak, 1991, Decentralized Control of Complex System, Academic Press, Boston, MA. Serra Negra, SP - ISSN