Minimization of the worst case peak-to-peak gain via dynamic programming: state feedback case

Considers the problem of designing a controller that minimizes the worst case peak-to-peak gain of a closed-loop system. In particular, we concentrate on the case where the controller has access to the state of a linear plant and it possibly knows the maximal disturbance input amplitude. We apply the principle of optimality and derive a dynamic programming formulation of the optimization problem. Under mild assumptions, we show that, at each step of the dynamic program, the cost to go has the form of a gauge function and can be recursively determined through simple transformations. We study both the finite horizon and the infinite horizon case under different information structures. The proposed approach allows us to encompass and improve earlier results based on viability theory. In particular, we present a computational scheme alternative to the standard bisection algorithm, or gamma iteration, that allows us to compute the exact value of the worst case peak-to-peak gain for any finite horizon. We show that the sequence of finite horizon optimal costs converges, as the length of the horizon goes to infinity, to the infinite horizon optimal cost. The sequence of such optimal costs converges from below to the optimal performance for the infinite horizon problem. We also show the existence of an optimal state feedback strategy that is globally exponentially stabilizing and derive suboptimal globally exponentially stabilizing strategies from the solutions of finite horizon problems.

[1]  W. Rudin Principles of mathematical analysis , 1964 .

[2]  Munther A. Dahleh,et al.  State feedback e 1 -optimal controllers can be dynamic , 1992 .

[3]  T. Georgiou,et al.  L2 State-feedback Control with a Prescribed Rate of Exponential Convergence , 1997, IEEE Trans. Autom. Control..

[4]  N. Elia,et al.  Controller design with multiple objectives , 1997, IEEE Trans. Autom. Control..

[5]  M. Sznaier,et al.  Persistent disturbance rejection via static-state feedback , 1995, IEEE Trans. Autom. Control..

[6]  M. Dahleh,et al.  Minimization of the maximum peak-to-peak gain: the general multiblock problem , 1993, IEEE Trans. Autom. Control..

[7]  J. Pearson,et al.  l^{1} -optimal feedback controllers for MIMO discrete-time systems , 1987 .

[8]  Murti V. Salapaka,et al.  Controller design to optimize a composite performance measure , 1995 .

[9]  M. Dahleh,et al.  A necessary and sufficient condition for robust BIBO stability , 1988 .

[10]  Minimax controller design in uniform metric , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[11]  J. B. Pearson,et al.  ℓ1-optimal Control of Multivariable Systems with Output Norm Constraints , 1991, Autom..

[12]  D. Bertsekas,et al.  On the minimax reachability of target sets and target tubes , 1971 .

[13]  J. Shamma Optimization of the l∞-induced norm under full state feedback , 1996, IEEE Trans. Autom. Control..

[14]  J. Shamma Nonlinear state feedback for l 1 optimal control , 1993 .

[15]  Mathukumalli Vidyasagar,et al.  Optimal rejection of persistent bounded disturbances , 1986 .

[16]  Olof J. Staffans The four-block model matching problem in l 1 and infinite-dimensional linear programming , 1993 .

[17]  Peter M. Young,et al.  Infinite-dimensional convex optimization in optimal and robust control theory , 1997, IEEE Trans. Autom. Control..

[18]  Munther A. Dahleh,et al.  Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization , 1988 .

[19]  D. Luenberger Optimization by Vector Space Methods , 1968 .

[20]  N. Elia,et al.  A quadratic programming approach for solving the l1 multiblock problem , 1998, IEEE Trans. Autom. Control..

[21]  Petros G. Voulgaris Optimal H2/l1 control via duality theory , 1995, IEEE Trans. Autom. Control..

[22]  M. Dahleh,et al.  Control of Uncertain Systems: A Linear Programming Approach , 1995 .

[23]  A. Stoorvogel Nonlinear L/sub 1/ optimal controllers for linear systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[24]  M. Khammash,et al.  Performance robustness of discrete-time systems with structured uncertainty , 1991 .

[25]  M. Sznaier,et al.  A solution to MIMO 4-block l/sup 1/ optimal control problems via convex optimization , 1995, Proceedings of 1995 American Control Conference - ACC'95.