The authors work out a framework for evaluating the performance of a continuous-time nonlinear system when this is quantified as the maximal value at an output port under bounded disturbances - the disturbance problem. This is useful in computing gain functions and L ∞-induced norms, which are often used to characterize performance and robustness of feedback system. The approach is variational and relies on the theory of viscosity solutions of Hamilton-Jacobi equations. Convergence of Euler approximation schemes via discrete dynamic programming is established. The authors also provide an algorithm to compute upper bounds for value functions. Differences between the disturbance problem and the optimal control problem are noted, and a proof of convergence of approximations schemes for the control problem is given. Case studies are presented which assess the robustness of a feedback system and the quality of trajectory tracking in the presence of structured uncertainty.
Bibliographical noteFunding Information:
Manuscript received May 30, 1997; revised August 1, 1998. Recommended by Associate Editor, K. Zhou. This work was supported in part by AFOSR under Grant AF/F49620-96-1-0094 and NSF under Grant NSF/ECS-9016050. I. J. Fialho is with the Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455 USA. T. T. Georgiou is with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: firstname.lastname@example.org). Publisher Item Identifier S 0018-9286(99)04529-8.