W e consider the problem of designing feedback compensators for linear, lumped, timeinvariant plants, subject to various frequency and time-domain performance specifications. Our approach is made possible by the development of new algorithms for the constrained optimization of regular, uniformly locally Lipschitz functions in R w. We consider two design situations.

First, we consider the case in which a good nominal plant model is available. We show that complex design problems involving hard bound constraints on the frequency and time domain responses of a multi-input m ulti-output continuous time feedback system can be formulated as constrained optimization problems in H^∞. Since our formulation uses the aifine param etrization of all achievable input-output maps obtained from the stable coprime factors of the plant, the constrained H^∞ problem we obtain involves only convex functions. We use the continuity properties of these functions, as well as approximations of rational functions in H^∞ by polynomials, to show that we can obtain arbitrarily close approximations to a constrained problem in by solving a sequence of constrained non-differentiable optimization problems of increasing dimension in R^N. We present implementable algorithms to obtain solutions to the constrained problems in R^N. These algorithms require only the computation of the largest singular value of matrices of low dimension and the simulation of linear dynamical systems. The design philosophy we thus propose is to reduce complex feedback compensator design problems to high-dimensional convex constrained optimization problems in R^N that lead to high order compensators with good performance. If needed, one can reduce the compensator order once the design is completed.

Second, we consider the case in which a reliable nominal model of the plant is not available. We restrict our attention to discrete time single-input single-output plants (ARMA models) of known order but with unknown parameter vector. We show that, if the plant param eter vector is known to belong to a compact set in R^N, a worst-case optimal compensator design problem can be reduced to that of the minimization of a regular locally uniformly Lipschitz continuous function for which convergent algorithms exist. We show that, under suitable assumptions, given a sequence of decreasing compact uncertainty sets to which the plant param eter vector is known to belong, on-line worst-case sequential redesign of the feedback compensator leads to a uniformly asymptotically stable closed loop system with optimal performance. Next, under the assumption th at a bound on the magnitude of the plant output disturbance is known, we present a new identification scheme th at produces such a sequence of uncertainty sets. Using new persistency of excitation results, we show that under mild assumptions on the frequency spectra of the signals external to the loop, the uncertainty sets generated by the identification scheme converge to a small set, provided that the compensator is not updated too often. Furthermore, we show that with appropiate assumptions on the probability distribution of the plant output disturbance, this small set is reduced to a point i.e., the actual parameter vector, with probability 1