A linear matrix inequality approach to peak-to-peak gain minimization

In this paper we take a new approach to the problem of peak-to-peak gain minimization (the L1 or induced L∞ problem). This is done in an effort to circumvent the complexity problems of other approaches. Instead of minimizing the induced L∞ norm, we minimize the * -norm, the best upper bound on the induced L∞ norm obtainable by bounding the reachable set with inescapable ellipsoids. Controller and filter synthesis for * -norm minimization reduces to minimizing a continuous function of a single real variable. This function can be evaluated, in the most complicated case, by solving a Riccati equation followed by an LMI eigenvalue problem. We contend that synthesis is practical now, but a key computational question-is the function to be minimized convex?—remains open. The filters and controllers that result from this approach are at most the same order as the plant, as in the case of LQG and H∞ design.