Second-order nonsmooth optimization for H ∞ synthesis

Abstract

The standard way to compute H ∞ feedback controllers uses algebraic Riccati equations and is therefore of limited applicability. Here we present a new approach to the H ∞ output feedback control design problem, which is based on nonlinear and nonsmooth mathematical programming techniques. Our approach avoids the use of Lyapunov variables, and is therefore flexible in many practical situations.

Control design in the time and frequency domain using nonsmooth techniques

“…See [10,8] for stabilization problems, [4,5,20,3,11] for H ∞ synthesis, and [3,6] for design with IQCs. These techniques are in our opinion a valuable addition to the designer's toolkit:…”

Robust control via sequential semidefinite programming

“…The difficulty in (12) or (14) can be gauged by the fact that computation of µ ∆ , and therefore also r ∆ , is NP-hard (in the state-space dimension of F (P, K)), so that solving program (14) frontally is out of the question. Realistically, we may just be able to perform robustness analysis of a given controller over Q at reasonable cost, i.e., check whether r ∆ (F (P, K)) > 1.…”

An approximate Newton method for solving non-smooth equations with infinite max functions

“…Theorem 10 Let x * be a solution of (1) with F in the form (10). Assume that F is semismooth in a neighborhood of x * and that ∂F (x * ) is nonsingular.…”

“…Moreover, there is nonsmooth infinite max-operator in optimization programs which arise in automatic control applications for H ∞ controller synthesis (see e.g. Bompart, Noll and Apkarian, [1]). So, assume that the function F in (1) has the following form…”