Scenario optimization with certificates and applications to anti-windup design

摘要

In this paper, we introduce a significant extension, called scenario withcertificates (SwC), of the so-called scenario approach for uncertainoptimization problems. This extension is motivated by the observation that inmany control problems only some of the optimization variables are used in thedesign phase, while the other variables play the role of certificates. Examplesare all those control problems that can be reformulated in terms of linearmatrix inequalities involving parameter-dependent Lyapunov functions. Thesecontrol problems include static anti-windup compensator design for uncertainlinear systems with input saturation, where the goal is the minimization of thenonlinear gain from an exogenous input to a performance output. The maincontribution of this paper is to show that randomization is a useful tool,specifically for anti-windup design, to make the overall approach lessconservative compared to its robust counterpart. In particular, we demonstratethat the scenario with certificates reformulation is appealing because itprovides a way to implicitly design the parameter-dependent Lyapunov functions.Finally, to further reduce the computational cost of this one-shot approach, wepresent a sequential randomized algorithm for iteratively solving this problem.