Robust linear static anti-windup with probabilistic certificates

TL;DR

This paper introduces a probabilistic approach for designing robust static anti-windup controllers in saturated linear systems with uncertainties, utilizing a novel scenario with certificates method and a sequential randomized algorithm.

Contribution

It develops a new scenario with certificates framework for robust anti-windup design, enabling implicit Lyapunov function synthesis and reducing computational complexity.

Findings

Effective robust anti-windup design under probabilistic uncertainties.

The scenario with certificates approach removes convexity restrictions.

Numerical examples demonstrate the method's practicality and efficiency.

Abstract

In this paper, we address robust static anti-windup compensator design and performance analysis for saturated linear closed loops in the presence of nonlinear probabilistic parameter uncertainties via randomized techniques. The proposed static anti-windup analysis and robust performance synthesis correspond to several optimization goals, ranging from minimization of the nonlinear input/output gain to maximization of the stability region or maximization of the domain of attraction. We also introduce a novel paradigm accounting for uncertainties in the energy of the disturbance inputs. Due to the special structure of linear static anti-windup design, wherein the design variables are decoupled from the Lyapunov certificates, we introduce a significant extension, called scenario with certificates (SwC), of the so-called scenario approach for uncertain optimization problems. This extension is of independent interest for similar robust synthesis problems involving parameter-dependent Lyapunov functions. We demonstrate that the scenario with certificates robust design formulation is appealing because it provides a way to implicitly design the parameter-dependent Lyapunov functions and to remove restrictive assumptions about convexity with respect to the uncertain parameters. Subsequently, to reduce the computational cost, we present a sequential randomized algorithm for iteratively solving this problem. The obtained results are illustrated by numerical examples.