Stability and Performance Verification of Optimization-based Controllers

TL;DR

This paper introduces a sum-of-squares programming method to verify stability and performance of various optimization-based controllers for polynomial discrete-time systems, accommodating practical control scenarios.

Contribution

It develops a unified SOS-based framework to verify stability and performance of optimization-based controllers, including those with inexact models and soft constraints.

Findings

Applicable to a wide range of practical control problems.

Can verify stability and performance in stochastic and deterministic settings.

Handles controllers with fixed iteration solutions of first-order methods.

Abstract

This paper presents a method to verify closed-loop properties of optimization-based controllers for deterministic and stochastic constrained polynomial discrete-time dynamical systems. The closed-loop properties amenable to the proposed technique include global and local stability, performance with respect to a given cost function (both in a deterministic and stochastic setting) and the $\mathcal{L}_2$ gain. The method applies to a wide range of practical control problems: For instance, a dynamical controller (e.g., a PID) plus input saturation, model predictive control with state estimation, inexact model and soft constraints, or a general optimization-based controller where the underlying problem is solved with a fixed number of iterations of a first-order method are all amenable to the proposed approach. The approach is based on the observation that the control input generated by an optimization-based controller satisfies the associated Karush-Kuhn-Tucker (KKT) conditions which, provided all data is polynomial, are a system of polynomial equalities and inequalities. The closed-loop properties can then be analyzed using sum-of-squares (SOS) programming.