Robust MPC via Min-Max Differential Inequalities

TL;DR

This paper introduces a novel, computationally efficient min-max differential inequality framework for tube-based robust MPC applicable to linear and nonlinear systems with time-varying disturbances, avoiding discretization of control policies.

Contribution

It develops a new min-max differential inequality for support functions, enabling the construction of conservative yet tractable tube-based MPC without control policy discretization.

Findings

The framework scales linearly with prediction horizon length.

The approach yields a robust MPC scheme with ellipsoidal tubes based on LMI constraints.

Numerical case study demonstrates effectiveness on a spring-mass-damper system.

Abstract

This paper is concerned with tube-based model predictive control (MPC) for both linear and nonlinear, input-affine continuous-time dynamic systems that are affected by time-varying disturbances. We derive a min-max differential inequality describing the support function of positive robust forward invariant tubes, which can be used to construct a variety of tube-based model predictive controllers. These constructions are conservative, but computationally tractable and their complexity scales linearly with the length of the prediction horizon. In contrast to many existing tube-based MPC implementations, the proposed framework does not involve discretizing the control policy and, therefore, the conservatism of the predicted tube depends solely on the accuracy of the set parameterization. The proposed approach is then used to construct a robust MPC scheme based on tubes with ellipsoidal cross-sections. This ellipsoidal MPC scheme is based on solving an optimal control problem under linear matrix inequality constraints. We illustrate these results with the numerical case study of a spring-mass-damper system.