Computation of safe disturbance sets using implicit RPI sets

TL;DR

This paper introduces a method to compute disturbance sets for stable LTI systems that closely match output constraints using implicit RPI sets, enhancing efficiency and reducing conservativeness.

Contribution

It develops an implicit RPI set-based approach with a novel disturbance parameterization to optimize disturbance sets for output constraint matching.

Findings

Reduces conservativeness of disturbance sets.

Improves computational efficiency over existing methods.

Provides theoretical guarantees on approximation errors.

Abstract

Given a stable linear time-invariant (LTI) system subject to output constraints, we present a method to compute a set of disturbances such that the reachable set of outputs matches as closely as possible the output constraint set, while being included in it. This problem finds application in several control design problems, such as the development of hierarchical control loops, decentralized control, supervisory control, robustness-verification, etc. We first characterize the set of disturbance sets satisfying the output constraint inclusion using corresponding minimal robust positive invariant (mRPI) sets, following which we formulate an optimization problem that minimizes the distance between the reachable output set and the output constraint set. We tackle the optimization problem using an implicit RPI set approach that provides a priori approximation error guarantees, and adopt a novel disturbance set parameterization that permits the encoding of the set of feasible disturbance sets as a polyhedron. Through extensive numerical examples, we demonstrate that the proposed approach computes disturbance sets with reduced conservativeness improved computational efficiency than state-of-the-art methods.