On Complexity Bounds for the Maximal Admissible Set of Linear Time-Invariant Systems

TL;DR

This paper introduces two efficient methods to estimate upper bounds on the complexity of the maximal admissible set for linear time-invariant systems, aiding in faster computation and resource planning.

Contribution

It presents algebraic and geometric methods to compute upper bounds on MAS complexity, extending to systems with constant inputs, which was previously unknown.

Findings

The methods provide rigorous upper bounds on MAS complexity.

Numerical comparison demonstrates the effectiveness of the two approaches.

Extension to systems with constant inputs broadens applicability.

Abstract

Given a dynamical system with constrained outputs, the maximal admissible set (MAS) is defined as the set of all initial conditions such that the output constraints are satisfied for all time. It has been previously shown that for discrete-time, linear, time-invariant, stable, observable systems with polytopic constraints, this set is a polytope described by a finite number of inequalities (i.e., has finite complexity). However, it is not possible to know the number of inequalities apriori from problem data. To address this gap, this contribution presents two computationally efficient methods to obtain upper bounds on the complexity of the MAS. The first method is algebraic and is based on matrix power series, while the second is geometric and is based on Lyapunov analysis. The two methods are rigorously introduced, a detailed numerical comparison between the two is provided, and an extension to systems with constant inputs is presented. Knowledge of such upper bounds can speed up the computation of MAS, and can be beneficial for defining the memory and computational requirements for storing and processing the MAS, as well as the control algorithms that leverage the MAS.