The $\epsilon$-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems

TL;DR

This paper introduces the concept of $\epsilon$-capacity for gain matrices in discrete-time linear systems, providing a way to identify controllers that are robust to initial state disturbances.

Contribution

It defines the $\epsilon$-capacity set for gain matrices and proposes an algorithm to determine if a control law belongs to this set, enhancing robustness analysis.

Findings

Characterization of the $\epsilon$-capacity set for gain matrices.

An algorithm to verify if a control law is within the $\epsilon$-capacity set.

Numerical simulations demonstrating the approach.

Abstract

Discrete-time linear systems with perturbed initial state are considered. A disturbance that infects the initial state is said to be $\epsilon$-tolerable if the corresponding output signal is relatively insensitive to their effects. In this paper, we will define a new set that characterize each gain matrix K and the associated feedback control law ui=Kxi, this set will be called the $\epsilon$-capacity of the gain matrix K. The set of all possible gain matrix that makes the system insensitive to all disturbances is noted $\Phi\cdot$. The characterization of $\Phi\cdot$ is investigated, and we propose an algorithmic approach that allows to determine if a control law is belongs to $\Phi\cdot$ or not. Numerical simulations are given.