Exponential Stabilization of Linear Systems using Nearest-Action Control with Countable Input Set

TL;DR

This paper presents a method for stabilizing linear systems using a finite set of control actions selected via a nearest-action approach, achieving exponential or asymptotic stability under certain conditions.

Contribution

It introduces a nearest-action control strategy for linear systems with countable input sets, ensuring exponential stability and conditions for asymptotic stability.

Findings

Closed-loop system converges exponentially to the target.

Logarithmic scaling extension enables asymptotic stability.

Method is validated with a practical example.

Abstract

This paper studies stabilization of linear time-invariant (LTI) systems when control actions can only be realized in finitely many directions where it is possible to actuate uniformly or logarithmically extended positive scaling factors in each direction. Furthermore, a nearest-action selection approach is used to map the continuous measurements to a realizable action where we show that the approach satisfies a weak sector condition for multiple-input multiple-output (MIMO) systems. Using the notion of input-to-state stability, under some assumptions imposed on the transfer function of the system, we show that the closed-loop system converges to the target ball exponentially fast. Moreover, when logarithmic extension for the scaling factors is realizable, the closed-loop system is able to achieve asymptotic stability instead of only practical stability. Finally, we present an example of the application that confirms our analysis.