Perturbation theory of observable linear systems

TL;DR

This paper analyzes the asymptotic control of linear oscillators by modeling the system as a perturbed observable linear system, providing error estimates and demonstrating the control's effectiveness in approaching equilibrium.

Contribution

It introduces a perturbation-based framework for observable linear systems in control theory, enabling error estimation and analysis of system convergence.

Findings

The control drives the system towards equilibrium with positive speed.

Error in state recovery is bounded by the $L_1$-norm of perturbation.

The approach applies to damping multiple oscillators efficiently.

Abstract

The present work is motivated by the asymptotic control theory for a system of linear oscillators: the problem is to design a common bounded scalar control for damping all oscillators in asymptotically minimal time. The motion of the system is described in terms of a canonical system similar to that of the Pontryagin maximum principle. We consider the evolution equation for adjoint variables as a perturbed observable linear system. Due to the perturbation, the unobservable part of the state trajectory cannot be recovered exactly. We estimate the recovering error via the $L_1$-norm of perturbation. This allows us to prove that the control makes the system approach the equilibrium state with a strictly positive speed.