Actuator Dynamics Compensation in Stabilization of Abstract Linear Systems

TL;DR

This paper presents a systematic method for stabilizing linear systems with infinite-dimensional actuator dynamics using full state feedback, ensuring well-posedness and exponential stability in an abstract framework.

Contribution

It introduces a novel feedback design via an upper-block-triangle transform for cascade systems with infinite-dimensional components, without requiring Lyapunov functions.

Findings

Successfully stabilizes systems with PDE actuator dynamics

Provides a sufficient condition for the existence of compensators

Validates the approach through numerical simulations on heat equations

Abstract

This is the first part of four series papers, aiming at the problem of actuator dynamics compensation for linear systems. We consider the stabilization of a type of cascade abstract linear systems which model the actuator dynamics compensation for linear systems where both the control plant and its actuator dynamics can be infinite-dimensional. We develop a systematic way to stabilize the cascade systems by a full state feedback. Both the well-posedness and the exponential stability of the resulting closed-loop system are established in the abstract framework. A sufficient condition of the existence of compensator for ordinary differential equation (ODE) with partial differential equation (PDE) actuator dynamics is obtained. The feedback design is based on a novelly constructed upper-block-triangle transform and the Lyapunov function design is not needed in the stability analysis. As applications, an ODE with input delay and an unstable heat equation with ODE actuator dynamics are investigated to validate the theoretical results. The numerical simulations for the unstable heat system are carried out to validate the proposed approach visually.