Steady-state response assignment for a given disturbance and reference: Sylvester equation rather than regulator equations

TL;DR

This paper introduces a new framework for assigning steady-state responses in control systems using Sylvester equations, extending moment concepts from model reduction to active control.

Contribution

It proposes a novel design method for steady-state response assignment via Sylvester equations, including necessary conditions and a constructive synthesis procedure.

Findings

Provides necessary and sufficient conditions for moment assignability.

Develops a canonical form of the dynamic compensator.

Offers a constructive synthesis procedure for control design.

Abstract

Conventionally, the concept of moment has been primarily employed in model order reduction to approximate system by matching the moment, which is merely the specific set of steady-state responses. In this paper, we propose a novel design framework that extends this concept from "moment matching" for approximation to "moment assignment" for the active control of steady-state. The key observation is that the closed-loop moment of an interconnected linear system can be decomposed into the open-loop moment and a term linearly parameterized by the moment of the compensator. Based on this observation, we provide necessary and sufficient conditions for the assignability of desired moment and a canonical form of the dynamic compensator, followed by constructive synthesis procedure of compensator. This covers both output regulation and closed-loop interpolation, and further suggests using only the Sylvester equation, rather than regulator equations.