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

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.

Figures (2)

• Figure C1: Structural decomposition of interconnected dynamics on the invariant manifold \ref{['eq: Invariant manifold of closed-loop system']}. (a) The original interconnected system with the signal generator of equation $\dot \omega = S \omega$ ($S$), plant ($\Sigma$), and compensator ($K$). (b) From the perspective of the plant $\Sigma$, incoming signals from $S$ and $K$ of (a) can be regarded as a signal from the virtual signal generator $(S, [L^\top \ \ M_\mathrm{c}^\top]^\top)$, yielding the open-loop moment $M_\mathrm{cl}$. Note that to assign moment to $M_\mathrm{des}$, existence of virtual a signal generator $(S,M_\mathrm{c})$ which makes $M_\mathrm{cl} = M_\mathrm{des}$ is essential. (c) From the perspective of the compensator $K$, incoming signal from $\Sigma$ of (a) can be regarded as a signal from virtual signal generator $(S, M_\mathrm{cl})$, yielding the open-loop moment $M_\mathrm{c}$. Note that the synthesis problem can now be simplified to designing a compensator that generates the required moment $M_\mathrm{c}$ when driven by the virtual desired generator $(S, M_\mathrm{des})$.
• Figure E1: Disturbance rejection and tracking control of HiMAT aircraft model

Theorems & Definitions (6)

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