Realization, Internal Stability, and Controller Synthesis

TL;DR

This paper introduces a unified framework based on the realization-stability lemma that encompasses various controller synthesis methods and realizations, simplifying their relationships and enabling a general synthesis approach.

Contribution

It presents a unifying theoretical foundation that connects multiple controller synthesis methods and realizations through a simple lemma.

Findings

All existing methods are special cases of the realization-stability lemma.

The framework simplifies equivalence proofs among synthesis methods.

It enables a general controller synthesis formulation.

Abstract

We have witnessed the emergence of several controller parameterizations and the corresponding synthesis methods, including Youla, system level, input-output, and many other new proposals. Meanwhile, under the same synthesis method, there are multiple realizations to adopt. Different synthesis methods/realizations target different plants/scenarios. Except for some case-by-case studies, we don't currently have a unified framework to understand their relationships. To address the issue, we show that existing controller synthesis methods and realization proposals are all special cases of a simple lemma, the realization-stability lemma. The lemma leads to easier equivalence proofs among existing methods and enables the formulation of a general controller synthesis problem, which provides a unified foundation for controller synthesis and realization derivation.