Realization-Stability Lemma for Controller Synthesis

TL;DR

This paper introduces the realization-stability lemma, a unifying theoretical framework that simplifies understanding, comparison, and robust stability analysis of various controller synthesis methods and realizations.

Contribution

It presents a simple lemma that unifies existing controller synthesis methods and realizations, enabling easier proofs and a general approach for robust stability and controller design.

Findings

Unified framework for controller synthesis methods

Simplified proofs of equivalence among methods

Robust stability conditions derived from the lemma

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. Also, various robust results are proposed to deal with different perturbed system structures. 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 robust stability conditions for general system analysis. It also enables the formulation of a general controller synthesis problem, which provides a unified foundation for controller synthesis, realization derivation, and robust stability analysis.