[Paper Review] On the Parameterization of Stabilizing Controllers using Closed-loop Responses.

Yang Zheng, Luca Furieri+2 more|arXiv (Cornell University)|Sep 26, 2019

Stability and Control of Uncertain Systems30 references4 citations

TL;DR

This paper introduces two new convex parameterizations of internally stabilizing controllers for strictly proper LTI systems using closed-loop responses, building on existing frameworks like Youla, SLP, and IOP. By identifying four equivalent groups of stable closed-loop transfer matrices that characterize internal stability—two of which are newly proposed—it establishes four convex parameterizations, enhancing the theoretical foundation of closed-loop convexity for controller synthesis via convex optimization.

ABSTRACT

In this paper, we study the problem of parameterizing all internally stabilizing controllers for strictly proper linear time-invariant (LTI) systems using closed-loop responses. It is known that the set of internally stabilizing controllers $\mathcal{C}_{ ext{stab}}$ is non-convex, but it admits a convex representation using certain closed-loop maps. A classical result is the Youla parameterization, and two recent notions are the system-level parameterization (SLP) and input-output parameterization (IOP). This paper further examines all possible parameterizations of $\mathcal{C}_{ ext{stab}}$ using certain closed-loop responses. Our main idea is to revisit the external transfer matrix characterization of internal stability, which uncovers that only four groups of stable closed-loop transfer matrices are equivalent to internal stability: one of them is used in SLP, another one is a classical result and is used in IOP, and the other two are new, leading to two new parameterizations for $\mathcal{C}_{ ext{stab}}$. All these four parameterizations are convex in term of the respectively introduced parameters, allowing us to use convex optimization for controller synthesis. These results contribute to a more complete picture of the notion of \emph{closed-loop convexity} for parameterizing $\mathcal{C}_{ ext{stab}}$.

Motivation & Objective

• To provide a comprehensive characterization of all possible parameterizations of internally stabilizing controllers using closed-loop responses.
• To identify and formalize all groups of stable closed-loop transfer matrices that are equivalent to internal stability in LTI systems.
• To extend the existing framework of closed-loop convexity by introducing two novel parameterizations beyond Youla, SLP, and IOP.
• To enable convex optimization-based controller synthesis by ensuring all proposed parameterizations are convex in their respective parameters.

Proposed method

• Revisits the external transfer matrix characterization of internal stability to identify all possible sets of stable closed-loop transfer matrices that imply internal stability.
• Identifies four distinct groups of stable closed-loop transfer matrices that are equivalent to internal stability, with two groups being newly introduced.
• Derives convex parameterizations of the set of all internally stabilizing controllers based on each of the four groups of closed-loop maps.
• Establishes that each parameterization is convex in its respective parameter, enabling the use of convex optimization techniques for controller design.
• Compares the new parameterizations to classical frameworks such as Youla, SLP, and IOP, showing their equivalence in stability characterization but differences in parameterization structure.
• Uses mathematical analysis of transfer matrix relationships to prove the equivalence and convexity of each parameterization.

Experimental results

Research questions

• RQ1 What are all possible sets of stable closed-loop transfer matrices that are equivalent to internal stability in strictly proper LTI systems?
• RQ2 How can new convex parameterizations of stabilizing controllers be derived from previously unidentified groups of closed-loop responses?
• RQ3 What is the relationship between the newly proposed parameterizations and existing frameworks such as SLP and IOP?
• RQ4 Can all such parameterizations be made convex in their respective parameters to support convex optimization in controller synthesis?
• RQ5 What is the complete classification of closed-loop convex parameterizations for internally stabilizing controllers?

Key findings

• The paper identifies four distinct groups of stable closed-loop transfer matrices that are equivalent to internal stability in strictly proper LTI systems.
• Two of these groups are newly introduced, leading to two novel convex parameterizations of the set of internally stabilizing controllers.
• All four parameterizations are convex in their respective parameters, enabling the use of convex optimization for controller synthesis.
• The new parameterizations extend the theoretical framework of closed-loop convexity beyond existing methods such as Youla, SLP, and IOP.
• The results provide a complete classification of all possible convex parameterizations of stabilizing controllers using closed-loop responses.
• The framework unifies and generalizes prior approaches by showing that multiple closed-loop response sets can serve as valid convex parameterization variables.