[Paper Review] Gain and phase type multipliers for feedback robustness

Axel Ringh, Xin Mao+3 more|arXiv (Cornell University)|Mar 22, 2022

Stability and Control of Uncertain Systems6 citations

TL;DR

This paper establishes that robust feedback stability against structured uncertainties—specifically gain-type (magnitude) or phase-type (phase) uncertainties—necessitates the existence of corresponding structured multipliers: gain-type multipliers (zero diagonal blocks) for phase uncertainty robustness, and phase-type multipliers (zero off-diagonal blocks) for gain uncertainty robustness. The key contribution is a duality between uncertainty type and multiplier structure, generalizing the small-gain and small-phase theorems within the integral quadratic constraint framework.

ABSTRACT

It is known that the stability of a feedback interconnection of two linear time-invariant systems implies that the graphs of the open-loop systems are quadratically separated. This separation is defined by an object known as the multiplier. The theory of integral quadratic constraints shows that the converse also holds under certain conditions. This paper establishes that if the feedback is robustly stable against certain structured uncertainty, then there always exists a multiplier that takes a corresponding form. In particular, if the feedback is robustly stable to certain gain-type uncertainty, then there exists a corresponding multiplier that is of phase-type, i.e., its diagonal blocks are zeros. These results build on the notion of phases of matrices and systems, which was recently introduced in the field of control. Similarly, if the feedback is robustly stable to certain phase-type uncertainty, then there exists a gain-type multiplier, i.e., its off-diagonal blocks are zeros. The results are meaningfully instructive in the search for a valid multiplier for establishing robust closed-loop stability, and cover the well-known small-gain and the recent small-phase theorems.

Motivation & Objective

• To establish necessary conditions for robust feedback stability under structured uncertainty using multiplier theory.
• To clarify the duality between uncertainty type (gain or phase) and multiplier structure (phase-type or gain-type).
• To generalize the small-gain and small-phase theorems by showing that robustness implies existence of specific structured multipliers.
• To provide a theoretical foundation for restricting multiplier search to structured forms when robustness to specific uncertainty types is required.

Proposed method

• Utilizes quadratic graph separation via integral quadratic constraints (IQCs) with structured multipliers.
• Applies the Parseval-Plancherel theorem to relate time-domain integrals to frequency-domain inequalities.
• Employs singular value decomposition (SVD) and unitary transformations to analyze robustness under arbitrary unitary perturbations.
• Uses permutation matrices and eigenvalue analysis to derive necessary conditions on singular values for determinant non-vanishing.
• Proves equivalence between robust stability under unitary perturbations and existence of phase-type or gain-type multipliers.
• Leverages matrix phase theory and sectorial matrix analysis to define and characterize phase-type multipliers.

Experimental results

Research questions

• RQ1 Under what conditions does robust stability against phase-type uncertainty imply the existence of a phase-type multiplier?
• RQ2 Does robust stability against gain-type uncertainty necessitate a gain-type multiplier?
• RQ3 How do structured multipliers (gain-type or phase-type) relate to the robustness of feedback systems under specific uncertainty classes?
• RQ4 Can the small-gain and small-phase theorems be unified under a general framework of multiplier existence?
• RQ5 What is the necessary and sufficient condition for robust stability under arbitrary unitary perturbations in terms of multiplier structure?

Key findings

• Robust stability against arbitrary stable unitary (all-pass) perturbations implies the existence of a phase-type multiplier, i.e., a multiplier with zero diagonal blocks.
• Robust stability against arbitrary positive scalar gain perturbations implies the existence of a gain-type multiplier, i.e., a multiplier with zero off-diagonal blocks.
• For matrices A and B, the condition det(I + UAVB) ≠ 0 for all unitary U and V is equivalent to the existence of a phase-type multiplier satisfying quadratic separation.
• If the product of the largest singular values of A and B is less than 1, then robustness against all unitary perturbations holds, and a phase-type multiplier exists.
• The existence of a gain-type multiplier is necessary and sufficient for robustness to positive scalar uncertainties, generalizing the small-gain theorem.
• The results establish a converse to standard IQC theory: robustness to a specific uncertainty type implies the existence of a multiplier of the corresponding structured form.