KLAP: KYP lemma based low-rank approximation for $\mathcal{H}_2$-optimal passivation

TL;DR

The paper introduces KLAP, a novel passivity enforcement method for LTI systems using KYP lemma, formulating an efficient unconstrained optimization problem to minimally perturb systems for passivity, with proven global optimality properties.

Contribution

This work develops KLAP, a new passivation approach based on KYP lemma, providing an efficient optimization framework with global optimality guarantees for large-scale systems.

Findings

Effective passivation with minimal perturbation.

Unconstrained optimization is differentiable and scalable.

Global optimality proven in certain cases.

Abstract

We present a novel passivity enforcement (passivation) method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and the closely related Lur'e equations. The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the $\mathcal{H}_2$ or frequency-weighted $\mathcal{H}_2$ distance between the original non-passive and the resulting passive system. We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings. We show that any minimizer of the unconstrained problem yields the same passive system. Furthermore, we prove that, in the absence of a feedthrough term, every local minimizer is also a global minimizer. For cases involving a non-trivial feedthrough term, we analyze global minimizers in relation to the extremal solutions of the Lur'e equations, which can serve as tools for identifying local minima. To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a non-global local minimum. Numerical examples illustrate the effectiveness of the proposed method.