Towards Optimal Passive Feedback Control of LTI Systems under LQR Performance

TL;DR

This paper develops a method to design state-feedback for LTI systems that ensures passivity with minimal LQR cost, using convex approximations and gradient flows.

Contribution

It introduces a novel approach to approximate the set of passivating gains and optimize LQR performance via convex inner-approximations and projected gradient methods.

Findings

The passivating gain set is unbounded and possibly nonconvex.

The proposed convex polytope approximation enables efficient optimization.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

We study state-feedback design for continuous-time LTI systems with a control input and an external input-output pair. Our objective is to determine feedback gains that render the closed-loop system (strictly) passive with respect to the external port while minimizing the standard LQR cost in the disturbance-free case. The resulting constrained optimization problem is intractable due to bilinear matrix inequalities. We analyze the set of passivating gains, showing it is unbounded, possibly nonconvex, path-connected, and contractible. We propose an indirect approach, in which the set of passivating feedback gains is inner-approximated by a compact, convex polytope. A projected gradient flow is employed to compute a gain within this polytope that minimizes the LQR cost. Numerical examples illustrate the effectiveness of the method.