Towards Optimal Passive Feedback Control of LTI Systems under LQR Performance
For control engineers, this provides a tractable approach to a previously intractable bilinear optimization problem, though the method is incremental as it relies on inner approximations.
The paper tackles the problem of designing state-feedback controllers for LTI systems that enforce passivity while minimizing LQR cost. The proposed method uses a convex polytope inner-approximation and projected gradient flow to compute a suboptimal gain, with numerical examples demonstrating effectiveness.
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.