Two-Timescale Optimization Framework for Sparse-Feedback Linear-Quadratic Optimal Control

TL;DR

This paper develops a two-timescale optimization framework for sparse-feedback linear-quadratic control, incorporating convex relaxations and direct $ ext{l}_0$-penalty methods to enhance communication efficiency and stability guarantees.

Contribution

It introduces a novel two-timescale algorithm for sparse-feedback LQ control with $ ext{l}_1$ relaxation, accelerated convergence for quadratic relaxation, and a direct $ ext{l}_0$-penalty approach using BSUM.

Findings

The two-timescale algorithm effectively minimizes the $ ext{l}_1$-relaxed sparse-feedback LQ cost.

Accelerated convergence achieved with piecewise quadratic relaxation.

Direct $ ext{l}_0$-penalty method provides precise approximations and variational insights.

Abstract

A $\mathcal{H}_2$-guaranteed sparse-feedback linear-quadratic (LQ) optimal control with convex parameterization and convex-bounded uncertainty is studied in this paper, where $\ell_0$-penalty is added into the $\mathcal{H}_2$ cost to penalize the number of communication links among distributed controllers. Then, the sparse-feedback gain is investigated to minimize the modified $\mathcal{H}_2$ cost together with the stability guarantee, and the corresponding main results are of three parts. First, the $\ell_1$ relaxation sparse-feedback LQ problem is of concern, and a two-timescale algorithm is developed based on proximal coordinate descent and primal-dual splitting approach. Second, piecewise quadratic relaxation sparse-feedback LQ control is investigated, which exhibits an accelerated convergence rate. Third, sparse-feedback LQ problem with $\ell_0$-penalty is directly studied through BSUM (Block Successive Upper-bound Minimization) framework, and precise approximation method and variational properties are introduced.