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

TL;DR

The paper addresses designing an $ abla\mathcal{H}_2$-guaranteed sparse-feedback LQ controller under convex parameterization and convex-bounded uncertainty by introducing an $\\ell_0$ penalty on the gain. It develops a trio of optimization approaches: (i) an $\ell_1$-relaxation solved via a two-timescale proximal-coordinate-descent framework with primal-dual splitting, (ii) a piecewise-quadratic relaxation that achieves accelerated $O(1/k^2)$ convergence, and (iii) direct $\ell_0$-penalty optimization using BSUM with variational insights and epi-convergence. The work provides convergence guarantees, connects sparse-feedback design to distributed LQ control with fixed topology, and demonstrates the methods through numerical examples showing the tradeoff between sparsity and performance and the speedups from acceleration. This yields finite-dimensional, guaranteed sparse controller synthesis with practical implications for scalable, fault-tolerant distributed control.

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.

Figures (6)

• Figure 1: Relation between $\gamma$ and $N$
• Figure 2: Feasible gain with $\gamma=10$
• Figure 3: System response
• Figure 4: System response
• Figure 5: Convergence of Algorithm \ref{['alg2']}

Theorems & Definitions (42)

• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Definition 1
• Remark 6
• Lemma 1
• proof