Sparse Linear-Quadratic Feedback Design Using Affine Approximation

TL;DR

This paper introduces an iterative affine approximation method to solve sparse linear-quadratic control problems with $ ext{l}_0$ regularization, promoting sparsity in feedback controllers, which are otherwise NP-hard to solve.

Contribution

The paper proposes a novel affine approximation-based iterative algorithm for $ ext{l}_1$-relaxed sparse LQ control problems, improving solution quality and sparsity patterns.

Findings

Algorithm achieves comparable or superior performance to existing methods.

Numerical experiments demonstrate effective sparsity promotion.

Method handles nonconvex $ ext{l}_1$-regularized LQ problems efficiently.

Abstract

We consider a class of $\ell_0$-regularized linear-quadratic (LQ) optimal control problems. This class of problems is obtained by augmenting a penalizing sparsity measure to the cost objective of the standard linear-quadratic regulator (LQR) problem in order to promote sparsity pattern of the state feedback controller. This class of problems is generally NP hard and computationally intractable. First, we apply a $\ell_1$-relaxation and consider the $\ell_1$-regularized LQ version of this class of problems, which is still nonconvex. Then, we convexify the resulting $\ell_1$-regularized LQ problem by applying affine approximation techniques. An iterative algorithm is proposed to solve the $\ell_1$-regularized LQ problem using a series of convexified $\ell_1$-regularized LQ problems. By means of several numerical experiments, we show that our proposed algorithm is comparable to the existing algorithms in the literature, and in some cases it even returns solutions with superior performance and sparsity pattern.