Minimum-gain Pole Placement with Sparse Static Feedback

TL;DR

This paper addresses the classical minimum-gain eigenvalue assignment problem for LTI systems by incorporating sparsity constraints in static feedback, proposing new optimization and iterative algorithms for solutions.

Contribution

It formulates the sparse MGEAP as an equality-constrained optimization, provides an analytical characterization of local solutions, and develops iterative algorithms for sparse feedback design.

Findings

The proposed algorithms converge in most cases.

The analytical characterization offers geometric insights.

Heuristic methods effectively compute sparse solutions.

Abstract

The minimum-gain eigenvalue assignment/pole placement problem (MGEAP) is a classical problem in LTI systems with static state feedback. In this paper, we study the MGEAP when the state feedback has arbitrary sparsity constraints. We formulate the sparse MGEAP problem as an equality-constrained optimization problem and present an analytical characterization of its locally optimal solution in terms of eigenvector matrices of the closed loop system. This result is used to provide a geometric interpretation of the solution of the non-sparse MGEAP, thereby providing additional insights for this classical problem. Further, we develop an iterative projected gradient descent algorithm to obtain local solutions for the sparse MGEAP using a parametrization based on the Sylvester equation. We present a heuristic algorithm to compute the projections, which also provides a novel method to solve the sparse EAP. Also, a relaxed version of the sparse MGEAP is presented and an algorithm is developed to obtain approximately sparse local solutions to the MGEAP. Finally, numerical studies are presented to compare the properties of the algorithms, which suggest that the proposed projection algorithm converges in most cases.