Semidefinite Programming Approximation for a Matrix Optimization Problem over an Uncertain Linear System

TL;DR

This paper introduces a polynomial-time semidefinite programming approximation for a complex matrix optimization problem over uncertain linear systems, providing theoretical error bounds and analyzing solution quality through numerical experiments.

Contribution

It develops a novel SDP approximation model for MOPUL, addressing its NP-hardness and analyzing approximation error and solution feasibility.

Findings

The SDP approximation provides feasible solutions with bounded error.

Numerical results show robustness of the SDP method under noise perturbations.

The approach is applicable to specific system cases with promising performance.

Abstract

A matrix optimization problem over an uncertain linear system on finite horizon (abbreviated as MOPUL) is studied, in which the uncertain transition matrix is regarded as a decision variable. This problem is in general NP-hard. By using the given reference values of system outputs at each stage, we develop a polynomial-time solvable semidefinite programming (SDP) approximation model for the problem. The upper bound of the cumulative error between reference outputs and the optimal outputs of the approximation model is theoretically analyzed. Two special cases associated with specific applications are considered. The quality of the SDP approximate solutions in terms of feasibility and optimality is also analyzed. Results of numerical experiments are presented to show the influences of perturbed noises at reference outputs and control levels on the performance of SDP approximation.