A mesh‐based partitioning algorithm for decreasing conservatism in solving bilinear matrix inequality problems

Abstract

In this paper, a mesh‐based partitioning algorithm (MBPA) to solve bilinear matrix inequality (BMI) problems is proposed, where the main idea is to divide the solution space of a non‐convex BMI problem into smaller parts in the form of Simplexes through meshing the space. In each simplex, only the concave part of BMI constraints is approximated by piecewise affine matrix (PAM), which eventually leads to a convex sub‐problem. After that, the feasible solution can be easily obtained in each simplex through the linear matrix inequality (LMI) optimization. Although the proposed MBPA exploits PAM approximation, it relaxes several restrictive conditions used in other methods, so, it can be used as an effective approach for solving BMI problems. Various numerical examples on systems from COMPleib library to design static output feedback controller are provided to validate the proposed MBPA. Simulation results reveal the satisfactory performance of MBPA in several examined problems and provide acceptable performance compared to other existing BMI solution algorithms. Comparing the percentage of superiority of MBPA with some existing prominent BMI algorithms in several control problems including spectral abscissa optimization problems, H∞$$ {H}_{\infty } $$ and H2$$ {H}_2 $$ optimization problems for the closed‐loop systems of COMPleib show that the proposed algorithm is more successful than the previous ones.