# Method of Reduction of Variables for Bilinear Matrix Inequality Problems in System and Control Designs

## TL;DR

This paper introduces a variable reduction method for bilinear matrix inequality problems in system and control design, transforming complex problems into simpler unconstrained forms to improve solution efficiency.

## Contribution

A novel reduction of variables method combining classification, problem transformation, and hybrid search algorithms for BMI problems in control systems.

## Key findings

- Outperforms existing BMI solution methods on benchmark problems.
- Effectively addresses feasibility and multiobjective problems.
- Reduces decision variables, simplifying complex BMI problems.

## Abstract

Bilinear matrix inequality (BMI) problems in system and control designs are investigated in this paper. A solution method of reduction of variables (MRVs) is proposed. This method consists of a principle of variable classification, a procedure for problem transformation, and a hybrid algorithm that combines deterministic and stochastic search engines. The classification principle is used to classify the decision variables of a BMI problem into two categories: 1) external and 2) internal variables. Theoretical analysis is performed to show that when the classification principle is applicable, a BMI problem can be transformed into an unconstrained optimization problem that has fewer decision variables. Stochastic search and deterministic search are then applied to determine the decision variables of the unconstrained problem externally and explore the internal problem structure, respectively. The proposed method can address feasibility, single-objective, and multiobjective problems constrained by BMIs in a unified manner. A number of numerical examples in system and control designs are provided to validate the proposed methodology. Simulations show that the MRVs can outperform existing BMI solution methods in most benchmark problems and achieve similar levels of performance in the remaining problems.