Title: Method of Reduction of Variables for Bilinear Matrix Inequality Problems in System and Control Designs,
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.