Combining Convex-Concave Decompositions and Linearization Approaches for Solving BMIs, With Application to Static Output Feedback

• Published in: IEEE transactions on automatic control Vol. 57; no. 6; pp. 1377 - 1390
• Main Authors: Quoc Tran Dinh, Gumussoy, S., Michiels, W., Diehl, M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.06.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Bilinear matrix inequality (BMI); Computer science; control theory; systems; Control system analysis; Control system synthesis; Control theory. Systems; Convergence; Convex functions; convex-concave decomposition; Exact sciences and technology; Linear matrix inequalities; linear time-invariant system; Matrix decomposition; Optimization; Programming; semidefinite programming; static feedback controller design; Vectors; linear time-invariant system; Feedback regulation; Control synthesis; Semi definite programming; Matrix inequality; Bilinear form; Convex programming; Optimization; convex―concave decomposition; semidefinite programming; Linear time invariant system; static feedback controller design; Bilinear matrix inequality (BMI); Linear matrix inequality; Convex function; Bilinear transform; Output feedback; Linearization
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2011.2176154

A novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality (BMI) constraints. The key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings. At each iteration of the algorithm the concave part is linearized, leading to a convex subproblem. Applications to various output feedback controller synthesis problems are presented. In these applications, the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality (LMI) constraints. The performance of the algorithm has been benchmarked on the data from the COMPl e ib library.