LMI Properties and Applications in Systems, Stability, and Control Theory

Abstract

Linear matrix inequalities (LMIs) commonly appear in systems, stability, and control applications. Many analysis and synthesis problems in these areas can be solved as feasibility or optimization problems subject to LMI constraints. Although most well-known LMI properties and manipulation tricks (e.g., Schur complement, congruence transformation) can be found in standard references, many useful LMI properties are scattered throughout the literature. The purpose of this document is to collect and organize properties, tricks, and applications related to LMIs from a number of references together in a single document. Proofs of the properties presented in this document are not included when they can be found in the cited references in the interest of brevity. Illustrative examples are included whenever necessary to fully explain a certain property. Multiple equivalent forms of LMIs are often presented to give the reader a choice of which form may be best suited for a particular problem at hand. The equivalency of some of the LMIs in this document may be straightforward to more experienced readers, but the authors believe that some readers may benefit from the presentation of multiple equivalent LMIs.Comment: Main edits/additions Sec. 1: New discussion on SDPs and solvers/parsers Sec. 2: Linearization and Dualization Lemmas, Frobenius norm and nuclear norm, eigenvalue properties, spectral radius, range of a symmetric matrix, Douglas-Fillmore-Williams Lemma Sec. 3: Dilated and descriptor system results, D-Admissibility, transient bounds, output energy bounds Sec. 5: Discrete-time optimal filterin