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

TL;DR

This paper compiles and organizes key properties, tricks, and applications of linear matrix inequalities (LMIs) used in systems, stability, and control, serving as a comprehensive reference or 'LMI cookbook' for researchers and practitioners.

Contribution

It provides a consolidated collection of LMI properties, transformations, and applications, including multiple equivalent forms, with illustrative examples, filling a gap in scattered literature.

Findings

Comprehensive collection of LMI properties and tricks

Multiple equivalent LMI forms presented for problem suitability

Illustrative examples clarify complex properties

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, such as the Schur complement and the 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. In this sense, the document can be thought of as an "LMI encyclopedia" or "LMI cookbook." 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.