[Paper Review] LMI Properties and Applications in Systems, Stability, and Control Theory

Ryan J. Caverly, James Richard Forbes|arXiv (Cornell University)|Mar 20, 2019

Stability and Control of Uncertain Systems152 references76 citations

TL;DR

A comprehensive, community-sourced compilation of LMI properties, tricks, and applications for systems, stability, and control, with guidance on SDPs and numerical tools.

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.

Motivation & Objective

• Aggregate and organize widely scattered LMI properties and tricks from the literature for use in systems, stability, and control theory.
• Provide multiple equivalent LMI forms to aid problem-specific formulation and solution.
• Explain foundational concepts (definiteness, LMIs, SDPs) and practical numerical tools for solving LMIs.
• Illustrate how LMIs underpin analysis and synthesis problems in continuous and discrete-time systems.

Proposed method

• Present definitions and notation for LMIs, definiteness, and matrix inequalities.
• Catalog and group LMI properties and tricks (e.g., Schur complement, Young’s relation, projection lemma, dilation).
• Discuss semidefinite programming as the optimization framework for LMIs and duality concepts.
• Describe numerical tools including SDP solvers and LMI parsers and their interoperability.
• Show how LMIs arise in stability, dissipativity, and performance analysis within control theory.

Experimental results

Research questions

• RQ1 What LMI properties and tricks are most useful for reformulating control-theoretic problems?
• RQ2 How can different equivalent LMI forms be leveraged to suit specific analysis or synthesis tasks?
• RQ3 What role do LMIs and SDPs play in Lyapunov analysis, Bounded Real, H2/H infinity norms, and related lemmas?
• RQ4 What numerical tools (solvers, parsers) best support LMI-based optimization in practice?

Key findings

• LMIs provide a convex framework for feasibility and optimization problems in systems, stability, and control.
• A wide range of LMI manipulation techniques (e.g., Schur complement, congruence, Young’s relation, projection lemma) enable flexible reformulations.
• LMIs underpin a broad set of results including Lyapunov inequalities, Bounded Real Lemmas, KYP, dissipativity, and various norm and stability analyses.
• Numerical tools (SDP solvers and LMI parsers) facilitate converting matrix-form LMIs into standard SDP form for computation.
• The document groups and clarifies many properties and tricks to aid researchers in formulating and solving LMIs across continuous and discrete-time domains.