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

Ryan James Caverly, James Richard Forbes

TL;DR

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.

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.

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

TL;DR

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.

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.

Paper Structure

This paper contains 223 sections, 2 theorems, 403 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Introduction
  3. Notation
  4. Definitions and Fundamental LMI Properties
  5. Definiteness of a Matrix
  6. Matrix Inequalities and LMIs
  7. Relative Definiteness of a Matrix
  8. Strict and Nonstrict Matrix Inequalities
  9. Concatenation of LMIs
  10. Convexity of LMIs
  11. Semidefinite Programs (SDPs)
  12. Numerical Tools to Solve SDPs
  13. SDP Solvers
  14. LMI Parsers
  15. Properties and Tricks Aimed at Reformulating BMIs as LMIs
  16. ...and 208 more sections

Key Result

Theorem 1.2

Horn2013, BernsteinMatrixBook Consider the symmetric matrix $\mbf{A}\in \mathbb{S}^{n}$. The matrix $\mbf{A}$ is

Figures (2)

  • Figure 1: Block diagram of the generalized plant $\boldsymbol{\mathcal{P}}$ with the controller $\boldsymbol{\mathcal{K}}$.
  • Figure 2: Block diagram of the basic servo loop with plant $\mbf{G}_p(s)$, controller $\mbf{K}(s)$, and weighting transfer matrices $\mbf{W}_r(s)$, $\mbf{W}_d(s)$, and $\mbf{W}_n(s)$.

Theorems & Definitions (43)

  • Definition 1.1
  • Theorem 1.2
  • Example 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Example 1.3
  • Definition 1.7
  • ...and 33 more