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The Non-Strict Projection Lemma

T. J. Meijer, T. Holicki, S. J. A. M. van den Eijnden, C. W. Scherer, W. P. M. H. Heemels

Abstract

The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation.

The Non-Strict Projection Lemma

Abstract

The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation.
Paper Structure (17 sections, 12 theorems, 80 equations)

This paper contains 17 sections, 12 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Motivating example
  4. Main result
  5. Applications
  6. Marginal stability and stabilizability revisited
  7. Interpolation and the matrix S-procedure
  8. Matrix dilations
  9. Conclusions
  10. Preliminaries
  11. Proof of Theorem \ref{['thm:ns-proj-lem']}
  12. Proof of Corollary \ref{['cor:SPL']}
  13. Proof of Corollary \ref{['cor:Helmersson']}
  14. Proof of Proposition \ref{['prop:stab-conds']}
  15. Proof of Lemma \ref{['lem:interpolation']}
  16. ...and 2 more sections

Key Result

Lemma 1

Let $U\in\mathbb{C}^{m\times p}$ and $V\in\mathbb{C}^{n\times p}$ be arbitrary complex matrices and let $Q\in\mathbb{H}^{p}$ be Hermitian. Then, there exists a matrix $X\in\mathbb{C}^{m\times n}$ which satisfies the LMI if and only if

Theorems & Definitions (16)

  • Lemma 1
  • Example 1
  • Lemma 2
  • Theorem 1
  • Example 2
  • Corollary 1
  • Corollary 2
  • Example 3
  • Remark 1
  • Proposition 1
  • ...and 6 more