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On Integral Linear Constraints on Convex Cones

Emil Vladu, Alexandre Megretski, Anders Rantzer

Abstract

In this paper, we consider integral linear constraints and the dissipation inequality with linear supply rates for certain sets of trajectories confined pointwise in time to a convex cone which belongs to a finite-dimensional normed vector space. Such constraints are then shown to be satisfied if and only if a bounded linear functional exists which satisfies a conic inequality. This is analogous to the typical situation in which a quadratic supply rate over the entire space is related to a linear matrix inequality. A connection is subsequently drawn precisely to linear-quadratic control: by proper choice of cone, the main results can be applied to produce a known L1-gain analogue to the bounded real lemma in positive systems theory, as well as a non-strict version of the Kalman-Yakubovich-Popov Lemma in linear-quadratic control.

On Integral Linear Constraints on Convex Cones

Abstract

In this paper, we consider integral linear constraints and the dissipation inequality with linear supply rates for certain sets of trajectories confined pointwise in time to a convex cone which belongs to a finite-dimensional normed vector space. Such constraints are then shown to be satisfied if and only if a bounded linear functional exists which satisfies a conic inequality. This is analogous to the typical situation in which a quadratic supply rate over the entire space is related to a linear matrix inequality. A connection is subsequently drawn precisely to linear-quadratic control: by proper choice of cone, the main results can be applied to produce a known L1-gain analogue to the bounded real lemma in positive systems theory, as well as a non-strict version of the Kalman-Yakubovich-Popov Lemma in linear-quadratic control.

Paper Structure

This paper contains 6 sections, 10 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Applications
  5. Proofs
  6. Conclusions

Key Result

Lemma 1

Given a matrix $Q \in \mathcal{S}^{n + m}$ such that $Q \succeq 0$ with corresponding partition then $\mathrm{Im} (Q_{nm}) \subseteq \mathrm{Im} (Q_{nn})$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (39)

  • Lemma 1
  • proof
  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Definition 1
  • Corollary 2
  • proof
  • ...and 29 more