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Induced Norm Analysis of Linear Systems for Nonnegative Input Signals

Yoshio Ebihara, Noboru Sebe, Hayato Waki, Dimitri Peaucelle, Sophie Tarbouriech, Victor Magron, Tomomichi Hagiwara

Abstract

This paper is concerned with the analysis of the $L_p\ (p\in[1,\infty), p=\infty)$ induced norms of continuous-time linear systems where input signals are restricted to be nonnegative. This norm is referred to as the $L_{p+}$ induced norm in this paper. It has been shown recently that the $L_{2+}$ induced norm is effective for the stability analysis of nonlinear feedback systems where the nonlinearity returns only nonnegative signals. However, the exact computation of the $L_{2+}$ induced norm is essentially difficult. To get around this difficulty, in the first part of this paper, we provide a copositive-programming-based method for the upper bound computation by capturing the nonnegativity of the input signals by copositive multipliers. Then, in the second part of the paper, we derive uniform lower bounds of the $L_{p+}\ (p\in[1,\infty), p=\infty)$ induced norms with respect to the standard $L_{p}$ induced norms that are valid for all linear systems including infinite-dimensional ones. For each linear system, we finally derive a computation method of the lower bounds of the $L_{2+}$ induced norm that are larger than (or equal to) the uniform one. The effectiveness of the upper/lower bound computation methods are fully illustrated by numerical examples.

Induced Norm Analysis of Linear Systems for Nonnegative Input Signals

Abstract

This paper is concerned with the analysis of the induced norms of continuous-time linear systems where input signals are restricted to be nonnegative. This norm is referred to as the induced norm in this paper. It has been shown recently that the induced norm is effective for the stability analysis of nonlinear feedback systems where the nonlinearity returns only nonnegative signals. However, the exact computation of the induced norm is essentially difficult. To get around this difficulty, in the first part of this paper, we provide a copositive-programming-based method for the upper bound computation by capturing the nonnegativity of the input signals by copositive multipliers. Then, in the second part of the paper, we derive uniform lower bounds of the induced norms with respect to the standard induced norms that are valid for all linear systems including infinite-dimensional ones. For each linear system, we finally derive a computation method of the lower bounds of the induced norm that are larger than (or equal to) the uniform one. The effectiveness of the upper/lower bound computation methods are fully illustrated by numerical examples.
Paper Structure (24 sections, 9 theorems, 68 equations, 3 figures)

This paper contains 24 sections, 9 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries, Motivations, and Problem Settings
  3. Preliminaries on Signals, Norms, and Cones
  4. Motivations: Relevance of $L_{p+}$ Induced Norm to Dynamical System Analysis and Synthesis
  5. Evaluation of Difference of Positive Systems
  6. Stability Analysis of Dynamical Systems Driven by Neural Network Controllers
  7. Problem Settings
  8. Novel Aspects Over Ebihara_ECC2022Ebihara_IFAC2023
  9. Main Results for $L_{2+}$ Upper Bound Analysis Problem
  10. Upper Bound Computation of $L_{2+}$ Induced Norm by Positive Filters
  11. Concrete Construction of Positive Filters
  12. Main Results for Uniform Lower Bound Analysis Problem
  13. Main Results for $L_{2+}$ Lower Bound Analysis Problem: Single-Input Case
  14. Reduction to Semi-Infinite Programming Problem
  15. Effective Lower Bound Computation Methods
  16. ...and 9 more sections

Key Result

proposition 1

The system $G$ given by eq:GLTI is externally positive if and only if its impulse response is nonnegative. In addition, it is internally positive if and only if $A\in\bbM^{n}$, $B\in\bbR_+^{n\times \nw}$, $C\in\bbR_+^{\nz\times \nw}$, and $D\in\bbR_+^{\nz\times \nw}$.

Figures (3)

  • Figure 1: Dynamical Systems Driven by NN Controllers (left) and Nonlinear Feedback System Representation (right).
  • Figure 2: Computed Upper Bounds $\oogam_{a,\alpha,N}$.
  • Figure 3: Computed Lower Bounds $\upsilon_N(G)$.

Theorems & Definitions (17)

  • definition 1: Farina_2000Kaczorek_2001
  • proposition 1: Farina_2000Kaczorek_2001
  • remark 1
  • theorem 1: Ebihara_ECC2022
  • theorem 2
  • remark 2
  • theorem 3: Ebihara_IFAC2023
  • lemma 1
  • theorem 4
  • remark 3
  • ...and 7 more