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Improved Small-Signal L2 Gain Analysis for Nonlinear Systems

Amy Strong, Reza Lavaei, Leila J. Bridgeman

Abstract

TheL2-gain characterizes a dynamical system's input-output properties, but can be difficult to determine for nonlinear systems. Previous work designed a nonconvex optimization problem to simultaneously search for a continuous piecewise affine (CPA) storage function and an upper bound on the small-signal L2-gain of a dynamical system over a triangulated region about the origin. This work improves upon those results by establishing a tighter upper-bound on a system's gain using a convex optimization problem. By reformulating the relationship between the Hamilton-Jacobi inequality and L2-gain as a linear matrix inequality and then developing novel LMI error bounds for a triangulation, tighter gain bounds are derived and computed more efficiently. Additionally, a combined quadratic and CPA storage function is considered to expand the nonlinear systems this optimization problem is applicable to. Numerical results demonstrate the tighter upper bound on a dynamical system's gain.

Improved Small-Signal L2 Gain Analysis for Nonlinear Systems

Abstract

TheL2-gain characterizes a dynamical system's input-output properties, but can be difficult to determine for nonlinear systems. Previous work designed a nonconvex optimization problem to simultaneously search for a continuous piecewise affine (CPA) storage function and an upper bound on the small-signal L2-gain of a dynamical system over a triangulated region about the origin. This work improves upon those results by establishing a tighter upper-bound on a system's gain using a convex optimization problem. By reformulating the relationship between the Hamilton-Jacobi inequality and L2-gain as a linear matrix inequality and then developing novel LMI error bounds for a triangulation, tighter gain bounds are derived and computed more efficiently. Additionally, a combined quadratic and CPA storage function is considered to expand the nonlinear systems this optimization problem is applicable to. Numerical results demonstrate the tighter upper bound on a dynamical system's gain.
Paper Structure (11 sections, 7 theorems, 39 equations, 2 figures)

This paper contains 11 sections, 7 theorems, 39 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. CPA Functions
  4. L2 Stability Analysis
  5. MAIN RESULTS
  6. LMI Error Bounds
  7. L2-gain Analysis
  8. NUMERICAL EXAMPLE
  9. Pendulum
  10. Pendulum with Control Affine Input
  11. DISCUSSION

Key Result

Lemma 1

(Remark 9 giesl2014revised) Consider the triangulation $\mathcal{T} = \{\sigma_i\}_{i=1}^{m_{\mathcal{T}}},$ where $\sigma_i = \text{co}(\{\mathbf{x}_{i,j}\}_{j=0}^n)$, and a set $\mathbf{W} = \{W_{\mathbf{x}}\}_{\mathbf{x} \in \mathbb{E_{\mathcal{T}}}}\subset\mathbb{R},$ where $W(\mathbf{x}) = W_{\

Figures (2)

  • Figure 1: Triangulations of the region, $\Omega$, about the origin for dynamical system, $\mathcal{G}.$
  • Figure 2: The $\mathcal{L}_2$-gain bound was analyzed for an increasing numbers of simplexes over $\Omega$ for two systems -- a pendulum and a pendulum with a control affine input.

Theorems & Definitions (16)

  • Definition II.1
  • Definition II.2
  • Definition II.3
  • Lemma 1
  • Lemma 2
  • Definition II.4
  • Definition II.5
  • Theorem 1
  • Theorem 2
  • proof
  • ...and 6 more