Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Dissipativity Analysis of Nonlinear Systems

Amy K. Strong, Leila Bridgeman

TL;DR

The paper develops a general, convex-optimization-based framework to determine local dissipativity for smooth nonlinear control-affine systems by simultaneously optimizing the input-output cone and constructing CPA or quadratic storage functions. It provides theoretical bounds and LMI-based tools to enforce dissipativity over triangulated regions, with guarantees of feasibility under sufficiently strict local dissipativity. Numerical results on pendulum and a 3D polynomial system demonstrate tight conic bounds and gains that rival or surpass SOS-based and polynomial-restricted methods. This approach enables scalable, region-focused dissipativity analysis with constructive storage functions for robust composition via network dissipativity concepts.

Abstract

Dissipativity is an input-output (IO) characterization of nonlinear systems that enables compositional robust control through Vidyasagar's Network Dissipativity Theorem (VDNT). However, determining the dissipativity of a system is an involved and, often, model-specific process. We present a general method to determine the local dissipativity properties of smooth, nonlinear, control affine systems. We simultaneously search for the optimal IO characterization of a system and synthesize a continuous piecewise affine (CPA) storage function via a convex optimization problem. To do so, we reformulate the dissipation inequality as a matrix inequality (MI) and develop novel linear matrix inequality (LMI) bounds for a triangulation to impose the dissipativity conditions on the CPA storage function Further, we develop a method to synthesize a combined quadratic and CPA storage function to expand the systems the optimization problem is applicable to. Finally, we establish that our method will always find a feasible IO characterization and storage function given that the system is sufficiently strictly locally dissipative and demonstrate the efficacy of our method in determining the conic bounds and gain of various nonlinear systems.

Local Dissipativity Analysis of Nonlinear Systems

TL;DR

The paper develops a general, convex-optimization-based framework to determine local dissipativity for smooth nonlinear control-affine systems by simultaneously optimizing the input-output cone and constructing CPA or quadratic storage functions. It provides theoretical bounds and LMI-based tools to enforce dissipativity over triangulated regions, with guarantees of feasibility under sufficiently strict local dissipativity. Numerical results on pendulum and a 3D polynomial system demonstrate tight conic bounds and gains that rival or surpass SOS-based and polynomial-restricted methods. This approach enables scalable, region-focused dissipativity analysis with constructive storage functions for robust composition via network dissipativity concepts.

Abstract

Dissipativity is an input-output (IO) characterization of nonlinear systems that enables compositional robust control through Vidyasagar's Network Dissipativity Theorem (VDNT). However, determining the dissipativity of a system is an involved and, often, model-specific process. We present a general method to determine the local dissipativity properties of smooth, nonlinear, control affine systems. We simultaneously search for the optimal IO characterization of a system and synthesize a continuous piecewise affine (CPA) storage function via a convex optimization problem. To do so, we reformulate the dissipation inequality as a matrix inequality (MI) and develop novel linear matrix inequality (LMI) bounds for a triangulation to impose the dissipativity conditions on the CPA storage function Further, we develop a method to synthesize a combined quadratic and CPA storage function to expand the systems the optimization problem is applicable to. Finally, we establish that our method will always find a feasible IO characterization and storage function given that the system is sufficiently strictly locally dissipative and demonstrate the efficacy of our method in determining the conic bounds and gain of various nonlinear systems.

Paper Structure

This paper contains 6 sections, 3 theorems, 15 equations, 4 figures.

Table of Contents

  1. $\mathcal{L}_2$ gain of Pendulum
  2. Three dimensional polynomial system
  3. Conclusion
  4. Function Error Bounds
  5. LMI Bound
  6. Analytical Conic Bound

Key Result

Lemma 18

Consider $\zeta: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m},$ where $\zeta \in \mathcal{C}^2$. Let $\sigma \vcentcolon= \text{co}\{x_j\}_{j=0}^n$ be an n-simplex in $\mathbb{R}^n.$ If $\sum_{j=0}^n\lambda_jx_j \in \sigma$, then where

Figures (4)

  • Figure 1: The analytical upper (b) and lower (a) conic bounds of \ref{['eq:nonlinearConic']} are determined via Problem \ref{['prob:LCA']} for an increasing number of simplices. We are able to determine conic bounds of the system that closely match the analytical bounds.
  • Figure 2: The $\mathcal{L}_2$ gain of the pendulum found on a triangulation over $\mathcal{X} \supseteq \Omega$ for an increasing number of simplices. As the triangulation becomes more fine, the gain found by Problem \ref{['prob:LCA']} becomes closer to the analytical bound.
  • Figure 3: An example of a triangulation over $\mathcal{X}\supseteq \Omega$, where $\Omega$ is an RPI set denoted by the red line. The set $\Omega$ was found using lavaei2025L2 and is valid for inputs where $\left\lvert u\right\rvert< 0.1942.$
  • Figure 4: An example of a storage function over $\mathcal{X} \supseteq \Omega$ found by Problem \ref{['prob:LCA']}.

Theorems & Definitions (3)

  • Lemma 18
  • Corollary 19
  • Lemma 20