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Feedback Stability Analysis via Dissipativity with Dynamic Supply Rates

Sei Zhen Khong, Chao Chen, Alexander Lanzon

TL;DR

This work introduces a general framework of dissipativity with dynamic supply rates for nonlinear systems, extending classical static dissipativity and IQC-based methods. It defines supply rates as outputs of a causal operator $\Xi$ and couples them with potentially independent auxiliary systems $\Phi$ to obtain a storage-function-based dissipation inequality, enabling Lyapunov, asymptotic, and exponential stability analysis for nonlinear feedback interconnections. The main contributions include Lyapunov stability results with complementary dynamic supply rates, asymptotic stability under strict dissipativity, and specialisations to $(\Psi,\Pi,\Upsilon,\Omega)$-dissipativity, terminal-cost dissipation, and affine-nonlinear systems, showing how dynamic multipliers can reduce conservatism. The framework unifies and extends small-gain, passivity, and IQC approaches, clarifying relationships with IQC theory and offering a practical coupling test for interconnections described by dynamic dissipativity. A numerical example demonstrates the applicability to nonlinear feedback with a clear path to broader extensions in large-scale networks and hybrid settings.

Abstract

We propose a general notion of dissipativity with dynamic supply rates for nonlinear systems. This extends classical dissipativity with static supply rates and dynamic supply rates of miscellaneous quadratic forms. The main results of this paper concern Lyapunov and asymptotic stability analysis for nonlinear feedback dissipative systems that are characterised by dissipation inequalities with respect to compatible dynamic supply rates but involving possibly different and independent auxiliary systems. Importantly, dissipativity conditions guaranteeing stability of the state of the feedback systems, without concerns on the stability of the state of the auxiliary systems, are provided. The key results also specialise to a simple coupling test for the interconnection of two nonlinear systems described by dynamic (Psi, Pi, Upsilon, Omega)-dissipativity, and are shown to recover several existing results in the literature, including small-gain, passivity indices, static (Q, S, R)-dissipativity, dissipativity with terminal costs, etc. Comparison with the input-output approach to feedback stability analysis based on integral quadratic constraints is also made.

Feedback Stability Analysis via Dissipativity with Dynamic Supply Rates

TL;DR

This work introduces a general framework of dissipativity with dynamic supply rates for nonlinear systems, extending classical static dissipativity and IQC-based methods. It defines supply rates as outputs of a causal operator and couples them with potentially independent auxiliary systems to obtain a storage-function-based dissipation inequality, enabling Lyapunov, asymptotic, and exponential stability analysis for nonlinear feedback interconnections. The main contributions include Lyapunov stability results with complementary dynamic supply rates, asymptotic stability under strict dissipativity, and specialisations to -dissipativity, terminal-cost dissipation, and affine-nonlinear systems, showing how dynamic multipliers can reduce conservatism. The framework unifies and extends small-gain, passivity, and IQC approaches, clarifying relationships with IQC theory and offering a practical coupling test for interconnections described by dynamic dissipativity. A numerical example demonstrates the applicability to nonlinear feedback with a clear path to broader extensions in large-scale networks and hybrid settings.

Abstract

We propose a general notion of dissipativity with dynamic supply rates for nonlinear systems. This extends classical dissipativity with static supply rates and dynamic supply rates of miscellaneous quadratic forms. The main results of this paper concern Lyapunov and asymptotic stability analysis for nonlinear feedback dissipative systems that are characterised by dissipation inequalities with respect to compatible dynamic supply rates but involving possibly different and independent auxiliary systems. Importantly, dissipativity conditions guaranteeing stability of the state of the feedback systems, without concerns on the stability of the state of the auxiliary systems, are provided. The key results also specialise to a simple coupling test for the interconnection of two nonlinear systems described by dynamic (Psi, Pi, Upsilon, Omega)-dissipativity, and are shown to recover several existing results in the literature, including small-gain, passivity indices, static (Q, S, R)-dissipativity, dissipativity with terminal costs, etc. Comparison with the input-output approach to feedback stability analysis based on integral quadratic constraints is also made.
Paper Structure (16 sections, 10 theorems, 74 equations, 3 figures)

This paper contains 16 sections, 10 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dissipativity with Dynamic Supply Rates
  3. Examples of Dynamic Dissipativity
  4. Main Results on Feedback Stability
  5. Definitions
  6. Problem Formulation and Preliminaries
  7. Lyapunov Stability
  8. Asymptotic Stability
  9. Exponential Stability
  10. Specialisation of the Main Results
  11. $(\Psi$, $\Pi$, $\Upsilon$, $\Omega)$-dissipativity
  12. Dissipation with Terminal Costs
  13. Dissipation for Affine-nonlinear Systems
  14. Relations with Integral Quadratic Constraint Theory
  15. A Numerical Example
  16. ...and 1 more sections

Key Result

Lemma 3

Let $\Sigma$ be given by eq: OLSystem and the auxiliary system $\Phi$ be given by eq: aux. Then $\Sigma$ is $\Xi$-dissipative with a $C^1$ storage function $S$ if for all $t\geq 0$, $u\in \mathscr{U}$, $x(0) \in \mathcal{X}$ and $\bar{x} \in \mathcal{\bar{X}}$, where $\xi(t)=\Xi(u, y, \bar{x})(t)$ and $x$, $z$, $y$ satisfying eq: OLSystem and eq: aux. Similarly, $\Sigma$ is $\Xi'$-dissipative if

Figures (3)

  • Figure 1: A physical system $\Sigma$ with state variable $x$, input $u$ and output $y$. It is associated with a dynamic supply rate $\xi=\Xi(u, y, \bar{{x}})$ and an auxiliary system $\Phi$ with state variable $z$ and output $\phi$. The dotted line signifies that the initial condition of $\Xi$ may be taken to be a function of $\bar{{x}}=x(0)$.
  • Figure 2: Feedback configuration of $\Sigma_1$ and $\Sigma_2$.
  • Figure 3: Simulation results of the state trajectories of $\Sigma_1 \| {\Sigma}_2$.

Theorems & Definitions (26)

  • Definition 1: Dynamic supply rate
  • Definition 2: Dynamic dissipativity
  • Lemma 3
  • Example 4: Input-output negative imaginariness
  • Example 5
  • Example 6
  • Example 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 16 more