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Convex Equilibrium-Free Stability and Performance Analysis of Discrete-Time Nonlinear Systems

Patrick J. W. Koelewijn, Siep Weiland, Roland Tóth

TL;DR

The paper addresses stability and performance analysis of discrete-time nonlinear systems without relying on a single equilibrium by introducing universal shifted and incremental notions. It develops velocity- and differential-based frameworks that relate equilibrium-free properties to (i) time-difference dynamics and (ii) differential dynamics, respectively, and shows how these can be formulated as convex LPV/DPV problems. By allowing state-dependent storage functions, the work reduces conservatism in incremental analyses and provides finite-dimensional LMIs via DPV embedding, enabling computable tests for USS/USAS, USP, USD, IS, and ID. An illustrative Duffing oscillator example demonstrates substantial improvements when using a state-dependent storage matrix over a constant one, highlighting practical applicability for equilibrium-free stability and performance guarantees in DT nonlinear systems.

Abstract

This paper considers the equilibrium-free stability and performance analysis of discrete-time nonlinear systems. We consider two types of equilibrium-free notions. Namely, the universal shifted concept, which considers stability and performance w.r.t. all equilibrium points of the system, and the incremental concept, which considers stability and performance between trajectories of the system. In this paper, we show how universal shifted stability and performance of discrete-time systems can be analyzed by making use of the time-difference dynamics. Moreover, we extend the existing results for incremental dissipativity for discrete-time systems based on dissipativity analysis of the differential dynamics to more general state-dependent storage functions for less conservative results. Finally, we show how both these equilibrium-free notions can be cast as a convex analysis problem by making use of the linear parameter-varying framework, which is also demonstrated by means of an example.

Convex Equilibrium-Free Stability and Performance Analysis of Discrete-Time Nonlinear Systems

TL;DR

The paper addresses stability and performance analysis of discrete-time nonlinear systems without relying on a single equilibrium by introducing universal shifted and incremental notions. It develops velocity- and differential-based frameworks that relate equilibrium-free properties to (i) time-difference dynamics and (ii) differential dynamics, respectively, and shows how these can be formulated as convex LPV/DPV problems. By allowing state-dependent storage functions, the work reduces conservatism in incremental analyses and provides finite-dimensional LMIs via DPV embedding, enabling computable tests for USS/USAS, USP, USD, IS, and ID. An illustrative Duffing oscillator example demonstrates substantial improvements when using a state-dependent storage matrix over a constant one, highlighting practical applicability for equilibrium-free stability and performance guarantees in DT nonlinear systems.

Abstract

This paper considers the equilibrium-free stability and performance analysis of discrete-time nonlinear systems. We consider two types of equilibrium-free notions. Namely, the universal shifted concept, which considers stability and performance w.r.t. all equilibrium points of the system, and the incremental concept, which considers stability and performance between trajectories of the system. In this paper, we show how universal shifted stability and performance of discrete-time systems can be analyzed by making use of the time-difference dynamics. Moreover, we extend the existing results for incremental dissipativity for discrete-time systems based on dissipativity analysis of the differential dynamics to more general state-dependent storage functions for less conservative results. Finally, we show how both these equilibrium-free notions can be cast as a convex analysis problem by making use of the linear parameter-varying framework, which is also demonstrated by means of an example.
Paper Structure (29 sections, 22 theorems, 93 equations, 4 figures)

This paper contains 29 sections, 22 theorems, 93 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Equilibrium-Free Stability and Performance
  3. Nonlinear system
  4. Equilibrium-free stability
  5. Equilibrium-free dissipativity
  6. Velocity Analysis
  7. The DT velocity form and velocity dissipativity
  8. Induced universal shifted stability
  9. Induced universal shifted dissipativity
  10. Differential Analysis
  11. The differential form
  12. Induced incremental stability
  13. Induced incremental dissipativity
  14. The relation between velocity and differential dissipativity
  15. Convex Equilibrium-Free Analysis
  16. ...and 14 more sections

Key Result

Theorem 1

The nonlinear system given by 8_eq:nonlinsys is USS, if there exists a function $V_\mathrm{s}:{\mathbb{R}}^{n_\mathrm{x}}\!\times\mathscr{W}\to{{\mathbb{R}}_0^+}$ with $V_\mathrm{s}(\cdot,w_*)\in\mathcal{C}_{1}\xspace$ and $V_\mathrm{s}(\cdot,w_*)\in\mathcal{Q}_{x_*}$ for every $(x_*,w_*)\in\mathop{ holds for every $(x_*,w_*)\in\mathop{\mathrm{\pi}}\nolimits_\mathrm{x_*,w_*}\mathscr{E}$ and for al

Figures (4)

  • Figure 1: The invariant tube $\mathbb{X}_{\tilde{x},\gamma}$ for incremental invariance.
  • Figure 2: Overview of the results and their connections.
  • Figure 3: State trajectories for input $w$ with initial condition $x_{0}$ ( ) and for input $\tilde{w}$ with initial condition $\tilde{x}_{0}$ ( ).
  • Figure 4: The cumulative incremental supply plus the initial storage $\sum_{\tau=0}^{t}s_\mathrm{i} (w(\tau), \tilde{w}(\tau),z(\tau), \tilde{z}(\tau)) + {\mathcal{V}_\mathrm{i}}(x(0), \tilde{x}(0))$ ( ) and the incremental storage ${\mathcal{V}_\mathrm{i}}(x(t+1),\tilde{x}(t+1))$ ( ) for the trajectories generated by the inputs $w$ and $\tilde{w}$.

Theorems & Definitions (33)

  • Definition 1: Universal shifted stability Koelewijn2023
  • Theorem 1: Universal shifted Lyapunov stability
  • Theorem 2: Universal shifted invariance
  • Definition 2: Incremental stability
  • Theorem 3: Incremental Lyapunov stability Tran2016Tran2018:
  • Theorem 4: Incremental invariance
  • Definition 3: Universal shifted dissipativity
  • Definition 4: Incremental dissipativity
  • Definition 5: Discrete-time velocity form
  • Definition 6: Velocity stability
  • ...and 23 more