L-Functions Certify Set Attractivity for Discrete-Time Uncertain Nonlinear Switched Systems

TL;DR

The paper addresses certifying attractivity of target sets for discrete-time uncertain nonlinear switched systems by introducing $L$-functions, which are Lyapunov-like certificates that guarantee robust local attractivity under admissible switching and disturbances. It establishes a constructive link between robust control contractive sets (RCCS) and $L$-functions, showing that the existence of a RCCS implies an $L$-function exists and ensures attractivity within a level set $\ ext{\mathcal{L}}_R(\\Omega)$; conversely, an $L$-function provides a verifiable decrease condition along system trajectories. A detailed construction is given via the $\\kappa(x)$-function and the $h$-function, yielding a piecewise-constant $L$-function that decreases by at least $\\eta_R(d(x,\\Omega))$ in the relevant region. The framework is illustrated with linear examples and a nonlinear AMR-dynamics case study, demonstrating practical applicability to biology and control design, and it provides a method to approximate the domain of attraction through robust controllable sets. Overall, this work offers a tractable, set-based certificate for attractivity under uncertainty with potential impact on biological modeling and robust switching control.

Abstract

We introduce the class of L-functions to certify the attractivity of sets for uncertain nonlinear switched systems in discrete time. The existence of an L-function associated with a set guarantees the robust local attractivity of that set under the system dynamics. We propose a constructive method for obtaining piecewise-continuous L-functions based on contractive sets for the system, and show that the existence of a robust control contractive set for the dynamics implies the existence of an appropriate L-function, and hence the robust local attractivity of the set itself. We illustrate the proposed framework through examples that elucidate the theoretical concepts, and through the case study of a nonlinear switched system modelling antimicrobial resistance, which highlights the relevance of the approach to the analysis of biological systems.

Figures (5)

• Figure 1: A.$\phi(x,\sigma^k,w^k)$ is the state reached at time $k$ from the initial state $x\in\mathbb{X}\xspace$, if the switching law $\sigma^k=\{\sigma(1),\dots,\sigma(k)\}$ is applied and the realised uncertainty sequence is $w^k=\{w(1),\dots,w(k)\}$, while $\boldsymbol{\Phi}(x, \sigma^k)$ is the set of all states that can be reached at time $k$ from $x$, if the switching law $\sigma^k$ is applied, for all possible uncertainty sequences in $\mathbb{W}\xspace^k$. B. A RCCS $\Omega$ is contained in the interior of $\mathbb{C}(\Omega)$; for all $k \ge 1$, the controllable sets satisfy the nested property $\mathbb{C}^{k-1}(\Omega) \subseteq \operatorname{int}(\mathbb{C}^{k}(\Omega))$, while function $h(k)$ quantifies the distance between the boundaries of $\mathbb{C}^{k-1}(\Omega)$ and of $\mathbb{C}^{k}(\Omega)$.
• Figure 2: Example \ref{['example1']}. A. The set $\tilde{\mathbb{D}}(\Omega_0)=\bigcup_{j=1}^{6} \Omega_j$, in light blue, satisfies $\Omega_0 \subset \text{int}(\tilde{\mathbb{D}}(\Omega_0))$, with the boundary of $\Omega_0$ denoted by the black contour. Set $\mathbb{W}\xspace$ is in gray, and the non-empty difference $\Omega_0 \ominus \mathbb{W}$ in orange. B. Controlled trajectories under the switching law $\sigma(x)$, for $100$ different random uncertainty sequences $w^{50} \in \mathbb{W}^{50}$. C. The switching law $\sigma(x)$.
• Figure 3: Example \ref{['example2']}. Given $\Omega_0$, whose boundary is denoted by the black contour, the set $\mathbb{C}(\Omega_0)$ in orange is a RCCS. For the blue set $\mathbb{C}^2(\Omega_0)=\bigcup_{j=1}^2\Omega_j$, it holds $\mathbb{C}(\Omega_0)\subset \text{int}(\mathbb{C}^2(\Omega_0))$ as expected. The domain of attraction $\mathbb{D}(\Omega_0)$ is approximated by $\tilde{\mathbb{D}}(\Omega_0) = \bigcup_{k=1}^6 \mathbb{C}^k(\Omega_0)$ in grey.
• Figure 4: The grid in the domain $D_0$ (with $b_{\max}\le 1.6\times10^6$) contains $11,325$ initial points uniformly distributed in the three cases. Blue points denote initial conditions $x_i\in D_0$ for which $X^+_{i,\sigma} \subseteq \Omega_0$ after one step for some $\sigma\in\{1,2\}$, to assess whether $\Omega_0 \subseteq \text{int}(\mathbb{C}(\Omega_0))$ holds for the set $\Omega_0$ associated with the considered value $b_0$. For $b_0=1,000$ (A.) and $b_0=10,000$ (B.), the inclusion holds, and hence the considered $\Omega_0$ is a RCCS. For $b_0=120,000$ (C.), the inclusion does not hold for the adopted grid of points.
• Figure 5: Approximation of $\mathbb{D}(\Omega_0)$ in the domain $D_0$ (with $b_{\max}\le 1.6\times10^6$) via $\tilde{\mathbb{D}}(\Omega_0)=\bigcup_{k=1}^{1500}\mathbb{C}^k(\Omega_0)$, where $\Omega_0$ is the set associated with $b_0= 100{,}000$, for which the inclusion $\Omega_0 \subseteq \text{int}(\mathbb{C}(\Omega_0))$ holds. A. The triangular RCCS $\Omega_0$ and its approximated domain of attraction $\mathbb{C}^{1500}(\Omega_0)$ are the sets enclosed in the black and blue contours, respectively. B. The RCCS $\Omega_0$ is shown along with $\mathbb{C}^{k}(\Omega_0)$ for $k=1,\ldots,1500$. C. The RCCS $\Omega_0$ is shown along with a zoom into the first controllable sets, confirming the expected inclusion $\mathbb{C}^{k-1}(\Omega)\subseteq \text{int}(\mathbb{C}^k(\Omega))$.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5
• Proposition 1
• Remark 2
• Proposition 2
• Definition 6