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Underapproximating Safe Domains of Attraction for Discrete-Time Systems Using Implicit Representations of Backward Reachable Sets

Mohamed Serry, Jun Liu

TL;DR

The paper tackles the challenge of certifying the safe domain of attraction (DOA) for general discrete-time nonlinear systems under state constraints, where traditional Lyapunov-based methods are often conservative or low-dimensional. It introduces an iterative approach using implicit representations of safe backward reachable sets, producing a sequence of monotone safe ROAs whose union exactly equals the safe DOA $\mathcal{D}_{0}^{\mathcal{X}}$. A key contribution is the implicit, sublevel-set representation with $\mathcal{V}_{k}=\{x\mid v_{k}(x)\le 1\}$ and $v_{k+1}(x)=\max\{\theta(x),\,v_k(f(x))\}$, enabling efficient pointwise verification and scalable computation. The initial safe ROA is obtained via a quadratic Lyapunov function around the origin, and the method is demonstrated on two nonlinear numerical examples (two-machine and cart-pole), illustrating monotone expansion and safe attraction verification. The framework provides certifiable, high-dimensional safe DOA underapproximations with potential extensions to robust DOA and null-controllability for perturbed or controlled discrete-time systems, using Bellman-type equations.

Abstract

Analyzing and certifying stability and attractivity of nonlinear systems is a topic of research interest that has been extensively investigated by control theorists and engineers for many years. Despite that, accurately estimating domains of attraction for nonlinear systems remains a challenging task, where available estimation approaches are either conservative or limited to low-dimensional systems. In this work, we propose an iterative approach to accurately underapproximate safe (i.e., state-constrained) domains of attraction for general discrete-time autonomous nonlinear systems. Our approach relies on implicit representations of safe backward reachable sets of safe regions of attraction, where such regions can be be easily constructed using, e.g., quadratic Lyapunov functions. The iterations of our approach are monotonic (in the sense of set inclusion), where each iteration results in a safe region of attraction, given as a sublevel set, that underapproximates the safe domain of attraction. The sublevel set representations of the resulting regions of attraction can be efficiently utilized in verifying the inclusion of given points of interest in the safe domain of attraction. We illustrate our approach through two numerical examples, involving two- and four-dimensional nonlinear systems.

Underapproximating Safe Domains of Attraction for Discrete-Time Systems Using Implicit Representations of Backward Reachable Sets

TL;DR

The paper tackles the challenge of certifying the safe domain of attraction (DOA) for general discrete-time nonlinear systems under state constraints, where traditional Lyapunov-based methods are often conservative or low-dimensional. It introduces an iterative approach using implicit representations of safe backward reachable sets, producing a sequence of monotone safe ROAs whose union exactly equals the safe DOA . A key contribution is the implicit, sublevel-set representation with and , enabling efficient pointwise verification and scalable computation. The initial safe ROA is obtained via a quadratic Lyapunov function around the origin, and the method is demonstrated on two nonlinear numerical examples (two-machine and cart-pole), illustrating monotone expansion and safe attraction verification. The framework provides certifiable, high-dimensional safe DOA underapproximations with potential extensions to robust DOA and null-controllability for perturbed or controlled discrete-time systems, using Bellman-type equations.

Abstract

Analyzing and certifying stability and attractivity of nonlinear systems is a topic of research interest that has been extensively investigated by control theorists and engineers for many years. Despite that, accurately estimating domains of attraction for nonlinear systems remains a challenging task, where available estimation approaches are either conservative or limited to low-dimensional systems. In this work, we propose an iterative approach to accurately underapproximate safe (i.e., state-constrained) domains of attraction for general discrete-time autonomous nonlinear systems. Our approach relies on implicit representations of safe backward reachable sets of safe regions of attraction, where such regions can be be easily constructed using, e.g., quadratic Lyapunov functions. The iterations of our approach are monotonic (in the sense of set inclusion), where each iteration results in a safe region of attraction, given as a sublevel set, that underapproximates the safe domain of attraction. The sublevel set representations of the resulting regions of attraction can be efficiently utilized in verifying the inclusion of given points of interest in the safe domain of attraction. We illustrate our approach through two numerical examples, involving two- and four-dimensional nonlinear systems.
Paper Structure (11 sections, 6 theorems, 7 equations, 3 figures, 1 algorithm)

This paper contains 11 sections, 6 theorems, 7 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Problem Setup
  4. Safe Backward Reachable Sets and the Domain of Attraction
  5. Backward Reachable Sets: Implicit Representations
  6. Efficient pointwise evaluation
  7. Initial Safe Region of Attraction
  8. Numerical Examples
  9. Two-machine system
  10. Cart-pole system
  11. Conclusion

Key Result

Lemma 1

Given $g,h\colon \mathbb{R}^{n}\rightarrow\mathbb{R}$, $f\colon \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, and $c\in \mathbb{R}$, we have $f^{-1}(\Psi(g,c))\cap \Psi(h,c)= \Psi(\tilde{g},c),$ where $\tilde{g}(\cdot)\mathrel{:=} \max\{h(\cdot),g(f(\cdot))\}.$

Figures (3)

  • Figure 1: Estimates of the safe DOA of the two-machine system inside $\mathcal{X}$ (red), and generated trajectories starting from $\mathcal{X}$ (blue, green, and magenta).
  • Figure 2: $x_{1}-x_{2}$ (left) and $x_{3}-x_{4}$ (right) cross-sections, with $(x_{3},x_{4})=(0,0)$ and $(x_{1},x_{2})=(0,0)$, respectively, of the safe ROAs of the closed-loop cart-pole system inside $\mathcal{X}$ (red).
  • Figure 3: Profile of $\theta$ along the trajectories starting from $x_{0}^{(1)}$ and $x_{0}^{(2)}$.

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Definition 1: Safe region of attraction
  • Definition 2: Safe backward reachable sets
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 7 more