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Safe Stabilization using Nonsmooth Control Lyapunov Barrier Function

Jianglin Lan, Eldert van Henten, Peter Groot Koerkamp, Congcong Sun

TL;DR

The paper tackles safe stabilization of continuous-time systems in the presence of multiple bounded unsafe regions by introducing a nonsmooth control Lyapunov barrier function (NCLBF) that unifies Lyapunov and barrier concepts without gradient cancellation. It provides a constructive design framework for single and multiple unsafe sets, employing a max-based V(x) constructed from $L(x)=\|x\|^2$ and $B(x)=\eta_2-\eta_1\|x-x_c\|^2$, partitioning the state space into regions and deriving piecewise continuous controllers that ensure $\overline{D}_{\mathcal{F}} V(x) \le -\rho(\|x\|)$. Extensions to multiple unsafe sets use $V(x)=\max(L(x), \max_i B_i(x))$ with disjointness constraints, yielding a single NCLBF that certifies $\mathcal{KL}$-stability and safety under region-based control laws. The approach is validated through linear and nonlinear simulations, demonstrating safe convergence to the origin while avoiding all unsafe regions and showing advantages over smooth CLBF-based methods.

Abstract

This paper addresses the challenge of safe stabilization, ensuring the system state reach the origin while avoiding unsafe regions. Existing approaches relying on smooth Lyapunov barrier functions often fail to guarantee a feasible controller. To overcome this limitation, we introduce the nonsmooth Control Lyapunov Barrier Function (NCLBF), which ensures the existence of a safe and stabilizing controller. We provide a systematic framework for designing NCLBF and feedback control strategies to achieve safe stabilization in the presence of multiple bounded unsafe regions. Theoretical analysis and simulations of both linear and nonlinear systems demonstrate the effectiveness and superiority of our approach compared to the existing smooth functions method.

Safe Stabilization using Nonsmooth Control Lyapunov Barrier Function

TL;DR

The paper tackles safe stabilization of continuous-time systems in the presence of multiple bounded unsafe regions by introducing a nonsmooth control Lyapunov barrier function (NCLBF) that unifies Lyapunov and barrier concepts without gradient cancellation. It provides a constructive design framework for single and multiple unsafe sets, employing a max-based V(x) constructed from and , partitioning the state space into regions and deriving piecewise continuous controllers that ensure . Extensions to multiple unsafe sets use with disjointness constraints, yielding a single NCLBF that certifies -stability and safety under region-based control laws. The approach is validated through linear and nonlinear simulations, demonstrating safe convergence to the origin while avoiding all unsafe regions and showing advantages over smooth CLBF-based methods.

Abstract

This paper addresses the challenge of safe stabilization, ensuring the system state reach the origin while avoiding unsafe regions. Existing approaches relying on smooth Lyapunov barrier functions often fail to guarantee a feasible controller. To overcome this limitation, we introduce the nonsmooth Control Lyapunov Barrier Function (NCLBF), which ensures the existence of a safe and stabilizing controller. We provide a systematic framework for designing NCLBF and feedback control strategies to achieve safe stabilization in the presence of multiple bounded unsafe regions. Theoretical analysis and simulations of both linear and nonlinear systems demonstrate the effectiveness and superiority of our approach compared to the existing smooth functions method.
Paper Structure (11 sections, 6 theorems, 33 equations, 8 figures)

This paper contains 11 sections, 6 theorems, 33 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem description and preliminaries
  3. NCLBF-based control for a single unsafe set
  4. Construction of NCLBF
  5. Properties of NCLBF
  6. NCLBF-based Control Design
  7. Handling multiple unsafe sets
  8. Illustrative examples
  9. Linear System with A Single Unsafe Set
  10. Nonlinear System with Multiple Unsafe Sets
  11. Conclusions

Key Result

Lemma 2.1

For system eq:sys1 with disjoint unsafe state sets $\mathcal{O}_i \subset \mathcal{X}$, $i \in [1,N]$, where $0 \notin \mathcal{O} := \cup_{i=1}^N \mathcal{O}_i$, if $V(x)$ is an NCLBF under the state-feedback controller $u = \kappa(x)$ and $x(0) \in \mathcal{X} \setminus \mathcal{O}$, then the cont

Figures (8)

  • Figure 1: A 2-D illustration of the proposed NCLBF design. The state regions $\mathcal{R}_1$, $\mathcal{R}_2$ and $\mathcal{R}_3$ are defined in \ref{['thm3:3 regions']}. The unsafe boundary $\partial \mathcal{O}$ has a radius $\sqrt{r}$ centred at $x_c$, while $\mathcal{R}_3$ has a radius $\sqrt{\bar{r}}$ centred at $\bar{x}_c$. The buffer width $b_w$ is the shortest distance between these boundaries.
  • Figure 2: A simple 2-D illustration shows the shrinking of the region (the sphere $\|x - \bar{x}_c\|^2 = \bar{r}$) by removing $\hat{\mathcal{R}}_3 = \left\{ x \in \mathcal{R}_3 \mid \|x\|^2 < \phi(x_c) \right\}$. The greyscale (darker indicates higher values) represents the NCLBF $V(x)$ in the state space $\mathcal{X} := \mathcal{R}_2 \cup \mathcal{R}_3 \cup \mathcal{R}_1 \cup \mathcal{O}$. The state regions $\mathcal{R}_1$, $\mathcal{R}_2$ and $\mathcal{R}_3$ are defined in \ref{['thm3:3 regions']}, and $H$ represents a tangent plane at $\tilde{x}$.
  • Figure 3: Trajectories for five initial states: Section \ref{['subsec:sim1']}, NCLBF.
  • Figure 4: $V(x)$ for five initial states: Section \ref{['subsec:sim1']}, NCLBF.
  • Figure 5: Trajectories for five initial states: Section \ref{['subsec:sim1']}, CLBF.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Theorem 3.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 4.2