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Funnel-based Control for Reach-Avoid-Stay Specifications

Ratnangshu Das, Pushpak Jagtap

TL;DR

This work addresses controller synthesis for control-affine nonlinear systems to achieve reach-avoid-stay specifications by introducing a funnel-based control framework with a circumvention mechanism around unsafe sets, yielding a closed-form controller. The method constructs time-varying funnel bounds $\rho_i(t)$ and normalized/transformed errors $e_i$ and $\varepsilon_i$, delivering a control law $u(x,t) = -g(x)^T(g(x)g(x)^T)^{-1}(k\xi(x,t)\varepsilon(x,t) - \frac{1}{2}\dot{\rho}_d(t)e(x,t))$ that drives the state to the target ${\mathbf{T}}$ while respecting state constraints. To handle obstacles, a circumvention function $\beta^{\hat{j}}_i(t)$ is integrated into an adaptive funnel, with updates to funnel bounds via $\gamma_L,\gamma_U$ and an adaptive law on $\alpha(t)$; the resulting closed-loop law $\hat{u}(x,t) = -g(x)^T(g(x)g(x)^T)^{-1}(\hat{k}\hat{\xi}\hat{\varepsilon} - \frac{1}{2}\dot{\gamma}_d\hat{e})$ ensures safe navigation around ${\mathcal{U}}^{\hat{j}}$ and convergence to ${\mathbf{T}}$. The approach further extends to general unsafe sets via Algorithm 1, with randomized path exploration around obstacles, and is validated through simulations of a mobile robot in a 2D arena. Overall, the paper provides a practical, scalable, closed-form solution for reach-avoid-stay using adaptive funnels and circumvention in nonlinear safety-critical control.

Abstract

The paper addresses the problem of controller synthesis for control-affine nonlinear systems to meet reach-avoid-stay specifications. Specifically, the goal of the research is to obtain a closed-form control law ensuring that the trajectories of the nonlinear system, reach a target set while avoiding all unsafe regions and adhering to the state-space constraints. To tackle this problem, we leverage the concept of the funnel-based control approach. Given an arbitrary unsafe region, we introduce a circumvent function that guarantees the system trajectory to steer clear of that region. Subsequently, an adaptive funnel framework is proposed based on the target, followed by the construction of a closed-form controller using the established funnel function, enforcing the reach-avoid-stay specifications. To demonstrate the efficacy of the proposed funnel-based control approach, a series of simulation experiments have been carried out.

Funnel-based Control for Reach-Avoid-Stay Specifications

TL;DR

This work addresses controller synthesis for control-affine nonlinear systems to achieve reach-avoid-stay specifications by introducing a funnel-based control framework with a circumvention mechanism around unsafe sets, yielding a closed-form controller. The method constructs time-varying funnel bounds $\rho_i(t)$ and normalized/transformed errors $e_i$ and $\varepsilon_i$, delivering a control law $u(x,t) = -g(x)^T(g(x)g(x)^T)^{-1}(k\xi(x,t)\varepsilon(x,t) - \frac{1}{2}\dot{\rho}_d(t)e(x,t))$ that drives the state to the target ${\mathbf{T}}$ while respecting state constraints. To handle obstacles, a circumvention function $\beta^{\hat{j}}_i(t)$ is integrated into an adaptive funnel, with updates to funnel bounds via $\gamma_L,\gamma_U$ and an adaptive law on $\alpha(t)$; the resulting closed-loop law $\hat{u}(x,t) = -g(x)^T(g(x)g(x)^T)^{-1}(\hat{k}\hat{\xi}\hat{\varepsilon} - \frac{1}{2}\dot{\gamma}_d\hat{e})$ ensures safe navigation around ${\mathcal{U}}^{\hat{j}}$ and convergence to ${\mathbf{T}}$. The approach further extends to general unsafe sets via Algorithm 1, with randomized path exploration around obstacles, and is validated through simulations of a mobile robot in a 2D arena. Overall, the paper provides a practical, scalable, closed-form solution for reach-avoid-stay using adaptive funnels and circumvention in nonlinear safety-critical control.

Abstract

The paper addresses the problem of controller synthesis for control-affine nonlinear systems to meet reach-avoid-stay specifications. Specifically, the goal of the research is to obtain a closed-form control law ensuring that the trajectories of the nonlinear system, reach a target set while avoiding all unsafe regions and adhering to the state-space constraints. To tackle this problem, we leverage the concept of the funnel-based control approach. Given an arbitrary unsafe region, we introduce a circumvent function that guarantees the system trajectory to steer clear of that region. Subsequently, an adaptive funnel framework is proposed based on the target, followed by the construction of a closed-form controller using the established funnel function, enforcing the reach-avoid-stay specifications. To demonstrate the efficacy of the proposed funnel-based control approach, a series of simulation experiments have been carried out.
Paper Structure (13 sections, 6 theorems, 31 equations, 3 figures)

This paper contains 13 sections, 6 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notations
  4. System Definition
  5. Problem Formulation
  6. Controller for Reachability Specification
  7. Extension to Reach-Avoid-Stay Specification
  8. Design of Circumvent Function
  9. Adaptive Funnel Design
  10. Controller Design
  11. Extension to Tackle General Unsafe sets
  12. Simulation Results
  13. Conclusion

Key Result

Theorem III.1

Consider the control-affine system $\mathcal{S}$ given in (eqn:sysdyn) with Assumptions assum:lip. Given a target set ${\mathbf{T}}$, the funnel constraints $\rho_{i,U}(t)$ and $\rho_{i,L}(t)$ (eqn:ppc), the control strategy will drive the state trajectory $x(t)$, to the target set ${\mathbf{T}}$ in finite time, i.e., $\exists t \in {\mathbb{R}}_0^+ : x(t) \in {\mathbf{T}}$. Here, $k$ is any posi

Figures (3)

  • Figure 1: Funnel Design. (a) Reachability funnel to obtain $\underline{t}$ and $\overline{t}$. (b) Introduction of circumvent function. (c) Funnel adapted around circumvent function.
  • Figure 2: 3D visualization.
  • Figure 3: (a) Starting from three different initial states, we have three different trajectories: T1 (blue circle), T2 (green diamond), and T3 (magenta square). Adaptive funnel framework with controlled system trajectories for (b) T1, (c) T2, and (d) T3.

Theorems & Definitions (13)

  • Remark II.2
  • Theorem III.1
  • proof
  • Remark IV.1
  • Lemma IV.2
  • proof
  • Theorem IV.3
  • proof
  • Lemma IV.4
  • Lemma IV.5
  • ...and 3 more