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Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit-time optimal control

Ivan Yegorov, Peter M. Dower, Lars Grüne

TL;DR

The paper addresses global stabilization of deterministic nonlinear systems by constructing CLFs through exit-time optimal control. By solving an exit-time problem on a sublevel set $\Omega_c$ of a local CLF and concatenating its value function with the local CLF, the authors obtain a global CLF on the domain $\mathcal D_0$, enabling a curse-of-dimensionality-free stabilization framework. When a suitable local CLF is unavailable, they propose a small-ball exit-time approach that preserves convergence to the original CLF on compact subsets, with practical stabilization results. Numerical experiments in 2D demonstrate strong agreement with local CLFs and control strategies, while a 6D PVTOL example shows the limits of purely characteristics-based methods and the value of direct numerical approaches within MPC; the work also discusses sparse-grid considerations and future directions for scalable high-dimensional implementations.

Abstract

This paper studies the problem of constructing control Lyapunov functions (CLFs) and feedback stabilization strategies for deterministic nonlinear control systems described by ordinary differential equations. Many numerical methods for solving the Hamilton-Jacobi-Bellman partial differential equations specifying CLFs typically require dense state space discretizations and consequently suffer from the curse of dimensionality. A relevant direction of attenuating the curse of dimensionality concerns reducing the computation of the values of CLFs and associated feedbacks at any selected states to finite-dimensional nonlinear programming problems. In this work, exit-time optimal control is used for that purpose. First, we state an exit-time optimal control problem with respect to a sublevel set of an appropriate local CLF and establish that, under a number of reasonable conditions, the concatenation of the corresponding value function and the local CLF is a global CLF in the whole domain of asymptotic null-controllability. This leads to a curse-of-dimensionality-free approach to feedback stabilization. We also investigate the formulated optimal control problem. A modification of these constructions for the case when one does not find a suitable local CLF is provided as well. Supporting numerical simulation results that illustrate our development are subsequently presented and discussed. Furthermore, it is pointed out that the curse of complexity may cause significant issues in practical implementation even if the curse of dimensionality is mitigated.

Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit-time optimal control

TL;DR

The paper addresses global stabilization of deterministic nonlinear systems by constructing CLFs through exit-time optimal control. By solving an exit-time problem on a sublevel set of a local CLF and concatenating its value function with the local CLF, the authors obtain a global CLF on the domain , enabling a curse-of-dimensionality-free stabilization framework. When a suitable local CLF is unavailable, they propose a small-ball exit-time approach that preserves convergence to the original CLF on compact subsets, with practical stabilization results. Numerical experiments in 2D demonstrate strong agreement with local CLFs and control strategies, while a 6D PVTOL example shows the limits of purely characteristics-based methods and the value of direct numerical approaches within MPC; the work also discusses sparse-grid considerations and future directions for scalable high-dimensional implementations.

Abstract

This paper studies the problem of constructing control Lyapunov functions (CLFs) and feedback stabilization strategies for deterministic nonlinear control systems described by ordinary differential equations. Many numerical methods for solving the Hamilton-Jacobi-Bellman partial differential equations specifying CLFs typically require dense state space discretizations and consequently suffer from the curse of dimensionality. A relevant direction of attenuating the curse of dimensionality concerns reducing the computation of the values of CLFs and associated feedbacks at any selected states to finite-dimensional nonlinear programming problems. In this work, exit-time optimal control is used for that purpose. First, we state an exit-time optimal control problem with respect to a sublevel set of an appropriate local CLF and establish that, under a number of reasonable conditions, the concatenation of the corresponding value function and the local CLF is a global CLF in the whole domain of asymptotic null-controllability. This leads to a curse-of-dimensionality-free approach to feedback stabilization. We also investigate the formulated optimal control problem. A modification of these constructions for the case when one does not find a suitable local CLF is provided as well. Supporting numerical simulation results that illustrate our development are subsequently presented and discussed. Furthermore, it is pointed out that the curse of complexity may cause significant issues in practical implementation even if the curse of dimensionality is mitigated.

Paper Structure

This paper contains 19 sections, 20 theorems, 151 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Global extension of a local CLF via exit-time optimal control
  3. Problem statement and preliminary considerations
  4. Main result
  5. Investigation of the exit-time optimal control problem
  6. Approximation of a global CLF in case when an appropriate local CLF is not available
  7. A curse-of-dimensionality-free approach to CLF approximation and feedback stabilization
  8. Numerical simulations
  9. Conclusion
  10. Proofs of some auxiliary results
  11. Proof of Lemma 2.12
  12. Proof of Proposition 2.29
  13. Further details on implementing the curse-of-dimensionality-free approach to CLF approximation and feedback stabilization
  14. Construction of a local CLF
  15. Using the characteristics based representation of the value function
  16. ...and 4 more sections

Key Result

Proposition 2.11

Let Assumptions Ass_1, Ass_3 and Ass_7 hold. There exists a constant $C_4 > 0$ satisfying that is , the Petrov condition holds for the right-hand side of (Eq_1) on $l_c$.

Figures (8)

  • Figure 1: The exit-time optimal control problem (\ref{['Eq_21']}), whose target set is a level set of a local CLF.
  • Figure 2: The exit-time optimal control problem (\ref{['Eq_75']}) (or , equivalently , (\ref{['Eq_77']})), whose target set is the closed ball with center $x = 0_n$ and sufficiently small radius $\delta \in (0, r]$.
  • Figure 3: The Kruzhkov transformed functions $v(\cdot), v_{\mathrm{loc}}(\cdot)$ and their difference for $U = [-1.2, 1.2]$ in Example \ref{['Exa_45']}. Some approximated level sets of $v(\cdot)$ are shown as well. In order to see the graph of the difference between $v(\cdot)$ and $v_{\mathrm{loc}}(\cdot)$ clearer, the scale of the vertical axis in the third subfigure is modified as compared to that in the first two subfigures.
  • Figure 4: The feedback control strategies $u^*(\cdot), u^*_{\mathrm{loc}}(\cdot)$ (corresponding to $v(\cdot), v_{\mathrm{loc}}(\cdot)$, respectively) and their difference for $U = [-1.2, 1.2]$ in Example \ref{['Exa_45']}. In order to see the graphs clearer, we do not fix the same scale for the vertical axes in the subfigures.
  • Figure 5: The shooting state error, shooting time, and shooting value replacement indicator for $U = [-1.2, 1.2]$ in Example \ref{['Exa_45']}. Different scales are used for the vertical axes in the subfigures.
  • ...and 3 more figures

Theorems & Definitions (49)

  • Remark 2.2
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • Proposition 2.11
  • Lemma 2.12
  • Lemma 2.13
  • ...and 39 more