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Synthesizing Control Lyapunov-Value Functions for High-Dimensional Systems Using System Decomposition and Admissible Control Sets

Zheng Gong, Hyun Joe Jeong, Sylvia Herbert

TL;DR

The paper tackles the challenge of constructing Control Lyapunov-Value Functions (CLVFs) for high-dimensional nonlinear systems under input bounds by introducing a decomposition framework with Admissible Control Sets/Signals (ACS/ACSS). Subsystem CLVFs are computed in low dimensions and reconstructed into a full-dimensional CLVF, with exact reconstruction guaranteed under no shared-controls or certain shared-control conditions, and a Lipschitz CLF used when exact reconstruction is not possible. A quadratic-programming (QP) controller is provided to enforce exponential stabilization on the applicable domain. The approach yields substantial computational savings and scales to a 10D quadrotor example, validating both exact and inexact reconstruction scenarios and offering practical paths for higher-dimensional CLVF synthesis and online deployment.

Abstract

Control Lyapunov functions (CLFs) play a vital role in modern control applications, but finding them remains a problem. Recently, the control Lyapunov-value function (CLVF) and robust CLVF have been proposed as solutions for nonlinear time-invariant systems with bounded control and disturbance. However, the CLVF suffers from the ''curse of dimensionality,'' which hinders its application to practical high-dimensional systems. In this paper, we propose a method to decompose systems of a particular coupled nonlinear structure, in order to solve for the CLVF in each low-dimensional subsystem. We then reconstruct the full-dimensional CLVF and provide sufficient conditions for when this reconstruction is exact. Moreover, a point-wise optimal controller can be obtained using a quadratic program. We also show that when the exact reconstruction is impossible, the subsystems' CLVFs and their ``admissible control sets'' can be used to generate a Lipschitz continuous CLF. We provide several numerical examples to validate the theory and show computational efficiency.

Synthesizing Control Lyapunov-Value Functions for High-Dimensional Systems Using System Decomposition and Admissible Control Sets

TL;DR

The paper tackles the challenge of constructing Control Lyapunov-Value Functions (CLVFs) for high-dimensional nonlinear systems under input bounds by introducing a decomposition framework with Admissible Control Sets/Signals (ACS/ACSS). Subsystem CLVFs are computed in low dimensions and reconstructed into a full-dimensional CLVF, with exact reconstruction guaranteed under no shared-controls or certain shared-control conditions, and a Lipschitz CLF used when exact reconstruction is not possible. A quadratic-programming (QP) controller is provided to enforce exponential stabilization on the applicable domain. The approach yields substantial computational savings and scales to a 10D quadrotor example, validating both exact and inexact reconstruction scenarios and offering practical paths for higher-dimensional CLVF synthesis and online deployment.

Abstract

Control Lyapunov functions (CLFs) play a vital role in modern control applications, but finding them remains a problem. Recently, the control Lyapunov-value function (CLVF) and robust CLVF have been proposed as solutions for nonlinear time-invariant systems with bounded control and disturbance. However, the CLVF suffers from the ''curse of dimensionality,'' which hinders its application to practical high-dimensional systems. In this paper, we propose a method to decompose systems of a particular coupled nonlinear structure, in order to solve for the CLVF in each low-dimensional subsystem. We then reconstruct the full-dimensional CLVF and provide sufficient conditions for when this reconstruction is exact. Moreover, a point-wise optimal controller can be obtained using a quadratic program. We also show that when the exact reconstruction is impossible, the subsystems' CLVFs and their ``admissible control sets'' can be used to generate a Lipschitz continuous CLF. We provide several numerical examples to validate the theory and show computational efficiency.
Paper Structure (14 sections, 4 theorems, 33 equations, 4 figures, 1 algorithm)

This paper contains 14 sections, 4 theorems, 33 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. CLF and CLVF
  4. System decomposition and ACS
  5. Problem Formulation and Approach
  6. Exact Reconstruction of the CLVF
  7. Inexact Reconstruction: ACS and lipschitz continuous CLF
  8. Optimal QP Controller
  9. Numerical Example
  10. 2D System (Lemma \ref{['lemma: exact_decomposition']})
  11. 3D System (Theorem \ref{['thrm: exact_decomposition']})
  12. 3D System (Theorem \ref{['thrm: inexact_decomposition']})
  13. 10D Quadrotor
  14. Conclusion

Key Result

Lemma 1

Assume there are no common controls after decomposition. Then, $\bar{V}^\infty_\gamma (x) = \max (V^\infty_{\gamma,1} (z_1) ,V^\infty_{\gamma,2} (z_2)) = V_\gamma^\infty (x)$ , where $z_1 = \mathcal{P}_{x,1} (x)$, $z_2 = \mathcal{P}_{x,2} (x)$, and $\mathcal{D}_\gamma = \mathcal{P}^{-1}_{x,1}(

Figures (4)

  • Figure 1: Original and reconstructed CLVFs for a nonlinear 2D system \ref{['eqn: nonlinear2D']}. Top, CLVFs for the two subsystems with two different rates of exponential stabilizability ($\gamma)$. Bottom, comparison of the reconstructed CLVFs and original CLVFs for different $\gamma$, along with their regions of exponential stabilizability (ROES). Both reconstructed CLVFs are identical to the original CLVFs, and the corresponding ROESs also match, validating Lemma \ref{['lemma: exact_decomposition']}.
  • Figure 2: Original and reconstructed CLVFs with $\gamma=0.1$ for system \ref{['eqn: multlinear3D']}. Top left, $S_\gamma$ for the reconstructed ACS produced by Alg. \ref{['algo:exact_decomposition']}. Top right, comparison of the reconstructed and original CLVF projected into $x_1-x_2$ plane. Bottom, different level sets of original and reconstructed CLVFs. The reconstructed CLVF is identical to the original CLVF in $\mathcal{S}_\gamma$, validating Theorem \ref{['thrm: exact_decomposition']}. The computation time for the original system's CLVF is 102s on a grid of $[-2,-2,-2]$ to $[2,2,2]$ with 61 grid points on each dimension. On the same grid, the computation time for the subsystems' CLVFs is 3.37s, and the computation of $\mathcal{S}_\gamma$ takes 0.86s. Combined, the decomposition speeds computation time by 30x.
  • Figure 3: Top: The black stars denote the states with empty ACS (and ACSS). Left, $\mathcal{S}_\gamma$ (consists only of the origin) computed from Alg. \ref{['algo:exact_decomposition']}, shown in green. Right, $\bar{\mathcal{S}}_\gamma$. Bottom left: trajectory simulation using the QP \ref{['eqn:QP']}. Bottom right: decay of the CLF value along the trajectory. When initialized inside $\bar{\mathcal{S}}_\gamma$, the system can be stabilized to the origin, validating Theorem \ref{['thrm: inexact_decomposition']}. The computation time for the subsystems' CLVFs is 15.5s with a grid from $[-2,-2]$ to $[2,2]$ and 101 nodes on each dimension. The computation of $\mathcal{S}_\gamma$ takes 3.26s.
  • Figure 4: Since the CLVF is a 10D function, we do not visualize it, but show instead a trajectory that is stabilized to the origin and the value decay along this trajectory. The computation for the X/Y subsystem takes 1374.51s, and 56.63s for the Z subsystem. The direct computation for the 10D CLVF is not tractable.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Theorem 2
  • ...and 5 more