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Parameter-Dependent Control Lyapunov Functions for Stabilizing Nonlinear Parameter-Varying Systems

Pan Zhao

TL;DR

This work generalizes Lyapunov-based stabilization to nonlinear parameter-varying (NPV) systems via parameter-dependent CLFs (PD-CLFs). It shows how a PD-CLF yields a min-norm control law through a robust quadratic program and develops a convex SOS-based framework to jointly synthesize a PD-CLF and a PD controller while maximizing the PD region of stabilization, all while handling input constraints. The approach preserves nonlinear dynamics without linearization, offering an alternative to LPV methods and enabling gain-scheduled stabilization across varying parameters. Numerical results on a 2D toy example and a 2D rocket-landing scenario demonstrate enlarged stabilization regions, reduced control effort, and robustness to parameter variations.

Abstract

This paper introduces the concept of parameter-dependent (PD) control Lyapunov functions (CLFs) for gain-scheduled stabilization of nonlinear parameter-varying (NPV) systems. It shows that given a PD-CLF, a min-norm control law can be constructed by solving a robust quadratic program. For polynomial control-affine NPV systems, it provides convex conditions, based on the sum of squares programming, to jointly synthesize a PD-CLF and a PD controller while maximizing the PD region of stabilization. Input constraints can be straightforwardly incorporated into the synthesis procedure. Unlike traditional linear parameter-varying (LPV) methods that rely on linearization or over-approximation to get an LPV model, the proposed framework fully captures the nonlinearities of the system dynamics. The theoretical results are validated through numerical simulations, including a 2D rocket landing case study under varying mass and inertia.

Parameter-Dependent Control Lyapunov Functions for Stabilizing Nonlinear Parameter-Varying Systems

TL;DR

This work generalizes Lyapunov-based stabilization to nonlinear parameter-varying (NPV) systems via parameter-dependent CLFs (PD-CLFs). It shows how a PD-CLF yields a min-norm control law through a robust quadratic program and develops a convex SOS-based framework to jointly synthesize a PD-CLF and a PD controller while maximizing the PD region of stabilization, all while handling input constraints. The approach preserves nonlinear dynamics without linearization, offering an alternative to LPV methods and enabling gain-scheduled stabilization across varying parameters. Numerical results on a 2D toy example and a 2D rocket-landing scenario demonstrate enlarged stabilization regions, reduced control effort, and robustness to parameter variations.

Abstract

This paper introduces the concept of parameter-dependent (PD) control Lyapunov functions (CLFs) for gain-scheduled stabilization of nonlinear parameter-varying (NPV) systems. It shows that given a PD-CLF, a min-norm control law can be constructed by solving a robust quadratic program. For polynomial control-affine NPV systems, it provides convex conditions, based on the sum of squares programming, to jointly synthesize a PD-CLF and a PD controller while maximizing the PD region of stabilization. Input constraints can be straightforwardly incorporated into the synthesis procedure. Unlike traditional linear parameter-varying (LPV) methods that rely on linearization or over-approximation to get an LPV model, the proposed framework fully captures the nonlinearities of the system dynamics. The theoretical results are validated through numerical simulations, including a 2D rocket landing case study under varying mass and inertia.

Paper Structure

This paper contains 14 sections, 9 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parameter-Dependent CLFs for Stabilizing NPV Systems
  4. Constructing min-norm control law via robust quadratic programs
  5. Robustness of min-norm control law against input gain uncertainty
  6. Local analysis and PD region of stabilization
  7. Convex synthesis of PD-CLFs Using SOS Programming
  8. Preliminaries about SOS optimization
  9. Convex Synthesis of a PD-CLF and a PD Controller
  10. QP-based control law vs pre-defined control law
  11. Simulation Results
  12. Numerical example
  13. Rocket landing under varying mass and inertia
  14. Conclusions

Key Result

Lemma 1

If $V(x,\theta)$ is a PD-CLF according to def:pd-clf and both $\frac{\partial V}{\partial x}$ and $\alpha$ are locally Lipschitz, then the control law $u^*(x,\theta)$ from solving eq:min-norm-formulation is locally Lipschitz.

Figures (4)

  • Figure 1: Geometric illustration of a PD-CLF $V(x,\theta)$ for a one-dimensional system with a single parameter ($n=n_\theta=1$ ) in the $x$-$\theta$ plane and an exemplary trajectory of $(x(t),\theta(t),V(t))$ (denoted by the red line) under stabilizing inputs generated by $V$. Notice that the trajectory will eventually approach the $\theta$-axis (corresponding to $x=0$) and stay there regardless of the trajectory of $\theta$.
  • Figure 2: PD region of stabilization (PD-ROS) (left) and its projection onto $x$-planes with different $\theta$-values together with the robust ROS (right). Robust ROS overlaps with the smallest projection of the PD-ROS, corresponding to $\theta=0.0.05$
  • Figure 3: Trajectories of states (top), Lyapunov function (middle) and control input (bottom) starting from four initial points under the pre-defined control law \ref{['eq:control-law-Y-X']} (left) and the min-norm control law \ref{['eq:min-norm-formulation']} (right).
  • Figure 4: Trajectories of states (top) and control inputs (bottom) starting from four initial points under the pre-defined control law \ref{['eq:control-law-Y-X']} (left) and the min-norm control law \ref{['eq:min-norm-formulation']} (right).

Theorems & Definitions (27)

  • Remark 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 17 more