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Robust Control using Control Lyapunov Function and Hamilton-Jacobi Reachability

Chun-Ming Yang, Pranav A. Bhounsule

TL;DR

The paper addresses robustness of Model Predictive Control (MPC) under model uncertainty by fusing a Control Lyapunov Function (CLF) approach with Hamilton-Jacobi (HJ) reachability. It leverages a linearized model with additive disturbance to compute a Region of Attraction (ROA) and a nonlinear model to compute a Backward Reachable Set (BRS), enabling estimation of a worst-case disturbance bound $w_{max}$. By overlaying the CLF-derived invariant set with the HJ-derived safe set, the method yields a conservative yet principled disturbance bound and safety guarantees for the nominal controller. The framework is demonstrated in simulation on a 2D quadcopter tracking a trajectory and a 2D quadruped height/velocity regulation task under unknown payload, with the robust MPC outperforming nominal MPC in tracking while maintaining safety constraints.

Abstract

The paper presents a robust control technique that combines the Control Lyapunov function and Hamilton-Jacobi Reachability to compute a controller and its Region of Attraction (ROA). The Control Lyapunov function uses a linear system model with an assumed additive uncertainty to calculate a control gain and the level sets of the ROA as a function of the uncertainty. Next, Hamilton-Jacobi reachability uses the nonlinear model with the modeled uncertainty, which need not be additive, to compute the backward reachable set (BRS). Finally, by juxtaposing the level sets of the ROA with BRS, we can calculate the worst-case additive disturbance and the ROA of the nonlinear model. We illustrate our approach on a 2D quadcopter tracking trajectory and a 2D quadcopter with height and velocity regulation in simulation.

Robust Control using Control Lyapunov Function and Hamilton-Jacobi Reachability

TL;DR

The paper addresses robustness of Model Predictive Control (MPC) under model uncertainty by fusing a Control Lyapunov Function (CLF) approach with Hamilton-Jacobi (HJ) reachability. It leverages a linearized model with additive disturbance to compute a Region of Attraction (ROA) and a nonlinear model to compute a Backward Reachable Set (BRS), enabling estimation of a worst-case disturbance bound . By overlaying the CLF-derived invariant set with the HJ-derived safe set, the method yields a conservative yet principled disturbance bound and safety guarantees for the nominal controller. The framework is demonstrated in simulation on a 2D quadcopter tracking a trajectory and a 2D quadruped height/velocity regulation task under unknown payload, with the robust MPC outperforming nominal MPC in tracking while maintaining safety constraints.

Abstract

The paper presents a robust control technique that combines the Control Lyapunov function and Hamilton-Jacobi Reachability to compute a controller and its Region of Attraction (ROA). The Control Lyapunov function uses a linear system model with an assumed additive uncertainty to calculate a control gain and the level sets of the ROA as a function of the uncertainty. Next, Hamilton-Jacobi reachability uses the nonlinear model with the modeled uncertainty, which need not be additive, to compute the backward reachable set (BRS). Finally, by juxtaposing the level sets of the ROA with BRS, we can calculate the worst-case additive disturbance and the ROA of the nonlinear model. We illustrate our approach on a 2D quadcopter tracking trajectory and a 2D quadcopter with height and velocity regulation in simulation.
Paper Structure (19 sections, 31 equations, 4 figures)

This paper contains 19 sections, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Model Predictive Control (MPC)
  4. Hamilton-Jacobi Reachability Analysis (HJ)
  5. Control Lyapunov Function (CLF)
  6. Example 1: 2D Quadcopter
  7. Nonlinear Model:
  8. Linearized Model:
  9. Robust control:
  10. Example 2: 2D Quadruped
  11. Nonlinear Model:
  12. Linearized Model:
  13. Robust Control:
  14. Hamilton-Jacobi Reachability:
  15. Results and Discussion
  16. ...and 4 more sections

Figures (4)

  • Figure 1: (a) Control Lyapunov function (CLF) uses the linear dynamics to compute the controller and the level set for the Region of Attraction (ROA) for a given maximum disturbance $w_i$ (red ellipses); (b) Hamilton-Jacobi (HJ) Reachability uses the nonlinear dynamics with the disturbance $w$ to compute the Backward Reachable set (gray region); (c) Superimposing the BRS with level sets enables computation of the worst case disturbance (here $w_2=w_{max}$) and the safe set (red ellipse)
  • Figure 2: (a) 2D quadcopter (b) 2D quadruped
  • Figure 3: Bicopter is tracking the figure 8 curve in the presence of bounded disturbance. (a) Figure 8 tracking in the y-z plane. (b) Horizontal tracking errors with invariant set. (c) Vertical tracking errors with invariant set.
  • Figure 4: For the 2D quadruped, two separate experiments are conducted. (a) simulation snapshot of quadruped while loading an unknown 5 kg object, along with the vertical height tracking result. (b) vertical height tracking errors with invariant set (c) simulation snapshot of quadruped while pushing an unknown 5 kg object along with the forward velocity tracking result (d) forward velocity tracking errors with invariant set