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Near-Optimal Constrained Feedback Control of Nonlinear Systems via Approximate HJB and Control Barrier Functions

Milad Alipour Shahraki, Laurent Lessard

Abstract

This paper presents a two-stage framework for constrained near-optimal feedback control of input-affine nonlinear systems. An approximate value function for the unconstrained control problem is computed offline by solving the Hamilton--Jacobi--Bellman equation. Online, a quadratic program is solved that minimizes the associated approximate Hamiltonian subject to safety constraints imposed via control barrier functions. Our proposed architecture decouples performance from constraint enforcement, allowing constraints to be modified online without recomputing the value function. Validation on a linear 2-state 1D hovercraft and a nonlinear 9-state spacecraft attitude control problem demonstrates near-optimal performance relative to open-loop optimal control benchmarks and superior performance compared to control Lyapunov function-based controllers.

Near-Optimal Constrained Feedback Control of Nonlinear Systems via Approximate HJB and Control Barrier Functions

Abstract

This paper presents a two-stage framework for constrained near-optimal feedback control of input-affine nonlinear systems. An approximate value function for the unconstrained control problem is computed offline by solving the Hamilton--Jacobi--Bellman equation. Online, a quadratic program is solved that minimizes the associated approximate Hamiltonian subject to safety constraints imposed via control barrier functions. Our proposed architecture decouples performance from constraint enforcement, allowing constraints to be modified online without recomputing the value function. Validation on a linear 2-state 1D hovercraft and a nonlinear 9-state spacecraft attitude control problem demonstrates near-optimal performance relative to open-loop optimal control benchmarks and superior performance compared to control Lyapunov function-based controllers.
Paper Structure (14 sections, 5 theorems, 13 equations, 5 figures, 2 tables)

This paper contains 14 sections, 5 theorems, 13 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Dynamics and Objectives
  4. Hamilton--Jacobi--Bellman (HJB) Equation
  5. Relaxed HJB Formulation
  6. Value Function Approximation
  7. Safety via Control Barrier Functions (CBFs)
  8. GHJB-CBF-QP Controller
  9. Formulation
  10. Theoretical Properties
  11. Application Examples
  12. Example 1 (Linear): 1D Hovercraft
  13. Example 2 (Nonlinear): Spacecraft Attitude Control
  14. Conclusion and Future Work

Key Result

Theorem 1

Under the above assumptions: (i) Problem eq:relaxed has a nonempty feasible set. (ii) For any feasible $V$, the control $\bar{u}(x) = -\frac{1}{2} R^{-1} g^\mathsf{T} \nabla V$ is globally asymptotically stabilizing. (iii) $V$ upper-bounds the cost: $V(x_0) \geq J(x_0, \bar{u})$ for all $x_0$. (iv)

Figures (5)

  • Figure 1: Simulated trajectories (position, velocity, input) for the 1D hovercraft for various controllers. All trajectories are the same for this example.
  • Figure 2: Representative attitude RPs, angular velocity, reaction wheel angular momentum, and control torque simulation responses for various controllers. Only the 3rd components of $\sigma$, $\omega$, $h_w$, and $u$ are shown.
  • Figure 3: Representative attitude RPs, angular velocity, reaction wheel angular momentum, and control torque simulation responses for various controllers. Only the 2nd components of $\sigma$, $\omega$, $h_w$, and $u$ are shown.
  • Figure 4: Pointing constraint $B(\sigma)$ simulation response for various controllers. Since $B(\sigma) > 0$, we are assured that the pointing constraint is satisfied during the trajectory.
  • Figure 5: Sensor boresight trajectory on the celestial sphere. The red region denotes the exclusion zone of half-angle $\theta = \ang{15}$ around the bright object. All constrained controllers maintain $B(\sigma) \geq 0$ throughout the maneuver.

Theorems & Definitions (12)

  • Theorem 1: Relaxed Solution Properties jiang2015global
  • Definition 1: Relative Degree xiao2021high
  • Definition 2: HOCBF ames2019controlxiao2021high
  • Theorem 2: HOCBF Forward Invariance ames2019controlxiao2021high
  • Remark 1: No comparison function
  • Proposition 1: Optimality Recovery
  • proof
  • Proposition 2: Safety-Performance Decoupling
  • proof
  • Remark 2: Optimality Gap Decomposition
  • ...and 2 more