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Cooptimizing Safety and Performance with a Control-Constrained Formulation

Hao Wang, Adityaya Dhande, Somil Bansal

TL;DR

This paper addresses cooptimizing safety and performance for autonomous systems by converting a state-constrained optimal control problem into a time-dependent, control-constrained problem via Hamilton-Jacobi reachability. It proves the equivalence of the two formulations and shows the value function is a viscosity solution to an HJB-PDE under the control-constrained view, enabling practical synthesis through safe-control sets. A 2D case study demonstrates consistent safety-performance improvements over baselines, with clear tradeoffs in offline/online computation. Limitations include scalability to high-dimensional systems and differentiability assumptions, with future work pointing to deep-learning value-function approximations and smooth overapproximations to address these issues.

Abstract

Autonomous systems have witnessed a rapid increase in their capabilities, but it remains a challenge for them to perform tasks both effectively and safely. The fact that performance and safety can sometimes be competing objectives renders the cooptimization between them difficult. One school of thought is to treat this cooptimization as a constrained optimal control problem with a performance-oriented objective function and safety as a constraint. However, solving this constrained optimal control problem for general nonlinear systems remains challenging. In this work, we use the general framework of constrained optimal control, but given the safety state constraint, we convert it into an equivalent control constraint, resulting in a state and time-dependent control-constrained optimal control problem. This equivalent optimal control problem can readily be solved using the dynamic programming principle. We show the corresponding value function is a viscosity solution of a certain Hamilton-Jacobi-Bellman Partial Differential Equation (HJB-PDE). Furthermore, we demonstrate the effectiveness of our method with a two-dimensional case study, and the experiment shows that the controller synthesized using our method consistently outperforms the baselines, both in safety and performance.

Cooptimizing Safety and Performance with a Control-Constrained Formulation

TL;DR

This paper addresses cooptimizing safety and performance for autonomous systems by converting a state-constrained optimal control problem into a time-dependent, control-constrained problem via Hamilton-Jacobi reachability. It proves the equivalence of the two formulations and shows the value function is a viscosity solution to an HJB-PDE under the control-constrained view, enabling practical synthesis through safe-control sets. A 2D case study demonstrates consistent safety-performance improvements over baselines, with clear tradeoffs in offline/online computation. Limitations include scalability to high-dimensional systems and differentiability assumptions, with future work pointing to deep-learning value-function approximations and smooth overapproximations to address these issues.

Abstract

Autonomous systems have witnessed a rapid increase in their capabilities, but it remains a challenge for them to perform tasks both effectively and safely. The fact that performance and safety can sometimes be competing objectives renders the cooptimization between them difficult. One school of thought is to treat this cooptimization as a constrained optimal control problem with a performance-oriented objective function and safety as a constraint. However, solving this constrained optimal control problem for general nonlinear systems remains challenging. In this work, we use the general framework of constrained optimal control, but given the safety state constraint, we convert it into an equivalent control constraint, resulting in a state and time-dependent control-constrained optimal control problem. This equivalent optimal control problem can readily be solved using the dynamic programming principle. We show the corresponding value function is a viscosity solution of a certain Hamilton-Jacobi-Bellman Partial Differential Equation (HJB-PDE). Furthermore, we demonstrate the effectiveness of our method with a two-dimensional case study, and the experiment shows that the controller synthesized using our method consistently outperforms the baselines, both in safety and performance.
Paper Structure (10 sections, 3 theorems, 12 equations, 2 figures, 1 table)

This paper contains 10 sections, 3 theorems, 12 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Background: Hamilton-Jacobi Reachability Analysis
  4. Method
  5. State and Time-Dependent Control-Constrained Optimal Control Problem
  6. Solving Control-Constrained Optimal Control Problem
  7. Characterizing the Set of Safe Controls
  8. Synthesizing the Cooptimization Controller
  9. Case Study
  10. Conclusion

Key Result

Theorem 1

Let us denote the optimal value of Prob. prob:state_constrained_opt_ctrl_prob and Prob. prob:ctrl_constrained_opt_ctrl_prob, at state $x\in\mathcal{X}$ and time $t\in[0,T]$, by $V_1(x,t)$ and $V(x,t)$. Then $V_1(x,t) = V(x,t) \ \forall x\in\mathcal{X} \ \text{and} \ \forall t\in[0,T]$.

Figures (2)

  • Figure 1: An illustration of the set of safe controls using Definition 1 of the boat system used in the case study. The control set $\mathcal{U}$ is given by $\{[u_1,u_2]\in\mathbb{R}^2| \ ||[u_1,u_2]||_2\leq 1\}$. The green-shaded region is the set of safe controls at the current state $x$ and time $t$, and the pink-shaded region is the set of controls that would lead the system into eventual violation of the state constraint. The location of the system relative to the boundary of the safe set, denoted using solid red line, is also illustrated in the top right subfigure. (Left) When the system is in the interior of the safe set, any admissible controls are permitted/safe. (Right) When the system is on boundary or outside the safe set, only admissible controls that lead to small decrease in the safety value are permitted.
  • Figure 2: Trajectories from initial state $[-2.58, 0.77]^\top$. Costs incurred are (i) Our method: 3.51, (ii) State constrained method, $(z_{res} = 70)$: 3.62, (iii) State constrained method, $(z_{res} = 210)$: 3.52, (iv) MPPI (horizon = 20): Violates safety constraint (v) MPPI (horizon = 100): 4.55, (vi) MPPI with safety filtering: 4.95, (vii) MPC: Violates safety constraint

Theorems & Definitions (8)

  • Definition 1: Set of Safe Controls
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 2: Set of Safe Controls Using HJ Reachability
  • Proposition 1
  • proof