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Safety-Constrained Optimal Control for Unknown System Dynamics

Panagiotis Kounatidis, Andreas A. Malikopoulos

Abstract

In this paper, we present a framework for solving continuous optimal control problems when the true system dynamics are approximated through an imperfect model. We derive a control strategy by applying Pontryagin's Minimum Principle to the model-based Hamiltonian functional, which includes an additional penalty term that captures the deviation between the model and the true system. We then derive conditions under which this model-based strategy coincides with the optimal control strategy for the true system under mild convexity assumptions. We demonstrate the framework on a real robotic testbed for the cruise control application with safety distance constraints.

Safety-Constrained Optimal Control for Unknown System Dynamics

Abstract

In this paper, we present a framework for solving continuous optimal control problems when the true system dynamics are approximated through an imperfect model. We derive a control strategy by applying Pontryagin's Minimum Principle to the model-based Hamiltonian functional, which includes an additional penalty term that captures the deviation between the model and the true system. We then derive conditions under which this model-based strategy coincides with the optimal control strategy for the true system under mild convexity assumptions. We demonstrate the framework on a real robotic testbed for the cruise control application with safety distance constraints.

Paper Structure

This paper contains 22 sections, 3 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Organization
  5. Problem Formulation
  6. Modeling framework
  7. Original optimal control problem for the actual system
  8. Model-based surrogate problem with penalized cost
  9. Hamiltonian Analysis and Optimality Conditions
  10. Hamiltonian for the actual system
  11. Hamiltonian for the model-based penalized problem
  12. Constrained Hamiltonian minimization. Existence and uniqueness
  13. Equivalence Results
  14. Convex analysis preliminaries
  15. General equivalence results
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose Assumptions assum:one--assum:two hold. Then, for almost every $t \in [0,T]$, the sets of minimizers are nonempty, closed, and convex. If, in addition, for almost every $t$ the Hamiltonians are strictly convex in $u$ on $\mathcal{U}$ (e.g., $\alpha$-strongly convex), then these minimizers are unique almost everywhere.

Figures (2)

  • Figure 2: Position trajectories (Figure \ref{['fig:state_equiv']}) and snapshots during one run (Figures \ref{['fig:second']} and \ref{['fig:third']}).
  • Figure 3: Equivalence of control inputs

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • Theorem 3
  • proof