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Solvability of the Inverse Optimal Control problem based on the minimum principle

Afreen Islam, Guido Herrmann, Joaquin Carrasco

Abstract

In this paper, the solvability of the Inverse Optimal Control (IOC) problem based on two existing minimum principal methods, is analysed. The aim of this work is to answer the question regarding what kinds of trajectories, that is depending on the initial conditions of the closed-loop system and system dynamics, of the original optimal control problem, will result in the recovery of the true weights of the reward function for both the soft and the hard-constrained methods [1], [2]. Analytical conditions are provided which allow to verify if a trajectory is sufficiently conditioned, that is, holds sufficient information to recover the true weights of an optimal control problem. It was found that the open-loop system of the original optimal problem has a stronger influence on the solvability of the Inverse Optimal Control problem for the hard-constrained method as compared to the soft-constrained method. These analytical results were validated via simulation.

Solvability of the Inverse Optimal Control problem based on the minimum principle

Abstract

In this paper, the solvability of the Inverse Optimal Control (IOC) problem based on two existing minimum principal methods, is analysed. The aim of this work is to answer the question regarding what kinds of trajectories, that is depending on the initial conditions of the closed-loop system and system dynamics, of the original optimal control problem, will result in the recovery of the true weights of the reward function for both the soft and the hard-constrained methods [1], [2]. Analytical conditions are provided which allow to verify if a trajectory is sufficiently conditioned, that is, holds sufficient information to recover the true weights of an optimal control problem. It was found that the open-loop system of the original optimal problem has a stronger influence on the solvability of the Inverse Optimal Control problem for the hard-constrained method as compared to the soft-constrained method. These analytical results were validated via simulation.
Paper Structure (17 sections, 9 theorems, 51 equations, 4 figures)

This paper contains 17 sections, 9 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Soft-constrained method
  4. Hard-constrained method
  5. Motivation for the analysis of solvability of the minimum principle based IOC methods
  6. Solvability of the IOC problem for soft and hard-constrained methods for a general nonlinear primary problem
  7. Solvability of the IOC problem for the soft-constrained method
  8. A more convenient arrangement of the Riccati equation
  9. Time-varying observability of the secondary problem
  10. Solvability of the IOC problem for the hard-constrained method
  11. Solvability of the IOC problem for soft and hard-constrained methods for an infinite horizon Linear Time Invariant primary problem
  12. Solvability of the IOC problem using the soft-constrained method for an infinite horizon LTI Primary Optimal Control Problem
  13. Analysis for different types of trajectories using soft-constrained method for an infinite horizon second-order LTI primary problem
  14. Solvability of the IOC problem using the hard-constrained method for an Infinite Horizon LTI Primary Problem
  15. Analysis for different kinds of trajectories of second-order primary problems using hard-constrained methods
  16. ...and 2 more sections

Key Result

Theorem 1

It is assumed that $C(t)$, $A(t)$ and $B(t)$ exist and are bounded within the interval $[t_0,\,t_f]$. There is an interval $[t_0,\,T]$ for $0<T\leq t_f$, where the pair $(C(t),\,A(t))$ is observable. Under this condition, $P(0)$ is positive definite and a unique solution to the IOC problem exists.

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Theorem 2
  • Remark 5
  • Remark 6
  • Theorem 3
  • Remark 7
  • ...and 7 more