Estimating unknown dynamics and cost as a bilinear system with Koopman-based Inverse Optimal Control

TL;DR

This paper introduces a Koopman-based inverse optimal control method that approximates unknown nonlinear system dynamics and costs as a bilinear system, enabling more tractable analysis and control design.

Contribution

It develops a novel approach combining Koopman theory and inverse optimal control to handle unknown dynamics via bilinear system approximation.

Findings

Effective approximation of unknown dynamics as bilinear systems

Application of inverse LQR theory to Koopman-based models

Successful validation through simulations and robotic experiments

Abstract

In this work, we address the challenge of approximating unknown system dynamics and costs by representing them as a bilinear system using Koopman-based Inverse Optimal Control (IOC). Using optimal trajectories, we construct a bilinear control system in transformed state variables through a modified Extended Dynamic Mode Decomposition with control (EDMDc) that maintains exact dynamical equivalence with the original nonlinear system. We derive Pontryagin's Maximum Principle (PMP) optimality conditions for this system, which closely resemble those of the inverse Linear Quadratic Regulator (LQR) problem due to the consistent control input and state independence from the control. This similarity allows us to apply modified inverse LQR theory, offering a more tractable and robust alternative to nonlinear Inverse Optimal Control methods, especially when dealing with unknown dynamics. Our approach also benefits from the extensive analytical properties of bilinear control systems, providing a solid foundation for further analysis and application. We demonstrate the effectiveness of the proposed method through theoretical analysis, simulation studies and a robotic experiment, highlighting its potential for broader applications in the approximation and design of control systems.