Koopman Operator Based Time-Delay Embeddings and State History Augmented LQR for Periodic Hybrid Systems: Bouncing Pendulum and Bipedal Walking

TL;DR

This paper extends time-delay embedding techniques to model and control periodic hybrid systems like bouncing pendulums and bipeds using Koopman operator theory, enabling linear control design for complex nonlinear systems.

Contribution

It introduces a novel approach to model hybrid systems as linear systems via time-delay embeddings and develops a state history augmented LQR for effective control.

Findings

Successfully models hybrid systems as linear systems using time-delay embeddings.

Designs an LQR controller utilizing current and past states for hybrid systems.

Provides open-source code for implementation and validation.

Abstract

Time-delay embedding is a technique that uses snapshots of state history over time to build a linear state space model of a nonlinear smooth system. We demonstrate that periodic non-smooth or hybrid system can also be modeled as a linear state space system using this approach as long as its behavior is consistent in modes and timings. We extend time-delay embeddings to generate a linear model of two periodic hybrid systems: the bouncing pendulum and the simplest walker with control inputs. This leads to a state history augmented linear quadratic regulator (LQR) which uses current and past state history for feedback control. Example code can be found at https://github.com/Chun-MingYang/koopman-timeDelay-lqr.git