用时延嵌入法建模周期性混合系统，实现更优的反馈控制。
Koopman Operator Based Time-Delay Embeddings and State History Augmented LQR for Periodic Hybrid Systems: Bouncing Pendulum and Bipedal Walking
- 通过状态历史时延嵌入构建线性模型，适配非光滑周期系统
- 在弹跳摆和双足行走系统上实现状态历史增强的LQR控制
- 适合研究机器人周期运动控制与非线性系统线性化方法者
时延嵌入技术利用状态随时间的历史快照，为非线性光滑系统构建线性状态空间模型。我们证明，只要系统在各模式下的行为和触发时机保持一致，周期性非光滑或混合系统同样可通过此方法建模为线性状态空间系统。本文将该方法拓展至两个周期性混合系统：带控制输入的弹跳摆和最简双足行走模型。由此生成的状态历史增强型线性二次调节器（LQR）结合当前与过去状态历史进行反馈控制。相关示例代码可在 https://github.com/Chun-Ming-Yang/koopman-timeDelay-lqr.git 获取。
原文摘要 · Abstract (English)
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
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