透過線上激發輸入序列演算法加速歐拉─拉格朗日動態系統的稀疏識別

Abstract

在當代基於模型的控制方法中，由於準確的系統建模可以顯著提升控制效果，因此系統識別在其中扮演了關鍵角色。近期結合稀疏性的數據驅動識別方法，例如非線性動態稀疏識別（SINDy）、擴展拉格朗日-SINDy（xL-SINDy）及其變體，在歐拉─拉格朗日系統的識別上展現出極大潛力。然而，這些方法往往忽略了激發設計對識別效果的影響。 本研究提出線上最佳擴展拉格朗日動態稀疏識別框架（簡稱 OOL-SINDy），該方法可線上即時生成最佳激發輸入，以提升識別準確性。在每一個時間步長內，OOL-SINDy 透過已蒐集的資料估測系統，並藉由最小化克拉梅爾─拉奧下界（Cramér–Rao bound）來計算下一個輸入。此線上最佳化機制能加速收斂，並在較短時間內獲得更準確的模型，優於傳統的激發輸入方法。值得一提的是，OOL-SINDy 為全數據驅動的方法，不需要太多系統的先驗知識，並能在線上自我調整。多項模擬以及實作於反應輪系統上的硬體實驗皆驗證了此方法的有效性與強健性。

System identification is fundamental to modern model-based control, as accurate models enable superior control performance. Recent data-driven identification methods that incorporate sparsity, such as \textit{Sparse Identification of Nonlinear Dynamical Systems (SINDy)}, \textit{Extended Lagrangian-SINDy (xL-SINDy)}, and their variants, have shown great promise in identifying Euler–Lagrangian systems. However, these approaches often overlook the impact of excitation trajectory design. This study introduces the \textit{Online-Optimized xL-SINDy (OOL-SINDy)} framework, which generates optimal excitation signals in real time to improve model identification. At each time step, OOL-SINDy estimates the system from collected data and computes the next input by minimizing the Cramér–Rao bound of the parameter estimates. This online optimization accelerates convergence and yields more accurate models in less time compared to traditional excitation signals. Notably, OOL-SINDy is fully data-driven, requires minimal prior knowledge of the system, and adapts online. Extensive simulation studies and hardware experiments on a reaction wheel system confirm the effectiveness and robustness of the proposed framework.