基于状态-参数转移张量的系统辨识解析梯度与海森矩阵求值
Analytical Gradient and Hessian Evaluation for System Identification using State-Parameter Transition Tensors
AI总结：
本文利用爱因斯坦求和约定构造状态-参数转移矩阵与张量，解析推导代价函数的梯度和海森矩阵，用于系统参数与初始条件辨识，并在两个真实动态系统基准数据集上验证了其优于MATLAB拟牛顿法工具箱的鲁棒性。
摘要：
本文利用爱因斯坦求和约定，通过求解一组常微分方程来合成状态转移矩阵和参数转移矩阵。此外，针对系统辨识问题，本文证明了代价函数的梯度和海森矩阵可以利用相同的矩阵和张量度量进行解析构造。随后，本文提出一个通用的基于梯度的优化问题，用于辨识未知系统参数和未知初始条件。在此过程中，代价函数的解析梯度和海森矩阵利用这些状态转移矩阵和参数转移矩阵推导得出。通过使用来自真实动态系统的两个广泛使用的基准数据集，本文证明了所提方法在辨识未知系统参数和未知初始条件方面，比现有的基于传统拟牛顿法的系统辨识工具箱（可在MATLAB中使用）具有更强的鲁棒性。在现有工具箱中，梯度和海森信息是通过有限差分法推导得到的，与本文提出的解析方法相比，更容易受到数值误差的影响。
英文摘要：
In this work, the Einstein notation is utilized to synthesize state and parameter transition matrices, by solving a set of ordinary differential equations. Additionally, for the system identification problem, it has been demonstrated that the gradient and Hessian of a cost function can be analytically constructed using the same matrix and tensor metrics. A general gradientbased optimization problem is then posed to identify unknown system parameters and unknown initial conditions. Here, the analytical gradient and Hessian of the cost function are derived using these state and parameter transition matrices. The more robust performance of the proposed method for identifying unknown system parameters and unknown initial conditions over an existing conventional quasi-Newton method-based system identification toolbox (available in MATLAB) is demonstrated by using two widely used benchmark datasets from real dynamic systems. In the existing toolbox, gradient and Hessian information, which are derived using a finite difference method, are more susceptible to numerical errors compared to the analytical approach presented. Keywords: Gradient-based Optimization, Transition matrix and tensors, Gradient and Hessian, System identification.
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