一维弹性振动系统的参数辨识与状态重构

The objective of the project is to the algorithm design for simultaneous identification of the coefficient and initial value of one-dimensional elastic oscillating system. Since the state in distributed parameter system is infinite-dimensional, it is difficult to measure them directly by sensors and modern measuring technology. And hence algorithm design becomes necessary from the point view of control theory. For the identification problem of one-dimensional oscillating system, it used to deal with the identification of coefficient or the reconstruction of system state. For the simultaneous identification of them, however, only identifiability is available. The main thoughts of the project is as follows: First, an algorithm is designed to extract the spectral data from the input and output, and then the coefficient identification can be solved by the known inverse eigenvalue theory; Second, we can transform the reconstruction of initial state to a fixed point problem, and then investigate the convergence of the iterative algorithm and its convergence rate. The boundary control and measurement data that we choose in this project is easy to realize in engineering, which establishes a foundation for the practical application of the algorithm.

本项目研究一维弹性振动系统中参数与初值的同时辨识算法设计问题。分布参数系统由于状态是无穷维的，目前的传感器和测量技术难以对其进行直接测定。因此，从控制理论的角度设计相应的辨识算法成为必然。对一维振动系统辨识问题的研究，过去处理的大多是单一的参数辨识或是状态重构，对两者同时进行辨识仅有可辨识性结果，并无重构算法。本项目旨在根据系统的输入输出，设计具体的辨识算法同时估计参数和初值。主要研究思路如下：首先，设计算法由输入和输出恢复系统的谱数据，根据已有的特征值反问题理论完成参数的辨识；然后，将初值的辨识转化为不动点问题来求解，并研究迭代算法的收敛性和收敛速度。本项目中尽量选取工程中易于实现的点控制、点测量，从而为算法的实际应用奠定了基础。

本项目中，我们主要研究了一维波动方程的参数辨识和状态重构算法设计问题。首先，我们证明了系统的精确可控性，并深入研究了指数（三角）函数族的Riesz基性质，由此证明了系统的谱可控性，这是Neumann边界控制、Dirichlet边界观测条件下应用边界控制方法的前提；然后，借助于自伴算子的变分原理，通过构造响应算子和连接算子，设计出了提取系统谱数据的算法，进而应用边界控制方法给出了重构系统参数的算法；最后，我们将辨识初值问题转化为不动点问题，在一定条件下初步给出了理论上的状态重构算法。我们设计算法的思想具有一定的普适性，可将其推广应用于其它一些系统，如热传导方程、薛定谔方程等。