Riemannian optimal identification method for linear systems with symmetric positive-definite matrix
Riemannian optimal identification method for linear systems with symmetric positive-definite matrix
复制标题
对称正定矩阵线性系统黎曼最优辨识方法
DOI：
10.1109/tac.2019.2957350
复制
发表时间：
2020
影响因子：
6.8
通讯作者：
and T. Damm
中科院分区：
文献类型：
--
作者：
K. Sato;H. Sato;and T. Damm
本文发展了线性连续时间对称系统的辨识方法，如电力网络系统、多智能体网络系统和建筑物内的温度动态。为此，我们针对相应的离散时间系统提出了三个系统辨识问题。第一个是最小二乘问题，我们希望最小化对称正定矩阵流形和两个欧氏空间的乘积流形上的真实输出和模型输出之间的平方误差之和。在第二个问题中，为了降低搜索维度，将乘积流形替换为正交群在指定群作用下的商集。在第三个问题中，将第一个问题中的对称正定矩阵流形替换为只有正对角元的矩阵流形。特别地，我们研究了第二个问题中的商几何。对于这三个问题，我们提出了黎曼共轭梯度法，并使用一个流行的子空间方法来选择初始点。通过数值模拟和与求解最小二乘问题最流行的方法之一的高斯-牛顿法的比较，证明了所提方法的有效性。
This article develops identification methods for linear continuous-time symmetric systems, such as electrical network systems, multiagent network systems, and temperature dynamics in buildings. To this end, we formulate three system identification problems for the corresponding discrete-time systems. The first is a least-squares problem in which we wish to minimize the sum of squared errors between the true and model outputs on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. In the second problem, to reduce the search dimensions, the product manifold is replaced with the quotient set under a specified group action by the orthogonal group. In the third problem, the manifold of symmetric positive-definite matrices in the first problem is replaced by the manifold of matrices with only positive diagonal elements. In particular, we examine the quotient geometry in the second problem. We propose Riemannian conjugate gradient methods for the three problems, and select initial points using a popular subspace method. The effectiveness of our proposed methods is demonstrated through numerical simulations and comparisons with the Gauss-Newton method, which is one of the most popular approach for solving least-squares problems.