Riemannian Optimal Model Reduction of Stable Linear Systems
Riemannian Optimal Model Reduction of Stable Linear Systems
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DOI：
10.1109/access.2019.2892071
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发表时间：
2018-03
期刊：
影响因子：
3.9
通讯作者：
Kazuhiro Sato
中科院分区：
文献类型：
--
作者：
Kazuhiro Sato
本文提出了一种在稳定矩阵集和两个欧氏空间的乘积集上，求原系统和约化系统的传递函数之间的$H^{2}$误差范数最小化问题的方法。也就是说，我们开发了一种从所有渐进稳定的线性系统中识别最优约化系统的方法。然而，很难开发出一种算法来解决这个问题，因为稳定矩阵的集合是高度非凸的。为了克服这个问题，我们证明了这个问题可以转化为一个易于处理的黎曼优化问题的产品流形上的一组反对称矩阵，流形的对称正定矩阵，和两个欧几里得空间。我们的问题的最优解构造的约化系统的渐近稳定性被保留。为了求解约化问题，推导了黎曼梯度和Hessian算子，并发展了一种黎曼信赖域方法。在所提出的方法中的初始点选择使用的平衡截断（BT）方法的输出。数值实验表明，该方法在$H^{2}$范数方面大大改进了BT和其他方法的结果，并为最小化$H^{\infty }$误差范数问题提供了全局近优解.此外，我们表明，我们的方法提供了一个更好的减少模型比BT和其他方法的频率响应的观点。
In this paper, we develop a method for solving the problem of minimizing the $H^{2}$ error norm between the transfer functions of the original and reduced systems on the product set of the set of stable matrices and two Euclidean spaces. That is, we develop a method for identifying the optimal reduced system from all the asymptotically stable linear systems. However, it is difficult to develop an algorithm for solving this problem, because the set of stable matrices is highly non-convex. To overcome this issue, we show that the problem can be transformed into a tractable Riemannian optimization problem on the product manifold of the set of skew-symmetric matrices, the manifold of the symmetric positive-definite matrices, and two Euclidean spaces. The asymptotic stability of the reduced systems constructed using optimal solutions to our problem is preserved. To solve the reduced problem, the Riemannian gradient and Hessian are derived, and a Riemannian trust-region method is developed. The initial point in the proposed approach is selected using the output from the balanced truncation (BT) method. The numerical experiments demonstrate that our method considerably improves the results given by BT and other methods in terms of the $H^{2}$ norm and also provides the reduced systems that are globally near-optimal solutions to the problem of minimizing the $H^{\infty }$ error norm. Moreover, we show that our method provides a better reduced model than the BT and other methods from the viewpoint of the frequency response.