论文标题
通过哈密顿系统，矩阵分解和优化解决矩阵近感问题
Solving matrix nearness problems via Hamiltonian systems, matrix factorization, and optimization
论文作者
论文摘要
在这些讲座中，我们回顾了我们最近的著作，这些著作解决了找到最近稳定系统到不稳定系统的各种问题。引入后，我们提供了一些初步背景，即定义 - 港口港口系统和耗散性汉密尔顿系统及其属性，简要讨论矩阵因素化，并描述我们在这些注释中将使用的优化方法。在第三章中，我们介绍了解决标准连续线性时间不变（LTI）系统稳定性距离的方法。主要思想是依靠稳定系统作为耗散的哈密顿系统的表征。我们展示了如何将这个想法推广到计算最接近的$ω$稳定矩阵，其中需要的系统矩阵$ a $需要属于相当一般的$ω$。我们还展示了如何使用这些想法来计算最小值静态反馈，也就是说，通过选择线性依赖于$ x（t）$（静态反馈）或$ y（t）$（静态输出反馈）的适当输入$ u（t）$来稳定系统。在第四章中，我们提出了解决被动距离的方法。主要思想是依靠稳定系统作为哈米尔顿港系统的表征。我们还会在更多详细信息中讨论计算最近稳定矩阵对的特殊情况。在上一章中，我们专注于离散的LTI系统。类似地，对于连续情况，我们提出了一个参数化，该参数允许有效计算最近的稳定系统（用于矩阵和矩阵对），从而可以计算稳定性的距离。我们展示了如何在数据驱动的系统标识中使用此想法，即给定一组输入输出对，标识系统$ a $。
In these lectures notes, we review our recent works addressing various problems of finding the nearest stable system to an unstable one. After the introduction, we provide some preliminary background, namely, defining Port-Hamiltonian systems and dissipative Hamiltonian systems and their properties, briefly discussing matrix factorizations, and describing the optimization methods that we will use in these notes. In the third chapter, we present our approach to tackle the distance to stability for standard continuous linear time invariant (LTI) systems. The main idea is to rely on the characterization of stable systems as dissipative Hamiltonian systems. We show how this idea can be generalized to compute the nearest $Ω$-stable matrix, where the eigenvalues of the sought system matrix $A$ are required to belong a rather general set $Ω$. We also show how these ideas can be used to compute minimal-norm static feedbacks, that is, stabilize a system by choosing a proper input $u(t)$ that linearly depends on $x(t)$ (static-state feedback), or on $y(t)$ (static-output feedback). In the fourth chapter, we present our approach to tackle the distance to passivity. The main idea is to rely on the characterization of stable systems as port-Hamiltonian systems. We also discuss in more details the special case of computing the nearest stable matrix pairs. In the last chapter, we focus on discrete-time LTI systems. Similarly as for the continuous case, we propose a parametrization that allows efficiently compute the nearest stable system (for matrices and matrix pairs), allowing to compute the distance to stability. We show how this idea can be used in data-driven system identification, that is, given a set of input-output pairs, identify the system $A$.
