寻找最接近的正实系统
Finding the nearest positive-real system
AI总结：
本文提出一种新算法，用于寻找最接近给定非正实系统的正实系统，通过快速梯度法实现，不依赖哈密顿矩阵的谱性质，可扰动所有系统矩阵。
摘要：
线性时不变（LTI）动态系统中的正实性概念，等价于被动性，是系统与控制理论中最古老的概念之一。本文考虑了寻找最接近非正实系统的正实系统的问题：给定一个由$E \dot{x}=Ax+Bu$和$y=Cx+Du$定义的LTI控制系统，最小化$(Δ_E,Δ_A,Δ_B,Δ_C,Δ_D)$的Frobenius范数，使得$(E+Δ_E,A+Δ_A,B+Δ_B,C+Δ_C,D+Δ_D)$成为正实系统。我们首先证明，一个系统是扩展严格正实的当且仅当它可以表示为严格能量守恒系统。这使我们能够将最近正实系统问题转化为一个具有简单凸可行集的优化问题。然后，我们使用快速梯度法获得给定非正实系统的附近正实系统，并在多个示例中展示了算法的行为。据我们所知，这是第一个不依赖相关哈密顿矩阵或笔的谱性质、允许扰动所有系统矩阵，并且不对原始给定系统做任何假设的计算附近正实系统的算法。
英文摘要：
The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive-real (PR) system to a non PR system: given an LTI control system defined by $E \dot{x}=Ax+Bu$ and $y=Cx+Du$, minimize the Frobenius norm of $(Δ_E,Δ_A,Δ_B,Δ_C,Δ_D)$ such that $(E+Δ_E,A+Δ_A,B+Δ_B,C+Δ_C,D+Δ_D)$ is a PR system. We first show that a system is extended strictly PR if and only if it can be written as a strict port-Hamiltonian system. This allows us to reformulate the nearest PR system problem into an optimization problem with a simple convex feasible set. We then use a fast gradient method to obtain a nearby PR system to a given non PR system, and illustrate the behavior of our algorithm on several examples. This is, to the best of our knowledge, the first algorithm that computes a nearby PR system to a given non PR system that (i) is not based on the spectral properties of related Hamiltonian matrices or pencils, (ii) allows to perturb all matrices $(E,A,B,C,D)$ describing the system, and (iii) does not make any assumption on the original given system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
