稀疏状态空间线性控制器的实现
Realizations of Linear Systems with Prescribed Sparsity Patterns
AI总结：
本文提出一种新方法，用于寻找线性系统的稀疏状态空间实现（如控制器）。通过传感器运动神经科学建模问题，研究了在给定线性时不变系统（如控制器）的传递函数形式和期望稀疏模式下，能否找到合适的稀疏状态空间实现。该问题非凸，但提出精确求解方法，利用代数几何工具解决多变量多项式方程。
摘要：
本文提出了一种新方法，用于寻找线性系统的稀疏状态空间实现（如控制器）。稀疏控制器常用于分布式控制，其中控制器通过某些稀疏惩罚项进行合成。在这里，受传感器运动神经科学建模问题的启发，我们研究了一个互补问题：给定一个线性时不变系统（如控制器）的传递函数形式和期望的稀疏模式，能否找到合适的稀疏状态空间实现？该问题高度非凸，但我们提出了一种精确求解方法。我们证明该问题可以转化为从模态实现找到适当相似变换，进而转化为求解多变量多项式方程系统。最后，我们利用代数几何工具（即Gröbner基）来精确求解该问题。我们提供了寻找实值和复值稀疏实现的算法，并在多个示例上展示了其有效性。
英文摘要：
This paper solves the following problem: given a transfer function and desired sparsity patterns for its state-space realization, how can we find a realization of the function that adheres to them? While sparse controllers are studied in distributed control, their sparsity patterns are generally designed to match those of the plant. In contrast, we are interested in \textit{arbitrary} sparsity patterns, which are relevant to modeling problems in sensorimotor neuroscience. Though this problem is highly nonconvex, we solve it exactly. We first show that the problem reduces to finding an appropriate similarity transform from the modal realization, which in turn reduces to solving a system of multivariate polynomial equations. We then leverage tools from algebraic geometry (Gröbner basis, moment method) to solve the system. Algorithms are provided for both real- and complex-valued realization problems, and their efficacy is demonstrated on several examples.
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。
