Minimum-Gain Pole Placement With Sparse Static Feedback
Minimum-Gain Pole Placement With Sparse Static Feedback
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DOI：
10.1109/tac.2020.3018615
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发表时间：
2018-05
影响因子：
6.8
通讯作者：
Vaibhav Katewa;F. Pasqualetti
中科院分区：
文献类型：
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作者：
Vaibhav Katewa;F. Pasqualetti
最小增益特征值配置/极点配置问题(MGEAP)是具有静态反馈的线性定常系统的经典问题。在本文中，我们研究了状态反馈具有任意稀疏约束时的MGEAP。我们将稀疏MGEAP问题表示为等式约束优化问题，并利用闭环系统的特征向量矩阵给出了其局部最优解的解析刻画。这一结果被用来提供非稀疏MGEAP解的几何解释，从而为这个经典问题提供了更多的见解。在此基础上，利用基于Sylvester方程的参数化法，提出了一种迭代投影梯度下降算法来获得稀疏MGEAP的局部解。提出了一种计算投影的启发式算法，为稀疏特征值/极点配置问题的求解提供了一种新的方法。同时，给出了稀疏MGEAP的一个松弛形式，并开发了一个算法来获得MGEAP的近似稀疏局部解。最后通过数值实验比较了两种算法的性能，结果表明所提出的投影算法在大多数情况下都是收敛的。
The minimum-gain eigenvalue assignment/pole placement problem (MGEAP) is a classical problem in linear time-invariant systems with static state feedback. In this article, we study the MGEAP when the state feedback has arbitrary sparsity constraints. We formulate the sparse MGEAP problem as an equality-constrained optimization problem and present an analytical characterization of its locally optimal solution in terms of eigenvector matrices of the closed-loop system. This result is used to provide a geometric interpretation of the solution of the nonsparse MGEAP, thereby providing additional insights for this classical problem. Furthermore, we develop an iterative projected gradient descent algorithm to obtain local solutions for the sparse MGEAP using a parameterization based on the Sylvester equation. We present a heuristic algorithm to compute the projections, which also provides a novel method to solve the sparse eigenvalue/pole assignment problem. Also, a relaxed version of the sparse MGEAP is presented and an algorithm is developed to obtain approximately sparse local solutions to the MGEAP. Finally, numerical studies are presented to compare the properties of the algorithms, which suggest that the proposed projection algorithm converges in most cases.