This paper addresses the robustness of large-scale closed-loop structured systems in the sense of arbitrary pole placement when subject to failure of feedback links. Given a structured system with input, output, and feedback matrices, we first aim to verify whether the closed-loop structured system is robust to the simultaneous failure of any subset of feedback links of cardinality at most γ. Subsequently, we address the associated design problem in which given a structured system with input and output matrices, we need to design a sparsest feedback matrix that ensures the robustness of the resulting closed-loop structured system to simultaneous failure of at most any γ feedback links. We first prove that the verification problem is NP-complete even for irreducible systems and the design problem is NP-hard even for so-called structurally cyclic systems. We also show that the design problem is inapproximable to factor $(1-o(1\left)\right)\log n,$ where n denotes the system dimension. Then we propose algorithms to solve both the problems: a pseudo-polynomial algorithm to address the verification problem of irreducible systems and a polynomial-time O(log n)-optimal approximation algorithm to solve the design problem for a special feedback structure, so-called back-edge feedback structure.

本文讨论了当反馈链路失效时任意极点布置意义上的大型闭环结构系统的鲁棒性。给定具有输入，输出和反馈矩阵的结构化系统，我们首先旨在验证闭环结构化系统是否对最多γ个基数的反馈链路的任何子集同时失效具有鲁棒性。随后，我们解决了相关的设计问题，在给定具有输入和输出矩阵的结构化系统的情况下，我们需要设计一个最稀疏的反馈矩阵，以确保最终闭环结构化系统的鲁棒性，以使最多任何γ反馈链路同时失效。我们首先证明，即使对于不可约系统，即使对于所谓的结构循环系统，设计问题也很难解决。我们还表明，设计问题与因子无关$（\mathrm{1个}-Ø（\mathrm{1个}））\mathrm{日志}ñ，$其中n表示系统尺寸。然后，我们提出了解决这两个问题的算法：用于解决不可约系统验证问题的伪多项式算法和用于解决特殊反馈结构的设计问题的多项式时间O（log  n）-最佳逼近算法，因此-称为后端反馈结构。