  This paper addresses the resilience 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
resilient to simultaneous failure of any subset of feedback links of a
specified cardinality. 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 resilience of the resulting
closed-loop structured system to simultaneous failure of any subset of feedback
links of a specified cardinality. 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))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.
