Sparsest Feedback Selection for Structurally Cyclic Systems With Dedicated Actuators and Sensors in Polynomial Time

Abstract

This paper deals with optimal feedback selection problem in linear time-invariant (LTI) structured systems for arbitrary pole placement, an important open problem in structured systems. Specifically, we solve the sparsest feedback selection problem for LTI structured systems. In this paper, we consider structurally cyclic systems with dedicated inputs and outputs. For this class of systems, we prove that the sparsest feedback selection problem is equivalent to the strong connectivity augmentation problem in graph theory. We present an $O(n^2)$ algorithm to find a sparsest feedback matrix for structured systems when every state is actuated by a dedicated input and every state is sensed by a dedicated output, where $n$ denotes the number of states in the system. If the inputs and the outputs are such that not every state is actuated by a dedicated input and/or not every state is sensed by a dedicated output, then we provide an $O(n^3)$ algorithm for the sparsest feedback selection problem. These results show that sparsest feedback selection with dedicated i/o is a specific case of the optimal feedback selection problem that is solvable in polynomial time. We later analyze the sparsest feedback selection problem for structurally cyclic systems when both the input and the output are dedicated and the feedback pattern is constrained . When some of the feedback links are forbidden, we prove that the problem is NP-hard. The results in this paper along with the previously known results conclude that the optimal feedback selection problem is polynomial-time solvable only for the dedicated input–output case without forbidden feedback links and also without weights for the feedback links.