Optimal Feedback Selection for Structurally Cyclic Systems with Dedicated Actuators and Sensors

TL;DR

This paper addresses the problem of selecting the sparsest feedback in structurally cyclic linear systems with dedicated actuators and sensors, providing a polynomial-time solution and correcting previous misconceptions about its computational complexity.

Contribution

It proves that the sparsest feedback selection problem is of linear complexity for structurally cyclic systems with dedicated inputs and outputs, correcting prior errors in the literature.

Findings

The problem is solvable in linear time for the specified systems.

Previous hardness proofs contained errors, now clarified with counter-examples.

Provides a polynomial-time algorithm for feedback selection in these systems.

Abstract

This paper solves the sparsest feedback selection problem for linear time invariant structured systems, a long-standing open problem in structured systems. We consider structurally cyclic systems with dedicated inputs and outputs. We prove that finding a sparsest feedback selection is of linear complexity for the case of structurally cyclic systems with dedicated inputs and outputs. This problem has received attention recently but key errors in the hardness-proofs have resulted in an erroneous conclusion there. This is also elaborated in this brief paper together with a counter-example.