A networked small-gain theorem based on discrete-time diagonal stability

• DOI: 10.48550/arxiv.2411.03380
• Publication Date: 2024-01-01
• Authors: Ofir, Ron; Margaliot, Michael
• Keywords: Optimization and Control (math.OC); FOS: Electrical engineering, electronic engineering, information engineering; FOS: Mathematics; Systems and Control (eess.SY); Electrical Engineering and Systems Science - Systems and Control; Mathematics - Optimization and Control

We present a new sufficient condition for finite-gain $L_2$ input-to-output stability of a networked system. The condition requires a matrix, that combines information on the $L_2$ gains of the sub-systems and their interconnections, to be discrete-time diagonally stable (DTDS). We show that the new result generalizes the standard small gain theorem for the negative feedback connection of two sub-systems. An important advantage of the new result is that known sufficient conditions for DTDS can be applied to derive sufficient conditions for networked input-to-output stability. We demonstrate this using several examples. We also derive a new necessary and sufficient condition for a matrix that is a rank one perturbation of a Schur diagonal matrix to be DTDS.