Distributed control of a class of interconnected systems in a string

The study addresses the stabilisation problem of interconnected systems, using the contraction theory framework. A sufficient condition for the stability of closed loop system is derived using contraction. Neighbouring systems in the string arrangement are considered to have a physical interconnection between them. The proposed decentralised control technique presents a simple-way of establishing the stability of the overall system in comparison to the existing Lyapunov-based approaches. Here, the conditions for stability are reflected in terms of bounds on controller parameters. The uniform negative definiteness of the associated Jacobian of the system in a differential framework is ensured by identifying the location of eigenvalues, using the Gerschgorin theorem. Results are verified by considering the case of a mass-spring interconnected system in a string, where controllers of individual systems are designed using its local information and information from its nearest neighbours. Numerical results establish the efficacy of the proposed approach. The analysis presented here could be useful in addressing the stabilisation and coordination behaviour of complex physically interconnected systems.