Probabilistic synchronisation of pinning control

Abstract

This paper is concerned with synchronization of complex stochastic dynamical networks in the presence of noise and functional uncertainty. A probabilistic control method for adaptive synchronization is presented. All required probabilistic models of the network are assumed to be unknown therefore estimated to be dependent on the connectivity strength, the state and control values. Robustness of the probabilistic controller is proved via the Liapunov method.Furthermore, based on the residual error of the network states we introduce the definition of stochastic pinning controllability. A coupled map lattice with spatiotemporal chaos is taken as an example to illustrate all theoretical developments. The theoretical derivation is complemented by its validation on two representative examples.

Towards probabilistic synchronisation of local controllers

“…Proof : The first part of the theorem can be proven by adapting the proof of theorem 1 in Herzallah (2012). The second part of the theorem can be proven by considering the partitioned form of the state, control, covariance and Riccati matrix as stated above.…”

“…This is typically because of the large number of nodes where it becomes impossible to add controllers to all nodes. Therefore, the FPD has been extended in Herzallah (2012) to control complex stochastic dynamical networks via pinning control. Optimized probabilistic pinning control laws have the same form as specified by equation (7), but inputs are applied only to a fraction of nodes in the network.…”

Scalable Harmonization of Complex Networks With Local Adaptive Controllers

“…The reported pair of experiments compares the proposed adaptive distributed probabilistic control with the global probabilistic pinning control [39]. In these experiments, the lattice is initiated by X = X 0 and the control aim is to keep X t as close as possible to the origin.…”

“…, X label the lattice sites z j , and X is the system size. For detailed description of the CML, the readers are referred to [39] where CML has been used to illustrate theoretical developments for probabilistic pinning control of complex dynamical networks.…”

“…In a comparative experiment, one controller is designed using the probabilistic pinning control methodology, where the length 4 lattice is controlled using two control signals that are placed next to each other at the sides of the lattice [39], thus yielding the following controlled version of CML,…”