Disturbance observer-based PID synchronization control of complex networks with probabilistic interval time-varying delays and multiple disturbances

In this article, the disturbance observer-based proportional–integral–derivative (PID) control protocol is investigated for synchronization of complex networks with probabilistic interval time-varying delays and multiple disturbances. For the sake of reducing conservatism, a method of probabilistic interval time-varying delay is proposed by introducing two certain intervals with known probability distributions. Moreover, two different disturbances are taken into account. One of the disturbances is produced by an exogenous system which acts through the input channel, while the other is usual norm-bounded disturbance. Simultaneously, a disturbance observer is designed to estimate and compensate the disturbance generated by the exogenous system. By constructing an appropriate Lyapunov–Krasovskii function, a sufficient criterion is obtained to ensure both the exponential synchronization and prescribed $H_{\infty }$ performance index. Finally, the simulation examples are employed to demonstrate the validity of the theoretical results.