This article discusses the issue of disturbance rejection and anti-windup control for a class of complex systems with both saturating actuators and diverse types of disturbances. At the input port, to better characterize those irregular disturbances, exogenous dynamic neural network (DNN) models with adjustable weight parameters are first introduced. A novel disturbance observer-based adaptive control (DOBAC) technique is then established, which realizes the dynamic monitoring for the unknown input disturbance. To handle the system disturbance with a bounded norm, the attenuation performance is concurrently analyzed by optimizing the L1 gain index. Moreover, the PI-type dynamic tracking controller is proposed by integrating the polytopic description of the saturating input with the estimation of the input disturbance. The favorable stability, tracking, and robustness performances of the augmented system are achieved within a given domain of attraction by employing the convex optimization theory. Finally, using DNN-based modeling for three kinds of different irregular disturbances, simulation studies for an A4D aircraft model are conducted to substantiate the superiority of the designed algorithm.