Fixed-time synchronization of multilayered complex dynamic networks via quantized variable-gain saturated control

This paper studies fixed-time (FxT) quantitative synchronization of multilayered complex dynamic networks (CDNs). First, a new FxT stability theorem is established and two new estimations of the settling time of stability are acquired, which are more accurate than the existing ones. Then, by using the improved theorem, several sufficient conditions ensuring FxT synchronization of a multilayered CDN are derived via two classes of innovative quantized variable-gain saturated controllers, and some high-precision estimates of the synchronous settling time (SST) are attained. In general, the saturation function used to replace the signum function is nonlinear and even includes the odd-even requirement on the exponent, and no quantization is involved in the developed FxT controllers. However, in our design, the classical linear saturation function is employed and the input signals of the controllers are also quantized. Therefore, our proposed strategies are more practical and easier to implement. Besides, different from the traditional 2-norm/1-norm-based approaches, a novel nonsmooth Lyapunov function is constructed so that the resultant estimates of the SST are unrelated to either the network size or the node's dimension, which are less conservative and closer to the actual synchronized time. Lastly, some numerical examples are given to validate the theoretical results.