Performance Benefits of Robust Nonlinear Zeroing Neural Network for Finding Accurate Solution of Lyapunov Equation in Presence of Various Noises

In the previous work, a finite-time zeroing neural network (ZNN) has been established to find the accurate solution of Lyapunov equation in the presence of no noises. In order to further improve the convergence speed of ZNN and suppress various noises encountered in real applications, in this paper, two robust nonlinear zeroing neural networks (RNZNNs) are designed by adding two novel nonlinear activation functions (AFs) for finding the solution of the Lyapunov equation in the presence of various noises. Unlike the previous ZNN activated by known AFs (e.g., linear activation function, bipolar sigmoid activation function, and power activation function), the proposed two RNZNN models possess predefined-time convergence (instead of finite-time convergence) even in the presence of various noises. The greatest advantage of the predefined-time convergence is independent to initial states of a dynamic system, which is much superior to the finite-time convergence related to initial states, and tremendously modifies the convergence performance. In addition, the predefined-time convergence of the RNZNN models for solving the Lyapunov equation are mathematically proved in detail under various external noises. The simulation comparisons further verify the superiority of the proposed RNZNN models for finding the solution of the Lyapunov equation.