Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation - Bolin Liao; Cheng Hua; Xinwei Cao; Vasilios N. Katsikis; Shuai Li (Mathematics, 2022)
Complex time-dependent Lyapunov equation (CTDLE), as an important means of stability analysis of control systems, has been extensively employed in mathematics and engineering application fields. Recursive neural networks (RNNs) have been reported as an effective method for solving CTDLE. In the previous work, zeroing neural networks (ZNNs) have been established to find the accurate solution of time-dependent Lyapunov equation (TDLE) in the noise-free conditions. However, noises are inevitable in the actual implementation process. In order to suppress the interference of various noises in practical applications, in this paper, a complex noise-resistant ZNN (CNRZNN) model is proposed and employed for the CTDLE solution. Additionally, the convergence and robustness of the CNRZNN model are analyzed and proved theoretically. For verification and comparison, three experiments and the existing noise-tolerant ZNN (NTZNN) model are introduced to investigate the effectiveness, convergence and robustness of the CNRZNN model. Compared with the NTZNN model, the CNRZNN model has more generality and stronger robustness. Specifically, the NTZNN model is a special form of the CNRZNN model, and the residual error of CNRZNN can converge rapidly and stably to order <semantics>10−5</semantics> when solving CTDLE under complex linear noises, which is much lower than order <semantics>10−1</semantics> of the NTZNN model. Analogously, under complex quadratic noises, the residual error of the CNRZNN model can converge to <semantics>2∥A∥F/ζ3</semantics> quickly and stably, while the residual error of the NTZNN model is divergent.