New Noise-Tolerant ZNN Models With Predefined-Time Convergence for Time-Variant Sylvester Equation Solving

Sylvester equation is often applied to various fields, such as mathematics and control systems due to its importance. Zeroing neural network (ZNN), as a systematic design method for time-variant problems, has been proved to be effective on solving Sylvester equation in the ideal conditions. In this paper, in order to realize the predefined-time convergence of the ZNN model and modify its robustness, two new noise-tolerant ZNNs (NNTZNNs) are established by devising two novelly constructed nonlinear activation functions (AFs) to find the accurate solution of the time-variant Sylvester equation in the presence of various noises. Unlike the original ZNN models activated by known AFs, the proposed two NNTZNN models are activated by two novel AFs, therefore, possessing the excellent predefined-time convergence and strong robustness even in the presence of various noises. Besides, the detailed theoretical analyses of the predefined-time convergence and robustness ability for the NNTZNN models are given by considering different kinds of noises. Simulation comparative results further verify the excellent performance of the proposed NNTZNN models, when applied to online solution of the time-variant Sylvester equation

Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation

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 10−5 when solving CTDLE under complex linear noises, which is much lower than order 10−1 of the NTZNN model. Analogously, under complex quadratic noises, the residual error of the CNRZNN model can converge to 2∥A∥F/ζ3 quickly and stably, while the residual error of the NTZNN model is divergent

Discrete-time zeroing neural network for solving time-varying Sylvester-transpose matrix inequation via exp-aided conversion

Time-varying linear matrix equations and inequations have been widely studied in recent years. Time-varying Sylvester-transpose matrix inequation, which is an important variant, has not been fully investigated. Solving the time-varying problem in a constructive manner remains a challenge. This study considers an exp-aided conversion from time-varying linear matrix inequations to equations to solve the intractable problem. On the basis of zeroing neural network (ZNN) method, a continuous-time zeroing neural network (CTZNN) model is derived with the help of Kronecker product and vectorization technique. The convergence property of the model is analyzed. Two discrete-time ZNN models are obtained with the theoretical analyses of truncation error by using two Zhang et al.’s discretization (ZeaD) formulas with different precision to discretize the CTZNN model. The comparative numerical experiments are conducted for two discrete-time ZNN models, and the corresponding numerical results substantiate the convergence and effectiveness of two ZNN discrete-time models

Proposing, developing and verification of a novel discrete-time zeroing neural network for solving future augmented Sylvester matrix equation

In this paper, a novel discrete-time advance zeroing neural network (DT-AZNN) model is proposed, developed and investigated for solving future augmented Sylvester matrix equation (F-ASME). First of all, based on the advance zeroing neural network (AZNN) design formula, a novel continuous-time advance zeroing neural network (CT-AZNN) model is shown for solving continuous-time augmented Sylvester matrix equation (CT-ASME). Secondly, a recently published discretization formula is further investigated with the optimal sampling gap of the discretization formula proposed. Then, for solving F-ASME, a novel DT-AZNN model is proposed based on the discretization formula. Theoretical analyses on the convergence property and the perturbation suppression performance of the DT-AZNN model are provided. Moreover, comparative numerical experimental results are conducted to prove the effectiveness and robustness of the proposed DT-AZNN model for solving F-ASME

Design and Comprehensive Analysis of a Noise-Tolerant ZNN Model With Limited-Time Convergence for Time-Dependent Nonlinear Minimization

Zeroing neural network (ZNN) is a powerful tool to address the mathematical and optimization problems broadly arisen in the science and engineering areas. The convergence and robustness are always co-pursued in ZNN. However, there exists no related work on the ZNN for time-dependent nonlinear minimization that achieves simultaneously limited-time convergence and inherently noise suppression. In this article, for the purpose of satisfying such two requirements, a limited-time robust neural network (LTRNN) is devised and presented to solve time-dependent nonlinear minimization under various external disturbances. Different from the previous ZNN model for this problem either with limited-time convergence or with noise suppression, the proposed LTRNN model simultaneously possesses such two characteristics. Besides, rigorous theoretical analyses are given to prove the superior performance of the LTRNN model when adopted to solve time-dependent nonlinear minimization under external disturbances. Comparative results also substantiate the effectiveness and advantages of LTRNN via solving a time-dependent nonlinear minimization problem

Zhang Neural Networks for Online Solution of Time-Varying Linear Inequalities

In this chapter, a special type of recurrent neural networks termed “Zhang neural network” (ZNN) is presented and studied for online solution of time-varying linear (matrix-vector and matrix) inequalities. Specifically, focusing on solving the time-varying linear matrix-vector inequality (LMVI), we develop and investigate two different ZNN models based on two different Zhang functions (ZFs). Then, being an extension, by defining another two different ZFs, another two ZNN models are developed and investigated to solve the time-varying linear matrix inequality (LMI). For such ZNN models, theoretical results and analyses are presented as well to show their computational performances. Simulation results with two illustrative examples further substantiate the efficacy of the presented ZNN models for time-varying LMVI and LMI solving