The Lyapunov equation is widely employed in the engineering field to analyze stability of dynamic systems. In this paper, based on a new evolution formula, a novel finite-time recurrent neural network (termed finite-time Zhang neural network, FTZNN) is proposed and studied for solving a nonstationary Lyapunov equation. In comparison with the original Zhang neural network (ZNN) model for a nonstationary Lyapunov equation, the convergence performance has a remarkable improvement for the proposed FTZNN model and can be accelerated to finite time. Besides, by solving the differential inequality, the time upper bound of the FTZNN model is computed theoretically and analytically. Simulations are conducted and compared to validate the superiority of the FTZNN model to the original ZNN model for solving the nonstationary Lyapunov equation. At last, the FTZNN model is successfully applied to online tracking control of a wheeled mobile manipulator.

Solving Sylvester equation is a common algebraic problem in mathematics and control theory. Different from the traditional fixed-parameter recurrent neural networks, such as gradient-based recurrent neural networks or Zhang neural networks, a novel varying-parameter recurrent neural network, [called varying-parameter convergent-differential neural network (VP-CDNN)] is proposed in this paper for obtaining the online solution to the time-varying Sylvester equation. With time passing by, this kind of new varying-parameter neural network can achieve super-exponential performance. Computer simulation comparisons between the fixed-parameter neural networks and the proposed VP-CDNN via using different kinds of activation functions demonstrate that the proposed VP-CDNN has better convergence and robustness properties.

In this paper, a distributed scheme is proposed for the cooperative motion generation in a distributed network of multiple redundant manipulators. The proposed scheme can simultaneously achieve the specified primary task to reach global cooperation under limited communications among manipulators and optimality in terms of a specified optimization index of redundant robot manipulators. The proposed distributed scheme is reformulated as a quadratic program (QP). To inherently suppress noises originating from communication interferences or computational errors, a noise-tolerant zeroing neural network (NTZNN) is constructed to solve the QP problem online. Then, theoretical analyses show that, without noise, the proposed distributed scheme is able to execute a given task with exponentially convergent position errors. Moreover, in the presence of noise, the proposed distributed scheme with the aid of NTZNN model has a satisfactory performance. Furthermore, simulations and comparisons based on PUMA560 redundant robot manipulators substantiate the effectiveness and accuracy of the proposed distributed scheme with the aid of NTZNN model.

The so-called zeroing neural network (ZNN) is an effective recurrent neural network for solving dynamic problems including the dynamic nonlinear equations. There exist numerous unperturbed ZNN models that can converge to the theoretical solution of solvable nonlinear equations in infinity long or finite time. However, when these ZNN models are perturbed by external disturbances, the convergence performance would be dramatically deteriorated. To overcome this issue, this paper for the first time proposes a finite-time convergent ZNN with the noise-rejection capability to endure disturbances and solve dynamic nonlinear equations in finite time. In theory, the finite-time convergence and noise-rejection properties of the finite-time convergent and noise-rejection ZNN (FTNRZNN) are rigorously proved. For potential digital hardware realization, the discrete form of the FTNRZNN model is established based on a recently developed five-step finite difference rule to guarantee a high computational accuracy. The numerical results demonstrate that the discrete-time FTNRZNN can reject constant external noises. When perturbed by dynamic bounded or unbounded linear noises, the discrete-time FTNRZNN achieves the smallest steady-state errors in comparison with those generated by other discrete-time ZNN models that have no or limited ability to handle these noises. Discrete models of the FTNRZNN and the other ZNNs are comparatively applied to redundancy resolution of a robotic arm with superior positioning accuracy of the FTNRZNN verified.