On Design,Analysis And Application Of Recurrent Neural Network Models For Solving Two Kinds Of Matrix Equations In System And Control

Considerable attention has been given to real-time solution to linear matrix equations(such as Stein equation,Sylvester equation,matrix Moore-Penrose inverse,etc.)because of their important significance in theoretical research and broad range of applications in system and control and other fields.Traditional methods for solving matrix equations are iteration-based numerical algorithms.These algorithms have various types and iterative forms,and may fit the need of real-time computation when the solving task is in small dimension.However,they are not suitable and effective enough for large-scale real-time applications due to their high computational complexity(i.e.,(9)~3)arithmetic operations with9)denoted the dimension of the matrix).Unlike the serial processing mechanism of iterative algorithms,recurrent neural net-works(RNNs),which have numerous interconnected,parallel-structured neurons,can process information in parallel.In addition,a large amount of information is stored in neurons and their connection weights,endowing RNNs with the characteristic of distribut-ed storage.These two properties together with the feature of hardware implementability make RNNs have huge advantages as compared to iterative algorithms when solving large-scale real-time problems.Gradient-based neural network(GNN)and zeroing neural network(ZNN)are t-wo popular kinds of widely-studied RNNs for real-time solving matrix-related problems.However,the performance(e.g.,convergence and robustness)of existing RNN models is not good enough,and they generally can not take into account a fast convergence speed and strong noise-suppression capability simultaneously.To overcome these shortcomings,the real-time solutions of static Stein equation and Moore-Penrose inverse of dynamic matrix,which are widely encountered in system and control fields,are selected as the study object,and new RNN models for solving them are designed,analyzed,and applied to applications.The main work and novelties of this dissertation are as follows.(1)The convergence performance and robustness against noises of the GNN model activated by four different functions for solving static Stein matrix equation are theoreti-cally investigated and analyzed.As for the convergence performance,with the aid of the Kronecker product and vectorization technique,it is proven that,in the ideal case(i.e.without noise),under the activation of linear function,the GNN model has exponential convergence performance with the exponential convergence rate given as well;under the activation of power-sigmoid function,the GNN model can achieve faster convergence in comparison to the situation where linear activation function is used;under the activation of sp(sign-power)and gsbp(general sign-bi-power)functions,the GNN model can respec-tively achieve finite-and fixed-time convergence.More importantly,the upper bounds of finite-and fixed-time convergence are presented as well.As for the robustness,the upper bounds of steady-state solution error(SSSE)of the noise-disturbed GNN model under the activation of the presented four different functions are also estimated through detailed theoretical derivation.(2)The robust performance of the ZNN model activated by a combined function(lin-ear combination of sign-bi-power function and linear function)for solving Moore-Penrose inverse of dynamic full-rank matrix in the presence of noise is theoretically analyzed.By constructing a novel Lyapunov function,the mathematical relationship between the upper bound of SSSE and network parameters is obtained.It is shown that the SSSE can be arbitrarily small by adjusting network parameters.Meanwhile,by solving differential inequalities,the exponential convergence rate of the solution error to approach the upper bound and the corresponding time are estimated.(3)To overcome the limitations of existing ZNN models in terms of convergence performance and robustness,two varying-parameter ZNN(VPZNN)models,which are based on a new evolution formula of error function and two varying-parameter activation functions,are proposed for solving Moore-Penrose inverse of dynamic full-rank matrix.By use of Lyapunov stability theory,it is proved that the proposed VPZNN models have global finite-time convergence in the presence of zero noise,constant noise,and dynamic bounded or unbounded linear noise.At the same time,the finite convergence time is estimated.Both theoretical analyses and simulation results are demonstrated to substantiate the superior robustness and convergence performance of the proposed VPZNN models in comparison with the existing original ZNN(OZNN)and integration-enhanced ZNN(IEZNN)models.Furthermore,a simulation example is provided as an expansion of the proposed VPZNN model by solving the Moore-Penrose inverse of a time-varying matrix with discontinuous derivatives.Simulation results show that,during the entire simulation,the proposed VPZNN model works well and is able to effectively solve the given problem.(4)Different ZNN models are applied to the resolution of the minimum velocity norm(MVN)scheme of redundant robot manipulator.In order to be more in line with actual situation,the influence of dynamic noise is also considered in the solution schemes.Simulation results show that the proposed VPZNN model has the best tracking effect and has the potential for practical applications in inverse kinematics control of manipulators.Meanwhile,the superior convergence performance and robustness of the VPZNN model in comparison with OZNN and IEZNN models are illustrated again.