A recurrent neural network for solving Sylvester equation with time-varying coefficients

Abstract

Presents a recurrent neural network for solving the Sylvester equation with time-varying coefficient matrices. The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation. Theoretical results of convergence and sensitivity analysis are presented to show the desirable properties of the recurrent neural network. Simulation results of time-varying matrix inversion and online nonlinear output regulation via pole assignment for the ball and beam system and the inverted pendulum on a cart system are also included to demonstrate the effectiveness and performance of the proposed neural network.

Zhang neural network and its application to Newton iteration for matrix square root estimation

“…To solve for time-varying matrix square root by Zhang et al's method [11,[13][14][15], the following matrix-valued error function could be defined firstly (instead of using any scalar-valued norm-based energy function associated with GNN [16]): EðXðtÞ; tÞ :¼ X 2 ðtÞ À AðtÞ 2 R nÂn :…”

“…Note that such a ZNN model (3) methodically and systematically exploits the time-derivative information of problemmatrix [i.e., _ AðtÞ], and thus it could be more effective in solving the time-varying matrix problems [11,[13][14][15].…”

“…Being another important type of solution, many parallel-processing computational schemes, including various dynamic recurrent neural networks (RNN) solvers, have been developed, analyzed, and implemented on specific architectures [9][10][11][12][13]. The RNN (or termed neural dynamics) approach is now regarded as a powerful alternative to online computation because of its parallel distributed nature and, more importantly, the suitability of hardware realization [11][12][13][14][15][16].…”

“…Since 12 March 2001, a special class of recurrent neural networks has been proposed by Zhang et al [11,[13][14][15], for online matrix problem solving (e.g., matrix inverse). The design of Zhang neural networks (ZNN) is based on a matrix-valued error function [11,[13][14][15], instead of a scalar-valued norm-based energy function widely used in conventional gradient-based neural networks (GNN) [16].…”

“…The design of Zhang neural networks (ZNN) is based on a matrix-valued error function [11,[13][14][15], instead of a scalar-valued norm-based energy function widely used in conventional gradient-based neural networks (GNN) [16]. In addition, ZNN models are depicted generally in implicit dynamics [11,[13][14][15], instead of explicit dynamics, which is usually associated with GNN (or Hopfield-type neural networks) [16].…”