Design and analysis of a general recurrent neural network model for time-varying matrix inversion

Following the idea of using first-order time derivatives, this paper presents a general recurrent neural network (RNN) model for online inversion of time-varying matrices. Different kinds of activation functions are investigated to guarantee the global exponential convergence of the neural model to the exact inverse of a given time-varying matrix. The robustness of the proposed neural model is also studied with respect to different activation functions and various implementation errors. Simulation results, including the application to kinematic control of redundant manipulators, substantiate the theoretical analysis and demonstrate the efficacy of the neural model on time-varying matrix inversion, especially when using a power-sigmoid activation function.

I. Introduction

The problem of finding the inverse of a time varying matrix online arises in numerous fields of science, engineering, and business. It is usually an essential part of many solutions, e.g., as preliminary steps for optimization [1], signal processing [2], electromagnetic systems [3], and robot kinematics [4]. Since the mid-1980s, efforts have been directed toward computational aspects of fast matrix inversion and many algorithms have been proposed [5]–[8]. For many numerical methods, the minimal arithmetic operations are usually proportional to the cube of the matrix dimension [9], and consequently such algorithms performed on digital computers may not be efficient enough in large-scale online applications. In view of this, parallel computation schemes have been investigated for matrix inversion.