Recurrent neural dynamics for handling linear equation system with rank-deficient coefficient and disturbance existence

Abstract

In this paper, for handling discrete-form time-variant linear equation system (DF-TV-LES) with rank-deficient coefficient and disturbance existence, a luminous discrete-time recurrent neural dynamics (DTRND) method is presented. Firstly, the continuous-time recurrent neural dynamics (CTRND) method can be discretized to the DTRND method by using recently-developed 5-instant discretization formula. Secondly, aiming at the situations of rank-deficient coefficient and disturbance existence, corresponding handling methods are presented, respectively. Specifically, on the one hand, under the situation of rank-deficient coefficient, we present an effective method to compute the least-squares solution of DF-TV-LES; on the other hand, under the situation of disturbance existence, integral state of error function is introduced, and then the presented DTRND method possesses a certain performance for restraining different types of disturbances. Finally, comparative numerical experiment substantiates the superiority of the presented DTRND method for handling DF-TV-LES.

Introduction

The rank-deficient coefficient problem of linear equation system is a fundamental and significant issue, which is encountered in variety of scientific and engineering fields, such as sensorless control [1], signal processing [2], and robotic kinematics [3], [4]. Many numerical algorithms have been designed by researchers to handle the linear equation system under the rank-deficient coefficient situation. However, when such numerical algorithms are applied to handle time-variant problems, the obtained results may not be entirely satisfactory.

Differing from the traditional numerical algorithms, a special class of recurrent neural network (RNN), i.e., zeroing neural network (also termed, Zhang neural network, ZNN), has been thoroughly investigated as an efficient approach for solving various time-variant problems [5], [6], [7]. Benefiting from the development of RNN methods, an excellent approach called variant-parameter convergent-differential neural-network (VP-CDNN) has been proposed. It is a parallel processing approach with high-efficiency and high-precision. Its unique advantage is that it is a real-time solver without any pre-training, and its computation errors have super-exponential convergent rate and strong robustness. For instance, the authors presented a remarkable paper that skillfully used nontrivial mathematics in order to prove a result of clear practical significance for robotics [8]; in [9], a novel neural network algorithm was designed, which removed a long-standing limitation of “error upper bound” of classic solvers; in addition, in order to solve the joint-angular-drift problems of redundant robot manipulators, a novel variant-parameter convergent-differential neural network was proposed and exploited [10]. Based on recently-developed mentioned RNN model, recurrent neural dynamics (RND) method can be seen as a novel and powerful tool for solving the scientific and engineering problems [11], [12], [13].

In earlier studies [14], several continuous-time RND (CTRND) methods have been developed for solving continuous-form time-variant linear equation system (CF-TV-LES). Note that the corresponding methods are designed for solving problems in real time on digital circuits and numerical algorithm development [15]. However, the coefficient matrices of CF-TV-LES may change with time, and the stringent nonsingular condition may not be satisfied at any time instant. In other words, the coefficient matrices may become rank-deficient at some time.

Moreover, disturbance always exists in practical applications, and it is desirable to obtain a method with disturbance restraint module [16], [17], [18]. Note that most of the algorithms are designed to study the situations of rank-deficient coefficient and disturbance existence, respectively. For example, in [19], Qiu et al. proposed and developed two new-type RNN methods to compute the least-squares solution of dynamic linear equation system with rank-deficient coefficient; in [20], two neurodynamic algorithms had been proposed and investigated for online future matrix inversion with sometimes-singular situation of the coefficient; in [21], in the presence of disturbance, the advanced neural network models still possessed excellent performance for solving time-variant equation systems. Evidently, when solving a linear equation system, the rank-deficient coefficient and disturbance may coexist. Consequently, it is necessary to find an effective method to handle the linear equation system with rank-deficient coefficient and disturbance existence simultaneously.

In addition, generally speaking, continuous-form time-variant system and discrete-form time-variant system can be seen as two standpoints of the same system in the time dimension, and they may be applicable in the different problems or in the different environments. However, in digital hardware (e.g., digital circuit and digital computer), the continuous-time models may be more difficult for implementation industrially than the discrete-time models. Thus, discretization approaches have attracted significant interest of researches. Among the various types of discretization approaches, the Euler-type discretization formula is viewed as a fundamental one-step-ahead approach and is widely applied in decades [22]. As a continuation of Euler-type discretization formula, the Taylor-type discretization formula has been designed and applied specially for solving discrete-form time-variant problems [23], [24], [25]. In addition, there are many discretization approaches that can be adopted to discrete the CTRND method with different characteristics [12], [26].

In this paper, for handling discrete-form time-variant linear equation system (DF-TV-LES) with rank-deficient coefficient and disturbance existence, a luminous discrete-time recurrent neural dynamics (DTRND) method is presented, which is obtained by discretizing the CTRND method and using a recently-developed one-step-ahead numerical differentiation rule called 5-instant discretization formula. Then, aiming at the situations of rank-deficient coefficient and disturbance existence, corresponding handling methods are presented, respectively. On the one hand, under the situation of rank-deficient, we can obtain an effective method to compute the least-squares solution of DF-TV-LES. On the other hand, under the situation of disturbance existence, integral state of error function is introduced, and the presented DTRND method possesses a certain performance for restraining different types of disturbances. Comparative results of numerical experiment substantiate the superiority of the presented DTRND method under the situations of rank-deficient coefficient and disturbance existence.

Before ending this section, the main contributions of this paper are listed as follows.

• Under the framework of DTRND method, aiming at DF-TV-LES with rank-deficient coefficient and disturbance existence, corresponding handling methods are presented, respectively.

• The theoretical and comparative numerical results show that the presented DTRND has the highest computational precision. Under the situation of rank-deficient coefficient and disturbance existence, such DTRND method can obtain the least-squares solution of DF-TV-LES in the presence of disturbance. It is quite different from the existing studies.

• Comparative illustration of numerical experiment is conducted to prove the efficacy and superiority of the DTRND method for solving DF-TV-LES with rank-deficient coefficient and disturbance existence.