Continuous and discrete zeroing neural network for a class of multilayer dynamic system

Abstract

Multilayer dynamic system is widely used in industry and other fields. Different from common systems, multilayer dynamic system has complex structure leading to challenges for research. In this paper, we study zeroing neural network(ZNN) models for a class of multilayer dynamic system(MLDS). In the case of continuous time, continuous ZNN models for continuous MLDS (MLDS-linear-ZNN, MLDS-nonlinear-ZNN and MLDS-noise-tolerant-ZNN) are proposed based on ZNN design method with theoretical analysis. In the discrete case concurrently, discrete ZNN models (MLDS-linear-mDZNN, MLDS-nonlinear-mDZNN and MLDS-noise-tolerant-mDZNN) with m-step ZeaD formula, a new Zhang et al. discretization formula presented in previous paper, are put forward and corresponding discrete algorithms are obtained. Finally, numerical experiments are carried out to verify the superiority and maneuverability of ZNN models for MLDS proposed in this paper.

Introduction

Neural networks (NNs) have been regarded as the key to solving various difficult problems in science and engineering fields with the heuristics of the biological nervous system [1], [2], [3] and references therein. In the past decade NNs have been investigated comprehensively and developed rapidly in a wide range of areas like image processing [4] and industrial automation [5]. Zeroing neural network (ZNN), as a class of NNs, can be applied for solving time-varying problems. In 1993, [6] took the initiative in putting forward a recurrent neural network(RNN). Zhang et al. proposed a novel class of RNN, termed zeroing neural network (ZNN) (or Zhang neural network after its inventors) to analyze and solve time-varying problems [7]. Researchers proposed various kinds of ZNN models for solving problems with different highlights such as matrix equation solving [8], [9], [10], time-varying matrix pseudoinversion [11], [12], robot manipulator [13], [14], [15], root-finding [16] and so on.

Especially, ZNN method can be applied to solve time-varying multilayer dynamic system. Multilayer dynamic system has more than two layers different from common system with signal layer. It plays an important role in industrial engineering. A typical application is manipulator operation: the manipulator tracking control with additional constraints can be transformed into solving multilayer dynamic system problems. Researchers have made progress along this direction and proposed various continuous and discrete ZNN models [13], [17], [18]. Different-Level Dynamic Linear System(DLDLS) was first formulated in [13]. Hybrid Multilayered Time-Varying Linear System(HMTVLS) was studied in [17]. And Future Different-Layer Nonlinear and Linear Equation System(DLNLES) was investigated in [18]. Continuous and discrete ZNN models were proposed in [13], [17], [18] to solve the DLDLS, HMTVLS and DLNLES respectively.

Note that, these models were all adopted the linear ZNN design method in the above mentioned papers [13], [17], [18]. Actually, at the beginning of the study, conventional ZNN models usually adopt the original linear ZNN design method. And with further research, researchers have tried to find new ZNN design method to achieve better convergence speed of ZNN models. Thereupon, activation functions have been applied in ZNN models [19], [20], [21], [22], [23]. In previous studies, researchers presented various ZNN models with different activation functions for different time-varying problems. [19] selected power-sum activation function to construct ZNN models for time-varying QP problem solving. [20] used hyperbolic sine activation function in the ZNN model for time-varying matrix square roots finding. [21] presented a new activation function with two tunable parameters. [22] adopted Sign-Bi-power (SBP) activation function for time-varying Sylvester equation. It has been shown that the selection of nonlinear functions can speed up the convergence of ZNN models [23]. Moreover, some nonlinear activation functions will bring finite convergence. Both theoretical analysis and simulation result in the above literature were provided to verify the outstanding performance of these nonlinear activation functions [19], [20], [21], [22], [23].

Besides, conventional ZNN models usually assume free of noise in the solving process [24]. However, noise actually exists in real time. [25] firstly proposed an integration-enhanced Zhang neural network(IEZNN) based on the novel integration-enhanced design formula and proved its noise resistance. Since the first empirical research, the noise-tolerant design formula has been widely used to construct ZNN models for solving time-varying problems with noise-suppression [26], [27], [28], [29], [30], [31]. In this paper, we both adopt linear and nonlinear ZNN design method to propose ZNN models for solving a class of time-varying continuous and discrete multilayer dynamic system(MLDS). Further, we take the noise pollution into account and propose noise-tolerant ZNN model.

The rest of this paper is organized as follows. Section 2 presents the continuous and discrete problem formulation of MLDS respectively. In Section 3, three kinds of ZNN models for continuous MLDS are discussed through theoretical analysis. In Section 4, ZNN models for discrete MLDS with m-step ZeaD formula are proposed. Section 5 is the numerical simulation and Section 6 is the conclusion of this paper. The major contributions to this paper are as follows.

• Different from common systems, MLDS with two levels is formulated and investigated.

• Linear, nonlinear and noise-tolerant ZNN models for continuous MLDS are presented and the convergence of solutions is analyzed.

• The existence of m-step ZeaD formula is proved. Linear, nonlinear and noise-tolerant ZNN models for discrete MLDS with m-step ZeaD formula are proposed and related algorithms are obtained for easier hardware realization.