ZNN for time-variant nonlinear inequality systems: A finite-time solution

Abstract

Since the solution of time-variant nonlinear inequality systems is trapped by the convergence performance of the models, this paper explores an enhanced nonlinear sign-bi-power activation function (AF) and further obtains a zeroing neural network (ZNN) model for solving time-variant nonlinear inequality systems, which is called nonlinear activated finite-time convergent zeroing neural network (NAFTCZNN). The strict theoretical analysis together with two theorems are given to demonstrate the enhanced convergence performance of the NAFTCZNN model. Furthermore, the stability and upper bound of convergent time of the NAFTCZNN model are analyzed and estimated in the theorems, which is more stable and less conservative than the ZNN models using the common sign-bi-power AFs. Numerical example results further validate the effectiveness and excellence of the NAFTCZNN model in terms of solving the time-variant nonlinear inequality systems.

Introduction

Taking the modality of $H(x)⩽0$ as an example to solve the nonlinear inequality system is extensively identified as an essential problem in scientific, mathematical research and engineering-project areas [1], [2], [3], [4]. For instance, in [1], the pure robot problem solving was transformed into solving some nonlinear inequality equations; in [2], nonlinear convex programming problems with nonlinear inequality constraints were studied in detail; in [3], [4], the stability conditions of the solution sets of linear and nonlinear inequality systems on closed convex sets were established in Banach space when the functions defining nonlinear inequality are disturbed. On the other hand, Chebyshev inequality and Cauchy-Schwartz inequality are kinds of inequalities that have important applications in many fields, such as linear algebra, mathematical analysis, probability theory, and vector algebra. This modality of inequality is considered to be one of the most important inequality in mathematics. Hence, to a certain degree, solving the nonlinear inequality systems may make a significant contribution in the applications of the mathematics and related computing or project fields.

In the past decades, many researchers have done a lot of work in analyzing and solving nonlinear inequality systems. For instance, in [5], Mayne proposed a modified Newton algorithm for solving a finite system of inequality in a finite number of iterations; in [6], by exploring the minimum 2-norm solution of the system in which nonlinear inequality system may be inconsistent, a new smooth function was introduced, so that the problem can be solved by a series of parameterized optimization techniques. These numerical iterative methods and advanced numerical algorithms were widely used in solving the nonlinear inequality systems in [7], [8]. However, due to the time of settling the nonlinear inequality system lies on the scale and structure of the problem, some commonly used numerical and iterative algorithms may only be suitable for small-scale calculations and may be faced with time-consuming and slow-speed limitations in the case of large-scale operations.

For settling the above problems caused by large-scale operations, neural network methods have been presented and extensively researched in recent decades. On the one hand, due to the characteristics of hardware implementation and parallel distribution, neural network methods have been extensively applied in scientific computation and optimization fields [9], [11], [12], and also have wide applications in actual areas [13], [14], [15], [16], [17], such as obstacle avoidance for kinematically redundant manipulators, dynamic system identification and control, stable path tracking of mobile robots, nonlinear system identification. On the other hand, with the rapid development of technology, many problems have been transformed from the original static system to the time-variant system. In [18], gradient neural network (GNN) was proposed to address the time-variant matrix equation. However, under the time-variant condition, the GNN cannot make the error function converge to zero any more, that is, this method is not valid for settling time-variant problems. Thus, for settling the convergence problem of the error function, zeroing neural network (ZNN) method was proposed in [19] and extensively investigated after that.

The ZNN is a dynamic neural network method in settling time-variant problems, which was derived from the study of the Hopfield neural network. Unlike the GNN, the ZNN method is broadly used in solving time-variant problems, such as matrix equation, quadratic programming, linear and nonlinear inequalities. For example, in [20], a concrete ZNN model was designed to settle the time-variant linear inequality; in [23], the problem of square root of time-variant matrix was settled by an enhanced ZNN model; in [21], the problem of matrix/function-vector inequality was solved by the improved ZNN method. Furthermore, with the in-depth research on the ZNN, the convergent rate of the model has become a key research problem, which leading many researchers to propose different AFs to accelerate the convergent rate. For example, in [22], a power AF was introduced to enable the ZNN model to achieve a stable state with a short period of time. With the further study about AFs, some special AFs were presented in [24], [25], [26], where the residual error of the ZNN model can converge to zero in a finite-time; in [27], a robust and fixed-time ZNN model activated by a novel function was study for computing time-variant nonlinear equation. In a word, the ZNN is a novel neural dynamic method which can generate accurate time-variant solutions of the problem to be solved.

Inspired by these studies, an enhanced nonlinear activated finite-time convergent zeroing neural network (NAFTCZNN) model is proposed to efficiently settle time-variant nonlinear inequality systems via exploring an elaborately constructed AF in this paper. Compared to other common AFs, the presented nonlinear enhanced AF can make the model have faster convergence rate and smaller convergent upper bound. In the part of theoretical analysis, two theorems are given to prove the stability of the NAFTCZNN model and to find the upper bound of finite-time convergence detailedly. In addition, the discussion about the solution sets of time-variant nonlinear inequality systems is provided, and the correctness of our theorems is verified by numerical simulation. Moreover, to further demonstrate the novelty and superiority of the NAFTCZNN model, we also contrast this study with other published works (i.e.,[28], [29], [30], [31]) and the comparison results are listed in Table 1, Table 2. Table 1 compares the differences of AFs when ZNN is used to settle different problems, while Table 2 focuses on the differences between the NAFTCZNN model proposed in this paper and the previous models for the settling nonlinear inequality, so as to highlight the advancement and superiority of the proposed model. As shown in Tables, it can be concluded that the NAFTCZNN model activated by the enhanced nonlinear AF can ensure the finite-time convergence, where the convergence rate is faster and the finite-time convergence upper bound is smaller.

The rest of this work is spread out in the following four sections. Section 2 shows the problem description and presents the original ZNN model. Section 3 focuses on the NAFTCZNN model activated by a novel elaborately constructed enhanced nonlinear AF. The global convergence and finite-time convergence of the NAFTCZNN model are proved in Section 4. In Section 5, simulation validation and comparison of the NAFTCZNN model are given. The conclusion comment of this article is drawn in Section 6. At the ending of this section, we introduce the main contributions of this work as below.

• This paper gives a solution method of time-variant nonlinear inequality systems by proposing an enhanced nonlinear activated finite-time convergent zeroing neural network (NAFTCZNN) model, which is the highlight of this paper.

• In theory, it is proved that the state solution of the NAFTCZNN model keeps unchange when the initial state is inside the initial solution set, and converges to a smaller upper bound of time-variant solution set in finite time when the initial state is outside the initial solution set.

• Numerically, a simulation example about the comparisons of models and parameters is given to adequately verify the excellent performance of the NAFTCZNN model for solving time-variant nonlinear inequality systems.