Solving perturbed time-varying linear equation and inequality problem with adaptive enhanced and noise suppressing zeroing neural network

Solving time-varying linear equation and inequality (TVLEI) problem has attracted extensive attention in numerous scientific and engineered fields. In this article, it is basically considered that the commonly used dynamics neural network in the virtual environment is inevitably interfered with by the variable measurement noises while dealing with the TVLEI problem. An adaptive enhanced and noise-suppressing zeroing neural network (AENSZNN) model is proposed as an improved algorithm for solving the TVLEI problem. An adaptive scale factor based on the residual error norm is designed to make the proposed AENSZNN model converge to the theoretical solution faster. Furthermore, the momentum enhancement terms added to the model enables the AENSZNN model to effectively solve the TVLEI problem in real-time under the obstruction of different measurement noises. Besides, theoretical results and numerical experiments indicate that the AENSZNN model has advantages in convergence accuracy and robustness to noises compared with the existing algorithms. Note that, the proposed AENSZNN model is successfully exploited for the estimation of mobile object localization.


I. INTRODUCTION
T HE problem of linear equation and inequality (LEI) generally exists in many fields of science and industry, such as control system [1], stability analysis [2], [3], data handling [4], information retrieval [5], etc. Many engineering and theoretical problems can be simplified as the inherent LEI system. For example, the robot control is modeled by a system of linear equations [6], [7]. In the redundancy robotic manipulators solution, the avoidance characteristics of joint physical constraints corresponds to a linear inequality system [8], [9]. By combining and solving the relevant combined linear equation and inequality system, robots can be successfully controlled without joint physical constraints or environmental obstacles [10]. The LEI problem can be described as the situation that if and only if the solution of C x = b and the solution of D x ≤ d exist at the same time and the intersection of their solutions is non-empty. Solving such a LEI problem is essentially finding simultane-ous solutions that satisfy equation and inequality. Obviously, the LEI problem has a great importance in the variant application, it becomes necessary and meaningful to develop an effective unified solution scheme [11]- [13]. Currently, a lot of studies have been proposed which aim at solving the LEI problem. In paper [14], Gol et al. study regularization and find the normal solutions of the linear equations and inequalities systems. Moreover, Li et al. analyze and study interval linear equations and inequalities, and demonstrate the feasibility and solvability of the problem in [15]. Besides, Cheng et al. [16] construct an optimal control method for Lyapunov inequality for linear elliptic equation. In paper [17], two kinds of discrete-time neural networks with fast convergence speed are proposed to solve the LEI problem by using the scaling techniques. Furthermore, Liang et al. [18] utilize the discrete-time zeroing neural network to solve the LEI problem and give the corresponding improved upper limit of the step size parameters. However, it is worth noting  [16] No No No No NA * NA * NA * NA * Recurrent neural network in [17] No No  No  No  Negligible  Bounded Bounded +∞ ‡ OZNN model in [23] Yes Yes No No Negligible Bounded Bounded +∞ ‡ The proposed AENSZNN (11) Yes Yes Yes Yes Negligible Negligible BS † BS † Note that the recurrent neural network in [17] is constructed to solve the static LEI problem. NA means that the item cannot be obtained from the algorithm in the corresponding papers. † BS denotes that the MSSRE of the corresponding situation is bounded tightly. ‡ +∞ denotes the positive infinity.
that the abovementioned methods are essentially presented to solve the time-invariant LEI problem, which leads to falling into low solution accuracy problem or even system collapse due to the lagging error.
In [19], a special kind of recurrent neural network termed zeroing neural network (ZNN) model is presented as an effective scheme to solve varieties of time-varying problems. By making full use of the time derivative information of time-varying parameters, the theoretical solution to the timevarying problems is tracked by the evolution formula. Taking advantage of the method of ZNN [20]- [22], the objective of effectively solving time-varying problems is achieved. Specifically, Xu et al. [23] furnish a ZNN model for ensuring the solution of the TVLEI problem, which takes full advantage of the ZNN model as well as the time-derivative information about the time-varying coefficients involved in the time-varying problems. Due to the sensitivity of the model to noises. Specifically, noises will lead to performance degradation, or even lead to the solving system collapse [24]. In view of the lack of convergence speed and poor anti-noise performance of ZNN models, many studies have presented [25], [26]. In [27], a dynamics zeroing neural network with intrinsic noises tolerance aims to suppress additive noises by adding an integral term so that it can converge globally to the theoretical solution precisely. Besides, Xiao et al. [28] present a unified ZNN, which simultaneously has both remarkable finite time convergence performance and inherent noises tolerance. In [29], a unified predefined-time convergent and robust ZNN model is put forward to obtain an exact solution in a specified finite time and accuracy can not be affected in the face of high amplitude perturbations. Moreover, by using special activation functions including the well-known sign-bi-power function in ZNN, Xiao et al. [30] recommend a finite-time Zhang neural network to make the convergence performance have a remarkable improvement. Despite the design and analysis of the accelerated ZNN models to find the optimum solution having acquired some achievements, the current research has not solved the existing problems. For instance, the learning speed of the model needs to be repeatedly and carefully adjusted. At this time, the model cannot make full use of the residual information in the solution system, which eventually leads to truncation errors or even crash the solution system. Therefore, this paper proposes an adaptive enhanced and noise-suppressing zeroing neural network (AENSZNN) model which can get the utmost out of the residual error information and momentum information of the solution system. By combining the residual error and the adaption scale factor, the AENSZNN model achieves a faster convergence rate and higher robustness than the previously proposed solution system. That is, this model can not only maintain high robustness under the various measurement noises disturbance but also speed up the convergence of the solving system. A good deal of various algorithms and neural network models have been put forward and applied in solving the TVLEI problem, the specific performance comparison results among them are exhibited in Table 1.
The remainder of this paper is segmented into the following five parts. Section II introduces the preliminary knowledge of the solution of the TVLEI problem. Further, the detailed description of the norm-based adaption factor design method and the AENSZNN model construction method to solve the TVLEI problem is settled in Section III. Next, Section IV strictly analyzes the global convergence and robustness of the AENSZNN model. Besides, an illustrative simulation example 1 (V-A) and the dynamic positioning simulative experiments based on the angle of arrival (AOA) positioning algorithm are provided to verify the effectiveness and superiority of the proposed AENSZNN model in Section V. Finally, Section VI concludes this article with concluding remarks. The main contributions of this paper are as follows: • As far as we know, this paper is the first work to investigate the time-varying linear equation system based on the norm-based adaption scale factor design framework at present. Furthermore, the proposed model achieves a better performance in convergence speed of the solving system and anti-noise ability over the state-of-the-art methods.
the time-varying matrices x(t) ∈ R n the vector which denotes the theoretical solution of the TVLEI problem b(t) ∈ R m , d(t) ∈ R p the time-varying smooth vectors an unknown vector need to be obtained the bounded random noises G(·) ∈ R n×n → R the adaptive scale factor and the parameter sup(·) the upper bound of the parameter

II. PROBLEM FOURMULATION AND PRELIMINARIES
The specific representation of the TVLEI problem can be depicted as where the time-varying matrices C(t) ∈ R m×n and D(t) ∈ R p×n , the time-varying smooth vectors b(t) ∈ R m , d(t) ∈ R p , and the vector x(t) ∈ R n denotes the theoretical solution of the TVLEI problem (1) which needs to be solved. A nonnegative relaxation variable Λ ·2 (t) ∈ R p is imported to further solving the TVLEI problem (1) [31]: where superscript ·2 denotes the square of each element of Λ ·2 (t) ∈ R p . When solving the equation (2) in this paper, Λ ·2 (t) ∈ R p is also an unknown vector to be obtained, by defining the diagonal matrix Finally, the TVLEI problem (1) is transformed as below: where the matrix W (t) ∈ R (m+p)×(n+p) and vector u(t) ∈ R (m+p) are arranged as the augmented vector y(t) = [ x T (t), Λ T (t)] T ∈ R n+p , with the superscript T denotes the transpose of a matrix or a vector.

III. AN ADAPTIVE ENHANCED AND NOISE SUPPRESSING ZNN MODEL
According to the equation (3) mentioned in Section Π, the corresponding error function is constructed as Based on the following OZNN model construction framework to make every element of δ(t) converge to zero, the following equation can be obtained: where γ > 0 ∈ R denotes the scale coefficient and η(·) : R n×n represents the activation function. Thus, applying the OZNN model to solve the TVLEI problem (1) can be derived as However, there will inevitably exist noises interference to the system in practical applications. Concretely speaking, parameter errors, calculation errors, external interference, and modeling errors can all be regarded as noises. Even the rounding error and truncation error in the numerical calculation can be also regarded as noises. Thus, to solve the TVLEI problem (1) in the case of online noises interference, the AENSZNN model is proposed and its evolution formula can be derived as follows: where the G(·) > 0 : R n×n → R represents the adaptive scale factor and the parameter β > 0. There are several schemes that can be applied to construct the G(·): • Exponential adaption scale factor scheme: where the parameters σ > 0 and ϕ > 1. • Power adaption scale factor scheme: where the parameter ω > 0.
Through combining the equation (8) and the equation (3), we VOLUME 4, 2016 can obtained: By observing the structure diagram of the AENSZNN model (11) in Fig. 1, we can view the AENSZNN model (11) as a broadly proportional-integral-derivative (PID) controller, which is differential, proportional, and integral control input section respectively as −Ẇ (t) The global convergence performance of the proposed AEN-SZNN model (11) is a significant criterion. To further analyze the convergence performance of the AENSZNN model (11) in the case of noises-free. The following theorem is given: Theorem 1: Given an any solvable TVLEI problem (1), the solution obtained by the AENSZNN model (11) always globally converges to the exact solution. That is to say, the residual error || δ(t)|| 2 of the AENSZNN model (11) synthesized by the error function (5) converges to zero within a finite time.
Proof: The ith subsystem of the AENSZNN model (11) is For further investigation, the following Lyapnuov candidate function [32] is formulated: Combining the equation (12) and G(·) > 0, the following conclusions can be drawn: which indicate that when δ i (t) = 0 or t 0 δ i (τ )dτ = 0, the Lyapnuov candidate function l i (t) > 0. If and only if δ i (t) = t 0 δ i (τ )dτ = 0. Therefore, the Lyapunov function candidate δ i (t) is positive definite. The derivative form of the function (12) can be expressed aṡ In other words, thel i (t) is negative definite for all the time t ∈ [0, +∞). Thus, according to the Lyapunov theory, the residual error || δ(t)|| 2 of the subsystem of the AENSZNN model (11) eventually converges to zero. It can be generalized and summarized that for each i ∈ 1, 2, ...n, the δ i (t) globally converges to zero. Generally speaking, the error function || δ(t)|| 2 globally converges to zero as time goes by. Thus, the proposed AENSZNN model (11) globally converges to the theoretical solution to the TVLEI problem (1) over finite time. The proof is thus completed.

IV. ROBUSTNESS ANALYSIS OF THE AENSZNN MODEL UNDER VARIOUS NOISES INFLUENCE
In practical application scenarios, the AENSZNN model (11) will unavoidably be influenced by various measurement noises, which will reduce the accuracy of the solving system or lead to the breakdown of the solving system. Therefore, considering the influence of measurement noises on the AENSZNN model (11) and further analyzing the various noises influence on the AENSZNN model (11), the formula of the AENSZNN model (11) interfered by measurement noises [33] is given as follows: The AENSZNN model (11) under the perturbations of constant noises, bounded random noises, and linear timevarying noises are provided severally. The following three theorems are given to analyze the robustness of the AEN-SZNN model (11).
Theorem 2: No matter how large amplitude of the constant noises Φ(t) = Φ ∈ R m+p perturbed AENSZNN model (11), Φ(t) can always globally converge to the theoretical solution of the TVLEI problem (1). Moreover, the ith subelement of x(t) in (11) globally converges to the theoretical solution of TVLEI problem (1).
Proof: On the basis of the definition of the Laplace transformation, the ith subelement of the mentioned AENSZNN model (11) can be written as where the parameterΦ i /s 2 represents the Laplace transformation of the linear time-varying noisesΦ i t term. In terms of lim t→∞ G(δ i (t)) = lim s→0 G(δ i (s)) = ψ > 0. Therefore, equation (17) can be further summarized: Taking the definition of Laplace final value theory and the Theorem 2 into account, we can obtain: And we get that when the parameter β → +∞ for lim t→∞ || δ(t)|| 2 = 0. The proof is completed.
When studying the AENSZNN model (11) for solving the TVLEI (1) problem with bounded random noises Φ(t) perturbation and analyzing the robustness of the AENSZNN model (11), the following theorem is proposed.

1) In the case of ψ 2 > 4β
Based on the method for solving second-order differential equations, equation (19) can be converted into: By the definition of the triangle inequality [35]: In short, the following inequality can be obtained: where sup(·) represents the upper bound of the parameter.

2) In the case of ψ 2 = 4β
Similar to the steps in ψ 2 > 4β, in this case, equation (19) can be written as It is worth noting that when ψ 2 = 4β the parameter ε 1 = −ψ/2. Theorem 1 in paper [36] proves the existence of the constants ω > 0 and ι > 0 that satisfy the following   (7), NCZNN model (26), PTCZNN model (27), and the proposed AENSZNN model (11) in the three vital indicators: maximum steady state residual (MSSRE), average steady state residual (ASSRE), and convergent time to solve the TVLEI problem (1) when the parameters G(t) = || δ(t)|| 2 + β, β = 10 and γ = 5. inequality: And then, in combination with the definition of the triangle inequality and inequality (23), equation (22) can be transformed renewedly into: The above inequality is simplified and rearanged to obtain: On the basis of the above facts we can reach the following conclusion: 6 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and Similar to the previous case of ψ 2 > 4β and ψ 2 = 4β, when ψ 2 < 4β, the following equation is satisfied: It can draw a conclusion that the proposed AENSZNN model (11) not only uses a competitive adaption scale factor to increase the convergence speed of the model, but also maintains excellent robustness under various noises disturbance by using the momentum information. Moreover, it is possible to solve the TVLEI problem (1) without being affected by the lagging error due to the consideration of the AENSZNN model (11) which utilizes the derivative term of the system. The proof is thus completed.

V. SIMULATION EXPERIMENTS AND COMPARISONS
In this section, the effectiveness and superiority of the proposed AENSZNN model (11) for the solving TVLEI problem (1) are proved through comparative numerical experiments. The simulations are carried out on a digital computer with Intel Core i7-8565U @1.80 GHz CPU, 16 GB memory, and Windows 10 operating system using MATLAB R2016b.

A. EXAMPLE 1
Based on formula (1), take the following time-varying matrices and vectors as a typical example for simulation experiments. Based on formula (1), take the following timevarying matrices and vectors as a typical example of simulation experiments: On the basis of the above theoretical analysis, such a TVLEI problem (1) can be transformed into a time-varying matrix-vector equation (2). The corresponding simulation results are shown in Figs. 2 and 4. Among them, when solving example 1 (V-A), the residual error || δ(t)|| 2 of the AENSZNN model (11) globally converges from the initial state to the theoretical solution. In addition, the following models are utilized to solve the TVLEI problem (1) and their specific evolution formula are as follows: • NCZNN model in [20].
where the Υ(·) represents the odd monotonically increasing activation function arrary.

B. NOISES FREE CONDITION
Specifically, Fig. 4 illustrates the simulation results of Example 1 (V-A) solved by the AENSZNN model (11) with noises free condition. There are thirty random initial states are set at the beginning of the experiment, and the simulation results synthesized by the AENSZNN model (11) based on the norm adaptive factor G(·) gradually approached the theoretical solution of the problem, the trajectory shape of the simulation result is exhibited in Fig. 4 (a) and (d). Among them, when solving example 1 (V-A), the AENSZNN model (11) globally converges from the initial state to the theoretical solution. What needs to be explained here is that the t(s) in all figures represents the length of time in seconds. From the Fig. 4 (b), it shows that the order of the residual || δ(t)|| 2 (11) generated by the AENSZNN model is 10 −6 , while other models converge to 10 −4 . Therefore, compared with the OZNN model (7), NCZNN model (26), and PTCZNN model (27), the AENSZNN model (11) proposed in dealing with the TVLEI problem (1) achieves higher computational accuracy. It is worth mentioning that because the AENSZNN model (11) integrates the norm-based adaption factor G(·), the target position at time t (xi,yi) the coordinates of the sensor node Q = x1 x2 · · · xn y1 y2 · · · yn the coordinate matrix of n sensor nodes αi(t) the incident angle of the sensor node the expected experimental results can be obtained without repeated adjustment of the convergence coefficient. Fig. 2 (a) and (b) illustrate the simulation results of the OZNN model (7), NCZNN model (26), and PTCZNN model (27) used to solve the TVLEI problem (1) in the case of the constant noises. In order to study and verify the global convergence and robustness of the AENSZNN model (11) in the case of constant noises, set the constant noises to Φ(t) = Φ = [2] 3 , and proceed with corresponding numerical simulation experiments are carried out. Specifically, the residual error || δ(t)|| 2 of the AENSZNN model (11) proposed in this paper converges to 10 −3 within 2s in the disturbance environment of constant noises. By comparison, the residual errors of the OZNN model (7), NCZNN model (26), and PTCZNN model (27) are still at the value of 1 with the parameter γ = 5.

C. NOISES CONDITION
For the sake of the robustness of the AENSZNN model (11) can be further studied under the disturbance of linear time-varying noises Φ(t) =Φt = [(2/5) × t] 3 and bounded random noises Φ(t) ∈ [0.5, 2] 3 . As can be seen from Fig. 2 (c)-(f) that the residual error || δ(t)|| 2 of the AENSZNN model (11) converges to of order 10 −1 and of order 10 −2 under the disturbance of linear noises and bounded random noises. In other words, even if the AENSZNN model (11) is interfered with by linear time-varying noises and bounded random noises, it still maintains high accuracy. At the same time, it can be seen that the residual error || δ(t)|| 2 obtained from the OZNN model (7), NCZNN model (26), and PTCZNN model (27) are relatively large. Therefore, when solving Example 1 (V-A) by the AENSZNN model (11) that uses norm-based adaption coefficient to regulate the convergence coefficient, the AENSZNN model (11) still maintains strong robustness despite constant noises disturbance.

D. AOA ALGORITHM
The angle of arrival (AOA) [37] dynamic positioning algorithm is a positioning scheme that relies on the angle of arrival between the measurement target and the sensor node. Specifically, the intersection of two rays passing through the target with the sensor node as the starting point is the site of the target. Moreover, Table 4 shows the arithmetic notations of the following problem descriptions.
The AOA dynamic positioning algorithm is a positioning scheme that relies on the angle of arrival between the measurement target and the sensor node. Specifically, as shown  (7), and proposed AENSZNN model (11) severally.
in the schematic diagram of Fig. 6, the intersection of the two rays e 1 (x 1 , y 1 ) and e 2 (x 2 , y 2 ) passing through the target with the sensor node as the starting point is the position q(x, y) where the dynamic target is located. In addition, for the convenience of notation and description, the Table 4 shows the arithmetic symbols described in the following problem. For the convenience of research, the incident angle can be expressed as By further derivating, the following equation can be obtained For further discussion, the equation (28) is expressed in the following form: where the Q(t) ∈ R n×2 , e(t) = [x(t), y(t)] T and a(t) ∈ R n . Moreover, to more intuitively observe the superiority of the proposed AENZNN model (11) compared to different models, the visual simulation results of the AENZNN model (11) and the OZNN model (7) while calculating the pseudoinverse of W (t) are shown in Fig. 8. It can be obviously seen that compared with the OZNN model (7), the proposed AENSZNN model (11) achieves better robust performance and obtains higher solution accuracy. Furthermore, in order 8 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and  (7), and the proposed AENSZNN model (11) to solve the dynamic 2-D positioning problem. Among them, (a)-(c) display the actual trajectory of the pseudo-inverse method, OZNN model (7), and AENSZNN model (11) and the resulting estimation trace fitting diagrams one by one.  to more intuitively observe the superiority of the proposed AENZNN model (11) relative to different models, the AEN-SZNN model (11) is compared with the classic OZNN model (7) and the pseudo-inverse iterative solution method for solving the time-varying problems while calculating the pseudoinverse of W (t). Fig. 5 (a)-(c) respectively show the actual trajectory and estimation trace fitting diagrams and steadystate residual error || δ(t)|| 2 convergent curves obtained by using the pseudo-inverse method, the OZNN model (7), and the AENSZNN model (11) proposed in this article as several solutions for solving two-dimensional dynamic positioning problem. It can be obviously seen that compared with the OZNN model (7) and the pseudo-inverse solution method, the proposed AENSZNN model (11) achieves a better robust performance and obtains a higher solution accuracy in Fig. 8.
Carefully observe the comparison result image, it is obvious that when the constant noises ϑ(t) is set as 5, the estimation trajectory obtained by the pseudo-inverse method deviates from the actual trajectory to a large extent; the forecast result of the OZNN model (7) deviates from the actual trajectory to a small extent. However, the AENSZNN model (11) can still obtain almost the same estimation trajectory results as the actual trajectory even under the interference of noises, and steady-state residual error || δ(t)|| 2 can converge faster.

VI. CONCLUSIONS AND FUTURE WORK
In this paper, we propose an adaptive enhanced and noisesuppressing zeroing neural network termed the AENSZNN model for solving the TVLEI problem (1) in the field of practical engineering. Furthermore, to study and confirm the reliability of applying the adaption scale factor to the ZNN model for the first time, the residual error || δ(t)|| 2 of exponential adaption scale factor (9) and power adaption scale factor (10) is shown in Fig. 3 (a) and (b). Specifically, the AENSZNN model adopts exponential adaption scale factor (9) and power adaption scale factor (10) accurately converge to 10 −6 , within 5 seconds. As a follow-up to this article, we will explore and give rein to the superiority of the AENSZNN model in response to TVLEI problem encountered in other practical application scenarios.