A varying-gain ZNN model with ﬁxed-time convergence and noise-tolerant performance for time-varying linear equation and inequality systems

—In this paper, a varying-gain zeroing (or Zhang) neural network (VG-ZNN) is proposed to obtain the online solution of the time-varying linear equation and inequality system. Distinguished from the ﬁxed-value design parameter in the original zeroing (or Zhang) neural network (ZNN) models, the design parameter of the VG-ZNN model is a nonlinear function that changes with time. The VG-ZNN model composed of the new time-varying design parameter we proposed can achieve ﬁxed-time convergence and tolerate time-varying bounded noise and time-varying derivable noise. The theoretical detailed analysis of the convergence and robustness of the VG-ZNN model are given. Numerical experiments are performed to conﬁrm the theoretical results, including numerical experiments of proposed VG-ZNN model and numerical comparison with the original ZNN models and the known VG-ZNN models. Furthermore, the VG-ZNN model is also applied to the control of six-link robot manipulator, thus showing the applicability of the proposed VG-ZNN model.


I. INTRODUCTION
S OLVING the systems of linear equation and inequality arises in numerous fields of science, engineering, and business, such as image restoration, computer tomography, system identification, and control system synthesis, all of these require solving a large linear equation and inequality system within a reasonable time frame [1]- [5]. There are some methods for solving it. For example, Castillo et al. proposed a Gamma algorithm in [6], abstract algorithm proposed by Spedicato et al. [7] and its variants [8], [9], and some recurrent neural networks (RNNs) [10]- [12]. They all can solve the linear equation and inequality system effectively, but it should be pointed out that these methods are proposed for solving static linear equation and inequality system and in current practical applications, the problem is always related to time. Therefore, the methods of solving the time-varying linear equation and inequality system is most in need of discussion.
At present, one of the most effective methods for solving time-varying problems is zeroing/Zhang neural network (ZNN) proposed by Zhang et al. in [13], such as [14]- [16]. Just like its name, this model uses an appropriate activation function to make each element of the matrix/vector valued error function converge to zero to achieve the purpose of solving the problem [17]. For example, Zhang et al. proposed the original ZNN model which can achieve exponential convergence to solve the time-varying matrix inversion in [18] and Li et al. proposed a finite-time ZNN model which activated by the sign-bi-power function and can achieve finitetime convergence to solve the time-varying Sylvester equation in [19], [20]. It is very necessary to point out that although the finite-time ZNN model improves the convergence time from infinite to finite, the upper bound of the convergence time is related to the initial value. That is to say, when the initial value is larger, the convergence time of the ZNN model may be longer [21]. Therefore, the main problem now is to eliminate the influence of the initial value on the upper bound of the convergence time. On the other hand, in the application of ZNN model, noise is inevitable, so the noisetolerant performance of ZNN models is another problem that needs to be solved. In fact, in order to solve the noise problem, Jin et al. once proposed a noise-tolerant ZNN model to solve the time-varying matrix inversion in [22], but the upper bound of the convergence time of this ZNN model is still related to the initial value. In fact, the above two problems were solved at the same time in [23], Xiao  Although the proposed ZNN models in [23], [24] can achieve fixed-time convergence and noise-tolerant performance, their design parameter is still fixed. That is to say, in practical applications, the design parameter need to be set as large as possible, so that the convergence time will be as short as possible, but this is difficult to implement in actual systems. Besides, in practical applications, the convergence parameters are always related to the reciprocal of capacitive parameters and inductance parameters, i.e., the convergence parameters change with time in the hardware system [25]. To this end, Zhang et al. proposed a varying-gain ZNN (VG-ZNN) model with design parameter as power function in [26] and exponential function in [27]. However, although the above two VG-ZNN models used time-varying design parameter, they still did not solve the two problems we mentioned above. Therefore, Xiao et al. improved the VG-ZNN model whose design parameter is exponential function, and proved that this VG-ZNN model can achieve fixed-time convergence and noise-tolerant performance in [28].
In this paper, inspired by [26]- [28], we proposed a new time-varying design parameter and proved that the resulting VG-ZNN model can achieve fixed-time convergence and noise-tolerant performance. It can also be regarded as the promotion of the VG-ZNN model proposed by Xiao et al. in [28]. The numerical experiments are performed to confirm the superiority of our VG-ZNN model over the known VG-ZNN models and original ZNN model proposed in [29] when solving the time-varying linear equation and inequality system. To the beat of our knowledge, this is the first time a VG-ZNN model has been used to solve this problem. Moreover, we apply the proposed VG-ZNN model to the control of robot manipulator. Physical experiment performed on a sixlink planar robot manipulator is presented to demonstrate physical realizability and effectiveness of the proposed VG-ZNN model.

II. ZNN MODELS
In this section, we first present the original ZNN model for solving time-varying linear equations and inequality systems, and then propose a new time-varying design parameter to form a VG-ZNN model for solving the same system.

A. Original ZNN model
In this paper, we consider the following time-varying linear equation and inequality system: where A(t) ∈ R m×n (with m < n), C(t) ∈ R p×n (with p < n), b(t) ∈ R m , and d(t) ∈ R p are known timevarying coefficient matrices and vectors, and x(t) ∈ R n is the unknown vector to be obtained. In order to ensure the existence of the solution, we assume that A(t) and C(t) are row-full-rank matrices. From [29], the equation (1) can be transformed into the following form by introducing a non-negative time-varying vector y .
It can be sorted into the following equation where the coefficient matrix M (t) ∈ R (m+p)×(n+p) , vector p(t) ∈ R m+p , and the unknown vector u(t) are defined as and u(t) = [x(t); y(t)].
To make each element of the error function going to zero with time, we consider the following ZNN model formula with the error function e(t) = M (t)u(t) − p(t) e(t) = −βF(e(t)) whereė(t) denotes the time derivative of e(t), β > 0 is a fixed design parameter, and F(·) : R m+p → R m+p denotes a vector-valued activation function array. Taking the derivative of the error function e(t) and then substituting it into the design formula yields the following original ZNN model for the time-varying linear equation and inequality system: with the coefficient matrices W (t) ∈ R (m+p)×(n+p) and Q(t) ∈ R (m+p)×(n+p) are defined as , Q(t) = Ȧ (t) 0 and W † (t) is the pseudoinverse of W (t).
For the original ZNN model (2), different kinds of activation function will lead to different performance, such as linear function, hyperbolic-sine function, and sign-bi-power function. The specific form of those three activation function is given below.
However, we still can not find the optimal activation function. The next lemma [27] will tell us how to choose the activation function that can make the ZNN model have a shorter convergence time. From Lemma 1 and Fig. 1, it can seen that the sign-bi-power function (k 1 = k 2 = k 3 = 1, η = 1/3, and ω = 3) is better than the hyperbolic-sine function (ι = 1) and the hyperbolicsine function (ι = 1) is better than the linear function. Thus, the sign-bi-power function is used in this paper.

B. Varying-gain ZNN Models
In order to solve the problem that the design parameter of the original ZNN model is fixed, ZNN model with timevarying design parameter has begun to be studied. According to the varying-gain ZNN (VG-ZNN) model theory, the following design formula can be given: where ν(t) is a time-varying smooth nonlinear function (linear function can be used as special cases) and ν(t) > 0 for t ∈ [0, +∞). Hence, the VG-ZNN model for time-varying linear equation and inequality can be obtained: (3) For the setting of the time-varying design parameter, Zhang et al. first proposed a power function design parameter [26] and then Xiao et al. proposed an exponential function design parameter with λ 1 > 0, and λ 2 > 0 in [28].
However, according to the nature of the power function f (t) = t α , α > 1, the value of the function in t ∈ [0, 1] is much smaller than t. In other words, when α > 1, the time-varying design parameter ϕ 1 (t) does not perform well. On the other hand, from Lemma 1, when ν(t) > 1, the VG-ZNN model (3) can converge faster than the original ZNN model (2). Thus, inspired by ϕ 1 (t) and ϕ 2 (t), the following new time-varying design parameter is given:

III. THEORETICAL ANALYSIS AND RESULTS
This section presents the theoretical analysis and results of the proposed VG-ZNN model (3) on solving time-varying linear equation and inequality system, including convergence time analysis and robustness analysis.
A. Fixed-time convergence of the VG-ZNN model Theorem 1 Given the time-varying linear equation and inequality system (1), if the VG-ZNN model (3) activated by the sign-bi-power function is used, then the state matrix x(t) starting from any initial state x(0) always converges to a time-varying theoretical solution x * (t). In particular, when the design parameter is set to ϕ(t), the VG-ZNN model can converge in the fixed time where the parameters β, r, k 1 , k 2 , η, ω, and λ 2 are the same as before.
Therefore, based on the Lyapunov stability theory [30], e i (t) globally converges to zero with time for each i. In other words, the error matrix e(t) globally converges to zero with time.
Next, let us probe the convergence time of the VG-ZNN model composed of the time-varying design parameter ϕ(t). We first prove the conclusion when r ≤ 1.
From (4), the following equation can be got: In order to study the convergence time of the VG-ZNN model, proceed in the following two steps: Integrating both sides of the equation (6) from 0 to t 1 , we get It is easy to get time t 1 to satisfy: Step 2. When |e i (t 1 )| = 1 → |e i (t 1 +t 2 )| = 0, the inequality (5) is equal tov Integrating both sides of the equation (7) from Then, the time t c is easily obtained: i.e., the error function e(t) globally converges to zero in the fixed-time t c when r ≤ 1.
The proof of the conclusion when r > 1 is similar to when r ≤ 1. But for the completeness, here we also give its proof.
Next studying the convergence time of the VG-ZNN model in two steps: Step 1. When |e i (0)| > 1 → |e i (t 1 )| = 1, the inequality (8) is equal tov Integrating both sides of the equation from 0 to t 1 , the time t 1 is easily obtained: ).
Integrating both sides of the equation from 0 to t 1 , the time t c is easily obtained: i.e., the error function e(t) globally converges to zero in the fixed-time t c when r > 1. The proof is thus completed.
It should be noted that, according to the nature of the exponential function f (x) = r x , r > 1,, as r increases, the faster the growth rate of the function will be. Combined with the lemma 1, the convergence time of the constructed VG-ZNN model will be shorter. We will verify this conclusion in the section (IV).
Remark 1 Using the proof method of theorem 1, it is easy to get that the original ZNN model (2) activated by the sign-bipower function also can achieve fixed-time convergence and the fixed convergence time is .

B. Noise-tolerant performance of the VG-ZNN model
In this subsection, we are going to prove that the VG-ZNN model can tolerate two types of noise, the time-varying bounded noise and the time-varying derivable noise. Supposing ∆n(t) is a time-varying noise with vector value, we further investigate the following VG-ZNN model   i (t) = 0, j ∈ {1, 2, · · · , κ}, then this VG-ZNN model also can converge.
There are two cases to discuss the size ofl i (t): (i) Ifl i (t) ≤ 0, then |e i (t)| will gradually decrease to zero, i.e. the VG-ZNN model can tolerant the noise ∆n(t).
(ii) Ifl i (t) > 0, |e i (t)| is monotonically increasing, that is, |e i (t)| will gradually increase as t increases and e i (t)∆n i (t) > 0.
So we only need to discuss the case (ii), that is On this basis, we can know It is easy to know that | − βν(t)F(e i (t)) + ∆n i (t)| will decrease as |e i (t)| increase, and |e i (t)| will stop increasing until −βν(t)F(e i (t)) + ∆n i (t) = 0. So, when t → ∞, |e i (t)| is bounded by where F −1 (·) is the inverse function of F(·). Due to in this paper, the F(·) is the sign-bi-power function, i.e., |F(x)| ≥ |k 3 x|, so |F −1 (x)| ≤ |x/k 3 |. Thus, the bound of |e i (t)| can be expressed as follows: Next, we will discuss two kinds of noise separately: For the time-varying bounded noise, there is a constant such that ∆n i (t) ≤ . In this case, inequality (10) can be indicated as When time t tends to infinity, since lim i.e., the VG-ZNN model (9) can tolerate the time-varying bound noise ∆n(t).
From the theorem 2, it can be obtained that the VG-ZNN model composed of the time-varying design parameter ϕ 1 (t), ϕ 2 (t), and ϕ(t) all can tolerate the time-varying bound noise and the VG-ZNN model composed of the time-varying design parameter ϕ 2 (t) and ϕ(t), r > 1 both can tolerate the time-varying derivable noise.

IV. SIMULATION VERIFICATION
In this section, for illustrating the above conclusions, numerical simulations are compared to substantiate the superior performance of VG-ZNN model composed of the time-varying design parameter ϕ(t) for time-varying linear equation and inequality system.
Considering the time-varying linear equation and inequality system with the following time-varying matrices: The corresponding simulation results are shown in Fig. 2 -Fig. 5. As illustrated in Fig. 2(a), when r ≤ 1, starting from five different initial states randomly selected in [−2, 2] 6×1 , the neural states x(t) of the VG-ZNN model always change with time. In Fig. 2(b), the residual error e(t) = ||M (t)u(t)−p(t)|| 2  all can diminish to zero within the fixed convergence time which is shown by a red circle in the figure. In order to verify the accuracy of the results, we also give out two error figures. In Fig. 2(c), the error of linear equation e 1 (t) = A(t)x(t)−b(t) can always diminish zero, and in Fig. 2(d), each element of the error of inequality e 2 (t) = C(t)x(t) − d(t) is always less than or equal to 0. Similarly, when r > 1, the results are shown in Fig. 3 and due to the limited space, only the simulation result of r = 4 is given. Fig. 4 described the convergence behaviors of the residual error ||e(t)|| 2 of all VG-ZNN models composed of different time-varying design parameter ν(t) in the ideal environment, i.e., ∆n i (t) = 0 and in the presence of some time-varying derivable noise. From Fig. 4(a), the error ||e(t)|| 2 generated  by ZNN models all can diminish to zero, but the VG-ZNN models we proposed have a shorter convergence time. In addition, it can be seen from Fig. 4 in the presence of noise. Moreover, it is worth noting that the error generated by the VG-ZNN model composed of ϕ 1 (t) = t 2 + 2 also can diminish to zero under the influence of noise ∆n i (t) = t. This is because ϕ 1 (t) = t 2 + 2 is 2-order derivable and lim t→+∞ 2t = +∞. It is also for this reason that in the presence of noise ∆n i (t) = t 2 , this VG-ZNN model can not converge to zero. For the noise ∆n i (t) = ln (1 + t), since lim t→+∞ ∆n (j) i (t) = 0, j ∈ {1, 2, · · · , +∞}, the error of the VG-ZNN models composed of ϕ(t), r > 1 also can diminish to zero. Fig. 5 show that when the noise ∆n(t) is time-varying bounded, the VG-ZNN composed of ϕ 1 (t), ϕ 2 (t), and ϕ(t) all can diminish to zero.
From Fig. 4 and Fig. 5, it can be seen that the convergence time of VG-ZNN models composed of ϕ(t), r > 1 becomes shorter as r increases which verifies the conclusion we made earlier. Besides, regardless of the presence or absence of noise, the convergence performance of the original ZNN model activated by the sign-bi-power function is better than the linear or hyperbolic sine function, and the VG-ZNN model is better than the original ZNN model.

V. ROBOT APPLICATION
In this section, the proposed VG-ZNN model is employed to a six-link robot manipulator by solving the time-varying linear equation and inequality system.
The purpose of robot manipulator control can be achieved by solving the following time-varying linear equation with boundary constraints [29], [31]- [34]: where θ(t) ∈ R n andθ(t) ∈ R n are joint-angle and jointvelocity vectors, respectively. J(θ(t)) ∈ R m×n is the Jacobian matrix, r d (t) ∈ R m is the desired path withṙ d (t) as its time derivative, andθ ± and θ ± correspond to the limits ofθ(t) and θ(t).

VI. CONCLUSION
In this paper, a varying-gain ZNN model with fixed-time convergence and noise-tolerant performance is proposed to solve the liner equation and inequality system. Its fixed-time convergence and noise-tolerant properties were theoretically analyzed. Some numerical comparisons with the known VG-ZNN models and the original ZNN models are also conducted. Moreover, we successfully apply the varying-gain ZNN model to robot manipulator control, which indicates the applicability of our new model in practice.