A Novel Error-Based Adaptive Feedback Zeroing Neural Network for Solving Time-Varying Quadratic Programming Problems

Abstract

This paper introduces a novel error-based adaptive feedback zeroing neural network (EAF-ZNN) to solve the time-varying quadratic programming (TVQP) problem. Compared to existing variable gain ZNNs, the EAF-ZNN dynamically adjusts the parameter to adaptively accelerate without increasing to very large values over time. Unlike adaptive fuzzy ZNN, which only considers the current convergence error, EAF-ZNN ensures regulation by introducing a feedback regulation mechanism between the current convergence error, the historical cumulative convergence error, the change rate of the convergence error, and the model gain parameter. This regulation mechanism promotes effective neural dynamic evolution, which results in high convergence rate and accuracy. This paper provides a detailed analysis of the convergence of the model, utilizing four distinct activation functions. Furthermore, the effect of changes in the proportional, integral, and derivative factors in the EAF-ZNN model on the rate of convergence is explored. To assess the superiority of EAF-ZNN in solving TVQP problems, a comparative evaluation with three existing ZNN models is performed. Simulation experiments demonstrate that the EAF-ZNN model exhibits a superior convergence rate. Finally, the EAF-ZNN model is compared with the other three models through the redundant robotic arms example, which achieves smaller position error.

1. Introduction

In the modern field of mathematical optimization, quadratic programming (QP) problems have played a crucial role. These problems find extensive applications across diverse domains including economics [1], physics [2], and computer vision [3]. While conventional methods like the interior point algorithm [4], Lagrange multiplier methods [5], and sequential quadratic programming (SQP) [6] exhibit distinct applicability, they require manual formulation and the adjustment of mathematical models. When these traditional methods deal with time-varying problems, the complexity of the modeling process increases significantly due to the real-time requirements of time-varying problems in practical applications. Consequently, the effective resolution of time-varying QP (TVQP) problems has captured the attention of researchers, prompting the exploration of innovative solutions to enhance computational efficiency.

In contrast to conventional numerical methods, neural-network-based approaches have increasingly garnered attention from researchers in recent years. Extensive research and exploration of recurrent neural networks (RNNs) [7,8,9] have revealed their practicality in various domains, including natural language processing (NLP) [10], bioinformatics [11], and robotics [12]. Continuous improvements to RNNs have resulted in gradient-based neural network (GNN) [13,14,15,16] models. However, when addressing TVQP problems, GNNs fall short due to their inability to rapidly track changing state variables and converge precisely to theoretically optimal time-varying solutions. Zhang et al. [17] leveraged the temporal derivative form of the error function and introduced the ZNN for tackling time-varying issues, encompassing domains such as time-varying Lyapunov equations [18], time-varying matrix inversion [19,20], time-varying linear matrix equations [21,22], and TVQP problems [23], among others. Nonetheless, the ZNN presents certain limitations, including a suboptimal convergence rate and the inherent challenge of configuring the convergence parameter.

In response to these deficiencies, Peng et al. [24] proposed the finite-time ZNN (FTZNN). This innovative framework adopts a ZNN architecture enhanced with two flexible activation functions, enabling the achievement of precise solutions to TVQP problems within a finite time. This stands in stark contrast to conventional ZNN approaches. At the same time, as an important role of ZNN, the activation function has also been studied by many scholars. For example, the power-sum activation function was proposed in [25] to solve TVQP problems. It further accelerates the convergence rate of the ZNN model for solving various time-varying problems. Zhang et al. [26] proposed the variable parameter convergent differential neural network (VP-CDNN). This innovative approach incorporates a time-dependent convergence parameter, which achieves an exponential convergence rate and heightened robustness compared to those of the ZNN. Based on the above research, a VGZNN model was proposed in [27], which can achieve finite-time convergence. However, the variable gain tends to positive infinity as time increases, rendering the existing variable gain unsuitable for long-term deployments. In light of this, the scheduled varying gain ZNN (SVGZNN) [23] was proposed, which adjusts the gain through gain constraints and the gain required for long-term deployment. In addition, Jia et al. [28] introduced a fuzzy control system into the zeroing neural network and proposed an adaptive fuzzy-type ZNN (AFT-ZNN), which was successfully applied to solve TVQP problems. The fuzzy parameter used can automatically adjust the corresponding convergence rate, so AFT-ZNN achieves better convergence. In [29], an AFRNN model with a fuzzy coefficient was also applied to redundant robot manipulators and achieved good results.

Although the above ZNN models have displayed promising performance in solving TVQP problems, there is still room for further improvement:

(1)

Most existing variable-gain ZNNs utilize time-varying gain to accelerate model convergence. In general, the larger the time-varying gain, the faster the model converges. However, the time-varying gain continues to increase to very large values over time, which is impractical in hardware implementation or some engineering applications. Furthermore, most existing time-varying ZNNs fail to consider the feedback regulation mechanism between the convergence error and the time-varying gain, degenerating the convergence rate and accuracy.

(2)

Adaptive fuzzy ZNN only considers the feedback regulation between the current convergence error and the model parameter gain via constructing the adaptive fuzzy control scheme. However, it failed to consider the feedback regulation mechanism between the current convergence error, historical cumulative convergence error, the change rate of convergence error, and the model parameter gain. This may degenerate the convergence rate and accuracy. Furthermore, due to the fuzzy scheme, the computational time of the adaptive fuzzy zeroing neural network is high.

To address the above limitations, we developed an error-based adaptive feedback zeroing neural network (EAF-ZNN), which can perform the effective neural dynamic evolution and obtain very competitive convergence performance, by establishing an error-based adaptive feedback parameter adjustment mechanism. The proposed adjustment mechanism is inspired by the PID controller [30], which can provide effective feedback control via rationally combining the current system output error, historical cumulative output error, and the change rate of output error. Similar to the PID controller, the proposed adjustment mechanism establishes an adaptive parameter adjustment relationship between the current convergence error, historical cumulative convergence error, the change rate of the convergence error, and the model gain parameter. Via error-based adaptive feedback parameter adjustment, the EAF-ZNN model can achieve effective neural dynamic evolution and iteration with the evolution formula and activation functions. Therefore, the proposed EAF-ZNN model can achieve high convergence rate and accuracy. Furthermore, this study investigated the impact of different values of the proportional factor, integral factor, and derivative factor on the convergence performance of the model. Additionally, a comprehensive theoretical analysis of the convergence of the EAF-ZNN model was conducted. To validate its superior convergence rate, we compared the model with the SVGZNN [23], VP-CDNN [26], and AFT-ZNN [28] models described in the literature. In addition, we applied the proposed EAF-ZNN model to a redundant robotic arm. In summary, the contributions of this article are as follows:

1.

To address TVQP problems with precision and efficiency, this paper introduces an innovative EAF-ZNN model. To the best of our knowledge, this is the first time that an error-based adaptive feedback parameter has been combined with ZNN.

2.

Different from previous ZNN models, the proposed EAF-ZNN model is improved in the following two aspects: (1) Compared with the variable-gain ZNN, the EAF-ZNN model realizes adaptive acceleration of the convergence rate. The EAF-ZNN model restricts the parameter to an appropriate range to effectively save hardware resources. (2) The EAF-ZNN model is more concise and intuitive in modeling than adaptive fuzzy ZNN. Additionally, the EAF-ZNN model establishes an adaptive parameter adjustment relationship among the current convergence error, historical cumulative convergence error, the change rate of convergence error, and the model gain parameter. This adaptive relationship enables the EAF-ZNN model to promptly respond to errors, resulting in a faster convergence rate and higher accuracy.

3.

The convergence and stability of the EAF-ZNN model are thoroughly analyzed and proved using four activation functions.

4.

The following four conclusions were drawn from the experimental results: (1) Within the range of constraint parameters, the larger the values of the proportional, integral, and derivative factors, the faster the convergence of the EAF-ZNN model. (2) The EAF-ZNN model demonstrates superior convergence across all four activation functions. Notably, under modified sign-bi-power activation, the model exhibits faster convergence and fewer residual errors than the alternative activation functions. (3) The EAF-ZNN model exhibits accelerated convergence in contrast to the VP-CDNN, SVGZNN, and AFT-ZNN models. (4) The EAF-ZNN model is applied to the redundant robotic arms experiment, and the position error is smaller than that of the other three models, which further verifies the effectiveness and practicability of the EAF-ZNN model.

The subsequent sections of this paper are structured as follows: Section 2 presents the description of the TVQP problem and the construction of the EAF-ZNN model. Section 3 provides a theoretical analysis of the convergence of the EAF-ZNN model. Section 4 illustrates and compares the simulation results. Section 5 applies the EAF-ZNN model to a redundant robotic arm. Section 6 discusses the specifics of the EAF-ZNN model and compares it with other models previously discussed in this paper. Finally, the whole paper is summarized. Figure 1 summarizes the main work of this article.

2. Programming and Design of the EAF-ZNN

In this section, we present the detailed solution technique used by the error-based adaptive feedback zeroing neural network (EAF-ZNN) solver used to solve the linear constrained TVQP problem. Firstly, the TVQP problem is formulated, and the problem is expressed in matrix form by combining Lagrange multiplier and convex optimization theory. Then, an error function is defined to measure the deviation from the optimal solution. The dynamic model is constructed based on the ZNN method, and a novel error adaptive feedback mechanism is used to adjust the feedback parameter to improve the convergence rate and stability. Finally, the activation function introduces nonlinearity to improve the ability of the model to deal with complex problems.

2.1. Problem Formulation

In modern mathematical theory, the overarching expression of the TVQP problem [31] can be articulated as

$$\begin{array}{cccc}\hfill & \min .\hfill & \hfill & \frac{1}{2}{x}^T\left(t\right)\mathcal{Q}\left(t\right)x\left(t\right)+{\mathcal{U}}^T\left(t\right)x\left(t\right)\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill & \mathrm{s}.\mathrm{t}.\hfill & \hfill & \mathcal{A}\left(t\right)x\left(t\right)=\mathcal{C}\left(t\right)\hfill \end{array}$$

(2)

where Equation (1) is mathematically a time-varying quadratic minimization (QM), but, in practical applications, the time-varying QM (1) is not sufficient to describe the target problem in many domains. In robot motion planning, consider a redundant robot manipulator motion tracking task, which is based on a robot motion planning scheme that requires a time-varying Equation (2) as a constraint. Thus, the combination of Equations (1) and (2) yields a TVQP problem. In the context of the TVQP problem, where the time variable is denoted as t ranging over the interval $\left[0,+\infty \right)$, several mathematical entities are considered to be smooth, including the time-varying symmetric matrix $\mathcal{Q}\left(t\right)\in {\mathbb{R}}^{n\times n}$, the matrix $\mathcal{A}\left(t\right)\in {\mathbb{R}}^{m\times n}$, and the vector $\mathcal{C}\left(t\right)\in {\mathbb{R}}^m$. The decision variable to be determined, denoted as $x\left(t\right)\in {\mathbb{R}}^n$, has a unique optimal solution because $\mathcal{Q}\left(t\right)$ is required to be positive definite. Additionally, it is assumed that $\mathcal{Q}\left(t\right)$, $\mathcal{U}\left(t\right)$, $\mathcal{A}\left(t\right)$, and $\mathcal{C}\left(t\right)$, along with their respective time derivatives, $\dot{\mathcal{Q}}\left(t\right)$, $\dot{\mathcal{U}}\left(t\right)$, $\dot{\mathcal{A}}\left(t\right)$, and $\dot{\mathcal{C}}\left(t\right)$, are all continuous and well defined over the given time interval. The Lagrange multipliers [32,33] were chosen to derive the equivalence equation, and the Lagrange function for the TVQP problem (1) is obtained as:

$$\begin{array}{cc}\hfill \mathcal{L}(x\left(t\right),\lambda \left(t\right),t)=\frac{1}{2}& x^T\left(t\right)\mathcal{Q}\left(t\right)x\left(t\right)+{\mathcal{U}}^T\left(t\right)x\left(t\right)\hfill \\ \hfill & +{\lambda }^T\left(t\right)(\mathcal{A}\left(t\right)x\left(t\right)-\mathcal{C}\left(t\right))\hfill \end{array}$$

(3)

where $\lambda \left(t\right)\in {\mathbb{R}}^m$ is a vector of Lagrange-multiplier. From the theory of convex optimization [34], its optimal characteristics satisfy the following equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{\partial \mathcal{L}\left(x\right(t),\lambda (t),t)}{\partial x\left(t\right)}=\mathcal{Q}\left(t\right)x\left(t\right)+\mathcal{U}\left(t\right)+{\mathcal{A}}^T\left(t\right)\lambda \left(t\right)=0\hfill \\ \frac{\partial \mathcal{L}\left(x\right(t),\lambda (t),t)}{\partial \lambda \left(t\right)}=\mathcal{A}\left(t\right)x\left(t\right)-\mathcal{C}\left(t\right)=0\hfill \end{array}\right.\end{array}$$

(4)

Equation (4) can be further expressed in matrix form as

$$\begin{array}{c}\hfill H\left(t\right)y\left(t\right)=\mathrm{g}\left(\mathrm{t}\right)\end{array}$$

(5)

where

$$\begin{array}{cc}\hfill & H\left(t\right)=\left[\begin{array}{cc}\mathcal{Q}\left(t\right)& {\mathcal{A}}^T\left(t\right)\\ \mathcal{A}\left(t\right)& 0\end{array}\right]\in {\mathbb{R}}^{(n+m)\times (n+m)},\hfill \\ \hfill & y\left(t\right)=\left[\begin{array}{c}x\left(t\right)\\ \lambda \left(t\right)\end{array}\right]\in {\mathbb{R}}^{n+m},\mathrm{g}\left(\mathrm{t}\right)=\left[\begin{array}{c}-\mathcal{U}\left(t\right)\\ \mathcal{C}\left(t\right)\end{array}\right]\in {\mathbb{R}}^{n+m}.\hfill \end{array}$$

(6)

Therefore, the theoretical solution of the TVQP problem (1) can be derived as

$$\begin{array}{c}\hfill y^{*}\left(t\right)=H^{-1}\left(t\right)\mathrm{g}\left(\mathrm{t}\right)\in {\mathbb{R}}^{n+m}\end{array}$$

(7)

2.2. EAF-ZNN Solver

Equation (5) is solved by defining a vector-type error function to obtain its optimal solution.

$$\begin{array}{c}\hfill E\left(t\right)=H\left(t\right)y\left(t\right)-\mathrm{g}\left(\mathrm{t}\right)\in {\mathbb{R}}^{n+m}\end{array}$$

(8)

According to the design of the ZNN, the solution of the TVQP problem (1) is naturally obtained when the setup error function tends to zero. Therefore, the dynamic formulation of the EAF-ZNN model can be constructed as

$$\begin{array}{c}\hfill \frac{dE\left(t\right)}{dt}=\dot{E}\left(t\right)=-(\mu +\theta )\Phi \left(E\left(t\right)\right)\end{array}$$

(9)

where $\mu$ is a positive design constant (i.e., $\mu >0$). In general, there are two common strategies for designing ZNN models with different convergence parameters. One approach involves employing a time-varying convergence parameter, while the other entails integrating adaptive convergence parameters generated by fuzzy logic systems into ZNN models. But, the relationship between the current convergence error, the historical cumulative convergence error, the change rate of the convergence error, and the model gain is not considered. This may affect the rate of convergence and accuracy. Inspired by the PID controller, the adaptive parameter regulation relationship between the current convergence error, the historical cumulative convergence error, the change rate of the convergence error, and the model gain parameter is established, and the error-based adaptive feedback parameter $\theta$ is introduced in order to track the error set:

$$\begin{array}{c}\hfill \epsilon \left(t\right)={∥E\left(t\right)∥}_F\end{array}$$

(10)

where ${∥·∥}_F$ represents the Frobenius norm [35]. And, $\theta$ is defined as

$$\begin{array}{c}\hfill \theta ={\mathcal{K}}_p\epsilon \left(t\right)+{\mathcal{K}}_i{\int }_0^t\epsilon \left(\tau \right)\mathrm{d}\left(\tau \right)+{\mathcal{K}}_d\frac{\mathrm{d}\epsilon \left(t\right)}{\mathrm{d}t}\end{array}$$

(11)

where ${\mathcal{K}}_p$ is the proportional factor, ${\mathcal{K}}_i$ is the integral factor, and ${\mathcal{K}}_d$ is the derivative factor. Adaptive feedback of the current convergence error, the historical accumulated error, and the change rate of the convergence error are realized by a linear combination of $\epsilon \left(t\right),{\int }_0^t\epsilon \left(\tau \right)\mathrm{d}\tau$, and $\mathrm{d}\epsilon \left(t\right)/\mathrm{d}t$.

In the realm of computer-based control systems, it is essential to recognize that computers inherently operate discretely. When implementing control using a computer, one must embrace a sampling control approach. Consequently, the integral and derivative components of the control Equation (11) need to be discretized. To mathematically represent $\theta$ in a digital context, we transition from continuous time points t to a series of discrete sampling time points. Additionally, we replace the integral operation with a summation approximation and the derivative operation with a first-order difference. This yields the following mathematical description of $\theta$:

$$\begin{array}{c}\hfill \theta ={\mathcal{K}}_p\epsilon \left(k\right)+{\mathcal{K}}_i\sum _{j=0}^k\epsilon \left(j\right)+{\mathcal{K}}_d\left[\epsilon \left(k\right)-\epsilon (k-1)\right]\end{array}$$

(12)

where $\epsilon \left(k\right)\mathrm{and}\epsilon (k-1)$ denote, respectively, the kth and ($k-1$)th output values. Parameters ${\mathcal{K}}_p,{\mathcal{K}}_i,\mathrm{and}{\mathcal{K}}_d$ are all greater than zero. They are the proportional factor, integral factor, and derivative factor, respectively, which are used to control the convergence rate and improve stability. After that, the EAF-ZNN model can be derived by substituting Equation (8) into Equation (9):

$$H\left(t\right)\dot{y}\left(t\right)=-(\mu +\theta )\Phi ((H\left(t\right)y\left(t\right)-\mathrm{g}\left(\mathrm{t}\right))+\dot{\mathrm{g}}\left(\mathrm{t}\right)-\dot{H}\left(t\right)y\left(t\right)$$

(13)

where $\mu +\theta$ is an adaptive variable parameter; the convergence rate of the model can be adjusted according to its value. Furthermore, $\Phi (·)$ represents a matrix-type mapping function, where each element of $\varphi (·)$ is a monotonically increasing odd function. In general, the activation function can introduce nonlinearity into neural network models, thus possesses the ability to handle nonlinear problems [36], which means that the network is able to learn and represent nonlinear relationships between inputs and outputs. Without activation functions, neural networks would be limited to learning linear mapping, which significantly limits their ability to express complex tasks. This paper considers the following four activation functions (as shown in Figure 2), defined as follows:

(1)

Linear activation function (LAF)

$$\Phi \left(\vartheta \right)=\vartheta$$

(14)

(2)

Power-sigmoid activation function (PSAF)

$$\begin{array}{c}\hfill \Phi \left(\vartheta \right)=\left\{\begin{array}{cc}{\vartheta }^p\hfill & \mathrm{if}\left|\vartheta \right|\ge 1\hfill \\ \frac{1+exp(-\chi )}{1-exp(-\chi )}·\frac{1-exp(-\chi \vartheta )}{1+exp(-\chi \vartheta )}\hfill & \mathrm{else}\hfill \end{array}\right.\end{array}$$

(15)

where $\chi >2$, $p\ge 3$.

(3)

Sign-bi-power activation function (SBPAF)

$$\Phi \left(\vartheta \right)=\frac{1}{2}{\mathrm{sgn}}^{\zeta }\left(\vartheta \right)+\frac{1}{2}{\mathrm{sgn}}^{\frac{1}{\zeta }}\left(\vartheta \right)$$

(16)

where $0<\zeta <1$.

(4)

Modified sign-bi-power activation function (MSBPAF)

$$\Phi \left(\vartheta \right)={\mathrm{sgn}}^{\zeta }\left(\vartheta \right)+\vartheta$$

(17)

where $0<\zeta <1$.

The symbol ${\mathrm{sgn}}^{\zeta }(·)$ mentioned above is defined as

$$\begin{array}{c}\hfill {\mathrm{sgn}}^{\zeta }\left(\vartheta \right)=\left\{\begin{array}{cc}{\left|\vartheta \right|}^{\zeta }\hfill & \mathrm{if}\vartheta >0\hfill \\ 0\hfill & \mathrm{if}\vartheta =0\hfill \\ -{\left|\vartheta \right|}^{\zeta }\hfill & \mathrm{if}\vartheta <0\hfill \end{array}\right.\end{array}$$

In order to describe the structure of the proposed EAF-ZNN, Equation (13) can be rewritten as

$$\begin{array}{c}\hfill \dot{y}\left(t\right)=(I-H\left(t\right))\dot{y}\left(t\right)-\dot{H}\left(t\right)y\left(t\right)-(\mu +\theta )\Phi (H\left(t\right)y\left(t\right)-\mathrm{g}\left(\mathrm{t}\right))+\dot{\mathrm{g}}\left(\mathrm{t}\right)\end{array}$$

(18)

where I denotes the identity matrix. According to reference [37], the dynamics equation for the ith neuron of Equation (18) can be written as the following explicit dynamics:

$$\begin{array}{cc}\hfill {\dot{y}}_i& =\sum _{j=1}^{n+m}(I_{ij}-h_{ij}){\dot{y}}_j-\sum _{j=1}^{n+m}{\dot{h}}_{ij}{y}_j\hfill \\ \hfill & -(\mu +\theta )\varphi \left(\sum _{j=1}^{n+m}{h}_{ij}{y}_j-{\mathrm{g}}_i\right)+{\dot{\mathrm{g}}}_i\hfill \end{array}$$

(19)

where $h_{ij}$ represents the ith-row and jth-column element of $H\left(t\right)$, and $I_{ij}$ represents the ith-row and jth-column element identity matrix. The topological structure of the EAF-ZNN model is shown in Figure 3. Specifically, the proposed model contains $5(n+m)$ summers, $n+m$ integral operators, and an $n+m$-dimension activation function $\Phi (·)$.

Figure 2. The four activation functions used in this simulation (i.e., LAF (14), PSAF (15) with $\chi =4,p=3$, SBPAF (16), and MSBPAF (17) with $\zeta =0.5$).

Figure 2. The four activation functions used in this simulation (i.e., LAF (14), PSAF (15) with $\chi =4,p=3$, SBPAF (16), and MSBPAF (17) with $\zeta =0.5$).

3. Convergence Analysis

Due to the TVQP problem (1) being time-varying, faster convergence and higher accuracy are required to obtain the optimal time-varying solution. In this section, the global asymptotic stability of the EAF-ZNN model (13) is proved, and the convergence of the model under four different activation functions is analyzed.

Theorem 1.

Given $\mu >0$, the EAF-ZNN model (13) for TVQP problem (1) achieves global asymptotic convergence if a monotonically increasing odd activation function is applied.

Proof. 

For Equation (11), it can be designed in the following form:

$$\begin{array}{c}\hfill \theta \left(t\right)={\mathcal{K}}_p\epsilon \left(t\right)+{\mathcal{K}}_i{\int }_0^t\epsilon \left(\tau \right)\mathrm{d}\left(\tau \right)+{\mathcal{K}}_d\frac{\mathrm{d}\epsilon \left(t\right)}{\mathrm{d}t}\end{array}$$

(20)

Applying the Laplace transform to Equation (20), we have

$$\begin{array}{c}\hfill \tilde{\theta }\left(s\right)={\mathcal{K}}_p\tilde{\epsilon }\left(s\right)+\frac{{\mathcal{K}}_i\tilde{\epsilon }\left(s\right)}{s}+{\mathcal{K}}_d[s\tilde{\epsilon }\left(s\right)-e\left(0\right)]\end{array}$$

(21)

According to the final value theorem, it can be obtained that

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}\theta \left(t\right)& =\underset{s\to 0}{lim}s\tilde{\theta }\left(s\right)\hfill \\ & =\underset{s\to 0}{lim}{\mathcal{K}}_{p}s\tilde{\epsilon }\left(s\right)+{\mathcal{K}}_i\tilde{\epsilon }\left(s\right)+{\mathcal{K}}_{d}s[s\tilde{\epsilon }\left(s\right)-e\left(0\right)]\hfill \\ & ={\mathcal{K}}_i\tilde{\epsilon }\left(0\right)\hfill \end{array}$$

(22)

From Equation (10), there is ${lim}_{t\to \infty }\theta \left(t\right)={\mathcal{K}}_i{∥\tilde{E}\left(0\right)∥}_F={\mathcal{K}}_i\sqrt{{\sum }_{i=1}^n{\sum }_{j=1}^m{\tilde{E}}_{ij}^2\left(0\right)}$. At this point, from the above analysis, it can be determined that $\theta$ converges globally to a constant. For convenience, this constant is set to $\rho$ ($\rho \ge 0$). As time increases, the error $E\left(t\right)$ becomes smaller and smaller, so we obtain $\theta \ge \rho$.

A Lyapunov function candidate is defined as $v\left(t\right)=\frac{1}{2}{∥E\left(t\right)∥}_2^2=\frac{1}{2}{E}^T\left(t\right)E\left(t\right)$. Taking the derivative of $v\left(t\right)$, we have the following equation:

$$\begin{array}{c}\hfill \dot{v}\left(t\right)=\frac{\mathrm{d}v\left(t\right)}{\mathrm{d}t}=E^T\left(t\right)\frac{\mathrm{d}E\left(t\right)}{\mathrm{d}t}\end{array}$$

(23)

After that, using Equation (9), one can obtain

$$\begin{array}{cc}\hfill \dot{v}\left(t\right)& =-(\mu +\theta )E^T\left(t\right)\Phi \left(E\left(t\right)\right)\hfill \\ & =-(\mu +\theta )\sum _{i=1}^{n+m}{E}_i\left(t\right)\varphi \left(E_i\left(t\right)\right)\hfill \\ & \le -(\mu +\rho )\sum _{i=1}^{n+m}{E}_i\left(t\right)\varphi \left(E_i\left(t\right)\right)\hfill \end{array}$$

(24)

where $E_i\left(t\right)$ represents the ith element of vector $E\left(t\right)$. $\Phi (·):{\mathbb{R}}^{m+n}\to {\mathbb{R}}^{m+n}$ represents the monotonically increasing array of singular activation function treatments, and $\varphi (·)$ represents the processing unit of array $\Phi (·)$. The monotonically increasing odd activation function $\varphi \left(E_i\right)$ ensures that

$$\begin{array}{c}\hfill E_i\left(t\right)\varphi \left(E_i\left(t\right)\right)\left\{\begin{array}{c}>0,\mathrm{if}{E}_i\left(t\right)\ne 0\hfill \\ =0,\mathrm{if}{E}_i\left(t\right)=0\hfill \end{array}\right.\end{array}$$

(25)

This implies that $\dot{v}\left(t\right)$ is less than zero at all times $t\in \left[0,+\infty \right)$. According to Lyapunov’s stability theorem [38], the online solution $y\left(t\right)$ approaches its theoretical solution $y^{*}\left(t\right)$ when $t⟶+\infty$. The EAF-ZNN model (13) is globally asymptotically convergent. □

Theorem 2.

Given $\mu >0,\chi >2,\mathrm{and}p\ge 3$, the EAF-ZNN model (13) can achieve a faster convergence rate than LAF (14) when activated by PSAF (15).

Proof. 

According to the definition of PSAF (15), we need to discuss two situations, as follows:

(1) When $\left|E_i\left(t\right)\right|\ge 1$, according to Theorem 1, the time derivative of $v\left(t\right)$ is

$$\begin{array}{c}\hfill \dot{v}\left(t\right)=-(\mu +\theta )\sum _{i=1}^{n+m}{\left|E_i\left(t\right)\right|}^{p+1}\le -(\mu +\rho )\sum _{i=1}^{n+m}{\left|E_i\left(t\right)\right|}^{p+1}\end{array}$$

(26)

If $\Phi (·)$ is LAF (14), then $\dot{v}\left(t\right)=-(\mu +\rho ){\sum }_{i=1}^{n+m}{\left|E_i\left(t\right)\right|}^2\le -(\mu +\rho ){\sum }_{i=1}^{n+m}{\left|E_i\left(t\right)\right|}^2$. However, when $p\ge 3$, ${\left|E_i\left(t\right)\right|}^{p+1}\gg {\left|E_i\left(t\right)\right|}^2$. Consequently, $E\left(t\right)$ under PSAF (15) experiences a more negative change. Therefore, we can conclude that PSAF (15) can make the EAF-ZNN model (13) converge faster convergence than LAF (14) in this case.

(2) When $\left|E_i\left(t\right)\right|<1$, from the definition of PSAF (15), $1+\exp (-\chi )/1-\exp (-\chi )$ is a positive number, and we have

$$\begin{array}{c}\hfill \left|\frac{1-\exp (-\chi E_i\left(t\right))}{1+\exp (-\chi E_i\left(t\right))}\right|>\left|E_i\left(t\right)\right|\end{array}$$

(27)

As in the first case, the analysis shows that $\dot{v}\left(t\right)$ is more negative. Thus, in this case, the EAF-ZNN model (13) converges faster with PSAF (15) than with LAF (14).

By analyzing the two scenarios mentioned above, we can deduce that the utilization of PSAF (15) accelerates the convergence of the EAF-ZNN model (13) in comparison to the use of LAF (14), which completes the proof. □

Theorem 3.

With $\mu >0$ and $\zeta \in \left(0,1\right)$, activating the EAF-ZNN model (13) with SBPAF (16) results in global convergence within finite time, with the upper limit on convergence time being

$$\begin{array}{c}\hfill t_b=\underset{i=1,2\dots ,n+m}{\max }\left\{\frac{2{\left|E_i\left(0\right)\right|}^{1-\zeta }}{(\mu +\rho )(1-\zeta )}\right\}\end{array}$$

(28)

where $E_i\left(0\right)$ represents the initial value of the $ith$ element of $E\left(t\right)$.

Proof. 

The kinetic expression synthesized from the EAF-ZNN model (13) and SBPAF (16) is given as

$$\dot{E}\left(t\right)=-(\mu +\theta )(\frac{1}{2}{\mathrm{sgn}}^{\zeta }\left(E\left(t\right)\right)+\frac{1}{2}{\mathrm{sgn}}^{\frac{1}{\zeta }}\left(E\left(t\right)\right))$$

(29)

where the ith independent subsystem (with $i=1,2,\dots ,n+m$) is given as

$$\dot{E_i}\left(t\right)=-(\mu +\theta )(\frac{1}{2}{\mathrm{sgn}}^{\zeta }\left(E_i\left(t\right)\right)+\frac{1}{2}{\mathrm{sgn}}^{\frac{1}{\zeta }}\left(E_i\left(t\right)\right))$$

(30)

where $0<\zeta <1$. Defining the Lyapunov power function $\Gamma \left(t\right)=E_i^2\left(t\right)$, we can obtain the time derivative of $\Gamma \left(t\right)$ as follows:

$$\begin{array}{cc}\hfill \dot{\Gamma }\left(t\right)& =-2(\mu +\theta )\Phi \left(E_i\left(t\right)\right)E_i\left(t\right)\hfill \\ & =-(\mu +\theta )({\left|E_i\left(t\right)\right|}^{\zeta +1}+{\left|E_i\left(t\right)\right|}^{\frac{1+\zeta }{\zeta }})\hfill \\ & \le -(\mu +\theta ){\left|E_i\left(t\right)\right|}^{\zeta +1}=-(\mu +\theta ){\Gamma }^{\frac{\zeta +1}{2}}\left(t\right)\hfill \\ & \le -(\mu +\rho ){\Gamma }^{\frac{\zeta +1}{2}}\left(t\right)\hfill \end{array}$$

(31)

Solve the inequality $\dot{\Gamma }\left(t\right)\le -(\mu +\rho ){\Gamma }^{\frac{\zeta +1}{2}}\left(t\right)$:

$$\begin{array}{cc}\hfill & {\int }_{\Gamma \left(0\right)}^{\Gamma \left(t\right)}{\Gamma }^{\frac{-\zeta -1}{2}}\left(t\right)d\Gamma \left(t\right)\le {\int }_0^t-(\mu +\rho )dt\hfill \\ \hfill & \frac{2}{1-\zeta }\left[{\Gamma }^{\frac{1-\zeta }{2}}\left(t\right)-{\Gamma }^{\frac{1-\zeta }{2}}\left(0\right)\right]\le -(\mu +\rho )t\hfill \end{array}$$

(32)

Because $\Gamma \left(t\right)=E_i^2\left(t\right)$ can be obtained, we have

$$\begin{array}{c}\hfill {\left|E_i\left(t\right)\right|}^{1-\zeta }\left(t\right)-{\left|E_i\left(t\right)\right|}^{1-\zeta }\left(0\right)\le \frac{-(1-\zeta )}{2}(\mu +\rho )t\end{array}$$

(33)

Combined with $\left|E_i\left(t\right)\right|>0$ and $0<\zeta <1$, we can further obtain

$$\begin{array}{cc}\hfill {\left|E_i\left(t\right)\right|}^{1-\zeta }\left(t\right)& \left\{\begin{array}{c}\le {\left|E_i\left(t\right)\right|}^{1-\zeta }\left(0\right)-\frac{1-\zeta }{2}(\mu +\rho )t,\mathrm{if}t<\frac{2{\left|E_i\left(0\right)\right|}^{1-\zeta }}{(\mu +\rho )(1-\zeta )}\hfill \\ =0,\mathrm{if}t\ge \frac{2{\left|E_i\left(0\right)\right|}^{1-\zeta }}{(\mu +\rho )(1-\zeta )}\hfill \end{array}\right.\hfill \end{array}$$

(34)

Hence, we can conclude that the EAF-ZNN model (13) converges globally in finite time, and the upper limit on the convergence time is

$$\begin{array}{c}\hfill t_b=\underset{i=1,2\dots ,n+m}{\max }\left\{\frac{2{\left|E_i\left(0\right)\right|}^{1-\zeta }}{(\mu +\rho )(1-\zeta )}\right\}\end{array}$$

(35)

The proof is concluded. □

Theorem 4.

Given $\mu >0\mathrm{and}0<\zeta <1$, activating the EAF-ZNN model (13) with MSBPAF (17) results in global convergence within finite time, with the upper limit on convergence time being

$$t_b=\frac{ln({\left|E_{max}\left(0\right)\right|}^{1-\zeta }+1)}{(1-\zeta )(\mu +\rho )}$$

(36)

where $\left|E_{max}\left(0\right)\right|=max\left(\left|E_i\left(0\right)\right|\right)$ denotes the absolute value of the largest element of the original error matrix $E\left(0\right)$.

Proof. 

From inequality $\left|E_{max}\left(t\right)\right|$ ≥ $\left|E_i\left(t\right)\right|,$ for $i=1,2,\dots ,n+m$, it follows that as long as $E_{max}\left(t\right)$ tends to zero, then $E_i\left(t\right)$ tends to zero as well. Because $-(\mu +\theta )\Phi \left(E_{max}\left(t\right)\right)\le -(\mu +\rho )\Phi \left(E_{max}\left(t\right)\right)$, we consider the following kinetic equation:

$$\begin{array}{cc}\hfill {\dot{E}}_{max}\left(t\right)& =-(\mu +\rho )\Phi \left(E_{max}\left(t\right)\right)\hfill \\ & =-(\mu +\rho )({\mathrm{sgn}}^{\zeta }\left(E_{max}\left(t\right)\right)+E_{max}\left(t\right))\hfill \end{array}$$

(37)

(1)

When $E_{max}\left(0\right)>0$, we have ${\dot{E}}_{max}\left(t\right)=-(\mu +\rho )({\left(E_{max}\left(t\right)\right)}^{\zeta }+E_{max}\left(t\right))$, which can be further expressed as

$$\begin{array}{cc}\hfill & {\left(E_{max}\left(t\right)\right)}^{-\zeta }{\dot{E}}_{max}\left(t\right)=-(\mu +\rho )(1+{\left(E_{max}\left(t\right)\right)}^{1-\zeta })\hfill \\ & \frac{\mathrm{d}{\left(E_{max}\left(t\right)\right)}^{1-\zeta }}{\mathrm{d}t}=(\mu +\rho )(\zeta -1)(1+{\left(E_{max}\left(t\right)\right)}^{1-\zeta })\hfill \\ & \mathrm{d}t=\frac{\mathrm{d}{\left(E_{max}\left(t\right)\right)}^{1-\zeta }}{(\mu +\rho )(\zeta -1)(1+\left({\left(E_{max}\left(t\right)\right)}^{1-\zeta }\right)}\hfill \end{array}$$

(38)

We take the integrals for both sides at the same time:

$$\begin{array}{cc}& {\int }_0^{t_b}d\mathrm{t}={\int }_{E_{max}\left(0\right)}^0\frac{\mathrm{d}{\left(E_{max}\left(t\right)\right)}^{1-\zeta }}{(\mu +\rho )(\zeta -1)(1+\left({\left(E_{max}\left(t\right)\right)}^{1-\zeta }\right)}\hfill \\ & t_b=\frac{ln({\left|E_{max}\left(0\right)\right|}^{1-\zeta }+1)}{(1-\zeta )(\mu +\rho )}\hfill \end{array}$$

(39)

(2)

When $E_{max}\left(0\right)<0$, the same result can be obtained using the same method, so it is not repeated in this paper.

(3)

When $E_{max}\left(0\right)=0$, it can be obtained that $t_b=0$.

Therefore, we can conclude that the EAF-ZNN model (13) achieves global convergence in finite time, and the upper limit on convergence time is

$$t_b=\frac{ln({\left|E_{max}\left(0\right)\right|}^{1-\zeta }+1)}{(1-\zeta )(\mu +\rho )}$$

(40)

The proof is complete. □

Theorem 5.

Given $\zeta =0.5$, when the EAF-ZNN model (13) is activated by MSBPAF (17), it exhibits faster convergence to zero within a finite time compared to its activation by SBPAF (16).

Proof. 

According to Theorems 3 and 4, we derive the upper limit on convergence time when activated by SBPAF (16) as

$$\begin{array}{c}\hfill t_{b1}=\frac{2{\left|E_{max}\left(0\right)\right|}^{1-\zeta }}{(\mu +\rho )(1-\zeta )}\end{array}$$

(41)

The upper limit on convergence time under activation by MSBPAF (17) is

$$t_{b2}=\frac{ln({\left|E_{max}\left(0\right)\right|}^{1-\zeta }+1)}{(1-\zeta )(\mu +\rho )}$$

(42)

Comparing the results for $t_{b1}$ and $t_{b2}$, we can easily obtain that $t_{b2}<t_{b1}$. Summarizing the above conclusion, we can infer that the EAF-ZNN model (13) exhibits the most rapid convergence rate with MSBPAF (17).

The proof is complete. □

4. Simulation Experiment

In this section, a numerical example is provided to demonstrate the effective performance of the EAF-ZNN model (13). The experiments were conducted using MATLAB 2022a with the built-in ODE solver ODE45, where the experimental stopping criterion relative error tolerance and absolute error tolerance were both set to $1\times {10}^{-5}$. We ensured that the accuracy of the numerical solution met our computational resiyrcs. Without loss of generality, consider the following TVQP problem:

$$\begin{array}{cc}\min .\hfill & \frac{1}{2}{x}^T\left(t\right)\mathcal{Q}\left(t\right)x\left(t\right)+{\mathcal{U}}^T\left(t\right)x\left(t\right)\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill & \mathrm{s}.\mathrm{t}.\hfill & \mathcal{A}\left(t\right)x\left(t\right)=\mathcal{C}\left(t\right)\hfill \end{array}$$

(44)

where

$$\begin{array}{cccc}\hfill & \mathcal{Q}\left(t\right)=\left[\begin{array}{cc}\sin \left(t\right)+2& \sin \left(t\right)\\ \cos \left(t\right)& \sin \left(t\right)+2\end{array}\right],\hfill & \hfill & \mathcal{U}\left(t\right)=\left[\begin{array}{c}-\sin \left(2t\right)\\ -\cos \left(2t\right)\end{array}\right],\hfill \\ & \mathcal{A}\left(t\right)=\left[\begin{array}{cc}\sin \left(3t\right)& \cos \left(3t\right)\end{array}\right],\hfill & \hfill & \mathcal{C}\left(t\right)=-\cos \left(2t\right),\hfill \\ \hfill & x\left(t\right)={\left[\begin{array}{cc}{x}_1\left(t\right)& x_2\left(t\right)\end{array}\right]}^T.\hfill \end{array}$$

According to Section 2, i.e.,

$$\begin{array}{cc}\hfill & H\left(t\right)=\left[\begin{array}{ccc}\sin \left(t\right)+2& \sin \left(t\right)& \sin \left(3t\right)\\ \cos \left(t\right)& \sin \left(t\right)+2& \cos \left(3t\right)\\ \sin \left(3t\right)& \cos \left(3t\right)& 0\end{array}\right],\hfill \\ \hfill & \mathrm{g}\left(t\right)={\left[\begin{array}{ccc}\sin \left(2t\right)& \cos \left(2t\right)& -\cos \left(2t\right)\end{array}\right]}^T,\hfill \end{array}$$

$y\left(t\right)={\left[\begin{array}{ccc}{x}_1\left(t\right)& x_2\left(t\right)& {\lambda }_1\left(t\right)\end{array}\right]}^T.$ In addition, for PSAF (15), $p=3$, and $\chi =4$. For SBPAF (16) and MSBPAF (17), $\zeta$ was set of $0.5$. The computational complexity of the numerical example provided is $O\left(N{n}^2\right)$, where N represents the number of iterations performed by ODE45, and n denotes the dimensionality of the state vector being solved. Figure 4 shows that the shapes of the objective and constraint functions as well as the minimum value and the minimum solution change with time t. Therefore, the QP problem can be regarded as a variable minimum problem.

In general, we analyzed the convergence experiments from three key perspectives. First, we used four different activation functions to investigate the effects of different values of ${\mathcal{K}}_p$, ${\mathcal{K}}_i$, and ${\mathcal{K}}_d$ on the EAF-ZNN model (13), as shown in Figure 5, Figure 6 and Figure 7. To investigate the effect of $\mu$ on the experiment, in Figure 8, we compare the convergence rate of the model with different $\mu$ values. Figure 9 compares the convergence rates of these four activation functions. Finally, we compared the rate of convergence of the residual errors to zero and the rate of convergence of the state solution with those of the VP-CDNN model [26], the SVGNN model [23], and the AFT-ZNN model [28], which effectively proved that the EAF-ZNN model (13) is superior, as shown in Figure 10, Figure 11, Figure 12 and Figure 13.

In Figure 5, a series of different ${\mathcal{K}}_p$ values are experimentally set to observe the residual errors ${∥H\left(t\right)y\left(t\right)-g\left(t\right)∥}_F$ of the EAF-ZNN model (13) under four different activation functions when ${\mathcal{K}}_d=0\mathrm{and}{\mathcal{K}}_i=0$. The experimental results show that in a limited range, the larger the ${\mathcal{K}}_p$ value, the faster the EAF-ZNN model (13) converges to 0. The same conclusion can be drawn from Figure 6 and Figure 7, where the larger the values of ${\mathcal{K}}_i$ and ${\mathcal{K}}_d$, the faster the EAF-ZNN model (13) converges to zero. For Figure 8, we set ${\mathcal{K}}_p=0,{\mathcal{K}}_i=0,\mathrm{and}{\mathcal{K}}_d=0$ to compare the effect of different $\mu$ values on the experiment, from which we can see that the larger $\mu$, the faster the convergence speed. A study [39] showed that the convergence rate of a recurrent neural network depends heavily on the design parameter. To obtain good performance, this parameter should theoretically be set to infinity, but this is not possible due to hardware limitations. In order to save hardware resources, we set ${\mathcal{K}}_p=2,{\mathcal{K}}_i=2,{\mathcal{K}}_d=2,\mathrm{and}\mu =4$ in the experiment. Figure 9 compares the convergence rates of the four activation functions LAF (14), PASF (15), SBPAF (16), and MSBPAF (17) under the same conditions. It is worth noting that PSAF (15) converges faster than LAF (14), while SBPAF (16) and MSBPAF (17) not only converge faster but also have a limited convergence time. Among them, MSBPAF (17) has the fastest convergence rate. The results shown in Figure 9 confirm the correctness of Theorems 1–5.

Finally, to verify the superior performance of the EAF-ZNN model (13) under different activation functions, the convergence of residual-errors ${∥H\left(t\right)y\left(t\right)-g\left(t\right)∥}_F$ and state solutions of four models under different activation functions (i.e., LAF (14), PSAF (15), SBPAF (16), MSBPAF (17)) are compared. Figure 10, Figure 11, Figure 12 and Figure 13 show that when using the above four activation functions, the convergence performance of both residual-errors and state solutions of EAF-ZNN is significantly better than that of VP-CDNN, SVGZNN, and AFT-ZNN with the same activation. Meanwhile, we also compare the computation time of different models in MATLAB under different activation functions, as shown in

Table 1. Computation times (in seconds) for different models under four different activation functions.

Activation Function	EAF-ZNN	VP-CDNN	SVGZNN	AFT-ZNN
LAF (14)	1.08902	4.1289	1.27519	28.42
PSAF (15)	1.35312	9.83002	1.53213	18.3
SBPAF (16)	1.75394	31.6752	1.685	19.9773
MSBPAF (17)	1.49094	28.7896	2.37265	16.0686

. From Table 1, we can see that the EAF-ZNN model (13) has a more obvious advantage in computing time compared to the other three models.

In conclusion, the above experimental results further validate the effectiveness and superiority of the EAF-ZNN model (13) in solving TVQP problems and the correctness of Theorems 1–5.

5. Robotic Trajectory Application

In this section, we aim to verify the applicability of the proposed EAF-ZNN model (13) in solving kinematic problems in redundant robotic arms (six-degree-of-freedom robotic arms). For this purpose, leaf-path-tracking simulation experiments and a physical experimentwereare conducted.

5.1. Leaf-Path Tracking

For this purpose, the inverse kinematic problem of the robotic arm should initially be formulated as a standard TVQP problem. For the redundant robotic arm, the inverse kinematics equations are as follows:

$$\begin{array}{c}\hfill r\left(t\right)=f\left(\phi \right(t\left)\right)\end{array}$$

(45)

where $r\left(t\right)\in {\mathbb{R}}^m$ and $\phi \left(t\right)\in {\mathbb{R}}^n$ represent the end-effector position and joint space vector, respectively. $f(·):{\mathbb{R}}^n\to {\mathbb{R}}^m$ represents the nonlinear mapping relationship between $r\left(t\right)$ and $\phi \left(t\right)$. Because $f(·)$ is a differentiable nonlinear function, it is challenging to find an analytical solution. Therefore, we linearize Equation (45),

$$\begin{array}{c}\hfill \dot{r}\left(t\right)=\mathcal{J}\left(\phi \left(t\right)\right)\left(\dot{\phi }\left(t\right)\right)\end{array}$$

(46)

where $\mathcal{J}\left(\phi \left(t\right)\right)=\partial r\left(t\right)/\partial \phi \left(t\right)\in {\mathbb{R}}^{m\times n}$ denotes the Jacobi matrix. $\dot{r}\left(t\right)$ and $\dot{\phi }\left(t\right)$ are the derivatives of $r\left(t\right)\in {\mathbb{R}}^m$ and $\phi \left(t\right)\in {\mathbb{R}}^n$, respectively.

As delineated in reference [26], the optimal tracking problem for robotic arm motion can be expressed as the following TVQP problem:

$$\min .\frac{1}{2}{\dot{\phi }}^T\left(t\right)\dot{\phi }\left(t\right)+q^T\left(t\right)\phi \left(t\right)$$

(47)

$$\mathrm{s}.\mathrm{t}.\mathcal{J}\left(\phi \left(t\right)\right)\dot{\phi }\left(t\right)=\dot{r}\left(t\right)$$

(48)

where $q\left(t\right)=0$. The robotic arm tracking problem can be equated to a general TVQP problem, and solving problem (47) can be equivalent to solving the problem (1). Therefore, problem (47) can be solved using the EAF-ZNN model (13).

The simulation shown in Figure 14a depicts the expected path and actual trajectory, and Figure 14b shows the motion trajectory of the robotic arm. From Figure 14a,b, it can be seen that the effector task is accomplished very well and the expected path is basically the same. Figure 14c,d show the end-effector states of the robot manipulator and joint angles of the end-effector, respectively. In Figure 14c, the robotic arm returns to its initial state. In Figure 14d, it can be seen that all the joint-angle curves are relatively smooth, implying that the robotic arm can track the task smoothly. From Figure 15, it can be seen that the position error between the end effector of the EAF-ZNN model (13) is less than $6\times {10}^{-10}$, which is much smaller than that of the other three models. For convenience of comparison, we give the integral absolute error (IAE) [40] of these four models in

Table 2. The integral absolute errors (IAES) of EAF-ZNN, VP-CDNN, SVGZNN, and AFT-ZNN for leaf-path tracking.

IAE	EAF-ZNN ($\times {10}^{-10}$)	VP-CDNN ($\times {10}^{-8}$)	SVGZNN ($\times {10}^{-7}$)	AFT-ZNN ($\times {10}^{-8}$)
${\int }_0^t\left|{\epsilon }_x\right|dt$ (m·s)	2.600	4.968	1.490	3.835
${\int }_0^t\left|{\epsilon }_y\right|dt$ (m·s)	1.641	3.741	1.627	6.974
${\int }_0^t\left|{\epsilon }_z\right|dt$ (m·s)	0.4577	2.199	1.685	4.773

. It was further verified that the EAF-ZNN model (13) has a smaller position error in the leaf-path-tracking task. In conclusion, these experimental results confirm the validity and precision of the EAF-ZNN model (13).

5.2. Physical Experiment

To substantiate the utility of the EAF-ZNN model (13), a leaf-path -racking experiment was conducted using the UR3 robotic arm. The proficient completion of the robotic manipulator arm task was captured in the snapshot obtained from Video S1 (Supplementary Material, including the video of the physical experiment, has been uploaded and is available for further reference), as shown in Figure 16. The video and figure illustrate the smooth and stable operation of the robotic arm throughout the entire execution process, resulting in the successful fulfillment of the assigned task. These empirical findings underscore the validity and practicality of the proposed EAF-ZNN model (13) in real-world scenarios.

6. Discussion

The results obtained in this study can be explained by the innovative design of the EAF-ZNN model (13). This model dynamically adjusts its parameters using a sophisticated feedback mechanism that incorporates the current convergence error, historical cumulative convergence error, the rate of change of the convergence error, and the model gain parameter. This feedback regulation mechanism ensures that the model maintains high convergence rates and accuracy, without the need for excessively large parameter values, over time. The adaptive nature of EAF-ZNN allows it to effectively handle TVQP problems by continuously fine tuning its response based on real-time error metrics.

Compared with the existing ZNN models, our proposed model has the following characteristics:

(1)

Variable-gain ZNNs like the VP-CDNN model [26] use time-varying gain for faster convergence, but the gain can increase to impractical levels over time. Different from time-varying ZNNs, our proposed EAF-ZNN model (13) addresses this by ensuring adaptive acceleration, enhancing early convergence while keeping the gain within a manageable range.

(2)

Adaptive fuzzy ZNN (i.e, the AFT-ZNN model [28]) and the EAF-ZNN model (13) adjust the convergence rate through an adaptive parameter. Different from adaptive fuzzy ZNN, the EAF-ZNN model (13) considers both the historical cumulative convergence error and the rate of change of the convergence error. The proposed method not only eliminates the steady-state error but also suppresses the rapidly changing error, thus improving the convergence rate and accuracy.

Experimentally, it was observed that increasing the values of ${\mathcal{K}}_p$, ${\mathcal{K}}_i$, ${\mathcal{K}}_d$, and $\mu$ within a limited range led to faster convergence rate of the EAF-ZNN model (13). Additionally, among the four activation functions evaluated (LAF (14), PASF (15), SBPAF (16), and MSBPAF (17)), MSBPAF (17) demonstrated the fastest convergence rate, which proves the correctness of Theorems 1–5.

A comparative evaluation with three existing ZNN models (VP-CDNN, SVGZNN, and AFT-ZNN) confirmed that the EAF-ZNN model (13) exhibits superior convergence rates. And, the computation time has certain advantages. Particularly, in the redundant robotic arm example, EAF-ZNN achieved a significantly smaller position error, demonstrating its practical efficacy and precision in real-world applications.

Despite these promising results, this study has several limitations. One notable limitation is the potential for overshoot due to the inclusion of an integral term in the feedback mechanism. This could lead to temporary instability or slower settling times in certain scenarios. Additionally, this study primarily focused on the convergence rate and did not extensively address the robustness of the model in the presence of external disturbances or noise. While the simulation experiments provided a comprehensive analysis of convergence properties, real-world applications might present unforeseen challenges that were not covered in this study.

The primary disadvantage of the EAF-ZNN model (13) is the potential for overshoot introduced by the integral component of the feedback mechanism. This issue can be mitigated in future research by refining the PID control parameters or introducing advanced control strategies that can predict and counteract the overshoot tendency. Moreover, the model’s performance under various disturbance conditions needs further exploration. Future studies should include robustness tests to assess how well the EAF-ZNN model (13) can handle external perturbations and maintain stable performance.

Future developments of this research could focus on several areas. First, enhancing the robustness of the EAF-ZNN model (13) to external disturbances is crucial, particularly for applications in dynamic and unpredictable environments. This may involve integrating more sophisticated control algorithms or adaptive mechanisms that can learn and adapt to varying disturbance patterns. Additionally, exploring the application of the EAF-ZNN model (13) to other complex, real-time optimization problems beyond the scope of TVQP could yield valuable insights and broader applicability.

Potential difficulties in future research include mathematical challenges in deriving and proving the stability and robustness of the enhanced model under diverse conditions. Experimentally, implementing the improved model in real-world systems, such as robotic arms or for pattern recognition, might encounter practical issues related to sensor accuracy, computational limitations, and real-time processing requirements. Addressing these challenges will require a multidisciplinary approach, combining theoretical advancements with practical engineering solutions.

7. Conclusions

This paper introduced an innovative approach, the error-based adaptive feedback ZNN (EAF-ZNN) model, for solving TVQP problems. The EAF-ZNN model regulates the convergence rate by injecting an error-based adaptive feedback parameter that combines the current convergence error, the historical cumulative convergence error, the change rate of the convergence error, and the model parameter gain. However, there may be overshoot due to the introduction of an integral term. In addition, the convergence of the algorithm under four different activation functions was proved using five detailed theorems. The simulation experiments not only explored the effects of different ${\mathcal{K}}_p$, ${\mathcal{K}}_i$, and ${\mathcal{K}}_d$ values on the convergence rate: we also verified that the EAF-ZNN model outperforms the VP-CDNN, SVGZNN, and AFT-ZNN models in terms of convergence rate through a series of comparison experiments. These results confirm that the EAF-ZNN model has stronger performance in solving TVQP problems. Finally, the EAF-ZNN model was applied to a redundant robotic arm, highlighting its effectiveness and accuracy in real-world applications. Given its adaptive advantages, our upcoming focus is exploring the robustness of the EAF-ZNN model in the face of disturbances, especially in the design of redundant robotic arms and controllers.