Event-Triggered Neural Sliding Mode Guaranteed Performance Control

Abstract

To solve the trajectory tracking control problem for a class of nonlinear systems with time-varying parameter uncertainties and unknown control directions, this paper proposed a neural sliding mode control strategy with prescribed performance against event-triggered disturbance. First, an enhanced finite-time prescribed performance function and a compensation term containing the Hyperbolic Tangent function are introduced to design a non-singular fast terminal sliding mode (NFTSM) surface to eliminate the singularity in the terminal sliding mode control and speed up the convergence in the balanced unit-loop neighborhood. This sliding surface guarantees arbitrarily small overshoot and fast convergence speed even when triggering mistakes. Meanwhile, we utilize the Nussbaum gain function to solve the problem of unknown control directions and unknown time-varying parameters and design a self-recurrent wavelet neural network (SRWNN) to handle the uncertainty terms in the system. In addition, we use a non-periodic relative threshold event-triggered mechanism to design a new trajectory tracking control law so that the conventional time-triggered mechanism has overcome a significant resource consumption problem. Finally, we proved that all the closed-loop signals are eventually uniformly bounded according to the stability analysis theory, and the Zeno phenomenon can be eliminated. The method in this paper has a better tracking effect and faster response and can obtain better control performance with lower control energy than the traditional NFTSM method, which is verified in inverted pendulum and ball and plate system.

1. Introduction

In most physical systems, some uncertainties may reduce the tracking accuracy and even lead to system instability, which has gained much attention from high-performance control research in recent years. It is well known that sliding mode control (SMC) techniques [1,2,3,4,5] are widely investigated in various nonlinear systems as they have some advantages, including robustness to parameter uncertainties and external disturbances, fast convergence, and desired signal tracking accuracy [6]. However, the conventional SMC uses a linear hyperplane, and the system is asymptotically converged to the equilibrium point considering the stability premise. However, the jitter phenomenon is the essential weakness of sliding mode control, and for this reason various jitter-reducing sliding mode controls are applied, such as integral terminal sliding mode [1,4], adaptive dynamic sliding mode [6], and non-singular terminal sliding mode [3]. To guarantee the convergence of the state variables to the equilibrium point in finite-time, a terminal sliding mode surface (TSM) is proposed in [7]. However, TSM suffers from both singularity problem and slow convergence. For this reason, a non-singular fast terminal sliding mode (NFTSM) control method is proposed in [8], which enables the tracking error to converge to zero quickly in finite-time. Although these control strategies achieve a gradual improvement in the temporary steady-state performance of the system’s position tracking control system, there is still room for further improvement of the system’s control performance in different aspects and to different degrees. Moreover, most of the controllers mentioned above [7,8] rely excessively on a priori known uncertainty of upper bounds in the design, which is usually difficult for obtaining information about an exact model’s uncertainties and external disturbances. When the uncertainties of the system are mainly from vital parameter uncertainties, the control performance of disturbances compensation methods is often poorer than that of non-linear adaptive control with strong learnability for parameter uncertainties [9]. Therefore, combining the adaptive control method of parameter uncertainty with disturbance compensation control is beneficial to obtain a better control effect.

The above studies based on sliding mode control [1,2,3,4,5,6,7,8] need to ensure not only that the control coefficients are known but also that the control direction is known. The direction of the system’s motion (also called the control direction) is determined by the sign in front of the control gain, which has an essential effect on the controller design. For this purpose, the Nussbaum gain technique [10,11,12] is utilized to solve such problems. The adaptive control of uncertain nonlinear systems with unknown control directions has been studied in [13], assuming that the unknown control gains and the uncertain parameters are constants. Still, the more challenging cases of time-varying unknown gains, time-varying uncertain parameters, and time-varying unknown disturbances have been rarely covered.

Prescribed performance control (PPC) techniques are widely used in nonlinear system research to simultaneously achieve the design requirements of stability, transient, and steady-state performance [14,15]. However, the tracking error of conventional PPC evolves strictly within a predefined envelope bounded by an exponentially decaying performance function [16], the tracking error converges only at an exponential rate, and the dynamic process is theoretically asymptotically convergent. Therefore, Li et al. [17] proposed a finite-time performance function (FTPF) to ensure that the tracking error converges within a specified time. Yet, this method is dependent on the initial conditions of the system. Hence, Liu et al. [18] developed a new FTPF with convergence time independent of initial conditions and design parameters was constructed, and the tracking error converged faster under the same parameters than Li et al. [17]. The above FTPF control studies [14,15,16,17,18] are executed in a time-triggered manner (i.e., control and update are periodic), which will generate a tremendous waste of resources, especially when a sufficiently small period is considered. Therefore, it is imperative to study an advanced control strategy that can not only balance transient and steady-state performance, but also save resources.

Event-triggered control (ETC), as a typical acyclic control method, has been widely studied for its ability to reduce the communication bandwidth between the controller and actuators, save communication resources, and ensure control effects [19,20,21]. Unlike periodic sampling, ETC sampling or execution is triggered by a specific “event”, resulting in information transfer and control updates only when requested by the system. Communication and computing resources are more effectively saved with the ETC control strategy. Through our literature survey, there is little research on how to deal with the event-triggered FTPF control problems of nonlinear systems with time-varying unknown gains, time-varying uncertain parameters, and time-varying unknown disturbance conditions. Coping with the time-varying unknown gains and time-varying uncertain parameters, ensuring that the tracking errors has been evolving within the prescribed boundary, and overcoming the Zeno phenomenon in trigger sampling are exceptionally challenging.

In addition, neural network-based perturbation compensation observers have been widely used in studying system modeling errors, parameter uncertainties, and external perturbations [2,7,10,11,18,21,22]. To ensure the accuracy of unknown perturbations estimation, the radial basis function (RBF) neural network used in the literature [2,18,21,22] requires a large selection of parameters such as the center and width of the basis function, and the gradient descent (GD) method used to train the network increases the complexity of online computation. For this reason, Zhang et al. [23] proposed a wavelet cerebellar neural network observer to compensate for system modeling errors, parameter uncertainties, and external perturbations. Zamfirache et al. [24] used policy iteration (PI) and grey wolf optimizer (GWO), and Roman et al. [25] used iterative feedback adjustment (IFT) algorithm to improve the training speed of NNs. However, the literature [23,24,25] suffers from high design complexity and computationally heavy problems, for which giving an accurate interference estimation method can further improve the application in engineering.

Motivated by the aforementioned discussions, the motivation for writing this paper can be described as follows: for a class of nonlinear systems with unknown dynamics and perturbations and unknown control directions, a control algorithm combining the neural slip pattern of finite-time performance functions with an event-triggered strategy is designed under a small number of assumptions using FTPF, Nussbaum gain technique, ETC, and SRWNN, to ensure Lyapunov stability, but still enhance the control performance to save resources. The main contributions are as follows:

(1)

To tackle the drawback that the tracking error of the conventional prescribed performance can only converge at an exponential rate, this paper designs a FTPF derivation scheme in which the convergence time is independent of the initial value of the system and the controller parameters, which ensures that the given FTPF limits the tracking error of the system.

(2)

We combine FTPF and tanh function to design a novel NFTSM sliding surface, which not only avoids the singularity problem, but also can adjust the convergence law index reasonably for the sliding surface and the transition error according to the distance between the sliding surface, the system state, and the equilibrium point. The result is a fast convergence dynamics model in the unit ring neighborhood of the equilibrium. The conventional SMC-based adaptive method cannot solve the dual problems of unknown control directions and time-varying unknown gains with uncertain time-varying parameters.

(3)

The traditional adaptive sliding mode control method is poor in solving the problems of unknown control directions, time-varying unknown gains, and time-varying uncertain parameters. For this reason, based on the improved NFTSM surface, we design a unified event-triggered control law by introducing the Nussbaum gain function and the ETC method, which not only saves communication resources but also ensures the boundedness of the uncertain time-varying parameters.

(4)

For the case where the random time-varying disturbances and nonlinear terms in the system are entirely unknown, it is assumed to be bounded, and then a self-recurrent wavelet neural network (SRWNN) is introduced for compensation, while the use of basis functions and memory functions ensures that the SRWNN has fast learning ability and better generalization ability to compound disturbances. The boundedness is not considered in the design of the controller, but the errors generated during the design process are compensated online using an adaptive law. Unlike Zamfirache et al. [24] and Roman et al. [25], here we use the minimum-learning-parameter (MLP) technique to update the parameters of the network in conjunction with the ideas of Shao et al. [15], which can significantly reduce the computational burden. Our proposed method improves several drawbacks of designing controllers in [22,26]. Finally, we applied the scheme to an inverted pendulum and ball and plate system model for validation.

2. Problem Formulation and Necessary Preliminaries

Assumption 1.

The lumped disturbance$L\left(x,\dot{x},t\right)$is continuously differentiable and has$\left|L\left(x,\dot{x},t\right)\right|$≤ ∆ for its Lipschitz constant ∆, where ∆ is an upper bound on this disturbance.

Assumption 2.

For Taylor higher-order terms and uncertainty δ is bounded and there exists |δ| ≤$\overline{\delta }$, where$\overline{\delta }$is a positive constant.

2.1. System Model

Consider the following nonlinear dynamical system:

$$\ddot{x}=F\left(x,\dot{x}\right)+G(x,\dot{x})u+d(x,\dot{x},t)$$

(1)

where x is n × 1 vector of state variables, u is m × 1 inputs vector, G is n × m matrix of inputs coefficients, F is n × 1 vector of system dynamics, and d is n × 1 vector of external disturbances. In our dynamical system, the system is described as fully driven if n = m and under-driven if m < n. To highlight the existence of uncertainties, the above system vectors $F\left(x,\dot{x}\right)$ and matrices $G(x,\dot{x})$ modified as follows:

$$\left\{\begin{array}{l}F\left(x,\dot{x}\right)=F_n\left(x,\dot{x}\right)+\Delta F\left(x,\dot{x}\right),\\ G(x,\dot{x})=G_n\left(x,\dot{x}\right)+\Delta G\left(x,\dot{x}\right),\end{array}\right.$$

(2)

where $F_n\left(x,\dot{x}\right)$ and $G_n\left(x,\dot{x}\right)$ are the nominal vector and input matrix, respectively; $\Delta F\left(x,\dot{x}\right)$ and $\Delta G\left(x,\dot{x}\right)$ are the uncertainty vector and matrix. We describe the total uncertainty $L\left(x,\dot{x},t\right)$ by Equation (3), and the final dynamical system obtained is shown in Equation (4).

$$L\left(x,\dot{x},t\right)=\Delta F\left(x,\dot{x}\right)+\Delta G\left(x,\dot{x}\right)+d\left(x,\dot{x},t\right)$$

(3)

$$\ddot{x}=F_n\left(x,\dot{x}\right)+G_n\left(x,\dot{x}\right)u+L\left(x,\dot{x},t\right)$$

(4)

where $G_n\left(x,\dot{x}\right)$ is a time-varying gain matrix of unknown size and positive and negative sign, which poses a significant challenge for control design.

In this paper, we focus on the unknown time-varying gain and unknown control direction problems associated with $G_n\left(x,\dot{x}\right)$, while considering the effect of unknown disturbance $L\left(x,\dot{x},t\right)$, making the application scenario of the system model of Equation (4) more general.

2.2. Novel FTPF

To achieve the prescribed performance, the error is defined as $e=x-x_d$. We introduce the smooth performance function $\rho (t):{\mathit{R}}^{+}\to {\mathit{R}}^{+}$ as:

$$\rho (t)=\left\{\begin{array}{l}{\left[{\rho }_0^l+\frac{h}{\varsigma }(1-e^{l\varsigma t})\right]}^{\frac{1}{l}}+{\rho }_{T^{\prime }} 0\le t<T\hfill \\ {\rho }_{T^{\prime }}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {t\ge T}^{}\hfill \end{array}\right.$$

(5)

where $\varsigma ,h>0$, 0 < l ≤ 1/n, is the parameter controlling the rate of convergence and n is the system order; $T=1/l\varsigma \cdot \ln (1+{\rho }_0^l/h)$ is the setup time. The following conditions are also satisfied by $\rho (t)$:

(1)

$\rho (t)$ is positive and monotonically decreasing, $t\in \left[0\hspace{1em}T\right)$;

(2)

$\underset{t\to T}{\lim }\rho (t)={\rho }_T,\dot{\rho }(t)\le 0;$

(3)

$\rho (t)={\rho }_T,\forall t\ge T$.

where ${\rho }_0$ and ${\rho }_T$ denoted the initial error range and the steady-state error range, respectively, and ${\rho }_0>{\rho }_T>0$; compared with the convergence time $\tilde{T}={\rho }_0^l/hl$ of Li et al. [17], we can obtain $T<\tilde{T}$. According to the derivation of Liu et al. [18], the error e(t) can be limited to $-\rho (t)<e(t)<\rho (t)$.

To obtain a system containing performance constraints, the error transformation ε we used was as follows:

$$\epsilon =\tan \left(\frac{\pi e(t)}{2\rho (t)}\right)$$

(6)

With the above definitions and error transformations, we can satisfy the performance constraint with the appropriate selected parameters, and obtain:

$$\left\{\begin{array}{l}e(t)=\frac{2}{\pi }\rho (t)\arctan (\epsilon ),\\ \dot{e}(t)=\frac{2}{\pi }\dot{\rho }(t)\arctan (\epsilon )+\frac{2}{\pi }\rho (t)\frac{\dot{\epsilon }}{1+{\epsilon }^2},\end{array}\right.$$

(7)

$$\left\{\begin{array}{c}\dot{\epsilon }=\frac{\pi (1+{\epsilon }^2)}{2\rho (t)}\left(\dot{e}(t)-\frac{2}{\pi }\dot{\rho }(t)\arctan (\epsilon )\right),\hfill \\ \ddot{\epsilon }=\frac{\pi (2\epsilon \dot{\epsilon }\rho (t)-\dot{\rho }(t)(1+{\epsilon }^2))}{2{\rho }^2(t)}\left(\dot{e}(t)-\frac{2}{\pi }\dot{\rho }(t)\arctan (\epsilon )\right)+\frac{\pi (1+{\epsilon }^2)}{2\rho (t)}\left(\ddot{e}(t)-\frac{2}{\pi }\ddot{\rho }(t)\arctan (\epsilon )-\frac{2}{\pi }\dot{\rho }(t)\frac{\dot{\epsilon }}{1+{\epsilon }^2}\right),\hfill \end{array}\right.$$

(8)

To facilitate the subsequent calculations, we define the intermediate variables as follows:

$$\begin{array}{l}{Z}_v=\frac{\pi (1+{\epsilon }^2)}{2\rho (t)},\\ Z_f=\frac{\pi (2\epsilon \dot{\epsilon }\rho (t)-\dot{\rho }(t)(1+{\epsilon }^2))}{2{\rho }^2(t)}\left(\dot{e}(t)-\frac{2}{\pi }\dot{\rho }(t)\arctan (\epsilon )\right)-Z_v\left(\frac{2}{\pi }\ddot{\rho }(t)\arctan (\epsilon )+\frac{2}{\pi }\dot{\rho }(t)\frac{\dot{\epsilon }}{1+{\epsilon }^2}\right),\end{array}$$

(9)

2.3. SRWNN Approximation

For the lumped disturbance $L\left(x,\dot{x},t\right)$ of system (4), the SRWNN is used to approximate the estimation. The SRWNN used in this paper has a detailed structure consisting of four layers, as shown in Figure 1. The recurrent structure in SRWNN uses interactive feedback and can obtain critical information from other rules.

The first layer is an input layer, and accepts the variables and transmits this directly to the next layer.

The second layer is the mother wavelet layer, as Equation (10):

$$v_{ij}{}^{(n)}=x_i{}^{(n)}+{\pi }_{ij}{}^{(n)}{\Phi }_{ij}{}^{(n-1)},z_{ij}={\sigma }_{ij}^2{(v_{ij}-m_{ij})}^2$$

(10)

where $m_{ij}$ is the wavelet expansion parameter; ${\sigma }_{ij}$ is the wavelet shifting parameter; ${\pi }_{ij}$ represents regression weights; ${\Phi }_{ij}{}^{(n-1)}$ represents the (n − 1)th output of the mother wavelet layer; the wavelet basis function of the wavelet layer is chosen as ${\Phi }_{ij}=(1-{\sigma }_{ij}^2{v}_{ij}^2)\exp (-z_{ij})$.

The third layer is the product layer, and the nodes in this layer are given by the product of the mother wavelet as follows:

$${\psi }_j=\prod _{i=1}^p(1-{{\sigma }_{ij}^{2}v}_{ij}^2)\exp (-{\sigma }_{ij}^2{(v_{ij}-m_{ij})}^2)$$

(11)

The fourth layer is the output layer. The node output is a linear combination of the results output from third layer, and the output node accepts the input values directly from the input layer. Thus, the production of SRWNN consists of each self-recursive wavelet and parameters as follows:

$$\widehat{y}=\sum _{j=1}^N{w}_j{\psi }_j+\sum _{i=1}^P{a}_i{x}_i$$

(12)

Thus, we can obtain:

$$L(x,\dot{x},t)={\mathit{w}}^{\ast \mathit{T}}\mathit{\psi }+{\mathit{a}}^{\ast \mathit{T}}\mathit{x}+\delta$$

(13)

where ${\mathit{w}}^{\ast }$ is the optimal parameter for w; ${\mathit{a}}^{\ast }$ is the optimal parameter for a; δ is the error of approximation. The optimal parameters cannot actually be obtained, so they are replaced by:

$$L(x,\dot{x},t)={\widehat{\mathit{w}}}^{\mathit{T}}\widehat{\mathit{\psi }}+{\widehat{\mathit{a}}}^{\mathit{T}}\mathit{x}$$

(14)

where $\mathit{a}\in {\mathit{R}}^P$, $\mathit{\sigma }\in {\mathit{R}}^{P\times N}$, $\mathit{\pi }\in {\mathit{R}}^{P\times N}$, $\mathit{m}\in {\mathit{R}}^{P\times N}$, $\mathit{w}\in {\mathit{R}}^P$, $\mathit{\psi }\in {\mathit{R}}^{P\times N}$, $\widehat{\mathit{w}}$ is an estimate of w, and $\hat{\mathit{a}}$ is an estimate of a.

2.4. Necessary Preliminaries

Definition 1.

Any continuous function is called a Nussbaum function if it is recognized by the following properties:

$$\left\{\begin{array}{l}\underset{\kappa \to \infty }{\lim }\sup \frac{1}{\kappa }{\displaystyle {\int }_0^{\kappa }N(\xi )d}\xi =+\infty ,\\ \underset{\kappa \to \infty }{\lim }\inf \frac{1}{\kappa }{\displaystyle {\int }_0^{\kappa }N(\xi )d}\xi =-\infty ,\end{array}\right.$$

(15)

Nussbaum functions are $\exp ({\xi }^2)\cos {\xi }^2$, ${\xi }^2\cos \xi$, and $({\xi }^2+2)\exp ({\xi }^2/2)\sin \xi +1$, typically.

Lemma 1

([27]). We consider V(•) and ξ(•) to be smooth functions satisfying $\left[0, t_f\right)$ and V(t) ≥ 0 as follows:

$$V(t)\le b/c+(V(0)-b/c)\exp (-ct)+\exp (-ct){\displaystyle {\int }_0^t(-G_{n}N(\xi )+1)\dot{\xi }}\exp (ct)d\tau$$

(16)

If Equation (14) holds, ${\displaystyle {\int }_0^t(-G_{n}N(\xi )+1)\dot{\xi }}d\tau$, V(•) and ξ(•) are in bounded on $[0, t_f)$, where c, b > 0 are constants, and $G_n$ is a bounded time-varying parameter.

Lemma 2

(Young’s inequality [28]). For ∀(x, y) ∈ R, if the real numbers r, c, d satisfy h > 0, c > 1, d > 1, (c − 1)(d − 1) = 1, then we have:

$$xy\le \frac{h^c}{c}|x{|}^c+\frac{1}{d{h}^d}|y{|}^d$$

(17)

Lemma 3

([29]). For ∀$\varsigma >0$ and x ∈ R, then

$$0\le |x|-x\tanh (x/\varsigma )\le \kappa \varsigma$$

(18)

where $\kappa$ satisfies $\kappa \exp (\kappa +1)=1$, and $\kappa \approx 0.2785$.

Lemma 4

([30]). For ∀$\delta \in \mathit{R}$ , the following inequality is established:

$$\tilde{\delta }\widehat{\delta }\le -0.5{\tilde{\delta }}^2+0.5{\delta }^2$$

(19)

where $\tilde{\delta }=\delta -\widehat{\delta }$.

3. Main Results

3.1. Controller Design

The following form of NFTSM surface s is designed as:

$$s=\epsilon +k_1{\mathrm{sig}}^{\gamma }{\epsilon }_{}+k_2{\mathrm{sig}}^{\beta }{\dot{\epsilon }}_{}+k_3|\epsilon {|}^m\tanh (k_4{\epsilon }^n)$$

(20)

where $1<\beta <2$, $\beta <\gamma$, ${\mathrm{sig}}^{\gamma }\epsilon =\ln (1+\gamma |\epsilon |)\mathrm{sign}(\epsilon )$, ${\mathrm{sig}}^{\beta }\dot{\epsilon }=|\dot{\epsilon }{|}^{\beta }\mathrm{sign}(\dot{\epsilon })$; 0 < m < 1, 0 < n < 1 and 1 < m + n; $k_1,k_2,k_3,k_4>0$; $\ln (\cdot )$ is the natural logarithm function; $\tanh (\cdot )$ is the hyperbolic tangent function, which satisfies $\tanh (x)=(e^x-e^{-x})/(e^x+e^{-x})$.

Remark 1.

Unlike the NFTSM surface in [31], ours is Lipshitz continuous holistically. There is no need to switch the NFTSM to a linear sliding mode surface to eliminate the singularities of the TSM controller when the generalized position state is close to equilibrium. Furthermore, when we build the non-singular terminal sliding mode surface (NFTSM), the controlled states converge quickly to the small neighborhood containing the equilibrium, which means that most of the dynamics of the classical terminal sliding mode (TSM) is preserved. As the system state moves away from the equilibrium point on the NFTSM surface (18),${\mathrm{sig}}^{\gamma }\epsilon$plays a dominant role in ensuring a faster convergence rate. As the system state approaches the equilibrium point,$|\epsilon {|}^m\tanh (k_4{\epsilon }^n)$allows the system tracking error to converge rapidly to zero. Thus, our proposed NFTSM surface provides for rapid convergence of the global error variables to zero.

Making $f(\epsilon )=k_3|\epsilon {|}^m\tanh (k_4{\epsilon }^n)$, then we have:

$$\dot{f}(\epsilon )=k_{3}m{\epsilon }^{m-1}\tanh (k_4{\epsilon }^n){\dot{\epsilon }}_{}+k_3{k}_{4}n{|\epsilon |}^{m+n-1}(1-{\tanh }^2(k_4{\epsilon }^n)){\dot{\epsilon }}_{}$$

(21)

Thus, the derivative for s is:

$${\dot{s}}_{}={\dot{\epsilon }}_{}+\gamma k_1{\dot{\epsilon }}_{}/(\gamma |\epsilon |+1)+\beta k_2{|{\dot{\epsilon }}_{}|}^{\beta -1}{Z}_v(F_n+G_n^{u+L})+\beta k_2{|{\dot{\epsilon }}_{}|}^{\beta -1}(Z_f-Z_v{\ddot{x}}_d)+\dot{f}(\epsilon )$$

(22)

Definitions:

$$\begin{array}{c}{F}_1={\dot{\epsilon }}_{}+\gamma k_1{\dot{\epsilon }}_{}/(\gamma |\epsilon |+1)+F_2(Z_f-Z_v{\ddot{x}}_d)+\dot{f}(\epsilon )\hfill \\ F_2=\beta k_2{|{\dot{\epsilon }}_{}|}^{\beta -1}\hfill \end{array}$$

(23)

Then Equation (20) is modified as:

$${\dot{s}}_{}=F_1+F_2{Z}_v(F_n+G_n^u+{\mathit{w}}^{\ast T}\mathit{\psi }+{\mathit{a}}^{\ast T}\mathit{x}+\delta )$$

(24)

The error bound δ is assumed to be a constant in the observation process. In the absence of a prophet about uncertainty and disturbances, an adaptive law is proposed by us to adjust the unknown upper bound $\overline{\delta }$ on δ. Assume $\overline{\delta }$ as follows:

$$\overline{\delta }=(b_1+b_2|\epsilon |+b_3|\dot{\epsilon }|+h)\tanh (\frac{S{F}_2{Z}_v}{\lambda })$$

(25)

where $b_1,b_2,b_3>0$ is the parameter to be adjusted, and $h,\lambda >0$ is a very small constant. The estimate of $\overline{\delta }$ can be obtained as follows:

$$\widehat{\delta }=({\widehat{b}}_1+{\widehat{b}}_2|\epsilon |+{\widehat{b}}_3|\dot{\epsilon }|+h)\tanh (\frac{s{F}_2{z}_v}{\lambda })$$

(26)

where ${\widehat{b}}_1,{\widehat{b}}_2\mathrm{and}{\widehat{b}}_3$ is the estimate of the $b_1,b_2$ and $b_3$.

Before designing the Nussbaum gain SRWNN sliding mode control law for the event-triggered mechanism, event-triggered control (ETC) is introduced as follows:

$$\begin{array}{c}v(t)=-(1+\tau )\left(\begin{array}{l}\mathsf{\Theta }\tanh \frac{F_2{Z}_{v}S\mathsf{\Theta }}{\lambda }+{\mu }_1\tanh \frac{S{F}_2{Z}_v{\mu }_1}{\lambda }+\\ \frac{F_1+F_2{Z}_v{F}_{n+{\mu }_{3}S}}{F_2{Z}_v}\end{array}\right),\\ u_T(t)=v(t_k),\forall t\in \left[t_k,t_{k+1}\right),\\ t_{k+1}=\inf \left\{t\in \mathit{R}\left|\right|P|\ge \tau |u_T|+{\mu }_2\right\},\end{array}$$

(27)

where P(t) = v(t) − u_T(t) is the event trigger error, 0 < τ < 1, ${\mu }_1>0$, ${\mu }_2>0$, and ${\mu }_1={\mu }_2/(1-\tau )$; t_k indicates the input update time. For time t ∈ [t_k, t_{k + 1}), $u_T$ treats it as a constant v(t_k). $u_T$ is the output signal generated by the relative threshold event trigger mechanism, which will hold the last updated value until the sampling error P violates the predefined relative threshold event trigger condition. When Equation (25) is activated, this moment is t_{k + 1}, and the control signal $u_T(t_{k+1})$ can be used in our system. Therefore, we can find the parameters satisfying ${\ell }_1$ and ${\ell }_2$ that Equation (26) holds.

$$v=(1+{\ell }_1\tau )u_T+{\ell }_2{\mu }_2$$

(28)

where ${\ell }_1\le 1$, ${\ell }_2\le 1$.

Remark 2.

In the traditional adaptive control framework, the output of the controller acts on the system at all times, regardless of whether the system needs to be tuned. This can be inappropriate when communication resources are limited. Therefore, ETC can reduce the number of data transmission and save communication resources in a better way.

Thus, the adaptive control law can be designed as follows:

$$\begin{array}{c}\mathsf{\Xi }=\frac{F_2{Z}_{v}S\widehat{w}\Vert \psi {\Vert }^2}{2a_w^2}+\frac{F_2{Z}_{v}S\widehat{a}\Vert x{\Vert }^2}{2a_a^2}+{\mu }_4{F}_2{Z}_{v}S+\frac{1}{2}({\widehat{b}}_1+{\widehat{b}}_1|\epsilon |+{\widehat{b}}_1|\dot{\epsilon }|+h)\tanh \frac{F_2{Z}_{v}S}{\lambda },\hfill \\ \mathsf{\Theta }=\mathsf{\Xi }\tanh \frac{F_2{Z}_{v}S\mathsf{\Xi }}{\lambda },\hfill \\ u=N(\xi )u_T,\hspace{1em}\dot{\xi }=-F_2{Z}_{v}S{u}_T,\hspace{1em}{u}_T=\frac{v-{\mathcal{l}}_2{\mu }_2}{1+{\mathcal{l}}_1\tau },\hfill \end{array}$$

(29)

Our parameter adaptive law is:

$$\begin{array}{l}\dot{\widehat{w}}={\eta }_1(\frac{{F_2}^2{Z_v}^2{S}^2\Vert \psi {\Vert }^2}{2a_w^2}-k_5\widehat{w}),\\ \dot{\widehat{a}}={\eta }_2(\frac{{F_2}^2{Z_v}^2{S}^2\Vert x{\Vert }^2}{2a_a^2}-k_6\widehat{a}),\\ {\dot{\widehat{b}}}_1={\eta }_3(\frac{1}{2}{F}_2{Z}_{v}S\tanh \frac{F_2{Z}_{v}S}{\lambda }-k_7{\widehat{b}}_1),\\ {\dot{\widehat{b}}}_2={\eta }_4(\frac{1}{2}{F}_2{Z}_{v}S|\epsilon |\tanh \frac{F_2{Z}_{v}S}{\lambda }-k_8{\widehat{b}}_2),\\ {\dot{\widehat{b}}}_3={\eta }_5(\frac{1}{2}{F}_2{Z}_{v}S|\dot{\epsilon }|\tanh \frac{F_2{Z}_{v}S}{\lambda }-k_9{\widehat{b}}_3),\end{array}$$

(30)

Remark 3.

Each element of w and a needs to be updated adaptively according to the universal approximation theorem to achieve high accuracy recognition. The number of nodes needs to be large enough, which leads to the problem of learning explosion and introduces a heavy computational load. To avoid the learning explosion problem, the learning dimension is significantly reduced by updating the norm$\Vert {\mathit{w}}^{\ast }\Vert$ and $\Vert {\mathit{a}}^{\ast }\Vert$ online with the help of the minimum learning parameter (MLP) technique. Moreover, SRWNN shows a better ability to handle uncertainty and analyze local details than traditional RBF in terms of basis function usage and memory function, improving recognition accuracy.

3.2. Stability Analysis

Theorem 1.

For the time-varying unknown gains, time-varying uncertain parameters, and time-varying unknown perturbations in nonlinear systems, with Equations (1) and (4) satisfying Assumptions 1 and 2, we construct the fixed-time performance function (5) by FTPF. For the NFTSM sliding mode surface of Equation (20), the ETC of Equation (27) and the control law of Equation (29), the design process of the controller of Equations (24)–(28), and the adaptive law of Equation (30) are taken, the tracking error evolves strictly within the predesignated FTPF range, and the signal$({\widehat{b}}_i-b_i)$,$\tilde{\mathit{w}}$,$\tilde{\mathit{a}}$and$\tilde{\delta }$are eventually uniformly bounded, and the Zeno phenomenon does not occur.

Proof. 

Firstly, proving the stability of the system by taking the Lyapunov function as follows:

$$V=\frac{1}{2}{s}^2+\frac{1}{2{\eta }_1}({\tilde{\mathit{w}}}^T{\tilde{\mathit{w}}}^{})+\frac{1}{2{\eta }_2}tr({\tilde{\mathit{a}}}^T{\tilde{\mathit{a}}}^{})+\sum _{i=1}^3\frac{1}{2{\eta }_{2+i}}{({\widehat{b}}_i-b_i)}^2$$

(31)

The derivative of Equation (27) is given:

$$\begin{array}{cc}\dot{V}& =S{F}_1+S{F}_2{Z}_v{F}_n+(-G_{n}N(\xi )+1)\dot{\xi }-\frac{S{F}_2{Z}_v{\ell }_2{\mu }_2}{1+{\ell }_1\tau }-\frac{1+\tau }{1+{\ell }_1\tau }(S{F}_1+S{F}_2{Z}_v{F}_n+{\mu }_3{S}^2)+S{F}_2{Z}_v{\mathit{w}}^{\ast T}\mathit{\psi }-\hfill \\ & \frac{1+\tau }{1+{\ell }_1\tau }(S{F}_2{Z}_v{\mu }_1\tanh \frac{S{F}_2{Z}_v{\mu }_1}{\lambda })+S{F}_2{Z}_v{\mathit{a}}^{\ast T}\mathit{x}-\frac{1+\tau }{1+{\ell }_1\tau }(S{F}_2{Z}_v\mathsf{\Theta }\tanh \frac{S{F}_2{Z}_v\mathsf{\Theta }}{\lambda })+S{F}_2{Z}_v\delta -\hfill \\ & \frac{1}{{\eta }_1}\tilde{w}\dot{\widehat{w}}-\frac{1}{{\eta }_2}\tilde{a}\dot{\widehat{a}}+{\displaystyle \sum _{i=1}^3}\frac{1}{{\eta }_{2+i}}({\widehat{b}}_i-b_i){\dot{\widehat{b}}}_i\hfill \end{array}$$

(32)

Since $0<1+{\ell }_1\tau <1+\tau$, $-\frac{S{F}_2{Z}_v{\mu }_2}{1-{\ell }_1\tau }\le \left|\frac{S{F}_2{Z}_v{\mu }_2}{1-\tau }\right|\le S{F}_2{Z}_v{\mu }_1$, applying the Young’s inequality yields:

$$\begin{array}{l}S{F}_2{Z}_v{\mathit{w}}^{\ast T}\mathit{\psi }-\frac{{F_2}^2{Z_v}^2{S}^2\widehat{w}\Vert \psi {\Vert }^2}{2a_w^2}\le \frac{a_w^2}{2},\\ S{F}_2{Z}_v{\mathit{a}}^{\ast T}\mathit{x}-\frac{{F_2}^2{Z_v}^2{S}^2\widehat{a}\Vert x{\Vert }^2}{2a_a^2}\le \frac{a_a^2}{2},\\ \frac{1}{2}S{F}_2{Z}_v\delta \le {\mu }_4{F_2}^2{Z_v}^2{S}^2+\frac{1}{16{\mu }_4}{\overline{\delta }}^2,\end{array}$$

(33)

Substituting Equation (31) into Equation (30) results in:

$$\begin{array}{cc}\dot{V}& \le -{\mu }_3{S}^2+(-G_{n}N(\xi )+1)\dot{\xi }+S{F}_2{Z}_v{\mathit{a}}^{\ast T}\mathit{x}-|S{F}_2{Z}_v{\mu }_1|-S{F}_2{Z}_v{\mu }_1\tanh \frac{S{F}_2{Z}_v{\mu }_1}{\lambda }+\hfill \\ & S{F}_2{Z}_v\delta -S{F}_2{Z}_v\mathsf{\Theta }+\kappa \lambda -\frac{1}{{\eta }_1}\tilde{w}\dot{\widehat{w}}-\frac{1}{{\eta }_2}\tilde{a}\dot{\widehat{a}}+{\displaystyle \sum _{i=1}^3}\frac{1}{{\eta }_{2+i}}({\widehat{b}}_i{-b}_i){\dot{\widehat{b}}}_i+S{F}_2{Z}_v{\mathit{w}}^{\ast T}\mathit{\psi }\hfill \\ & \le (-G_{n}N(\xi )+1)\dot{\xi }-({\mu }_3{S}^2)+S{F}_2{Z}_v{\mathit{w}}^{\ast T}\mathit{\psi }+S{F}_2{Z}_v{\mathit{a}}^{\ast T}\mathit{x}-S{F}_2{Z}_v\mathsf{\Xi }+3\kappa \lambda +\hfill \\ & \frac{1}{2}S{F}_2{Z}_v\delta +{\mu }_4{S}^2{F_2}^2{Z_v}^2+\frac{1}{16{\mu }_4}{\overline{\delta }}^2-\frac{1}{{\eta }_1}\tilde{w}\dot{\widehat{w}}-\frac{1}{{\eta }_2}\tilde{a}\dot{\widehat{a}}+{\displaystyle \sum _{i=1}^3}\frac{1}{{\eta }_{2+i}}({\widehat{b}}_i{-b}_i){\dot{\widehat{b}}}_i\hfill \end{array}$$

(34)

According to Lemmas 3 and 4, we obtain:

$$\begin{array}{l}|S{F}_2{Z}_v{\mu }_1|-S{F}_2{Z}_v{\mu }_1\tanh \frac{S{F}_2{Z}_v{\mu }_1}{\lambda }\le \kappa \lambda ,\\ S{F}_2{Z}_v{\mu }_1-S{F}_2{Z}_v{\mu }_1\tanh \frac{S{F}_2{Z}_v{\mu }_1}{\lambda }\le \kappa \lambda ,\\ -S{F}_2{Z}_v\mathsf{\Theta }\tanh \frac{S{F}_2{Z}_v\mathsf{\Theta }}{\lambda }\le -S{F}_2{Z}_v\mathsf{\Theta }+\kappa \lambda ,\\ |S{F}_2{Z}_v|\overline{\delta }-S{F}_2{Z}_v\overline{\delta }\tanh \frac{S{F}_2{Z}_v}{\lambda }\le \overline{\delta }\kappa \lambda ,\\ \tilde{w}\widehat{w}\le -\frac{1}{2}{\tilde{w}}^2+\frac{1}{2}{\widehat{w}}^2,\tilde{a}\widehat{a}\le -\frac{1}{2}{\tilde{a}}^2+\frac{1}{2}{\widehat{a}}^2,\\ ({\widehat{b}}_i-b_i){\widehat{b}}_i\le -\frac{1}{2}({\widehat{b}}_i-b_i{)}^2+\frac{1}{2}{b}_i^2,\end{array}$$

(35)

Substituting Equations (29), (30) and (32) into Equation (34) results in:

$$\begin{array}{ll}\dot{V}& \le -{\mu }_3{S}^2+(-G_{n}N(\xi )+1)\dot{\xi }+S{F}_2{Z}_v{\mathit{a}}^{\ast T}\mathit{x}+\frac{1}{2}S{F}_2{Z}_v\delta -\frac{1}{2}S{F}_2{Z}_v\widehat{\delta }-\\ & \frac{{F_2}^2{Z_v}^2{S}^2\widehat{w}\Vert \psi {\Vert }^2}{2a_w^2}-\frac{{F_2}^2{Z_v}^2{S}^2\widehat{a}\Vert x{\Vert }^2}{2a_a^2}-\frac{1}{{\eta }_1}\tilde{w}\dot{\widehat{w}}+\frac{1}{{16\mu }_4}{\overline{\delta }}^2-\\ & \frac{1}{{\eta }_2}\tilde{a}\dot{\widehat{a}}+{\displaystyle \sum _{i=1}^3}\frac{1}{{\eta }_{2+i}}({\widehat{b}}_i-b_i){\dot{\widehat{b}}}_i+S{F}_2{Z}_v{\mathit{w}}^{\ast T}\mathit{\psi }+3\kappa \lambda \\ & \le -{\mu }_3{S}^2+(-G_{n}N(\xi )+1)\dot{\xi }+S{F}_2{Z}_v{\mathit{w}}^{\ast T}\mathit{\psi }+S{F}_2{Z}_v{\mathit{a}}^{\ast T}\mathit{x}+3\kappa \lambda -\\ & \frac{{F_2}^2{Z_v}^2{S}^2\widehat{w}\Vert \psi {\Vert }^2}{2a_w^2}-\frac{{F_2}^2{Z_v}^2{S}^2\widehat{a}\Vert x{\Vert }^2}{2a_a^2}+{\displaystyle \sum _{i=1}^3}{k}_{6+i}(b_i-{\widehat{b}}_i){\widehat{b}}_i+\\ & \frac{1}{2}S{F}_2{Z}_v\delta -\frac{1}{2}S{F}_2{Z}_v\overline{\delta }+k_5\tilde{w}\widehat{w}+k_6\tilde{a}\widehat{a}\\ & \le -{\mu }_3{S}^2+(-G_{n}N(\xi )+1)\dot{\xi }-\frac{1}{2}{k}_5{\tilde{w}}^2-\frac{1}{2}{k}_6{\tilde{a}}^2-{\displaystyle \sum _{i=1}^3}\frac{k_{6+i}}{2}{({\widehat{b}}_i-b_i)}^2+\\ & \frac{a_w^2}{2}+\frac{a_a^2}{2}+\frac{1}{2}{k}_5{\widehat{w}}^2+\frac{1}{2}{k}_6{\widehat{a}}^2+\frac{1}{{16\mu }_4}{\overline{\delta }}^2+3\kappa \lambda +{\displaystyle \sum _{i=1}^3}\frac{k_{6+i}}{2}{b}_i^2\\ & \le -{\sigma }_{1}V+(-G_{n}N(\xi )+1)\dot{\xi }+{\sigma }_2\end{array}$$

(36)

where ${\sigma }_1=\min \left\{{\mu }_3, \frac{1}{2}{k}_5, \frac{1}{2}{k}_6, \frac{1}{2}{k}_7, \frac{1}{2}{k}_8, \frac{1}{2}{k}_9\right\}$, ${\sigma }_2=\frac{a_w^2}{2}+\frac{a_a^2}{2}+\frac{1}{2}{k}_5{\widehat{w}}^2+\frac{1}{2}{k}_6{\widehat{a}}^2+\frac{1}{{16\mu }_4}{\overline{\delta }}^2+$ $3\kappa \lambda +{\displaystyle \sum _{i=1}^3}\frac{k_{6+i}}{2}{b}_i^2\ge 0$.

Multiplying both sides of Equation (36) by $e^{{t\sigma }_1}$, yields:

$$e^{{t\sigma }_1}V\le -{\sigma }_{1}V{e}^{{t\sigma }_1}+(-G_{\mathrm{n}}N(\xi )+1)\dot{\xi }{e}^{{t\sigma }_1}+{\sigma }_2{e}^{{t\sigma }_1}$$

(37)

Equation (35) can be rewritten as:

$$\frac{d(V{e}^{{t\sigma }_1})}{dt}\le {\sigma }_2{e}^{{t\sigma }_1}+(-G_{n}N(\xi )+1)\dot{\xi }{e}^{{t\sigma }_1}$$

(38)

Integration of Equation (36) above [0, t]:

$$0\le V(t)\le {\sigma }_2/{\sigma }_1+(V(0)-{\sigma }_2/{\sigma }_1)e^{-{\sigma }_{1}t}+e^{-{\sigma }_{1}t}{\int }_0^t(-G_{n}N(\xi )+1)\dot{\xi }{e}^{{\sigma }_1\tau }d\tau$$

(39)

Since G_n is a time-varying parameter belonging to a closed interval, it follows from Lemma 1 that ${\int }_0^t(-G_{n}N(\xi )+1)\dot{\xi }d\tau$, V(t) and ξ(t) are bounded on $[0, t_f)$. Therefore, according to Lyapunov theorem, V(t) is consistent and eventually bounded, then the tracking error converges to zero, and the signals $({\widehat{b}}_i-b_i)$, $\tilde{w}$, $\tilde{a}$ and $\tilde{\delta }$ are all bounded. If the upper bound of $\exp (-{\sigma }_{1}t){\int }_0^t(-G_{n}N(\xi )+1)\dot{\xi }\exp ({\sigma }_{1}t)d\tau$ is D, taking Equation (31) into (39), we obtain:

$$\frac{1}{2}{s}^2\le {\sigma }_2/{\sigma }_1+(V(0)-{\sigma }_2/{\sigma }_1)\exp (-{\sigma }_{1}t)+D$$

(40)

Further, $|s|\le \sqrt{2{\sigma }_2/{\sigma }_1+2(V(0)-{\sigma }_2/{\sigma }_1)\exp (-{\sigma }_{1}t)+2D}$ is obtained, so that s can be made arbitrarily small by choosing the appropriate parameters. The conversion error ε is bounded according to our defining Equation (18), ensuring the tracking error is strictly within a prescribed range.

Now we prove the Zeno behavior. If t_s > 0 such that ∀k ∈ Z+ and t_{k + 1} − t_k ≥ t_s, assuming P(t) = v(t) − u_T(t), ∀t ∈ [t_k, t_{k + 1}) and P(t) ∈ R, then we obtain:

$$\frac{d|P(t)|}{dt}=\frac{d}{dt}|P(t)\cdot P(t){|}^{\frac{1}{2}}=\mathrm{sign}(P(t))\dot{P}(t)\le |\dot{v}(t)|$$

(41)

Suppose that $\dot{v}(t)$ is bounded and there is $|\dot{v}(t)|\le M$. By Equation (25), the condition for P(t_k) = 0 is $\underset{t\to t_{k+1}}{\lim }P(t)=\tau |u_T|+{\mu }_2$. Then, if ∀t ∈ [t_k, t_{k + 1}), a lower bound t_s, the execution time can be obtained satisfying $t_s={\mu }_2/M$. Therefore, our controller avoids the Zeno behavior. □

Remark 4.

A lower bound on the execution time is obtained for all k ∈ Z+ based on our proposed event-triggered control method t_{k + 1} − t_k ≥ ts > 0, which means the Zeno behavior does not occur and ensures the stability of the system (1).

4. Numerical Simulations

To verify the effectiveness and robustness of the proposed adaptive controller against disturbance event triggering, we apply it to the inverted pendulum model, ball, and plate system model, and compare our method with the unmodified NFTSM and the ANFTSM presented by Yao et al. [31].

The ANFTSM controller is designed as:

$$\begin{array}{l}{s}_{}=e+k_1{\mathrm{sig}}^{\gamma }e+k_2{\mathrm{sig}}^{\beta }\dot{e},\\ u=G_n^{-1}\left[-\frac{1}{k_2\beta }|\dot{e}{|}^{2-\beta }\mathrm{sign}(\dot{e})-\frac{k_1\gamma }{k_2\beta }|e{|}^{\gamma -1}|\dot{e}{|}^{2-\beta }\mathrm{sign}(\dot{e})-\right.\left.ks-(\widehat{D}+\eta )\mathrm{sign}(s)-F_n+{\ddot{x}}_d\right],\\ \dot{\widehat{D}}=\mu k_2\beta |\dot{e}{|}^{\beta -1}|s|,\end{array}$$

(42)

In this study, the integral absolute error (TAE), integral time absolute error (ITAE) and control energy factor (CE) are defined as performance evaluation metrics to better evaluate the performance of the controller [32], as follows:

$$\begin{array}{l}IAE={\int }_{t_r}^{t_c}{\displaystyle \sum _{i=1}^n}|x_i-x_{id}|dt,\\ ITAE={\int }_{t_r}^{t_c}{\displaystyle \sum _{i=1}^n}t\times |x_i-x_{id}|dt,\\ CE={\int }_{t_r}^{t_c}{\displaystyle \sum _{i=1}^m}{u}_i^{2}dt,\end{array}$$

(43)

where $t_r$ is the time after the system has fully converged and $t_c$ is the time of the entire control process.

Example 1. The inverted pendulum model of the trolley is as per Wang et al. [33]:

$$\begin{array}{l}{\dot{x}}_1=x_2,\\ {\dot{x}}_2=\frac{6g\sin x_1-3aml{x}_2^2\sin (2x_1)-6a\cos (x_1)u}{8l-6aml{\cos }^2{x}_1}+d,\\ a=\frac{l}{m+M},\end{array}$$

(44)

where u is the control input for the trolley force, taking the inverted pendulum mass $m=0.21kg$, the trolley mass $M=0.45kg$, the inverted pendulum length 2l = 0.65 m, and the gravitational acceleration $g=9.81m/s^2$, d is the disturbance taken as $0.1\sin t$. The desired trajectory is chosen as $y_d=\cos (1.2t)\sin (0.2t)$ and the Nussbaum function is $N(\xi )=({\xi }^2+2)\exp ({\xi }^2/2)\sin \xi +1$.

The simulation process parameters are set in

Table 1. Controller parameters setting.

Controllers	Parameters
Proposed method	$\begin{array}{l}\gamma =5, \beta =3.1/3, k_1=1, k_2=0.05, k_3=0.8, k_4=12.5,\\ m=17/19, n=17/33, {\mathsf{\ell }}_1=0.1, {\mathsf{\ell }}_2=0.1, {\mu }_1=2, \lambda =0.01,\\ {\mu }_2=0.5, {\mu }_3=2.5, {\mu }_4=0.05, a_a=0.1, a_w=0.1, {\eta }_1=0.05, \\ \tau =0.5, {\eta }_2=0.02, {\eta }_i=0.005, k_j=0.1, i=3,\cdots ,5, j=5,\cdots ,9\end{array}$
ANFTSM [31]	$\gamma =3, \beta =5/3, k=100, k_1=80, k_1=0.1, \mu =10, \eta =0.5$
Unmodified NFTSM	The value is the same as the proposed method

with each parameter initially being zero, and the FTPF is set to $l=2/13$, $h/\varsigma =1.3$, T = 0.754, $\varsigma =5$, ${\rho }_0=1.15$, and ${\rho }_T=0.05$. Our simulation results are shown in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 and

Table 2. Controller simulation results.

Controllers	tc/s	IAE/m·s	ITAE/m·s2	CE/N2·s
Proposed method	0.193	$4.98\times {10}^{-4}$	0.0085	$7.61\times {10}^{-4}$
ANFTSM [31]	0.488	$6.67\times {10}^{-2}$	0.8126	$4.62\times {10}^3$
Unmodified NFTSM	0.288	$8\times {10}^{-3}$	0.0519	$9.17\times {10}^{-4}$

.

We analyzed Equation (4) and (44) to obtain Equations (45) and (46):

$$\ddot{x}=F_n\left(x,\dot{x}\right)+G_n\left(x,\dot{x}\right)u+L\left(x,\dot{x},t\right)$$

(45)

$$\left\{\begin{array}{l}{F}_n=\frac{6g\sin x_1-3aml{x}_2^2\sin (2x_1)}{8l-6aml{\cos }^2{x}_1},\\ G_n=\frac{-6aml\cos x_1}{8l-6aml{\cos }^2{x}_1},\\ L=d,\end{array}\right.$$

(46)

This allows us to easily apply Example 1 to the controller designed in Section 3. From the literature [23,24,25] and the comparison method (ANFTSM), it can be seen that the most significant advantage of the method in this paper is that the positive and negative singularity of Gn need not be known in advance. Only the stability index $\dot{V}=-aV+(-G_{n}N(\xi )+1)\dot{\xi }+b$ in Lemma 1 needs to be satisfied, which is achieved by introducing the Nussbaum function to simplify the design of the controller. Another advantage is that we introduce a relative threshold event triggering mechanism, which can save a large amount of communication resources compared to time triggering.

Position tracking errors and driving force are essential manifestations of the algorithm performance. Figure 3 and Figure 4 show the position tracking error and driving force comparison of the three algorithms, respectively. In terms of position tracking error, ANFTSM tracking error is the largest, the maximum tracking error after 0.5 s is around 2 cm, unimproved NFTSM has a noticeable local maximum error around 0.46 s, about 5 mm or so, and the proposed method is below 0.5 mm. For the axis driving force shown in Figure 4, ANFTSM shows continuous fluctuations throughout the operation phase, and the local driving force reaches more than 20 N, which is caused by the time-varying perturbation d that makes the adaptive term coefficient change continuously. The other two algorithms have a large driving force at the initial moment, but the duration is within 0.5 s. Compared with Table 2, it can be seen that compared with other algorithms, the shortest regulation time of this paper’s algorithm is about 0.193 s and the maximum of ANFTSM is about 0.488 s. In addition, the results of TAE, ITAE, and CE are also consistent. The reason for the extensive consequences of the proposed method and unimproved NFTSM is that the addition of the compensation term $|\epsilon {|}^m\tanh (k_4{\epsilon }^n)$ reduces the effect of non-matching disturbances caused by the estimation error of SRWNN while improving the convergence speed and tracking accuracy of the control system.

From the simulation results, it is clear that our method not only has fast response speed and accuracy, but also the controller output is continuous and smooth throughout without jittering. The tracking error is strictly controlled within the prescribed evolution region, while the unimproved NFTSM shows poor results. The ANFTSM method has significant jitter and poor results are obtained. As can be seen from Table 2, the method in this paper has the best metrics and the ANFTSM is the worst. Moreover, ANFTSM requires sufficiently significant gains to achieve better performance as the system is in a steady state. In contrast, ours can be chosen arbitrarily. Figure 5 shows that we sample in steps of 0.001 s, which means that the conventional controller needs to be updated 20,000 times within 20 s, while ours only requires 530 times. This indicates that no Zeno behavior is exhibited under the ETC mechanism. The output of N(ξ) and ξ in Figure 6 shows that N(ξ) and ξ have converged to a stable range at about 0.2 s. Therefore, the proposed ETC-based coordinated control scheme is easy to implement and cost-effective for trajectory tracking control and communication resources, despite the parameter uncertainties and external disturbances.

Example 2.

The model of ball and plate system is as per Zakeri et al. [34]:

$$\begin{array}{l}(m+\frac{I_b}{r_b^2})\ddot{x}-m\left(x{\dot{\alpha }}^2+y\dot{\alpha }\dot{\beta }\right)+mg\sin \alpha =d_1\\ (m+\frac{I_b}{r_b^2})\ddot{y}-m\left(y{\dot{\beta }}^2+x\dot{\alpha }\dot{\beta }\right)+mg\sin \beta =d_2\end{array}$$

(47)

$$\begin{array}{l}\ddot{X}=F_n+G_{n}u+L,\\ X=\left[\begin{array}{c}x\\ y\end{array}\right], F_n=\mathbf{0}, G_{n}u=\frac{mg{r}_b^2}{m{r}_b^2+I_b}\left[\begin{array}{c}\sin \alpha \\ \sin \beta \end{array}\right],\\ L=\frac{m{r}_b^2}{m{r}_b^2+I_b}\left[\begin{array}{c}x{\dot{\alpha }}^2+y\dot{\alpha }\dot{\beta }+d_1\\ y{\dot{\beta }}^2+x\dot{\alpha }\dot{\beta }+d_2\end{array}\right], u=\left[\begin{array}{c}\alpha \\ \beta \end{array}\right],\end{array}$$

(48)

where x, y, α, and β are the displacement of the ball and plate system in the X-direction, the displacement in the Y-direction, the angle of rotation in the X-direction and the angle of rotation in the Y-direction, respectively. The mass of the sphere $m=0.263 \mathrm{kg}$, the radius $r=0.02 \mathrm{m}$, the inertia of rotation of the sphere $I_b=4.2\times {10}^{-5} \mathrm{kg}·{\mathrm{m}}^2$, and the acceleration of gravity $g=9.81 \mathrm{m}/{\mathrm{s}}^2$. d₁ and d₂ are the sum of the unknown disturbances in the X and Y directions, respectively, which comprise unknown dynamics, external load disturbances, and friction.

The aggregate disturbance is selected by analysis of the system as follows:

$$d_1=d_2=0.2\sin (t)e^{-0.01t}+0.1\sin (t)+0.1\cos (t)$$

(49)

The desired tracking trajectory is selected as:

$$\left\{\begin{array}{l}{x}_d=0.03{\sin }^3(t)\\ y_d=0.015\sin (t+\frac{\pi }{2})-0.005\sin (2t+\frac{\pi }{2})-0.002\sin (3t+\frac{\pi }{2})-0.001\sin (4t+\frac{\pi }{2})\end{array}\right.$$

(50)

Considering the symmetry of our system, the parameters of the two axis controllers are set exactly the same, and the parameters of the simulation process are set in

Table 3. Controller parameters setting.

Controllers	Parameters
Proposed method	$\begin{array}{l}\gamma =5, \beta =3.1/3, k_1=5, k_2=0.005, k_3=0.8, k_4=10,\\ m=12/13, n=7/11, {\mathsf{\ell }}_1=0.1, {\mathsf{\ell }}_2=0.1, {\mu }_1=2, \lambda =0.01,\\ {\mu }_2=0.5, {\mu }_3=2.4, {\mu }_4=0.05, a_a=0.1, a_w=0.1, {\eta }_1=0.02,\\ \tau =0.5, {\eta }_2=0.02, {\eta }_i=0.005, k_j=0.4, i=3,\cdots ,5, j=5,\cdots ,9\end{array}$
ANFTSM [31]	$\gamma =2, \beta =5/3, k=200, k_1=220, k_1=5, \mu =10 ,\eta =0.25$
Unmodified NFTSM	The value is the same as the proposed method

. FTPF is set to X-direction $l=2/13$, $h/\varsigma =1.1$, T = 0.502, $\varsigma =5$, ${\rho }_0=0.014$, ${\rho }_T=0.001$; Y-direction $l=2/13$, $h/\varsigma =1.1$, T = 0.571, $\varsigma =5$, ${\rho }_0=0.39$, ${\rho }_T=0.001$. The Nussbaum function is $N(\xi )=({\xi }^2+2)\exp ({\xi }^2/2)\sin \xi +1$.

Set the initial parameters of the system to x(0) = 0.03, y(0) = 0.015, and the rest of the parameters to 0. The simulation results are shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 and

Table 4. Controller simulation results.

Controllers	tc/s	IAE/m·s	ITAE/m·s2	CE/rad2·s
Proposed method	0.230	$1.47\times {10}^{-7}$	$1.41\times {10}^{-6}$	$1.07\times {10}^{-9}$
ANFTSM [31]	0.385	0.0066	0.0679	0.0664
Unmodified NFTSM	0.265	$1.65\times {10}^{-6}$	$1.68\times {10}^{-5}$	$9.31\times {10}^{-8}$

.

According to Equation (46), the system satisfies the canonical form of Equation (4) and designs a control scheme similar to Example 1. The control method in this paper is mainly based on sliding mode control, and we solve the performance effects arising from transients and steady states based on FTPF, design SRWNN to compensate the disturbances, and construct the control law under the Nussbaum function. Xu et al. [1] and Li et al. [2] used the terminal sliding mode for trajectory tracking control of the ball and plate system, which resulted in insufficient control accuracy as they required the assumption of $\sin \alpha \approx \alpha$, $\sin \beta \approx \beta$ before designing the control law. However, this paper uses the Nussbaum function for the overall solution without an approximation process, which improves the control accuracy significantly.

The system complexity of this example is low relative to Experiment 1, and the output of the amplitude by the controller is small. As can be seen from Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 and Table 4, both our method and the unmodified NFTSM have good control effects, while the ANFTSM produces larger jitter during convergence. As shown in Figure 7 and Figure 8, the proposed NFTSM control and unmodified NFTSM control have tracked the reference signals with extremely high precision. In terms of position tracking error, in the X-direction ANFTSM tracking error is the largest, and the maximum tracking error is around 8 cm after 0.8 s. Unimproved NFTSM has a noticeable local maximum error around 0.3 s, which is about 1 mm, and the proposed method is below 0.5 mm; for the y-direction, ANFTSM tracking error is obviously larger, and the tracking errors of the other two algorithms are less than 0.001 mm after 0.15 s; for the driving force of each axis shown in Figure 9, ANFTSM shows continuous fluctuations throughout the operation phase, and the local driving force reaches more than 0.0015 rad, and the other two algorithms show a larger driving force at the initial moment, about 0.26 rad, but the duration is within 1 s, which is due to the larger position error in the initial moment which makes the perturbation estimation value larger, caused by the other moments of the drive being smoother, but through the local magnification chart it can be seen that the algorithm in this paper appears as a small local drive fluctuation, which is due to the small perturbation estimation error and the fluctuation of the perturbation estimation error makes the adaptive term coefficient change generated. Because most of the perturbation is estimated by SRWNN, the perturbation estimation error is more minor and the fluctuation of the driving force is smaller. Compared with Table 4, it can be seen that the shortest regulation time of this paper’s algorithm is about 0.230 s and the maximum of ANFTSM is about 0.385 s, compared with other algorithms. In addition, TAE, ITAE, and CE are consistent with the results of the regulation time.

Unlike the simulation results, even if the proposed NFTSM manifold is employed to develop a first-order SMC, the resulting control algorithm does not cause serious chattering due to the high gain characteristics near the equilibrium and the adaptive law enhances the robustness of the lumped disturbance. In addition, the chattering-reduced ability of the developed NFTSM control can be found through the comparison between Figure 9A,B. ANFTSM needs to choose a more considerable control gain to obtain a more desirable control effect, which is challenging to implement in a practical system. The trigger results are shown in Figure 10 and Figure 11. Within 20 s, our proposed method is triggered 235 times and 186 times in X and Y directions, respectively, with far fewer controller updates than the temporal triggering of ANFTSM. Obviously, the experimental results also show that the event-triggered strategy can significantly reduce the communication frequency between the controller and the actuator, so as to alleviate the communication burden of the system. The outputs of N(ξ) and ξ in Figure 6 show that N(ξ) and ξ in both directions have converged to a stable range at about 0.3 s. Therefore, the ETC-based controller effectively saves communication resources, which sufficiently proves the effectiveness of our method.

The above analysis shows that the overall control performance of the proposed method is better than that of the comparison method for the following reasons:

(1)

Considering external non-matching perturbations, the comparison method needs to take an excessive upper uncertainty limit in the absence of a priori knowledge of uncertainty. The existence of the sign function causes the jitter problem, which is not conducive to practical engineering applications.

(2)

The convergence rate of FTPF in finite time introduced in this paper can be specified in advance.

(3)

By visualizing the motion constraints and the prescribed performance metrics as position tracking error bounds, the fundamental idea behind the FTPF design is to transform the original constrained tracking error dynamics into an equivalent “state-constrained” one. The dynamics of the sliding surface can be constrained, which means that the state constraints are satisfied.

(4)

Even in external time-varying disturbances, the introduced SRWNN and the adaptive law can be compensated quickly.

(5)

The sliding surface associated with the tracking error is still bounded by a given constant. With the addition of FTPF and the compensation term, the typical overshoot effects of the sliding mode can be bounded in a predetermined set.

(6)

Under the Lipschitz condition that $L\left(x,\dot{x},t\right)$ satisfies Assumption 1, the system state rapidly converges to the equilibrium point under event-triggered control and the control input data transfer is greatly reduced.

(7)

Furthermore, the underlying singularity problem in the NFTSM is avoided by properly choosing the performance bounds for the position tracking errors.

It should be emphasized above all that the proposed controller ensures the sliding surface’s boundedness that intrinsically guarantees the full state constraints for the dynamic systems. Moreover, the convergence of the sliding surface to the origin is satisfied. Thus, all the sliding-mode properties are preserved with the developed control scheme. Notice that in several works that combine the SMC and barrier Lyapunov function (BLF), this fact has not been proven (see [3,10,23,28]).

On the topic of recent papers, the authors found a research study developed by Shu et al. [26] that combined the SMC and the ETC. Even when in this work the output variable converges in a finite time to a predefined neighborhood of zero, the full state constraints for the dynamic system are not proven.

Overall, the NFTSM surface designed in this paper can maintain a fast descent speed throughout the descent process, which better balances the convergence characteristics near the equilibrium point and avoids the compensation input from being too large when the initial error is significant, and can ensure the tracking error evolves within a strictly finite time FTPF. In addition, the convergence speed of our method is acceptable, and the corresponding control force and convergence error are the smallest of all the compared methods. In addition, it does not suffer from violent chattering, contributing to the sustainability of the actuator.

5. Conclusions

In this paper, nonlinear systems’ trajectory tracking control problem with time-varying parameter uncertainties and unknown control directions are solved with relative threshold event triggering and Nussbaum gains techniques in the FTPF framework. Combining the SRWNN and NFTSM techniques, we propose an adaptive control scheme with prescribed performance event triggering. In addition, we introduce the FTPF technique and tanh function to improve the NFTSM sliding surface and construct SRWNN to compensate for the persistent disturbance and model uncertainty, and use the MLP technique to solve the learning explosion problem more concisely. Our adaptive controller based on the event triggering mechanism and Nussbaum technique ensures good control performance of the system under time-varying unknown gain, time-varying uncertain parameters, and disturbances and avoids Zeno behavior. Compared with the ANFTSM method, the developed scheme can reduce the control force and low gain parameter values, the controller output is continuously smooth throughout, and the conversion tracking error is guaranteed to be strictly within the prescribed performance constraints. Finally, the effectiveness of the proposed scheme is verified by simulation.