Echo State Network-Based Adaptive Event-Triggered Control for Stochastic Nonaffine Systems with Actuator Hysteresis

Abstract

This paper studies the problem of the event-triggered control of nonaffine stochastic nonlinear systems with actuator hysteresis. The echo state network (ESN) is introduced to approximate an unknown nonlinear function. The command filtering technology is used to avoid the derivation of the virtual controller in the controller design process and tries to solve the problem of complexity explosion in the traditional method. Based on Lyapunov’s finite-time stability theory, the proposed method verifies the stability of non-affine stochastic nonlinear systems. It is proved that the proposed controller method can guarantee that all of the signals in the closed-loop system are bounded, and the tracking error can converge to a minimal neighborhood of zero even if there exists an actuator hysteresis. The effectiveness of the proposed method is demonstrated by the simulation example. The simulation results show that the proposed method is effective.

1. Introduction

With the development of society, system control plays a vital role in many fields such as the military, as well as industrial production. Linear systems are not only theoretically studied but also widely applied [1]. However, in real life the system can be easily affected by uncertain factors. As a result, the system will be nonlinear [2]. With the passage of time, the traditional nonlinear system cannot meet the needs of social development. The research of complex systems has increasingly attracted scholars’ attention. Due to the influence of stochastic disturbances, the actual systems are embodied with randomness. Therefore, the study of stochastic nonlinear systems is of great significance and practical value. Regarding stochastic nonlinear systems, there are also many systems whose state variables or actual controls are not clearly represented. Therefore, the system often has a non-affine appearance.

Stochastic nonaffine nonlinear systems can be applied to many fields, including aerospace systems and robot operations. In [3,4], the unknown nonaffine input is transformed into a partially affine form using the mean value theorem, and all signals in a closed loop system are bounded. Therefore, the same method is used in this paper to deal with the nonaffine problem of control variables by using the mean value theorem and transform the nonaffine system into a simple standard stochastic nonlinear system.

For some nonlinear systems, backstepping control is a powerful and typical control method for parametric uncertainty [5,6]. The backstepping method was proposed in [7]. A controller was established by constructing the quadratic Lyapunov function. So far, many new algorithms have been created to solve the nonlinear system problems. The backstepping control strategy is an effective method for solving uncertain systems, and it has been widely used. However, the traditional backstepping design method also has disadvantages. The complexity of the control design and stability increase exponentially with the increase in the system order, that is, each calculation step of the virtual controller may lead to the “explosion of complexity”. In order to solve this problem, ref. [8] developed a dynamic surface control (DSC) technique that introduced a low-pass filter in each control input for the first time. The above DSC technique does not take the error of the first-order filter into consideration, which may affect the precision and accuracy of the system. The command filter control (CFC) adopted in this paper utilizes an error compensation mechanism at each step of the command filter to reduce the influence of filter errors.

The control of nonlinear systems with unknown hysteretic nonlinearity has always been a popular topic. Hysteretic nonlinearity is very common in the actuation of smart materials, such as piezoelectric materials and shape memory alloys. Nonlinearity and hysteresis have a remarkable influence on the system, causing it to become unstable [9]. Robust adaptive control and adaptive inversion control for a class of nonlinear systems with unknown hysteresis are studied in [10,11]. The approach presented in [12,13,14] cannot solve the control problem of nonlinear systems with a post-performance period. Therefore, it is a great challenge to solve the problem of stochastic nonaffine systems with actuator hysteresis.

In addition, with the development of the network, event-triggered control has become popular as it can effectively reduce the waste of communication and resources. The article [15] considers the problem of adaptive fuzzy switching event-triggered control for a class of nonaffine stochastic systems with periodic actuator faults. The paper [16] considers the event-triggered adaptive tracking control for RDE systems with coexisting parametric uncertainties and severe nonlinearities. In traditional control design, the output of the controller is transmitted to the actuators, which may generate redundant update signals and then cause waste of the communication network [17]. Thus, these issues motivate the development of event-triggered strategies. In [18], an event-triggered strategy is developed for nonlinear uncertain systems, in which the uncertain part is represented by the product of known functions and unknown parameters. In [19], for nonlinear uncertain systems, an event-triggered strategy using neural networks to approximate the uncertain part is proposed. The fixed threshold strategy and the relative threshold strategy are discussed in [20,21] since the threshold of the fixed threshold strategy is a constant but the control signal of the actual system is not static. If the amplitude of the control signal is too large, it will lead to system instability. A neural adaptive event-triggered strategy that greatly saves communication resources while ensuring system performance is proposed [22]. Therefore, this paper employs the relative threshold strategy. However, to the best of our knowledge, there are currently few methods to solve the event-triggered problem of stochastic nonaffine systems with actuator hysteresis, thus motivating our current work.

Based on the above discussion, a neural network control design for stochastic non-affine nonlinear systems with event triggering and actuator hysteresis is constructed, and the echo state network (ESN) is introduced to approximate the unknown nonlinear functions. In other papers, such as [23,24], the radial basis function neural network (RBFNN) structure is used to approximate the unknown functions. In [25], a recurrent neural network (RNN) is proposed to approximate the unknown function. However, the former can only rely on the current input, while the latter requires higher computational costs. This paper proposes a simple method to approximate the unknown functions by using the echo state network, which is better than other methods and can be trained easily.

(1) The ESN network is used to approximate the unknown function generated during the design process. Compared with RBFNN in [24], ESN has better stability than the RBFNN network without a complex training process. Compared with RNNS in [25], weight updating does not require a particularly high computational cost, and the training speed is faster than that of RNNS. Therefore, this paper uses the echo state network as a new approach to approximate the unknown function more simply and accurately in the controller design, which greatly reduces the calculation cost.

(2) This paper introduces the actuator hysteresis into the nonaffine system and uses the mean value theorem to transform the nonaffine stochastic nonlinear system into a stochastic nonlinear system. Compared with the nonlinear control problem of [12,13,14], actuator hysteresis has not been fully considered. This paper greatly reduces the complexity of the system and solves the problem of actuator hysteresis in nonlinear systems.

(3) Compared with [17] and [18], the problem of resource waste in the communication process is solved by designing an event-triggered controller, and the relative threshold is used to ensure the stability of the system in the design process.

The rest of the paper is as follows. In Section 2, more information of stochastic nonaffine systems and ESN will be further illustrated. Section 3 introduces the design method of the event trigger controller and provides the stability analysis of the system. Section 4 describes the simulation process to verify the effectiveness of the proposed method. In the end, the conclusions are given in Section 5.

2. Preliminaries

2.1. Problem Formulation

For the following stochastic nonlinear system

$$d\iota =f\left(\iota \right)dt+g\left(\iota \right)d\omega$$

(1)

where $\iota$ is the system state, $f\left(\iota \right)$ and $g\left(\iota \right)$ indicate the locally Lipschitz functions with $f\left(0\right)=g\left(0\right)=0$. $\omega$ represents an r-dimensional standard Wiener process.

Definition 1 ([26]). 

For any given function $V\left(\iota \right)$ in $C^2$, define the differential operator $\mathcal{L}$ as

$$\mathcal{L}V=\frac{\partial V}{\partial \iota }f+\frac{1}{2}Tr\left\{g^T\frac{{\partial }^{2}V}{\partial {\iota }^2}g\right\}$$

(2)

with Tr as the matrix trace.

Lemma 1 ([27]). 

For the stochastic nonlinear system (1), there is a Lyapunov function $V\left(\iota \right)$, ${\psi }_1(·)$ and ${\psi }_2(·)\in k_{\infty }$. When $0<k\le 1$, function $V\left(\iota \right)$ meets the following requirements:

$$\begin{array}{cc}\hfill {\psi }_1\left(∥\iota ∥\right)& \le V\left(\iota \right)\le {\psi }_2\left(∥\iota ∥\right)\hfill \\ \hfill \mathcal{L}V\left(\iota \right)& \le -\alpha V^k\left(\iota \right)+\Gamma \hfill \end{array}$$

(3)

when $k=1$, two normal numbers, α and Γ, exist. System (1) has a unique strong solution, all of the signals in the closed-loop system are bounded in probability, and the system satisfies

$$E\left[V\left(\iota \right)\right]\le V\left({\iota }_0\right)e^{-\alpha t}+\frac{\Gamma }{\alpha }$$

(4)

For the following class of stochastic nonaffine systems with actuator hysteresis:

$$\left\{\begin{array}{cc}\hfill d{x}_i& =f_i\left({\underline{x}}_i,x_{i+1}\right)dt+{\phi }_i^{\top }\left({\underline{x}}_i\right)d\omega \hfill \\ \hfill d{x}_n& =f_n\left({\underline{x}}_n,\varphi \left(v\right)\right)dt+{\phi }_n^{\top }\left({\underline{x}}_n\right)d\omega \hfill \\ \hfill y& =x_1,i=1,\dots ,n-1\hfill \end{array}\right.$$

(5)

where ${\underline{x}}_i=[x_1,x_2,\dots ,x_i]$, $v\in R$ is the system input, and $y\in R$ is the system output. $\varphi \left(v\right)$ denotes the hysteresis nonlinearities. $\omega$ is a r-dimensional standard Brownian motion defined in the complete probability space $(\Omega ,F,P)$. $\Omega$ denotes the sample space. F denotes the $\sigma$-field. P denotes the probability measure. $f_i\left(·\right)$ and ${\phi }_i(·)$ represent the unknown smooth functions.

The control signal v and the hysteresis type of nonlinearity $\varphi \left(v\right)$ in the Formula (5) are defined as

$$\frac{d\varphi }{dt}=\mu \left|\frac{dv}{dt}\right|(\pi v-\varphi )+\psi \frac{du}{dt}$$

(6)

where $\mu$, $\pi$, and $\varphi$ are designed parameters; $\pi >0$ is the slope of lines; and guarantee $\pi >\psi$.

Dynamics $\left(6\right)$ is used to simulate the backlash-like hysteresis, and additional parameters are put up as $\mu =0.05$, $\pi =0.2$, and $\psi =0.0055$, and the initial conditions as $v\left(0\right)=0$, $\varphi \left(0\right)=0$.

Moreover, Equation (6) can be properly handled as follows:

$$\begin{array}{cc}\hfill \varphi \left(v\right)& =\pi v\left(t\right)+\rho \left(v\right)\hfill \\ \hfill \rho \left(v\right)& =[{\varphi }_0-\pi v_0]e^{-\mu (v-v_0)\mathrm{sgn}\left\{\dot{v}\right\}}\hfill \\ \hfill & +e^{-\mu v\mathrm{sgn}\left\{\dot{v}\right\}}\underset{v_0}{\overset{v}{\int }}[\psi -\pi ]e^{-\mu v\mathrm{sgn}\left\{\dot{v}\right\}}d{x}_i\hfill \end{array}$$

where $v\left(0\right)=v_0$ and $\varphi \left(v_0\right)=\varphi \left(0\right)$. $\rho \left(v\right)$ are bounded and satisfy $\left|\rho \left(v\right)\right|\le \overline{\rho }$, and $\overline{\rho }$ is the unknown constant.

For the sake of simplicity, the time variable t will be omitted below. Based on mean value theorem, the smooth nonlinear function $f_i(·)$ can be changed into the form as

$$f_i({\underline{x}}_i,x_{i+1})=f_i({\underline{x}}_n,x_{i+1}^0)+h_{{\mu }_i}(x_{i+1}-x_{i+1}^0)$$

(7)

where ${\left.h_{{\mu }_i}=\frac{\partial f_i({\underline{x}}_i,x_{i+1})}{\partial x_{i+1}}\right|}_{x_{i+1}=x_{{\mu }_i}}$, $x_{{\mu }_i}={\mu }_i{x}_{i+1}+(1-{\mu }_i)x_{i+1}^0$, $0<{\mu }_i<1$, $i=1,\dots ,n-1$, $x_{n+1}=v$. With $x_{i+1}^0=0$, the original system (5) can be described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill d{x}_i& =\left(f_i({\underline{x}}_i,0)+h_{{\mu }_i}{x}_{i+1}\right)dt+{\phi }_i^T\left(x_i\right)d\omega \hfill & \\ \hfill d{x}_n& =(f_n({\underline{x}}_n,0)+h_{{\mu }_n}\pi v+h_{{\mu }_n}\rho )dt+{\phi }_n^T\left(x_n\right)d\omega \hfill \\ \hfill y& =x_1,i=1,\dots ,n-1\hfill \end{array}\right.\end{array}$$

(8)

Assumption A1. 

The sign of $h_{{\mu }_i}$ in $\left(8\right)$ is known for $i=1,\cdots ,n-1$. Without a loss of generality, and for the convenience of analysis and design, assume that

$$\begin{array}{c}0<c_i<h_{{\mu }_i}<{\overline{c}}_i,1\le i\le n-1\hfill \\ 0<c_n<\pi h_{{\mu }_n}<{\overline{c}}_n\hfill \end{array}$$

(9)

Assumption A2. 

The reference signal $y_d$ and its time derivatives up to the n th order are continuous and bounded.

2.2. Echo State Network

The following form of ESN continuous-time dynamics is given [28]

$$\dot{P}\left(Z\right)=-\lambda P\left(Z\right)+tanh(W_{in}^{*}u+W_d^{*}P\left(Z\right)+W_{fb}^{*}y)$$

(10)

where the activation function of the dynamic reservoirs $P\left(Z\right)$. $\lambda$ is a positive number representing the stored neuron’s leakage rate; $tanh(·)$ denotes a hyperbolic tangent function. $W_{in}^{*}$ and $W_d^{*}$ represent the input and internal connection weight matrices, respectively. $W_{fb}^{*}$ represents feedback connection weight matrices; u means the external input with K-dimensional. The output signal’s equation is defined as follows:

$$y=W^{*T}P\left(Z\right)$$

(11)

with $W^{*}\in R^{N\times 1}$ being the weight matrix of output.

It can be seen in [29] that the output of RNNs can be used to approximate any continuous function, which indicates that there is an ESN system, as shown in Equation (11) above. Therefore, as for any given continuous smooth function $f(·)$, the below inequality is true

$$\underset{Z\in \Omega }{sup}\left|f\left(Z\right)-W^{*T}P\left(Z\right)\right|\le \epsilon$$

(12)

In addition, the function $f\left(Z\right)$ can be approached by

$$f\left(Z\right)=W^{T}P\left(Z\right)+\delta ,\forall Z\in \Omega$$

(13)

where $W\in R^{N\times 1}$ is the ideal weight matrix satisfying $W=arg{min}_{W^{*}\in R^{N\times 1}}\{\underset{Z\in \Omega }{sup}f\left(x\right)-W^{*T}P\left(Z\right)\}$, and $\delta$ is bounded to satisfy $\left|\delta \right|\le \epsilon$. $P\left(Z\right)={[p_1\left(Z\right),\dots ,p_N\left(Z\right)]}^T$ stands for the activation function where $p_j\left(Z\right)$ is chosen as

$$p_j\left(Z\right)=\frac{r_j}{s_j+e^{-\frac{Z}{q_j}}}+u_j,j=1,\dots ,N$$

(14)

where $r_j$, $s_j$, $q_j$, and $u_j$ are the constant parameters, and $p_j\left(Z\right)$ is bounded by $0<p_j\left(Z\right)<u_m$ and $u_m=max\left\{\right|\left(r_j\right)/\left(s_j\right)+u_j|,|\left(r_j\right)/(s_j+1)+u_j\left|\right\}$.

On account of the fact the weight W is generally uncharted in reality, to guarantee the asymptotic tracking performance, the estimated value of W, (expressed as $\widehat{W}$), is employed and updated by designing adaptive laws online.

Remark 1. 

From the above discussions, it can be deduced that ESN is an optional tool for function approximation and can be replaced with any other approximation techniques, for instance, RBFNNs, FLSs, and others [30,31]. Compared to RBFNNs, ESN does not need to adjust the weights between the input layer and the hidden layer, and the training is simple and accurate.

Lemma 2. 

The hyperbolic function holds the following property [32]:

$$0\le \left|p\right|-ptanh\left(\frac{p}{q}\right)\le 0.2785q$$

(15)

where $-ptanh\left(\frac{p}{q}\right)\le 0$, $q>0$, and $p\in R$.

Lemma 3. 

The Young’s inequality is described as [33]:

$$ab\le \frac{{\mu }^n}{n}{\left|a\right|}^n+\frac{1}{p{\mu }^p}{\left|b\right|}^p$$

(16)

where $n>1$, $p>1$, $\mu >0$, and $(n/l)+(1/p)=1$.

3. Event-Triggered Adaptive Controller Design

In the backstepping design process, the following tracking errors are defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill e_1& =x_1-y_d\hfill \\ \hfill e_i& =x_i-x_{i,c}\hfill \end{array}\right.\end{array}$$

(17)

where $i=2,\dots ,n$, $y_d$ represents the reference signals, virtual controllers ${\alpha }_i$ are the input signals of the command filters, and the outputs signal of the filters are $x_{i,c}$. The command filters are defined as follows:

$${\dot{x}}_{2,c}{\omega }_n={\alpha }_1-x_{2,c},x_{1,c}\left(0\right)={\alpha }_1\left(0\right)$$

(18)

If the input signal ${\alpha }_1$ satisfies $\left|{\dot{\alpha }}_1\right|\le {\rho }_1$ and $\left|{\ddot{\alpha }}_1\right|\le {\rho }_2$ for all $0\le t$, where ${\rho }_1>0$, ${\rho }_2>0$, and $x_{2,c}\left(0\right)=0$, for any $\xi >0$, the filter design parameters ${\omega }_n>0$ exist, and the following inequality holds: $|x_{1,c}-{\alpha }_1|\le \xi$.

In order to reduce the influence of command filtering on error, the following new tracking error signals are defined by using the error compensation mechanism:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill v_1=e_1-q_1\\ \hfill v_i=e_i-q_i\end{array}\right.\end{array}$$

(19)

where $i=2,\dots ,n$, the compensating signals $q_i$ are designed as

$$\begin{array}{ccc}\hfill {\dot{q}}_1& =& -k_1{q}_1+h_{{\mu }_1}{q}_2+h_{{\mu }_1}(x_{2,c}-{\alpha }_1)\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill {\dot{q}}_i& =& -k_i{q}_i+h_{{\mu }_i}{q}_{i+1}+h_{{\mu }_i}(x_{i+1,c}-{\alpha }_i)\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill {\dot{q}}_n& =& -k_n{q}_n\hfill \end{array}$$

(22)

where $k_i>0$ are design constants and $q\left(0\right)=0$.

Step 1: The following can be obtained from (17)

$$\begin{array}{ccc}\hfill d{v}_1& =& (f_1({\underline{x}}_1,0)+h_{{\mu }_1}{x}_2)dt+{\phi }_1^T\left({\underline{x}}_1\right)d\omega -{\dot{y}}_d-{\dot{q}}_1\hfill \\ & =& (f_1({\underline{x}}_1,0)+h_{{\mu }_1}(e_2+x_{2,c})-{\dot{y}}_d-{\dot{q}}_1)dt\hfill \\ & & +{\phi }_1^T\left({\underline{x}}_1\right)d\omega \hfill \end{array}$$

(23)

Choose the Lyapunov candidate function as follows:

$$V_1=\frac{1}{4}{v}_1^4+\frac{1}{2r_1}{\tilde{\theta }}_1^2$$

(24)

where $r_1$ is a design constant, and ${\tilde{\theta }}_1={\theta }_1-{\widehat{\theta }}_1$, ${\widehat{\theta }}_1$ denotes the estimation of ${\theta }_1$.

According to (3) and Definition 1, we have

$$\begin{array}{ccc}\hfill L{V}_1& =& v_1^{3}d{v}_1-\frac{1}{r_1}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1\hfill \\ & =& v_1^3(f_1\left({\underline{x}}_1,0)+h_{{\mu }_1}(e_2+x_{2,c})-{\dot{y}}_d-{\dot{q}}_1\right)\hfill \\ & & +\frac{3}{2}{v}_1^2{\phi }_1^T{\phi }_1-\frac{1}{r_1}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1\hfill \end{array}$$

(25)

By using Young’s inequality, the following inequality holds:

$$\frac{3}{2}{v}_1^2{\phi }_1^T{\phi }_1\le \frac{3}{4}{v}_1^4{∥{\phi }_1∥}^4{l}_1^{-2}+\frac{3}{4}{l}_1^2$$

(26)

where $l_1>0$ is a designed constant. Substituting (26) into (25), we have

$$\begin{array}{c}\hfill L{V}_1\le v_1^3(f_1({\underline{x}}_1,0)+h_{{\mu }_1}(e_2+x_{2,c})-{\dot{y}}_d-{\dot{q}}_1)+\frac{3}{4}{v}_1^4{∥{\phi }_1∥}^4{l}_1^{-2}+\frac{3}{4}{l}_1^2-\frac{1}{r_1}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1\end{array}$$

(27)

where ${\overline{f}}_1\left(Z_1\right)=f_1({\underline{x}}_1,0)+\frac{3}{4}{v}_1{∥{\phi }_1∥}^4{l}_1^{-2}$ and $Z_1=\left[x\right]$. By the ESN (13), the uncharted nonlinear function ${\overline{f}}_1$ can be approximated to ${\overline{f}}_1\left(Z_1\right)=W_1^T{P}_1\left(Z_1\right)+{\delta }_1\left(Z_1\right)$, ${\delta }_1\left(Z_1\right)$ is the approximation error satisfying $\left|{\delta }_1\left(Z_1\right)\right|\le {\epsilon }_1$, and ${\epsilon }_1>0$.

By using Lemma 3, the following inequality holds:

$$\begin{array}{ccc}\hfill v_1^3{\overline{f}}_1& =& v_1^3(W_1^T{P}_1\left(Z_1\right)+{\delta }_1\left(Z_1\right))\hfill \\ & \le & \frac{1}{2}\frac{v_1^6{\theta }_1{P}_1^T{P}_1}{a_1^2}+\frac{1}{2}{a}_1^2+\frac{3}{4}{v}_1^4+\frac{1}{4}{\epsilon }_1^2\hfill \end{array}$$

(28)

where ${\theta }_1={∥W_1∥}^2$, $a_1>0$ is a designed parameter.

Substituting (28) into (27) yields,

$$\begin{array}{c}\hfill L{V}_1\le v_1^3\frac{v_1^3{\theta }_1{P}_1^T{P}_1}{2a_1^2}+\frac{3}{4}{v}_1+h_{{\mu }_1}(e_2+x_{2,c})-{\dot{y}}_d-{\dot{q}}_1+\frac{1}{2}{a}_1^2+\frac{1}{4}{\epsilon }_1^4+\frac{3}{4}{l}_1^2-\frac{1}{r_1}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1\end{array}$$

(29)

The compensating signal ${\dot{q}}_1$ can be designed as follows:

$${\dot{q}}_1=-k_1{q}_1+h_{{\mu }_1}{q}_2+h_{{\mu }_1}(x_{2,c}-{\alpha }_1)$$

(30)

with $k_1>0$.

Substituting (30) into (29) yields

$$\begin{array}{c}\hfill L{V}_1\le v_1^3(\frac{v_1^3{\theta }_1{P}_1^T{P}_1}{2a_1^2}+\frac{3}{4}{v}_1+h_{{\mu }_1}{v}_2-{\dot{y}}_d+k_1{q}_1+h_{{\mu }_1}{\alpha }_1)+\frac{1}{2}{a}_1^2+\frac{1}{4}{\epsilon }_1^4+\frac{3}{4}{l}_1^2-\frac{1}{r_1}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1\end{array}$$

(31)

By using Young’s inequality and Assumption 1, the inequality is as follows:

$$v_1^3{h}_{{\mu }_1}{v}_2\le \frac{3}{4}{v}_1^4{\overline{c}}_1^{-\frac{4}{3}}+\frac{1}{4}{v}_2^4$$

(32)

Substituting (32) and into (31) yields

$$\begin{array}{ccc}\hfill L{V}_1& \le & v_1^3(\frac{v_1^3{\theta }_1{P}_1^T{P}_1}{2a_1^2}+\frac{3}{4}{v}_1+\frac{3}{4}{v}_1{\overline{c}}_1^{-\frac{4}{3}}-{\dot{y}}_d+k_1{q}_1\hfill \\ \hfill & & +{\overline{c}}_1{\alpha }_1)+\frac{1}{4}{v}_2^4+\frac{1}{2}{a}_1^2+\frac{1}{4}{\epsilon }_1^4+\frac{3}{4}{l}_1^2-\frac{1}{r_1}{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1\hfill \end{array}$$

(33)

The virtual control signal is devised as follows:

$${\alpha }_1=\frac{1}{{\overline{c}}_1}(-k_1{e}_1-\frac{v_1^3{\widehat{\theta }}_1{P}_1^T{P}_1}{2a_1^2}+{\dot{y}}_d-\frac{3}{4}{v}_1{\overline{c}}_1^{-\frac{4}{3}}-\frac{3}{4}{v}_1)$$

(34)

By using (34), (33) can be rewritten as

$$\begin{array}{c}\hfill L{V}_1\le -k_1{v}_1^4+\frac{1}{4}{v}_2^4+\frac{1}{2}{a}_1^2+\frac{1}{4}{\epsilon }_1^4+\frac{3}{4}{l}_1^2+\frac{{\tilde{\theta }}_1}{r_1}(\frac{r_1{v}_1^6{P}_1^T{P}_1}{2a_1^2}-{\dot{\widehat{\theta }}}_1)\end{array}$$

(35)

The designed adaptive law is as follows:

$${\dot{\widehat{\theta }}}_1=\frac{r_1{v}_1^6{P}_1^T{P}_1}{2a_1^2}-{\sigma }_1{\widehat{\theta }}_1$$

(36)

Substituting (36) into (35) yields

$$L{V}_1\le -k_1{v}_1^4+\frac{1}{4}{v}_2^4+\frac{1}{2}{a}_1^2+\frac{1}{4}{\epsilon }_1^4+\frac{3}{4}{l}_1^2+\frac{{\sigma }_1}{r}{\tilde{\theta }}_1{\widehat{\theta }}_1$$

(37)

Step i $\left(2\le i\le n-1\right):$ Based on (17), the following formula comes into existence:

$$\begin{array}{ccc}\hfill d{v}_i& =& (f_i({\underline{x}}_i,0)+h_{{\mu }_i}{x}_{i+1})dt+{\phi }_i^T\left(x_i\right)d\omega -{\dot{x}}_{i,c}-{\dot{q}}_i\hfill \\ & =& (f_i({\underline{x}}_i,0)+h_{{\mu }_i}(x_{i+1,c}+e_{i+1})-{\dot{x}}_{i,c}-{\dot{q}}_i)dt+{\phi }_i^T\left(x_i\right)d\omega \hfill \end{array}$$

(38)

Select the following candidate Lyapunov function:

$$\begin{array}{c}\hfill V_i=V_{i-1}+\frac{1}{4}{v}_i^4+\frac{1}{2r_i}{\tilde{\theta }}_i^2\end{array}$$

(39)

where $r_i$ are designed constants, ${\tilde{\theta }}_i={\theta }_i-{\widehat{\theta }}_i$, $i=2,\dots ,n-1$, ${\widehat{\theta }}_i$ are expressed as an estimate of ${\theta }_i$, and ${\theta }_i={∥W_i∥}^2$. According to Definition 1 and (38), the conclusions can be drawn as follows:

$$\begin{array}{c}\hfill L{V}_i=L{V}_{i-1}+v_i^3(f_i({\underline{x}}_i,0)+h_{{\mu }_i}(x_{i+1,c}+e_{i+1})-{\dot{x}}_{i,c}-{\dot{q}}_i)+\frac{3}{2}{v}_i^2{\phi }_i^T{\phi }_i-\frac{1}{r_i}{\tilde{\theta }}_i{\dot{\widehat{\theta }}}_i\end{array}$$

(40)

According to Lemma 3, the following equation is true

$$\begin{array}{c}\hfill \frac{3}{2}{v}_i^2{\phi }_i^T{\phi }_i\le \frac{3}{4}{v}_i^4{∥{\phi }_i∥}^4{l}_i^{-2}+\frac{3}{4}{l}_i^2\end{array}$$

(41)

and $l_i(i=2,\cdots ,n)$ are the designed normal numbers. By using (41), (40) can be rewritten as follows:

$$\begin{array}{c}\hfill L{V}_i\le L{V}_{i-1}+v_i^3(f_i({\underline{x}}_i,0)+h_{{\mu }_i}(x_{i+1,c}+e_{i+1})-{\dot{x}}_{i,c}-{\dot{q}}_i)+\frac{3}{4}{v}_i^4{\overline{c}}_i^{-\frac{4}{3}}+\frac{3}{4}{l}_i^2-\frac{1}{r_i}{\tilde{\theta }}_i{\dot{\widehat{\theta }}}_i\end{array}$$

(42)

where ${\overline{f}}_i\left(Z_i\right)=f_i({\underline{x}}_i,0)+\frac{3}{4}{v}_i{∥{\phi }_i∥}^4{l}_i^{-2}$ and $Z_i={\underline{x}}_i$. For any given ${\epsilon }_i>0$, $W_i^T{P}_i\left(X_1\right)$ exists, such that ${\overline{f}}_i\left(Z_i\right)=W_i^T{P}_i+{\delta }_i\left(Z_i\right)$, $\left|{\delta }_i\left(Z_i\right)\right|\le {\epsilon }_i$, where $\left|{\delta }_i\left(Z_i\right)\right|$ denotes the approximation error.

By using Young’s inequality, it follows that

$$\begin{array}{cc}\hfill v_i^3{\overline{f}}_i& =v_i^3\left(W_i^T{P}_i\left(Z_i\right)\right)+{\delta }_i\left(Z_i\right))\hfill \\ \hfill & \le \frac{v_i^6{\theta }_i{P}_i^T{P}_i}{2a_i^2}+\frac{1}{2}{a}_i^2+\frac{3}{4}{v}_i^4+\frac{1}{4}{\epsilon }_i^2\hfill \end{array}$$

(43)

where ${\theta }_i={∥W_i∥}^2$ and $a_i$ is a positive constant.

By substituting (43) into (42) yields, we obtain that

$$\begin{array}{ccc}\hfill L{V}_i& \le & L{V}_{i-1}+v_i^3(\frac{v_i^3{\theta }_i{P}_i^T{P}_i}{2a_i^2}+\frac{3}{4}{v}_i+h_{{\mu }_i}(x_{i+1,c}+e_{i+1})-{\dot{x}}_{i,c}-{\dot{q}}_i)\hfill \\ & & +\frac{1}{2}{a}_i^2+\frac{1}{4}{\epsilon }_i^4+\frac{3}{4}{l}_i^2-\frac{1}{r_i}{\tilde{\theta }}_i{\dot{\widehat{\theta }}}_i\hfill \end{array}$$

(44)

The compensation signal ${\dot{q}}_i$ is designed as

$$\begin{array}{c}\hfill {\dot{q}}_i=-k_i{q}_i+h_{{\mu }_i}{q}_{i+1}+h_{{\mu }_i}(x_{i+1,c}-{\alpha }_i)\end{array}$$

(45)

with $k_i>0$, substituting (45) into (44), the following can be obtained:

$$\begin{array}{ccc}\hfill L{V}_i& \le & L{V}_{i-1}+v_i^3(\frac{v_i^3{\theta }_i{P}_i^T{P}_i}{2a_i^2}+\frac{3}{4}{v}_i+h_{{\mu }_i}{v}_{i+1}-{\dot{x}}_{i,c}+k_i{q}_i+h_{{\mu }_i}{\alpha }_i)\hfill \\ & & +\frac{1}{2}{a}_i^2+\frac{1}{4}{\epsilon }_i^4+\frac{3}{4}{l}_i^2-\frac{1}{r_i}{\tilde{\theta }}_i{\dot{\widehat{\theta }}}_i\hfill \end{array}$$

(46)

Based on Young’s inequality, we have

$$v_i^3{h}_{{\mu }_i}{v}_{i+1}\le \frac{3}{4}{v}_i^4{\overline{c}}_i^{-\frac{4}{3}}+\frac{1}{4}{v}_{i+1}^4$$

(47)

Substituting (47) into (46) yields

$$\begin{array}{ccc}\hfill L{V}_i& \le & L{V}_{i-1}+v_i^3(\frac{v_i^3{\theta }_i{P}_i^T{P}_i}{2a_i^2}+\frac{3}{4}{v}_i-{\dot{x}}_{i,c}+k_i{q}_i+{\overline{c}}_i{\alpha }_i+\frac{3}{4}{v}_i{\overline{c}}_i^{\frac{4}{3}})\hfill \\ \hfill & & +\frac{1}{4}{v}_{i+1}^4+\frac{1}{2}{a}_i^2+\frac{1}{4}{\epsilon }_i^4+\frac{3}{4}{l}_i^2-\frac{1}{r_i}{\tilde{\theta }}_i{\dot{\widehat{\theta }}}_i\hfill \end{array}$$

Design the virtual control signals as follows:

$${\alpha }_i=\frac{1}{{\overline{c}}_i}(-k_i{e}_i-\frac{v_i^3{\widehat{\theta }}_i{P}_i^T{P}_i}{2a_i^2}+{\dot{x}}_{i,c}-\frac{3}{4}{v}_i{\overline{c}}_i^{-\frac{4}{3}}-\frac{3}{4}{v}_i)$$

(48)

Using (48), (48) can be rewritten as

$$\begin{array}{c}\hfill L{V}_i\le \sum _{j=1}^i(-k_j{v}_j^4+\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2)+\frac{1}{4}{v}_{i+1}^4-\sum _{j=1}^{i-1}\frac{{\sigma }_j}{r_i}{\tilde{\theta }}_j{\widehat{\theta }}_j+\frac{{\tilde{\theta }}_i}{r_i}(\frac{v_i^3{r}_i{P}_i^T{P}_i}{2a_i^2}-{\dot{\widehat{\theta }}}_i)\end{array}$$

(49)

Design the adaptive laws as follows:

$${\dot{\widehat{\theta }}}_i=\frac{r_i{v}_i^6{P}_i^T{P}_i}{2a_i^2}-{\sigma }_i{\widehat{\theta }}_i$$

(50)

Substituting (50) into (49) yields

$$\begin{array}{ccc}\hfill L{V}_i& \le & \sum _{j=1}^i(-k_j{v}_j^4+\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2-\frac{{\sigma }_j}{r_j}{\tilde{\theta }}_j{\widehat{\theta }}_j)+\frac{1}{4}{v}_{i+1}^4\hfill \end{array}$$

(51)

Design of Event-Triggered Controller

The event-triggered strategy is described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill u\left(t\right)& =\zeta \left(t_k\right),\forall t\in [t_k,t_{k+1})\hfill \\ \hfill t_{k+1}& =inf\{t\in R\left|\left|\varrho \left(t\right)\right|>\Xi \left|u\left(t\right)\right|+d_1\right.\}\hfill \end{array}\right.\end{array}$$

(52)

where $\varrho \left(t\right)=\zeta \left(t\right)-u\left(t\right)$, and $d_1>0$ is the designed parameter.

Remark 2. 

Whensoever the event-triggered mechanism $t_{k+1}=inf\{t\in R\left|\left|\varrho \left(t\right)\right|\ge \Xi \left|u\left(t\right)\right|\right.+d_1\}$ is triggered, the time is marked as $t_{k+1}$ and the control value $u\left(t_{k+1}\right)$ is applied to the system. During the time $t\in [t_k,t_{k+1})$, the control signal holds as a constant.

From (52), we can draw a conclusion $\zeta \left(t\right)=(1+z_1\left(t\right)\Xi )u\left(t\right)+z_2\left(t\right)d_1$, $t_k\le t<t_{k+1}$, and $\left|z\left(t\right)\right|\le 1$ is the time-varying parameter. As a result, the equation is obtained as follows:

$$u\left(t\right)=\frac{\zeta \left(t\right)}{1+z_1\left(t\right)\Xi }-\frac{z_2\left(t\right)d_1}{1+z_1\left(t\right)\Xi }$$

(53)

Step n: From (17), we can obtain

$$\begin{array}{c}\hfill d{v}_n=(f_n({\underline{x}}_n,0)+h_{{\mu }_n}\pi u+h_{{\mu }_n}\rho -{\dot{x}}_{n,c}-{\dot{q}}_n)dt+{\phi }_n^T\left(x_n\right)d\omega \end{array}$$

(54)

Choose a Lyapunov function as

$$V_n=V_{n-1}+\frac{1}{4}{v}_n^4+\frac{1}{2r_n}{\tilde{\theta }}_n^2$$

where $r_n$ is a designed constant, and ${\tilde{\theta }}_n={\theta }_n-{\widehat{\theta }}_n$, ${\widehat{\theta }}_n$ is the estimation of ${\theta }_n$, and ${\theta }_n={∥W_n∥}^2$.

According to Definition 1 and (54), it can be deduced that

$$\begin{array}{c}\hfill L{V}_n=L{V}_{n-1}+v_n^3(f_n({\underline{x}}_n,0)+h_{{\mu }_n}\pi u+h_{{\mu }_n}\rho -{\dot{x}}_{n,c}-{\dot{q}}_n)+\frac{3}{2}{v}_n^2{\phi }_n^T{\phi }_n-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n\end{array}$$

(55)

Design the compensating signal ${\dot{q}}_n$ as follows:

$${\dot{q}}_n=-k_n{q}_n$$

(56)

According to (56), (55) can be rewritten as follows:

$$\begin{array}{c}\hfill L{V}_n\le L{V}_{n-1}+v_n^3{f}_n({\underline{x}}_n,0)+v_n^3{\overline{c}}_{n}u+v_n^3{h}_{{\mu }_n}\rho -v_n^3({\dot{x}}_{n,c}-k_n{q}_n)+\frac{3}{2}{v}_n^2{\phi }_n^T{\phi }_n-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n\end{array}$$

(57)

Substituting (53) into (57) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & v_n^3{f}_n({\underline{x}}_n,0)+L{V}_{n-1}-v_n^3({\dot{x}}_{n,c}-k_n{q}_n)+v_n^3{h}_{{\mu }_n}\pi (\frac{\zeta \left(t\right)}{1+z_1\left(t\right)\Xi }-\frac{z_2\left(t\right)d_1}{1+z_1\left(t\right)\Xi })\hfill \\ & & +v_n^3{h}_{{\mu }_n}\rho +\frac{3}{2}{v}_n^2{\phi }_n^T{\phi }_n-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n\hfill \end{array}$$

(58)

By Assumption 1, and $z_1\left(t\right)\in [-1,1],z_2\left(t\right)\in [-1,1]$, the inequalities can be held as

$$h_{{\mu }_n}\pi \frac{v_n^3\zeta \left(t\right)}{1+z_1\left(t\right)\Xi }\le {\overline{c}}_n\frac{v_n^3\zeta \left(t\right)}{1+\Xi }$$

(59)

$$-h_{{\mu }_n}\pi \frac{v_n^3{z}_2\left(t\right)d_1}{1+z_1\left(t\right)\Xi }\le \left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|$$

(60)

Substituting (59) and (60) into (58) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & v_n^3{f}_n({\underline{x}}_n,0)+{\overline{c}}_n\frac{v_n^3\zeta \left(t\right)}{1+\Xi }+\left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|+v_n^3{h}_{{\mu }_n}\rho \hfill \\ \hfill & & -v_n^3({\dot{x}}_{n,c}-k_n{q}_n)+\frac{3}{2}{v}_n^2{\phi }_n^T{\phi }_n-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n+L{V}_{n-1}\hfill \end{array}$$

(61)

where

$$\zeta \left(t\right)=-(1+\Xi )({\alpha }_{n}tan\frac{v_n^3{\overline{c}}_n{\alpha }_n}{ϵ}+{\overline{m}}_{1}tanh\frac{v_n^3{\overline{c}}_n{\overline{m}}_1}{ϵ})$$

(62)

and ${\overline{m}}_1>d_1/(1-\Xi )$.

Substituting (62) into (61) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & v_n^3{f}_n({\underline{x}}_n,0)-{\overline{c}}_n{v}_n^3({\alpha }_{n}tan\frac{v_n^3{\overline{c}}_n{\alpha }_n}{ϵ}+{\overline{m}}_{1}tanh\frac{v_n^3{\overline{c}}_n{\overline{m}}_1}{ϵ})+\left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|\hfill \\ \hfill & & +v_n^3{h}_{{\mu }_n}\rho -v_n^3({\dot{x}}_{n,c}-k_n{q}_n)+\frac{3}{2}{v}_n^2{\phi }_n^T{\phi }_n-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n+L{V}_{n-1}\hfill \end{array}$$

(63)

Based on Lemma 3, we can obtain

$$\begin{array}{ccc}\hfill L{V}_n& \le & v_n^3{f}_n({\underline{x}}_n,0)+{\overline{c}}_n{v}_n^3{\alpha }_n-\left|{\overline{c}}_n{v}_n^3{\overline{m}}_1\right|+\left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|+v_n^3{h}_{{\mu }_n}\rho \hfill \\ \hfill & & -v_n^3({\dot{x}}_{n,c}-k_n{q}_n)+\frac{3}{2}{v}_n^2{\phi }_n^T{\phi }_n-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n+L{V}_{n-1}+0.557ϵ\hfill \end{array}$$

(64)

Based on Young’s inequality, one has

$$\begin{array}{c}\hfill \frac{3}{2}{v}_n^2{\phi }_n^T{\phi }_n\le \frac{3}{4}{v}_n^4{∥{\phi }_n∥}^4{l}_n^{-2}+\frac{3}{4}{l}_n^2\end{array}$$

(65)

Substituting (65) into (64) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & v_n^3{f}_n({\underline{x}}_n,0)+{\overline{c}}_n{v}_n^3{\alpha }_n-\left|{\overline{c}}_n{v}_n^3{\overline{m}}_1\right|+\left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|-v_n^3({\dot{x}}_{n,c}-k_n{q}_n)\hfill \\ \hfill & & +v_n^3{h}_{{\mu }_n}\rho +\frac{3}{4}{v}_n^4{∥{\phi }_n∥}^4{l}_n^{-2}+\frac{3}{4}{l}_n^2-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n+L{V}_{n-1}+0.557ϵ\hfill \end{array}$$

(66)

During the process of designing virtual controllers, the unknown nonlinear function ${\overline{f}}_n(Z_n,0)$ is approximated by ESNs, where ${\overline{f}}_n\left(Z_n\right)=f_n({\underline{x}}_n,0)+\frac{3}{4}{v}_n{∥{\phi }_n∥}^4{l}_n^{-2}$$Z_n={\underline{x}}_n$. By using the universal approximation capability of ESN (13), for any given ${\epsilon }_n$, an ESN $W_n^T{P}_n\left(Z_n\right)$ always exist, such that ${\overline{f}}_n\left(Z_n\right)=W_n^T{P}_n+{\delta }_n$, where $Z_n={\underline{x}}_n$ and ${\delta }_n$ is the approxomation error satisfying $\left|{\delta }_n\left(Z_n\right)\right|\le {\epsilon }_n$.

Baesd on Lemma 3, the inequality holds that

$$\begin{array}{cc}\hfill v_n^3{\overline{f}}_n& =v_n^3\left(W_n^T{P}_n\left(Z_n\right)\right)+{\delta }_n\left(Z_n\right))\hfill \\ \hfill & \le \frac{v_n^6{\theta }_n{P}_n^T{P}_n}{2a_n^2}+\frac{1}{2}{a}_n^2+\frac{3}{4}{v}_n^4+\frac{1}{4}{\epsilon }_n^2\hfill \end{array}$$

(67)

where ${\theta }_n={∥W_n∥}^2$, $a_n$ is a given positive constant.

Substituting (67) into (66) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & \frac{v_n^6{\theta }_n{P}_n^T{P}_n}{2a_n^2}+\frac{3}{4}{v}_n^4+\frac{1}{2}{a}_n^2+\frac{1}{4}{\epsilon }_n^4+{\overline{c}}_n{v}_n^3{\alpha }_n-\left|{\overline{c}}_n{v}_n^3{\overline{m}}_1\right|+\left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|\hfill \\ \hfill & & +v_n^3{h}_{{\mu }_n}\rho -v_n^3({\dot{x}}_{n,c}-k_n{q}_n)+\frac{3}{4}{l}_n^2-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{}}\theta }_n+L{V}_{n-1}+0.557ϵ\hfill \end{array}$$

(68)

It can be known from Young’s inequality that the inequality holds

$$v_n^3{h}_{{\mu }_n}\rho \le \frac{3}{4}{v}_n^4{h}_{{\mu }_n}^{\frac{4}{3}}+\frac{1}{4}{\rho }^4\le \frac{3}{4}{v}_n^4{\overline{c}}_n^{\frac{4}{3}}+\frac{1}{4}{\overline{\rho }}^4$$

(69)

Substituting (51) and (69) into (68) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & \frac{v_n^6{\theta }_n{P}_n^T{P}_n}{2a_n^2}+\frac{3}{4}{v}_n^4+\frac{1}{2}{a}_n^2+\frac{1}{4}{\epsilon }_n^4+{\overline{c}}_n{v}_n^3{\alpha }_n-\left|{\overline{c}}_n{v}_n^3{\overline{m}}_1\right|+\left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|\hfill \\ \hfill & & +\frac{3}{4}{v}_n^4{\overline{c}}_n^{\frac{4}{3}}+\frac{1}{4}{\overline{\rho }}^4-v_n^3({\dot{x}}_{n,c}-k_n{q}_n)+\frac{3}{4}{l}_n^2-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n+0.557ϵ\hfill \\ \hfill & & +\sum _{j=1}^{n-1}(-k_j{v}_j^4+\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2+\frac{{\sigma }_j}{r_j}{\tilde{\theta }}_j{\widehat{\theta }}_j)+\frac{1}{4}{v}_n^4\hfill \end{array}$$

(70)

Design the virtual control signal as follows:

$${\alpha }_n=\frac{1}{{\overline{c}}_n}(-k_n{e}_n-\frac{v_n^3{\widehat{\theta }}_n{P}_n^T{P}_n}{2a_n^2}+{\dot{x}}_{n,c}-\frac{3}{4}{v}_n{\overline{c}}_n^{-\frac{4}{3}}-v_n)$$

(71)

Substituting (71) into (70) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & \frac{v_n^6{\tilde{\theta }}_n{P}_n^T{P}_n}{2a_n^2}+\frac{1}{2}{a}_n^2+\frac{1}{4}{\epsilon }_n^4-v_n^4{k}_n-\left|{\overline{c}}_n{v}_n^3{\overline{m}}_1\right|+\left|\frac{v_n^3{\overline{c}}_n{d}_1}{1-\Xi }\right|+\frac{1}{4}{\overline{\rho }}^4\hfill \\ & & +\frac{3}{4}{l}_n^2-\frac{1}{r_n}{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n+0.557ϵ+\sum _{j=1}^{n-1}(-k_j{v}_j^4+\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2+\frac{{\sigma }_j}{r_j}{\tilde{\theta }}_j{\widehat{\theta }}_j)\hfill \end{array}$$

(72)

We design the adaptive law as follows:

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_n=\frac{r_n{v}_n^6{P}_n^T{P}_n}{2a_n^2}-{\sigma }_n{\widehat{\theta }}_n\end{array}$$

(73)

Substituting (73) into (72) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & \sum _{j=1}^n(-k_j{v}_j^4+\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2+\frac{{\sigma }_j}{r_j}{\tilde{\theta }}_j{\widehat{\theta }}_j)+\frac{1}{4}{\overline{\rho }}^4+0.557ϵ\hfill \end{array}$$

(74)

By using Young’s inequality,

$$\frac{{\sigma }_j}{r_j}{\tilde{\theta }}_j{\widehat{\theta }}_j\le -\frac{{\sigma }_j{\tilde{\theta }}_j^T{\tilde{\theta }}_j}{2r_j}+\frac{{\sigma }_j{\theta }_j^T{\theta }_j}{2r_j}$$

(75)

Substituting (75) into (74) yields

$$\begin{array}{ccc}\hfill L{V}_n& \le & -\sum _{j=1}^n(k_j{v}_j^4+\frac{{\sigma }_j{\tilde{\theta }}_j^T{\tilde{\theta }}_j}{2r_j})+\sum _{j=1}^n(\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2+\frac{{\sigma }_j{\theta }_j^T{\theta }_j}{2r_j})+\frac{1}{4}{\overline{\rho }}^4+0.557ϵ\hfill \\ \hfill & \le & -\sum _{j=1}^n(k_j{v}_j^4+\frac{{\sigma }_j{\tilde{\theta }}_j^T{\tilde{\theta }}_j}{2r_j})+C\hfill \end{array}$$

(76)

where $C={\displaystyle \sum _{j=1}^n}(\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2+\frac{{\sigma }_j{\theta }_j^T{\theta }_j}{2r_j})+\frac{1}{4}{\overline{\rho }}^4+0.557ϵ$

Theorem 1. 

Consider a class of stochastic nonaffine nonlinear system (5) under Assumptions 1 and 2. The packaged unknown ${\overline{f}}_i$ can be approximated by the ESN networks by which the approximating error ${\delta }_i$ is bounded. Event-triggered strategy (52); adaptive controllers (33), (48), (62), and (73); and adaptive laws (36), (50), and (73) in the closed-loop system are bounded in probability. Specifically, the tracking error $y-y_d$ will be converged as

$${\Omega }_1=\{v_1\left(t\right)\in R|E[{∥y-y_d∥}^4]\le \frac{8C}{\eta },\forall t>T_1\}$$

(77)

with $\eta =min\{4k_1,\cdots ,4k_n,{\sigma }_1,\cdots ,{\sigma }_n\}$, and $C={\displaystyle \sum _{j=1}^n}(\frac{1}{2}{a}_j^2+\frac{1}{4}{\epsilon }_j^4+\frac{3}{4}{l}_j^2+\frac{{\sigma }_j{\theta }_j^T{\theta }_j}{2r_j})+\frac{1}{4}{\overline{\rho }}^4+0.557ϵ$.

Proof. 

Define $V_n={\displaystyle \sum _{i=1}^n}{v}_i$, it could be obtained from (76) that

$$\begin{array}{cc}\hfill L{V}_n& \le -\eta \sum _{i=1}^n(v_i^4-\frac{{\tilde{\theta }}_i^2}{2r_i})+C\hfill \\ \hfill & =-\eta V+C\hfill \end{array}$$

(78)

Furthermore, according to (55) and (78), we have

$$\frac{dE\left(V\right(t\left)\right)}{dt}\le -E\left(V\left(t\right)\right)+C,t\ge 0$$

(79)

$$0\le E\left(V\left(t\right)\right)\le (V\left(0\right)-\frac{C}{\eta })e^{-\eta t}+\frac{C}{\eta }$$

(80)

which means that

$$E\left(V\left(t\right)\right)\le \frac{C}{\eta },t\to \infty$$

(81)

$$\begin{array}{cc}\hfill E({∥v_1∥}^4)& \le E{({∥v_1∥}^2)}^2\hfill \\ \hfill & \le 24E\left(v_1\right)\hfill \\ \hfill & \le \frac{8C}{\eta }\hfill \end{array}$$

(82)

$$E({∥y-y_d∥}^4)\le E({∥v_1∥}^4)\le \frac{8C}{\eta },\forall t>T_1$$

(83)

which means that all of the signals in the closed-loop system are bounded in probability. □

Remark 3. 

An adaptive neural finite-time-event-triggered consensus tracking problem is studied for nonlinear multi-agent systems (MASs) under directed graphs [34]. In stochastic systems such as [35,36], there is no error graph, but the tracking graph cannot completely overlap. On the one hand, the tracking error cannot completely converge to zero because of different systems. On the other hand, the tracking error cannot completely converge to zero in finite time. Inspired by [34], the following research will help the stochastic system tracking error converge to zero.

4. Simulation Results

To illustrate the effectiveness of the proposed control method, we choose the following stochastic nonaffine nonlinear systems:

$$\begin{array}{ccc}\hfill d{x}_1& =& [x_2-0.1sin\left(x_1\right)+0.1sin\left(x_1^2{x}_2\right)]dt\hfill \\ \hfill & & +0.5sin\left(x_1\right)cos\left(x_2\right)d\omega \hfill \end{array}$$

(84)

$$\begin{array}{ccc}\hfill d{x}_2& =& [0.1x_{1}cos\left(x_2\right)+0.1sin\left(x_1{x}_2^2\right)+{\varphi }_1]dt\hfill \\ \hfill & & +0.5sin\left(2x_1{x}_2^2\right)d\omega \hfill \end{array}$$

(85)

$$\begin{array}{ccc}\hfill y_1& =& x_1\hfill \end{array}$$

(86)

where $x_1$, $x_2$ are the system states.

The command filter is devised as

$$\begin{array}{c}{\dot{x}}_{2,c}{\omega }_n={\alpha }_1-x_{2,c}\hfill \end{array}$$

(87)

We design the compensating signals as

$$\begin{array}{cc}\hfill d{q}_1& =-k_1{q}_1+{\overline{c}}_1(x_{2c}-{\alpha }_1)+{\overline{c}}_1{q}_2\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill d{q}_2& =-k_2{q}_2\hfill \end{array}$$

(89)

The virtual controllers ${\alpha }_1$, ${\alpha }_2$, and updating laws ${\theta }_1$, ${\theta }_2$ are designed as

$${\alpha }_1=\frac{1}{{\overline{c}}_1}(-k_1{e}_1-\frac{v_1^3{\widehat{\theta }}_1{P}_1^T{P}_1}{2a_1^2}+{\dot{y}}_d-\frac{3}{4}{v}_1{\overline{c}}_1^{-\frac{4}{3}}-\frac{3}{4}{v}_1)$$

(90)

$${\dot{\widehat{\theta }}}_1=\frac{r_1{v}_1^6{P}_1^T{P}_1}{2a_1^2}-{\sigma }_1{\widehat{\theta }}_1$$

(91)

$${\alpha }_2=\frac{1}{{\overline{c}}_2}(-k_2{e}_2-\frac{v_2^3{\widehat{\theta }}_2{P}_2^T{P}_2}{2a_2^2}+{\dot{x}}_{2,c}-\frac{3}{4}{v}_2{\overline{c}}_2^{-\frac{4}{3}}-v_2)$$

(92)

$${\dot{\widehat{\theta }}}_2=\frac{r_2{v}_2^6{P}_2^T{P}_2}{2a_2^2}-{\sigma }_2{\widehat{\theta }}_2$$

(93)

The event-triggered controller $\zeta \left(t\right)$ and actual controller u are designed as

$$\begin{array}{cc}\hfill \zeta \left(t\right)& =-(1+\Xi )({\alpha }_{2}tanh\frac{v_2^3{\overline{c}}_2{\alpha }_2}{\epsilon }+{\overline{m}}_{1}tanh\frac{v_2^3{\overline{c}}_2{\overline{m}}_2}{\epsilon })\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill u& =\zeta \left(t_k\right),k\in z^{+}\hfill \end{array}$$

(95)

and the event-triggered mechanism is as follows: $t_{k+1}=inf\{t\in R\left|\left|\varrho \left(t\right)\right|\ge \Xi \left|u\left(t\right)\right|\right.+d_1\}$.

We choose the initial condition and main parameters as follows: ${\varphi }_1=0.01$, $x_1=0.8$, $x_2=0.8$, ${\widehat{\theta }}_1=1$, ${\widehat{\theta }}_2=1$, $q_1=0.5$, $q_2=0.1$, $x_{2,c}=5$. The reference $y_d=0.85sin\left(t\right)$, $k_1=30$, $k_2=10$, ${\sigma }_1=1$, ${\sigma }_2=2$, ${\overline{c}}_1=2$, ${\overline{c}}_2=40$, ${\overline{m}}_1=18$, $a_1=2$, $a_2=20$, $\Xi =0.01$, $ϵ=6$, $d_1=5$, $r_1=0.05$, $r_2=0.05$, ${\mu }_1=5$, ${\pi }_1=0.3$, ${\psi }_1=0.005$, ${\omega }_n=0.005$.

The final simulation results are represented in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. The output of system $x_1$ can keep up with the desired trajectory $y_d$ within a very small error range, where T = 20 s; Figure 1 expresses the tracking diagram for reference $y_d$ and output y, and the error trajectory is shown in Figure 2. It can be clearly indicated that the tracking error $e_1$ is in a minimal neighborhood of zero. Figure 3 shows the triggered events. Figure 4 shows that the adaptive parameters ${\theta }_1$ and ${\theta }_2$, and the control signal is shown in Figure 5.

5. Conclusions

In this paper, a control scheme based on error compensation for command filtering is proposed to solve the tracking control problem of stochastic nonaffine nonlinear systems with event triggering and actuator hysteresis. Using ESN to approximate unknown functions is simpler than using RBF neural networks or other methods. The “explosion of complexity” is solved by command-filtering technology, and the error caused by command filtering is solved by the compensation signal. By utilizing the event-triggered control, the waste of communication resources is reduced. Considering the existence of hysteretic nonlinearity, the proposed control scheme can ensure that the signals in the closed loop are all bounded. Finally, the effectiveness of the method is proved by the simulation results.