Event-triggered adaptive neural finite-time tracking control for non-strict feedback stochastic nonlinear systems with unknown time-varying control directions and input nonlinearities

Abstract

This paper focuses on dynamic-surface-based event-triggered adaptive neural finite-time tracking control for a category of non-strict feedback stochastic nonlinear system subject to unknown time-varying control directions, and input nonlinearities. Firstly, an improved event-triggered mechanism is presented, which simultaneously considers tracking error variables and compensating for event errors in threshold value. Secondly, a novel Lemma is provided to address concurrently the issues of Nussbaum terms in stability analysis and stochastic system finite-time stability. Thirdly, neural networks are utilized to cope with unknown functions. In addition, the Nussbaum functions are applied to identify time-varying unknown control directions and input nonlinearities. Moreover, the errors of the dynamic surface technique are compensated successfully by error compensation signals. By combining the proposed Lemma 1 and event-triggered adaptive neural finite-time control scheme, finite-time stability of considered systems in probability can be guaranteed, the tracking error can drive to a small neighborhood of the origin in finite time, and the Zeno behavior can be excluded. Finally, the effectiveness of the designed control scheme has been demonstrated through a second-order numerical system and a mass-spring-damper example.

Control of stochastic nonlinear systems have attracted much attention in recent years.^1–6 Because many actual systems, such as robotic manipulators, ⁷ induction motors, ⁸ and so on, contain stochastic disturbances. Additionally, in actual implementations, the designed control strategy is expected to be finished in a finite time, such as the finite-time control method considered by ^9–11 for deterministic nonlinear systems. Thus, system stability is accomplished in a finite time with the assistance of a finite-time control approach. For stochastic nonlinear systems, the concept of finite-time stabilization was introduced in. ¹² Subsequently, the preceding approach was further expanded to various stochastic systems. In, ¹³ Li et al. investigated the stochastic nonlinear systems in the strict-feedback form withfull-state constraints and input saturation, and a new stability definition and the Lyapunov stability theorem were presented. In, ¹⁴ for a class of uncertain strict-feedback nonlinear systems with hysteresis, Zhang et al. developed the adaptive fuzzy finite-time control based on command filter. According to the Lyapunov stability theorem, a semi-globally finite-time stability theorem in probability was also proposed, in which the convergent time is shorter than the. ¹⁵ In, ¹⁶ Fang et al. explored the finite-time stabilization of high-order stochastic nonlinear systems in the strict-feedback form subjected to an output restriction. The pure feedback stochastic nonlinear systems with full state constraint has been studied in. ¹⁷

Furthermore, comparing pure-feedback, and strict feedback structures, the non-strict feedback is a more general form of feedback. In non-strict feedback stochastic nonlinear systems, all state variables are included in f _i (X) functions with X = [x₁, x₂, …, x _n ]. To this purpose, a series of studies on non-strict feedback systems has been done in.^18–21 The Chen et al. ¹⁹ overcame this obstacle by utilizing variable separation method and adaptive backstepping strategies. In, ¹⁸ Wang et al. deployed neural networks to handle non-strict functions based on their strong approximation capacity, which simplifies processing for non-strict feedback systems. In, ²⁰ Liu et al. considered the finite stability of the nonlinear systems in the non-strict feedback form with state constraints, where the unknown function is approximated by a fuzzy logic system. Moreover, in, ²¹ Wang et al. investigated the finite stability for non-strict feedback stochastic systems subject to input delay and output constraints. It further extends the range of applications for finite-time control. Although superb results were been achieved above-mentioned works, there is a hypothesis that control gains require know.

Among the mentioned above works, it is assumed that the sign and magnitude of the control gain are known a constant, which have been difficultly acquired in actual applications due to some sudden interferences and fast changes on system states, such as the autopilot design of time-varying ships ²² and multi-agent systems. ²³ Hence, research is valuable and meaningful for unknown control directions of the system. A class of Nussbaum functions was presented to identify the unknown coefficient in the equation in. ²⁴ Subsequently, combined with Nussbaum technology, an adaptive control scheme had been studied for strict-feedback nonlinear with unknown control directions in. ²⁵ Admittedly, uncertain time-varying control directions are more common compared with the constant case in many practical control applications. In, ²⁶ Ding investigated a new Nussbaum function to deal with unknown high-frequency gain. According to the new Nussbaum function, Chen et al. ²⁷ had focused on multi-input and multi-output (MIMO) systems with unknown time-varying control directions, and an adaptive fuzzy asymptotic control was achieved. Recently, Shahriari-Kahkeshi et al. had considered uncertain time-varying control directions for multi-agent systems with dead-zone nonlinearity in. ²⁸

Furthermore, there were some other works to research unknown control directions for stochastic nonlinear systems as follows. The stability of stochastic nonlinear large-scale systems in the strict-feedback form with unknown control directions was studied in, ²⁹ and a Theorem was proposed to directly analyze the stability of the system. Also, a Theorem was presented in, ³⁰ although the Nussbaum function under consideration is not the same as in. ²⁹ Further, an adaptive neural output feedback control was probed for non-strict feedback stochastic nonlinear systems with unknown backlash-like hysteresis and unknown control directions. In, ³¹ Xia et al. developed an observer-based event-triggered adaptive fuzzy control scheme for unmeasured stochastic nonlinear systems in the non-strict feedback structure with unknown control directions. Additionally, other studies of stochastic nonlinear systems with uncertain control directions may be encountered in.^32–35 It’s worth noting that the unknown time-varying control directions for stochastic systems with finite-time stability have yet to be fully solved. This is one of the driving forces behind the work in this paper.

In recent years, event-triggered control schemes have garnered considerable attention as a way to preserve communication resources and lessen the strain of communication. Qiu et al. proposed a fixed-threshold approach for communication constraint in, ³⁶ Note that the event-triggered controller only samples in practice when the trigger objectives are satisfied. In reality, the maximum trigger interval is difficult to predict ahead of time, which could lead to data redundancy problems. Therefore, in, ³⁷ Kong et al. designed a relative threshold strategy that varies dynamically with the input signal to improve the controller’s performance. Moreover, in, ³⁸ Wu et al. considered the tracking error variables in the threshold value based on the relative threshold strategy to improve the system’s tracking performance. In, ³⁹ Li et al. proposed a new event-triggering mechanism that reduced conservatism in control design and stability analysis while simultaneously considering the system’s finite-time stability. Recently, an event-triggered adaptive Fixed-Time tracking controller was designed for a family of nonlinear systems with a non-strict feedback form in Wang et al. ⁴⁰ . In addition, event-triggered adaptive fuzzy control had been considered for stochastic strict-feedback nonlinear systems with unknown backlash-like hysteresis and unmeasured states in Zhu et al. ⁴¹ . In, ⁴² Liu and Zhu further extended the event-triggered control strategy to stochastic strict-feedback nonlinear systems and had explored event-triggered adaptive neural control of stochastic nonlinear systems with time-varying delays and state restrictions. However, much of the works discussed above ignore the input nonlinearities and the uncertain control directions for stochastic non-strict feedback systems, as well as the fact that the system is stabilized in infinite time.

On the other hand, input nonlinearities can be encountered in many practical applications, and such as saturation, ¹³ dead-zone, backlash, and hysteresis ¹⁴ . If these input nonlinearities are neglected, which can cause system instability and degrade system performance. Recently, input nonlinearities had been investigated in.^43–45 In, ⁴⁴ Malek et al. had researched prescribed performance-based adaptive control for switched MIMO uncertain nonlinear systems in the strict-feedback form subject to unmodeled dynamics and input nonlinearities. In, ⁴³ Zouari et al. was developed a neural network controller for fractional-order systems with input nonlinearities and asymmetric time-varying Pseudo-state constraints. Furthermore, in, ⁴⁵ Kamalamiri et al. had considered input nonlinearities for non-strict feedback systems with output constraint, unknown control direction, and an adaptive finite-time neural control scheme was proposed. Nonetheless, in the above works, the system’s stochastic disturbances are underexplored and the system’s input is driven by time strategy leading to unnecessary waste on communication resources in the aforementioned studies. In reality, the “explosion of complexity” can occur during the traditional design controller process as a result of repetitive differential virtual control, such as.^46,47 Moreover, note that the dynamic surface control (DSC) was used to overcome this issue in ⁴⁸ and ⁴⁹ . It is universally recognized that dynamic surface technique can lead to greater errors. To end this, Farrell et al. proposed a compensation signal in ⁴⁹ to remove dynamic surface technique errors, similar work is also mentioned in. ¹⁴

The finite-time stability problems of the non-strict feedback stochastic systems were handled in Wang et al. ²¹ , but triggered project was neglected for controller to lead the wastes from the aspect of saving communication resources. Although Li et al. ³⁹ considered event-based design of finite-time control for the nonlinear systems, it cannot be directly utilized in this paper due to omitting unknown time-varying control directions, input nonlinearities, and stochastic disturbances on systems. Therefore, how to design event-based finite-time control for the non-strict feedback nonlinear stochastic systems with unknown time-varying control directions and input nonlinearities is still an important and open-end topic.

Motivated by the above-mentioned discussions, a class of event-based finite-time control strategies are investigated for nonlinear stochastic systems subject to unknown time-varying control directions and input nonlinearities in the non-strict feedback form. Nussbaum functions are applied to identify time-varying unknown control direction and input nonlinearities. Based on the backstepping technique, combining the dynamic surface control (DSC) technique and the MLP algorithm, appropriate controllers only containing one adjusted parameter are designed in this paper. The primary contributions of this paper are summarized in three aspects as follows

(1) An improved event-triggered mechanism is presented that considers both tracking error variables and compensating for event errors at the same time in threshold value. Hence, the control performance of the algorithm is enhanced and the conservatism is minimized in control design and stability analysis. Moreover, the proposed event-triggered mechanism not violates Zeno behavior.

(2) Based on the Lyapunov stability theorem, a new criterion of semi-globally finite-time stability in probability (SGFSP) is proposed in Lemma 1. Due to the existence of Nussbaum terms in the stability analysis, the currently available finite-time stability criterion cannot be used to directly analyze the stability of stochastic nonlinear systems with time-varying unknown control directions, such as. ⁵⁰ By Lemma1, an event-triggered adaptive neural finite-time tracking control strategy is successfully constructed for stochastic nonlinear systems in the non-strict feedback form with unknown time-varying control directions, and input nonlinearities.

(3) Both the dynamic surface technique and hyperbolic tangent functions have been incorporated into controller design to address ”complexity explosion” and ”singularity”. Meanwhile, an error compensation mechanism is built to eliminate the dynamic surface technique’s errors. Additionally, to alleviate computing load, just one adaptive law should be updated online.

Problem formulation and preliminaries

Stochastic finite-time stability theorem

Consider the normal stochastic nonlinear system

$$dx=f\left(x\right)dt+g^T\left(x\right)d\omega$$

(1)

where x ∈ Rn is the system state; ω denotes an r-dimensional standard Wiener process defined on a complete probability space $\mathrm{\Omega },\mathrm{ℱ},P$ where Ω is a sample space, $\mathrm{ℱ}$ is a σ-field and p is the probability measure; f(x): R ⁿ → R and g(x): R ⁿ → R^n×r are Borel measurable continuous functions.

Definition 1: ³⁰ For a $C^2$ function V(x) ∈ R ⁿ , denote LV as the differential operator of V respect to time, then, LV is defined for system equation (1) as following

$$LV\left(x\right)=\frac{\partial V\left(x\right)}{\partial t}+\frac{\partial V\left(x\right)}{\partial x}f\left(x\right)+\frac{1}{2}trace\left(g^T\frac{{\partial }^{2}V}{\partial x^2}g\right)$$

(2)

where $\frac{1}{2}\text{trace}\left(g^T\frac{{\partial }^{2}V}{\partial x^2}g\right)$ is positive semi-definite calling the Herssian term.

Definition 2: ¹³ For the system (1), there exists an unique solution x₀ ∈ R ⁿ , and the setting time $T\left(x_0,\overline{\epsilon }\right)$ is finite almost surely. For $t\ge T\left(x_0,\overline{\epsilon }\right)$ , such that $E\left(\Vert x\left(t;x_0\right)\Vert \right)<\overline{\epsilon }$ , Then, the solution of the system (1) is semi-globally finite-time stability in probability (SGFSP).

Definition 3: ²⁵ Any even differentiable function N(ξ): R → R is called a Nussbaum-type function if it has the following properties

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

(3)

Generally, Nussbaum-type function is constructed by FG which F is a nonnegative increasing function and $G$ is Trigonometric function, such as, $e^{{\xi }^2}\cos \left((\pi /2)\xi \right)$ , $\frac{a^2{b}^2+1}{b^2\sqrt{b^2+1}}{e}^{a\left|\xi \right|}\sin \left(\frac{\xi }{b}\right)$ ⁵¹ , and $e^{{\xi }^2/2}(k^2+2)\sin \left(\xi \right)$ ⁵² . In this paper, the $e^{{\xi }^2/2}(k^2+2)\sin \left(\xi \right)$ function be utilized to design the controllers.

Lemma 1: Let ξ(t) is a smooth function defined on (0, T) and N(ξ) is a category of Nussbaum-type function. Consider the stochastic nonlinear system (1). If there exist a Lyapunov function $V\left(x\right)\in C^2$ and $k_1,k_2\in k_{\infty }$ , such that

$$\begin{array}{l}{k}_1\left(\parallel x\parallel \right)\le V\left(x\right)\le k_2\left(\parallel x\parallel \right)\hfill \\ LV\le -C_1{V}^{\beta }\left(x\right)+\left(hN\left(\xi \right)+1\right)\dot{\xi }+C\hfill \end{array}$$

(4)

where $\left(hN\left(\xi \right)+1\right)\dot{\xi }={\displaystyle {\sum }_{i=1}^n\left(h_i\left(t\right)N\left({\xi }_i\right)+1\right)}\stackrel{.}{{\xi }_i},0<\beta <1$ and 0 < C < ∞ . C₁ is positive constants. Then, the system (1) is SGFSP with the settling time T as following

$$E\left(T\right)\le \frac{1}{\left(1-\beta \right)C_1\gamma }\left[E\left(V^{1-\beta }\left(x_0\right)\right)-{\left(\frac{C}{\left(1-\gamma \right)C_1}\right)}^{\frac{\left(1-\beta \right)}{\beta }}+\mathrm{{\rm E}}M\right]$$

(5)

where 0 < γ < 1, $\mathrm{ℰ}=\left[(1-\beta )(1-\gamma )C_1\right]/C$ and

Proof: From the inequality equation (4), we can deduce that for x₀ ∈ R ⁿ , V(x) ≥ 0 and $LV\le \overline{M}+C$ with $\left|\left(hN\left(\xi \right)+1\right)\dot{\xi }\right|<\overline{M}$ . Based on the Theorem 4.1 in, ⁵³ the system equation (1) has an unique solution.

According to the definition 2, we need to prove that there exists a finite settling time T, such that $E(\parallel x(t;x_0)\parallel )<\overline{\epsilon }$ , $t\ge T(x_0,\overline{\epsilon })$ and $T(x_0,\overline{\epsilon })<\infty$ .

From 0 < γ < 1, the following inequality can be obtained

$$LV\le -C_1\gamma V^{\beta }\left(x\right)-C_1\left(1-\gamma \right)V^{\beta }\left(x\right)+\left(hN\left(\xi \right)+1\right)\dot{\xi }+C$$

(6)

define $\mathrm{\Omega }=\left\{x|E{V}^{\beta }(x)\le \left(\frac{C}{(1-\gamma )C_1}\right)\right\}$ , letting the complementary set of Ω as $\overline{\mathrm{\Omega }}$ with $\overline{\mathrm{\Omega }}=\left\{x|E{V}^{\beta }(x)\ge \left(\frac{C}{(1-\gamma )C_1}\right)\right\}$ . Next, two cases be considered as x ∈ Ω and $x\in \overline{\mathrm{\Omega }}$ .

Case1: $x\in \overline{\mathrm{\Omega }}$ . According to equation (6), one can get that

$$LV\le -C_1\gamma V^{\beta }\left(x\right)+\left(hN\left(\xi \right)+1\right)\dot{\xi }$$

(7)

define τm = inf{t > 0, ∥ x(t) ∥≥ m} as the stopping time for ∀m > 0. It is obvious that when m → ∞, τ _m → ∞. Next, letting W(x) = V^1−β(x), t ∈ [0, T ∧ τ _m ]. It is noting that V^1−β(x) is a positive definite $C^2$ function with $E{V}^{1-\beta }(x)\ge {\left(\frac{C}{(1-\gamma )C_1}\right)}^{\frac{1-\beta }{\beta }}$ . By It $\widehat{o}$ ’s formula, one has

$$EW\left(x\left(T\hspace{0.17em}\vee \hspace{0.17em}\tau m\right),T\hspace{0.17em}\vee \hspace{0.17em}\tau m\right)=EW\left(x_0\right)+E{\displaystyle \underset{0}{\overset{T\hspace{0.17em}\vee \hspace{0.17em}\tau m}{\int }}LW\left(x\left(s\right),s\right)ds}$$

(8)

Moreover, according to the stochastic differential in Definition 1 and the Theorem 1 in, ¹³ one can deduce that

$$LW\le (1-\beta )V^{-\beta }LV+\frac{-\beta (1-\beta )}{2}{V}^{-\beta -1}\text{Tr}\left(h^T{\left(\frac{\partial w}{\partial x}\right)}^T\left(\frac{\partial w}{\partial x}\right)h\right)$$

(9)

where $\text{Tr}\left(h^T{\left(\frac{\partial w}{\partial x}\right)}^T\left(\frac{\partial w}{\partial x}\right)h\right)$ is positive semi-definite. Hence, one can get by equation (9) that

$$LW\le \left(1-\beta \right)V^{-\beta }LV$$

(10)

Then, substituting equation (7) into (10) yields that

$$LW\le -\left(1-\beta \right)C_1\gamma +\left(1-\beta \right)V^{-\beta }\left(hN\left(\xi \right)+1\right)\dot{\xi }$$

(11)

Bringing the above inequality into the second term on the right-hand side of the equation (8) inequality and using the definition $\overline{\mathrm{\Omega }}$ yield

$$E{\displaystyle \underset{0}{\overset{T\wedge \tau m}{\int }}-\left(1-\beta \right)C_1\gamma +\left(1-\beta \right)V^{-\beta }\left(hN\left(\xi \right)+1\right)\dot{\xi }ds\le }E{\displaystyle \underset{0}{\overset{T\wedge \tau m}{\int }}-\left(1-\beta \right)C_1\gamma +\mathrm{ℰ}\left(hN\left(\xi \right)+1\right)\dot{\xi }ds}$$

(12)

where $\mathrm{ℰ}=\left[(1-\beta )(1-\gamma )C_1\right]/C$ . Substituting the above inequality into equation (8) gets

$$\begin{array}{ccc}EW\left(x\left(T\wedge {\tau }_m\right),T\wedge {\tau }_m\right)& \le & EW\left(x_0\right)+E{\displaystyle \underset{0}{\overset{T\wedge \tau m}{\int }}-\left(1-\beta \right)C_1\gamma +\mathrm{ℰ}\left(hN\left(\xi \right)+1\right)\dot{\xi }ds}\\ EW\left(x\left(T\wedge {\tau }_m\right),T\wedge {\tau }_m\right)& \le & E{V}^{1-\beta }\left(x_0\right)-\left(1-\beta \right)C_1\gamma E\left(T\wedge {\tau }_m\right)+\mathrm{ℰ}EA\left(\xi \right)\end{array}$$

(13)

where . $N\left(\xi \right)$ is a category of Nussbaum functions with the normal form satisfying $N\left(\xi \right)=FG$ . Moreover, $F$ is a nonnegative increasing function and $G$ is Trigonometric function. The can be expressed difficultly as the explicit functions. Therefore, ξ boundedness is proved by the property of Trigonometric function which is necessarily positive or negative on certain intervals. Without loss of generality, let $\mathit{G}$ is positive on intervals of the $\overline{N}$ and negative on intervals $\underset{¯}{N}$ . Hence, according the above discussion we can further deduce that and . Next, the $A\left(\xi \right)$ boundedness be analyzed for h > 0 and h < 0.

Case 1.1: If h > 0, supposing that ξ(s) has no bound on s ∈ [0, T ∧ τ _m ). Namely, there must exist ξ(s) → ∞, such as $\underset{\xi \to \infty \wedge \overline{N}}{\text{lim}}A\left(\xi \right)=-\infty$ . Then, combining equation (13), one can get that

$$\begin{array}{ccc}E{V}^{1-\beta }\left(x\left(T\wedge {\tau }_m\right),T\wedge {\tau }_m\right)& \le & \begin{array}{l}E{V}^{1-\beta }\left(x_0\right)-\left(1-\beta \right)C_1\gamma E\left(T\wedge {\tau }_m\right)\\ +\mathrm{ℰ}E\left[\underset{\xi \to \infty \wedge \overline{N}}{\text{lim}}A\left(\xi \right)\right]\end{array}\end{array}$$

(14)

From equation (14), it can be derived that $E{V}^{1-\beta }\left(x\left(T\wedge {\tau }_m\right),T\wedge {\tau }_m\right)<0$ . However, there is a contradiction because $E{V}^{1-\beta }\left(x\left(T\wedge {\tau }_m\right),T\wedge {\tau }_m\right)$ is positive semi-definite. Hence, ξ(s) has bound on s ∈ [0, T ∧ τ _m ). Then, $A\left(\xi \right)$ must be bounded on [0, T ∧ τ _m ) with $\Vert A\left(\xi \right)\Vert \hspace{0.17em}\le M_1$ .

Case 1.2: If h < 0, Similar to the proving process of Case 1.1 to acquire $\Vert A\left(\xi \right)\Vert \le M_2$ . Admittedly, $A\left(\xi \right)$ respect to functions of z _i , v _i , θ, then, z _i , v _i , θ are limited by the parameters C, C₁, γ. Therefore, $A\left(\xi \right)$ tends to a small value when chooses appropriate parameters C, C₁, γ. From equation (14), one gets

$$E\left(T\wedge {\tau }_m\right)\le \frac{1}{\left(1-\beta \right)C_1\gamma }\left[E{V}^{1-\beta }\left(x_0\right)-E{V}^{1-\beta }\left(x\left(T\wedge {\tau }_m\right),T\wedge {\tau }_m\right)+\mathrm{ℰ}EM\right]$$

(15)

where M ≫ max{M₁, M₂}. It is obvious that letting m → ∞, then τ _m → ∞, combining equation (15), one can get that the finite settling time T is

$$E\left(T\right)\le \frac{1}{\left(1-\beta \right)C_1\gamma }\left[E\left(V^{1-\beta }\left(x_0\right)\right)-{\left(\frac{C}{\left(1-\gamma \right)C_1}\right)}^{\frac{\left(1-\beta \right)}{\beta }}+\mathrm{ℰ}M\right]$$

(16)

Then, we can deduce that there exists the finite settling time T. The trajectory of x(t) can reach the set Ω, i.e.,x(t) ∈ Ω for ∀t ≥ T with T satisfied $E(T)\le \frac{1}{(1-\beta )C_1\gamma }\left[E\left(V^{1-\beta }(x_0)\right)-{\left(\frac{C}{(1-\gamma )C_1}\right)}^{\frac{(1-\beta )}{\beta }}+\mathrm{ℰ}M\right]$ .

Case 2: x ∈ Ω. Similar to the proving process of Case1 can obtain $\Vert A\left(\xi \right)\Vert \hspace{0.17em}\le M$ and the trajectory of x(t) does not exceed the set Ω.

To conclude, combining the above proof can gain that the trajectory of x(t) enters set $\mathrm{\Omega }=\left\{x|E{V}^{\beta }(x)\le \left(\frac{C}{(1-\gamma )C_1}\right)\right\}$ when ∀t ≥ T. Since, the nonlinear system (1) is SGFSP.

Remark 1. Note that Lemma 1 provides an extremely effective tool to analyse the stabilization of the stochastic systems with uncertain time-varying control coefficients in a finite time. Additionally, Lemma 1 consinders a class of Nussbaum functions, broadening its applicability. While considers a similar lemma, the system’s finite-time stability was been ignored. Moreover, Lemma 1 can be also utilized to analyze the finite-time stability of deterministic systems.

Problem formulation

Consider a class of stochastic nonstrict-feedback nonlinear systems with the time-varying unknown control directions and input nonlinearities as follows

$$\{\begin{array}{c}d{x}_i=\left(h_i\left(t\right)x_{i+1}+f_i\left(X\right)\right)dt+g_i^T\left(X\right)d\omega \left(t\right),\\ d{x}_n=\left(h_n\left(t\right)u\left(v\left(t\right)\right)\right)+f_n\left(X\right)dt+g_n^T\left(X\right)d\omega \left(t\right)\\ y=x_1\end{array}$$

(17)

where $X={[x_1,\dots ,x_n]}^T\in R^n$ is the state vector of the system. u(v(t)), v(t) and y ∈ R are system output of actuator nonlinearities, system controller input and system output signal, respectively. ω is an r-dimensional standard Brownian motion defined on a complete probability space $(\mathrm{\Omega },\mathrm{ℱ},P)$ with Ω being a sample space, $\mathrm{ℱ}$ being a σ-field and P being a probability measure. h _i (t), f _i (·) and g _i (·) are time-varying unknown smooth nonlinear functions.

The control objective is to design an event-triggered adaptive finite-time controller for the system (17) such that

(1) all the signals in the closed loop system are SGFSP,

(2) the output y(t) follows the desired trajectory y _d as expected in the mean square sense,

(3) the Zeno behavior can be completely avoided for new event-triggered threshold.

Input nonlinearities

The output of actuator nonlinearities is describe as follows

$$u\left(v\left(t\right)\right)=m\left(t\right)v\left(t\right)+\mathrm{\varrho }\left(t\right)$$

(18)

where $0<m<\overline{m},0<\hspace{0.17em}\left|\mathrm{\varrho }\left(t\right)\right|\hspace{0.17em}<\overline{\mathrm{\varrho }}$ and $\overline{m},\overline{\mathrm{\varrho }}$ are positive unknown constants. As mentioned before, four conditions can be formulated by equation (18) such as saturation, ¹³ dead-zone, backlash, and hysteresis. ¹⁴

To end this section, some useful assumptions and lemmas are given for the design of controllers.

Assumption 1. ⁵⁴ The desired trajectories y _d and its first time derivatives y _d is known and bounded.

Assumption 2. The function $h_i\left({\overline{x}}_i\right)$ in equation (17) meets the following restrictions

$$\left|h_i\left(t\right)\right|\hspace{0.17em}\le {\overline{h}}_i$$

(19)

where ${\overline{h}}_i$ is the upper bound of h _i (t).

Lemma 2. ¹⁰ The trajectory of system $\dot{x}=f\left(x,u\right)$ is practical almost fast finite-time stable if there are some design constants ν₁, ν₂, 0 < ϕ < 1, and 0 < ν < ∞, which makes

$$\dot{V}\left(x\right)\le -{\nu }_{1}V\left(x\right)-{\nu }_2{V}^q\left(x\right)+\nu$$

(20)

hold. The settling time is T as following

$$T=\frac{1}{{\nu }_1\left(1-\varphi \right)}\text{ln}\left\{1-\left[{\left(\frac{\nu }{\left(1-\overline{\epsilon }\right){\nu }_2}\right)}^{\frac{\varphi }{1-\varphi }}-V^{1-\varphi }\left(x_0\right)\right]\frac{{\nu }_1}{\overline{\epsilon }{\nu }_2}\right\}$$

(21)

where $0<\overline{\epsilon }<1$ a constant.

Lemma 3. ⁵⁵ For $x_i\in R,i=\mathrm{1,2},\dots ,n,p\in \left(\mathrm{0,1}\right]$

$${\left({\displaystyle \sum _{i=1}^n\left|x_i\right|}\right)}^P\le {\displaystyle \sum _{i=1}^n|x_i{|}^p}\le n^{1-p}{\left({\displaystyle \sum _{i=1}^n\left|x_i\right|}\right)}^P$$

(22)

Lemma 4. ⁵⁶ For any positive real numbers c, d and any real-valued function ψ(x, y) > 0

$$\left|x{|}^c\right|y{|}^d\le \frac{c}{c+d}\psi |x{|}^{c+d}+\frac{d}{c+d}{\psi }^{-c/d}|y{|}^{c+d}$$

(23)

Lemma 5. ⁵⁷ (Young’s inequality) For ∀(x, y) ∈ R², the following inequality holds

$$xy\le \frac{p^n}{n}|x{|}^n+\frac{1}{m{p}^m}|y{|}^m$$

(24)

where p > 0, m > 1, n > 1 and (m − 1)(n − 1) = 1.

Lemma 6. ⁵⁸ For any stochastic process x(t), if there exists a integer p and a positive constant Θ such that E|x| ^p ≤ Θ, ∀t ≤ 0, then x(t) is said to be bounded in probability.

Lemma 7. ⁵⁹ For x ∈ R and o > 0, the tanh(·) function satisfies

$$0\le \left|x\right|-x\tanh \left(\frac{x}{o}\right)\le 0.2785o$$

(25)

Event-triggered control mechanism

To reduce the burden and waste of resources on communication, an event-triggered mechanism of the relative threshold is designed as follows

$$\begin{array}{l}\varpi \left(t\right)={\alpha }_n-\iota Z_0\text{tanh}\left(\frac{\iota Z_0{V}_n^3}{o}\right),\hfill \\ v\left(t\right)=\varpi \left(t_k\right)\forall t\in \left(t_k,t_{k+1}\right),k\in R^{+},\hfill \\ t_{k+1}=\inf \left\{t\in R^{+}\mathrm{\mid }\left|e\left(t\right)\right|\hspace{0.17em}\ge {\mu }_0\left|v\left(t\right)\right|+\mu Z_0+{\mu }_1\left|csch\left(v_n^3{Z}_{0}t+\mathit{∍}\right)\right|\right\}.\hfill \end{array}$$

(26)

Where e(t) = ϖ(t) − v(t) is triggering error. ϖ(t) denotes the actual controller and α _n is the virtual controller that will be designed later. $Z_0=1/\left[\left({\displaystyle {\sum }_{i=1}^n{z}_i}\right)+z_0\right]$ with z _i represented tracking errors and z₀ is designed parameter. In addition, 0 < μ₀ < 1, ι > μ > 0, μ₁ > 0, ϶ > 0 are designed parameters. When t ∈ [t _k , t_k+1), the v(t) maintains a unchanged input ϖ(t _k ). Moreover, t_k+1 is update time points for v(t), which system input v(t) changed as ϖ(t_k+1).

Remark 2. The event-triggered mechanism in this paper is inspired by pioneer works Wu et al. ³⁸ and Li et al. ³⁹ . Different from previous works of the relative threshold method in, ⁶⁰ an improved event-triggered mechanism is presented here based on the relative threshold strategy by incorporating tracking error function Z₀ with Hyperbolic Cosecant function csch(·) in threshold value. It’s worth mentioning that when the tracking error z _i grow larger, the trigger mechanism’s threshold shrinks, resulting in desired tracking performance and a shorter trigger period. Conversely, a longer trigger period is obtained when the error is smaller to save more communication resources. Moreover, in, ⁶⁰ Xing et al. considered an event-triggered mechanism to produce some conservatism in control design and stability analysis due to predetermined robust term ${\overline{m}}_1\text{tanh}(z_n{\overline{m}}_1/\epsilon )$ with ${\overline{m}}_1>[m_1/(1-\delta )]$ . To this end, the Hyperbolic Cosecant function csch(·) in this article is invoked to circumvent this problem. In addition, Zeno behavior of the proposed triggered mechanism can be excluded owing to the existence csch(϶). The detailed process of proof can be seen. As a result, the improved event-triggered mechanism can enhance desired tracking performance and reduce the conservatism for the proposed adaptive event-triggered control scheme to select appropriate parameters μ₀, ι, μ, μ₁ and ϶.

From equation (26), it can be deduced that $\left|\varpi (t)-v(t)\right|\le {\mu }_0\left|v(t)\right|+\mu Z_0+{\mu }_1\left|csch(v_n^3{Z}_{0}t+\mathit{∍})\right|t\in [t_k,t_{k+1})$ . Moreover, it can be obtained $\varpi \left(t\right)=(1+{\chi }_1\left(t\right){\mu }_0)v\left(t\right)+{\chi }_2\left(t\right)(\mu Z_0+{\mu }_1|csch(v_n^3{Z}_{0}t+\mathit{∍})\left|\right)$ with χ₂(t) = ±χ₁(t) and |χ₁(t)| ≤ 1 (The proving process as eqaution (50)–(52) in ⁴¹ ). Hence, we have

$$v\left(t\right)=\frac{\varpi \left(t\right)-{\chi }_2\left(t\right)\mu Z_0-{\chi }_2\left(t\right){\mu }_1\left|csch(v_n^3{Z}_{0}t+\mathit{∍})\right|}{1+{\chi }_1\left(t\right){\mu }_0}$$

(27)

Radial basis function neural network (RBF NN)

It is well known that RBFNN is widely used as a tool for modeling nonlinear functions since its good capability in function approximation in. ⁶¹ In structure, the RBFNN takes the form $W^T\left(Z\right)$ , where Z ∈ Ω _Z ∈ R ^q , q is the NN input dimension, and $W={[W_1,\dots ,W_l]}^T$ is the weight vector. The NN node number l > 1 and $S\left(Z\right)={[S_1,\dots ,S_l]}^T$ with S _i being chosen as the commonly used Gaussian functions, which is in the following form

$$S\left(Z\right)={\exp }^{\left(-{\left(Z-{\mu }_i\right)}^T\left(Z-{\mu }_i\right)/{\kappa }_i^2\right)}$$

(28)

where μ _i = [μ₁, …, μ _iq ] is the center of the receptive field and κi the width of the Gaussian function. It has been proved that the RBFNN can approximate any continuous function over a compact set Ω _Z ∈ R ^q arbitrary accuracy as

$$f\left(Z\right)=W^T\left(Z\right)S\left(Z\right)+\epsilon \left(Z\right)$$

(29)

where $\epsilon \left(Z\right)$ is the approximation error. The ideal weight vector W is an artificial quantity required for analytical purpose. W is defined as the value of $\widehat{W}$ which minimizes |ɛ| for all Z ∈ Ω _Z in a compact region, i.e.

$$W=\text{arg}\underset{\widehat{W}\in R^l}{\min }\left\{\text{sup}|f\left(Z\right)-W^{T}S\left(Z\right)|\right\}$$

(30)

Event-triggered adaptive finite-time controller design

With the backstepping design technique, the event-triggered adaptive finite-time NN control scheme is designed. First of all, define the tracking error variables as

$$\begin{array}{l}{z}_i=x_i-x_{i,c},i=1,\dots ,n,\hfill \end{array}$$

(31)

where x_1,c = y _d , and y _d is the reference signal. x _{i c} is the output of the following first-order filter with the intermediate controller signal as input signal α _i and α _i will be specified later. The dynamic surface is defined as following

$${\iota }_i{\dot{x}}_{i,c}={\alpha }_{i-1}-x_{i,c},{\alpha }_{i-1}\left(0\right)=x_{i,c}\left(0\right)$$

(32)

where ι _i > 0 is a design parameter, i = 2, …, n.

It is noticed that the errors of the filtering α_i−1 − x _{i c} can influence the control performance and thus, we introduce compensation signal q _i to eliminate the impact of the filtering α_i−1 − x _{i c} with q _i defined as follows

$$\{\begin{array}{l}{\dot{q}}_i=-c_i{q}_i+h_i\left(t\right)\left(x_{i+1,c}-{\alpha }_i\right)+h_i\left(t\right)q_{i+1}-l_i\tanh \left(\frac{q_i}{{\kappa }_{1,i}}\right),\hfill \\ {\dot{q}}_n=0,\hfill \end{array}$$

(33)

where l _i are design parameters, i = 1, …, n.

Next, the compensated tracking error signals are defined as follows

$$v_i=z_i-q_i,i=1,\dots ,n$$

(34)

In this paper, the virtual controller α _i and adaptive law $\widehat{\theta }$ are constructed as follows

$${\alpha }_i=N\left({\xi }_i\right){\overline{\alpha }}_i,\stackrel{.}{{\xi }_i}=v_i^3{\overline{\alpha }}_i,i=1,\dots ,n$$

(35)

$$\begin{array}{lll}{\overline{\alpha }}_1\hfill & =\hfill & {\lambda }_1{v}_1^{4\beta -3}\tanh \left(\frac{v_1^2}{{\kappa }_{\mathrm{2,1}}}\right)+\frac{3}{4}{v}_1+c_1{q}_1-{\dot{x}}_{1,c}+\frac{\widehat{\theta }}{2{\varsigma }_1^2}{v}_1^3{S}_1^T\left(Z_1\right)S_1\left(Z_1\right),\hfill \\ {\overline{\alpha }}_i\hfill & =\hfill & {\lambda }_i{v}_i^{4\beta -3}\tanh \left(\frac{v_i^2}{{\kappa }_{2,i}}\right)+v_i+c_i{q}_i-{\dot{x}}_{i,c}+\frac{\widehat{\theta }}{2{\varsigma }_i^2}{v}_i^3{S}_i^T\left(Z_i\right)S_i\left(Z_i\right),i=2,\dots ,n-1,\hfill \end{array}$$

(36)

$$\begin{array}{lll}{\overline{\alpha }}_n\hfill & =\hfill & {\lambda }_n{v}_n^{4\beta -3}\tanh \left(\frac{v_n^2}{{\kappa }_{2,n}}\right)+\frac{7}{4}{v}_n+\frac{1}{2}{v}_n^3-{\dot{x}}_{n,c}+\frac{\widehat{\theta }}{2{\varsigma }_n^2}{v}_n^3{S}_n^T\left(Z_n\right)S_n\left(Z_n\right),\hfill \\ \dot{\widehat{\theta }}\hfill & =\hfill & {\displaystyle \sum _{i=1}^n\frac{v_i^6{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)}{2{\varsigma }_i}}-\sigma \widehat{\theta },\hfill \end{array}$$

(37)

where $c_i,{\lambda }_i,{\varsigma }_i,\sigma ,{\kappa }_{2,i},0<\beta <1,\beta =\frac{4\overline{\aleph }-1}{4\overline{\aleph }+1}$ are positive design parameters and $\overline{\aleph }\ge 2$ is an integer. The parameter $\widehat{\theta }$ is the estimate of the unknown constant θ which is defined as

$$\theta =\max \left\{\mathrm{\Vert }{W}_i{\mathrm{\Vert }}^2,i=1,\dots ,n\right\}$$

(38)

where W _i are unknown weight parameter vector and will be specified later. The parameter estimation error is $\tilde{\theta }=\widehat{\theta }-\theta$ .

The detailed process of designing controllers is shown in the following

Step 1: According to v₁ = z₁ − q₁, z₁ = x₁ − x_1,c, one gets

$$\begin{array}{lll}d{v}_1\hfill & =\hfill & d{z}_1-d{q}_1=d{x}_1-d{x}_{1,c}-d{q}_1\hfill \\ d{v}_1\hfill & =\hfill & \left(h_1\left(t\right)x_2+f_1\left(X\right)-{\dot{x}}_{1,c}-{\dot{q}}_1\right)dt+g_1^T\left(t\right)d\omega \left(t\right)\hfill \end{array}$$

(39)

then, according to z₂ = x₂ − x_2,c yields that

$$d{v}_1=\left(h_1\left(t\right)\left(z_2+x_{2,c}\right)+f_1\left(X\right)-{\dot{x}}_{1,c}-{\dot{q}}_1\right)dt+g_1^T\left(t\right)d\omega \left(t\right)$$

(40)

Choose the Lyapunov function candidate as

$$V_{v_1}=\frac{1}{4}{v}_1^4$$

(41)

Then, combining the equation (2), the time derivative of $V_{v_1}$ is

$$\begin{array}{l}L{V}_{v_1}=v_1^3\left(h_1\left(t\right)z_2+h_1\left(t\right)\left(x_{2,c}-{\alpha }_1\right)+h_1\left(t\right){\alpha }_1+f_1\left(X\right)-{\dot{x}}_{1,c}\right)\\ +\frac{3}{2}{v}_1^2\text{Tr}\left\{g_1^T{g}_1)\right\}\end{array}$$

(42)

The compensating signal is defined as follow

$${\dot{q}}_1=-c_1{q}_1+h_1\left(t\right)\left(x_{2,c}-{\alpha }_1\right)+h_1\left(t\right)q_2-l_1\text{tanh}\left(\frac{q_1}{{\kappa }_{\mathrm{1,1}}}\right)$$

(43)

where κ_1,1 is designed parameter. Then, substituting equation (43) into (42) yield that

$$\begin{array}{l}L{V}_{v_1}=v_1^3\left(h_1\left(t\right)v_2+h_1\left(t\right){\alpha }_1+f_1\left(X\right)-{\dot{x}}_{1,c}+c_1{q}_1+l_1\tanh \left(\frac{q_1}{{\kappa }_{\mathrm{1,1}}}\right)\right)\\ +\frac{3}{2}{v}_1^{2}Tr\left\{g_1^T{g}_1)\right\}\end{array}$$

(44)

By utilizing Lemma 5, one has

$$h_1\left(t\right)v_1^3{v}_2\le \frac{3}{4}{h}_1^{4/3}{v}_1^4+\frac{1}{4}{v}_2^4$$

(45)

$$\frac{3}{2}{v}_1^2\text{Tr}\left\{g_1^T{g}_1)\right\}\le \frac{3}{4}{a}_1^2+\frac{3}{4a_1^2}{v}_1^4\mathrm{\Vert }{g}_1^T{g}_1){\mathrm{\Vert }}^4$$

(46)

where a₁ is designed parameter. Then, substituting equation (45)–(46) into (44) yield that

$$L{V}_{v_1}\le v_1^3\left(h_1\left(t\right){\alpha }_1+{\overline{f}}_1\left(Z_1\right)-{\dot{x}}_{1,c}+c_1{q}_1\right)+\frac{1}{4}{v}_2^4+\frac{3}{4}{a}_1^2$$

(47)

where $Z_1={\left[X\right]}^T$ . ${\overline{\phi }}_1\left(Z_1\right)=f_1(X)+l_1\text{tanh}(\frac{q_1}{{\kappa }_{\mathrm{1,1}}})+\frac{3}{4}{h}_1^{4/3}{v}_1+\frac{3}{4a_1^2}{v}_1\mathrm{\Vert }{g}_1^T{g}_1){\mathrm{\Vert }}^4$ , which is continuous. Thus, the nonlinear continuous function ${\overline{f}}_1\left(Z_1\right)$ can be approximated by RBFNN $W_1^T\left(Z_1\right)S_1\left(Z_1\right)$ such that

$${\overline{f}}_1\left(Z_1\right)=W_1^T\left(Z_1\right)S_1\left(Z_1\right)+{\epsilon }_1\left(Z_1\right)$$

(48)

By utilizing Lemma 5, one has

$$\begin{array}{l}{v}_1^3\left(W_1^T\left(Z_1\right)S_1\left(Z_1\right)+{\epsilon }_1\left(Z_1\right)\right)\le \frac{\theta }{2{\varsigma }_1^2}{v}_1^6{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)\\ +\frac{3}{4}{v}_1^4+\frac{{\varsigma }_1^2}{2}+\frac{1}{4}{\overline{\epsilon }}_1^4,\end{array}$$

(49)

where ${\overline{\epsilon }}_1^4$ is constant value. Then, substituting equation (48) and (49) into (47) yield that

$$\begin{array}{l}L{V}_{v_1}\le v_1^3\left(h_1\left(t\right){\alpha }_1+\frac{\theta }{2{\varsigma }_1^2}{v}_1^3{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)+c_1{q}_1+\frac{3}{4}{v}_1-{\dot{x}}_{1,c}\right)\\ +\frac{1}{4}{v}_2^4+\frac{3}{4}{a}_1^2+\frac{{\varsigma }_1^2}{2}+\frac{1}{4}{\overline{\epsilon }}_1^4\left(Z_1\right)\end{array}$$

(50)

Then, design the virtual controller α₁ as

$$\begin{array}{ll}{\alpha }_1\hfill & =N\left({\xi }_1\right){\overline{\alpha }}_1\hfill \\ {\overline{\alpha }}_1\hfill & ={\lambda }_1{v}_1^{4\beta -3}\text{tanh}\left(\frac{v_1^2}{{\kappa }_{\mathrm{2,1}}}\right)+\frac{3}{4}{v}_1+c_1{q}_1-{\dot{x}}_{1,c}+\frac{\widehat{\theta }}{2{\varsigma }_1^2}{v}_1^3{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)\hfill \end{array}$$

(51)

substituting equation (51) into (50) yield that

$$\begin{array}{l}L{V}_{v_1}{\le }_1^3\left(-{\lambda }_1{v}_1^{4\beta -3}\tanh \left(\frac{v_1^2}{{\kappa }_{\mathrm{2,1}}}\right)+h_{1}N\left({\xi }_1\right){\overline{\alpha }}_1+{\overline{\alpha }}_1-\frac{\tilde{\theta }}{2{\varsigma }_1^2}{v}_1^3{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)\right)\\ +\frac{1}{4}{v}_2^4+\frac{3}{4}{a}_1^2+\frac{{\varsigma }_1^2}{2}+\frac{1}{4}{\overline{\epsilon }}_1^4\left(Z_1\right)\end{array}$$

(52)

Define

$$\stackrel{.}{{\xi }_1}=v_1^3{\overline{\alpha }}_1$$

(53)

which yields

$$L{V}_{v_1}\le -{\lambda }_1{v}_1^{4\beta }\text{tanh}\left(\frac{v_1^2}{{\kappa }_{\mathrm{2,1}}}\right)+\left(h_1\left(t\right)N\left({\xi }_1\right)+1\right)\stackrel{.}{{\xi }_1}-\frac{\tilde{\theta }}{2{\varsigma }_1^2}{v}_1^6{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)+\frac{1}{4}{v}_2^4+c_1$$

(54)

where $c_1=\frac{3}{4}{a}_1^2+\frac{{\varsigma }_1^2}{2}+\frac{1}{4}{\overline{\epsilon }}_1^4\left(Z_1\right)$ . $\frac{1}{4}{v}_2^4$ will be removed in the next step.

Step 2 ≤ i ≤ n − 1: Choose the Lyapunov function candidate as

$$V_{v_i}=\frac{1}{4}{v}_i^4$$

(55)

Then, by following the similar analysis as equation (39) and (40), one has

$$\begin{array}{l}L{V}_{v_i}=v_i^3\left(h_i\left(t\right)z_{i+1}+h_i\left(t\right)\left(x_{i+1,c}-{\alpha }_i\right)+h_i\left(t\right){\alpha }_i+f_i\left(X\right)-{\dot{x}}_{i,c}\right)\\ +\frac{3}{2}{v}_i^{2}Tr\left\{g_i^T{g}_i)\right\}\end{array}$$

(56)

The compensating signal is defined as follow

$${\dot{q}}_i=-c_i{q}_i+h_i\left(t\right)\left(x_{i+1,c}-{\alpha }_i\right)+h_i\left(t\right)q_{i+1}-l_i\tanh \left(\frac{q_i}{{\kappa }_{1,i}}\right)$$

(57)

where κ_1,i is designed parameter. Then, substituting equation (57) into (56) yield that

$$L{V}_{v_i}\le v_i^3\left(h_i\left(t\right){\alpha }_i+{\overline{f}}_i\left(Z_i\right)-{\dot{x}}_{i,c}+c_i{q}_i\right)+\frac{1}{4}{v}_{i+1}^4+\frac{3}{4}{a}_i^2$$

(58)

where $Z_i={\left[X\right]}^T$ . ${\overline{\phi }}_i\left(Z_i\right)=f_i(X)+l_i\text{tanh}(\frac{q_i}{{\kappa }_{1,i}})+\frac{3}{4}{h}_i^{4/3}{v}_i+\frac{3}{4a_i^2}{v}_i\mathrm{\Vert }{g}_i^T{g}_i){\mathrm{\Vert }}^4$ , which is continuous. Thus, the nonlinear continuous function ${\overline{f}}_i\left(Z_i\right)$ can be approximated by RBFNN $W_i^T\left(Z_i\right)S_i\left(Z_i\right)$ such that

$${\overline{f}}_i\left(Z_i\right)=W_i^T\left(Z_i\right)S_i\left(Z_i\right)+{\epsilon }_i\left(Z_i\right)$$

(59)

By utilizing Lemma 5, one has

$$v_i^3\left(W_i^T\left(Z_i\right)S_i\left(Z_i\right)+{\epsilon }_i\left(Z_i\right)\right)\le \frac{\theta }{2{\varsigma }_i^2}{v}_i^6{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)+\frac{3}{4}{v}_i^4+\frac{{\varsigma }_i^2}{2}+\frac{1}{4}{\overline{\epsilon }}_i^4$$

(60)

where ${\overline{\epsilon }}_i^4$ is constant value. Then, substituting equation (59) and (60) into (58) yield that

$$\begin{array}{l}L{V}_{v_i}\le v_i^3\left(h_i\left(t\right){\alpha }_i+\frac{\theta }{2{\varsigma }_i^2}{v}_i^3{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)+c_i{q}_i+\frac{3}{4}{v}_i-{\dot{x}}_{i,c}\right)\\ +\frac{1}{4}{v}_2^4+\frac{3}{4}{a}_i^2+\frac{{\varsigma }_i^2}{2}+\frac{1}{4}{\overline{\epsilon }}_i^4\left(Z_i\right)\end{array}$$

(61)

Then, design the virtual controller α _i as

$$\begin{array}{lll}{\alpha }_i\hfill & =\hfill & N\left({\xi }_i\right){\overline{\alpha }}_i,\stackrel{.}{{\xi }_i}=v_i^3{\overline{\alpha }}_i,\hfill \\ {\overline{\alpha }}_i\hfill & =\hfill & {\lambda }_i{v}_i^{4\beta -3}\tanh \left(\frac{v_i^2}{{\kappa }_{2,i}}\right)+v_i+c_i{q}_i-{\dot{x}}_{i,c}+\frac{\widehat{\theta }}{2{\varsigma }_i^2}{v}_i^3{S}_i^T\left(Z_i\right)S_i\left(Z_i\right),\hfill \end{array}$$

(62)

substituting (62) into (61) yield that

$$\begin{array}{l}L{V}_{v_i}\le -{\lambda }_i{v}_i^{4\beta }\tanh \left(\frac{v_i^2}{{\kappa }_{2,i}}\right)+\left(h_i\left(t\right)N\left({\xi }_i\right)+1\right)\stackrel{.}{{\xi }_i}-\frac{\tilde{\theta }}{2{\varsigma }_i^2}{v}_i^6{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)\\ -\frac{1}{4}{v}_i^4+\frac{1}{4}{v}_{i+1}^4+c_i\end{array}$$

(63)

where $c_i=\frac{3}{4}{a}_i^2+\frac{{\varsigma }_i^2}{2}+\frac{1}{4}{\overline{\epsilon }}_i^4\left(Z_i\right)$ . $\frac{1}{4}{v}_{i+1}^4$ will be removed in the next step.

Step n: According to v _n = z _n − q _n , one has

$$d{v}_n=\left(h_n\left(t\right)u\left(v\left(t\right)\right)+f_n\left(X\right)-{\dot{x}}_{n,c}-{\dot{q}}_n\right)dt+g_n^T\left(t\right)d\omega \left(t\right)$$

(64)

Choose the Lyapunov function candidate as

$$V_{v_n}=\frac{1}{4}{v}_n^4$$

(65)

Then, we can obtain the time derivative of $V_{v_n}$

$$L{V}_{v_n}=v_n^3\left(h_n\left(t\right)\left\{m\left(t\right)v\left(t\right)+〉\left(t\right)\right\}+f_n\left(X\right)-{\dot{x}}_{n,c}\right)+\frac{3}{2}{v}_n^{2}Tr\left\{g_n^T{g}_n\right\}$$

(66)

By utilizing Lemma 5, one has

$$v_n^3\mathrm{\varrho }\le \frac{3}{4}{v}_n^4+\frac{1}{4}{\overline{\mathrm{\varrho }}}^{\mathrm{4}}$$

(67)

$$\frac{3}{2}{v}_n^{2}Tr\left\{g_n^T{g}_n\right\}\le \frac{3}{4}{a}_n^2+\frac{3}{4a_n^2}{v}_n^4\mathrm{\Vert }{g}_n^T{g}_n{\mathrm{\Vert }}^4$$

(68)

where a _n is designed parameter. Then, substituting equation (67) and (68) into (66) yield that

$$L{V}_{v_n}\le v_n^3\left(h_n\left(t\right)\overline{m}v\left(t\right)+{\overline{f}}_n\left(Z_n\right)-{\dot{x}}_{n,c}+\frac{3}{4}{v}_n\right)+\frac{3}{4}{a}_n^2+\frac{1}{4}{\overline{〉}}^4$$

(69)

where $Z_n={\left[X\right]}^T.{\overline{f}}_n\left(Z_n\right)=f_n\left(X\right)+\frac{3}{4a_n^2}{v}_n\mathrm{\Vert }{g}_n^T{g}_n{\mathrm{\Vert }}^4$ , which is continuous. Thus, the nonlinear continuous function ${\overline{f}}_n\left(Z_n\right)$ can be approximated by RBFNN $W_n^T\left(Z_n\right)S_n\left(Z_n\right)$ such that

$${\overline{f}}_n\left(Z_n\right)=W_n^T\left(Z_n\right)S_n\left(Z_n\right)+{\epsilon }_n\left(Z_n\right)$$

(70)

By utilizing Lemma 5, one has

$$v_n^3\left(W_n^T\left(Z_n\right)S_n\left(Z_n\right)+{\epsilon }_n\left(Z_n\right)\right)\le \frac{\theta }{2{\varsigma }_n^2}{v}_n^6{S}_n^T\left(Z_n\right)S_n\left(Z_n\right)\frac{3}{4}{v}_n^4+\frac{{\varsigma }_n^2}{2}+\frac{1}{4}{\overline{\epsilon }}_n^4$$

(71)

where ${\overline{\epsilon }}_n^4$ is constant value. Then, substituting equation (70) and (71) into (69) yield that

$$\begin{array}{l}L{V}_{v_n}\le v_n^3\left(h_n\left(t\right)\overline{m}v\left(t\right)-{\dot{x}}_{n,c}+\frac{3}{2}{v}_n+\frac{\theta }{2{\varsigma }_n^2}{v}_n^3{S}_n^T\left(Z_n\right)S_n\left(Z_n\right)\right)\\ +\frac{3}{4}{a}_n^2+\frac{1}{4}{\overline{\mathrm{\varrho }}}^4+\frac{{\varsigma }_n^2}{2}+\frac{1}{4}{\overline{\epsilon }}_n^4\left(Z_n\right)\end{array}$$

(72)

From equation (27), the actual control signal ϖ(t) as

$$\varpi \left(t\right)={\alpha }_n-\iota Z_0\tanh \left(\frac{\iota Z_0{V}_n^3}{o}\right)$$

(73)

where $Z_0=1/\left[\left({\displaystyle {\sum }_{i=1}^n{z}_i}\right)+{\mu }_0\right]$ . ι > μ > 0 and o are design parameters. α _n is the intermediate controller signal in equation (35). Then, by combining equation (73) and (27), it can be obtain

$$\begin{array}{l}{v}_n^3{h}_n\overline{m}v\left(t\right)=\frac{v_n^3{h}_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}\left({\alpha }_n-\iota Z_0\tanh \left(\frac{\iota Z_0{V}_n^3}{o}\right)\right)-\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}\mu v_n^3{Z}_0{\chi }_2\left(t\right)\\ -\frac{h_n\overline{m}|\csc h\left(v_n^3{Z}_{0}t+\text{'}\right)|}{1+{\chi }_1\left(t\right){\mu }_0}{v}_n^3{\chi }_2\left(t\right)\\ \le \frac{v_n^3{h}_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}{\alpha }_n-\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}\iota v_n^3{Z}_0\tanh \left(\frac{\iota Z_0{V}_n^3}{o}\right)\\ +\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}\left|\mu v_n^3{Z}_0\right|+\frac{h_n\overline{m}|\csc h(\text{'})|}{1+{\chi }_1\left(t\right){\mu }_0}\left|v_n^3\right|\end{array}$$

(74)

According to Lemma 5 and Lemma 7, it follows that

$$-\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}\iota v_n^3{Z}_0\tanh \left(\frac{\iota Z_0{V}_n^3}{o}\right)+\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}\left|\mu v_n^3{Z}_0\right|\le \frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}0.2785o$$

(75)

$$\frac{h_n\overline{m}|\csc h(\text{'})|}{1+{\chi }_1\left(t\right){\mu }_0}\left|v_n^3\right|\le \frac{1}{2}{v}_n^6+\frac{1}{2}{\left[\frac{h_n\overline{m}|\csc h(∍)|}{1+{\chi }_1\left(t\right){\mu }_0}\right]}^2$$

(76)

Bringing the above inequalities into equation (74) yields

$$v_n^3{h}_n\overline{m}v\left(t\right)\le v_n^3\left(\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}{\alpha }_n+\frac{1}{2}{v}_n^3\right)+\frac{h_n\overline{m}}{1+{\mu }_0}0.2785o+\frac{1}{2}{\left[\frac{h_n\overline{m}|\csc h(∍)|}{1+{\chi }_1\left(t\right){\mu }_0}\right]}^2$$

(77)

then, substituting equation (77) into (72) yield that

$$\begin{array}{l}L{V}_{v_n}\le v_n^3\left(\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}{\alpha }_n+\frac{1}{2}{v}_n^3-{\dot{x}}_{n,c}+\frac{3}{2}{v}_n+\frac{\theta }{2{\varsigma }_n^2}{v}_n^3{S}_n^T\left(Z_n\right)S_n\left(Z_n\right)\right)\\ +\frac{3}{4}{a}_n^2+\frac{1}{4}{\overline{〉}}^4+\frac{{\varsigma }_n^2}{2}+\frac{1}{4}{\overline{\epsilon }}_n^4\left(Z_n\right)+\frac{h_n\overline{m}}{1+{\mu }_0}0.2785o+\frac{1}{2}{\left[\frac{h_n\overline{m}|\csc h(∍)|}{1+{\chi }_1\left(t\right){\mu }_0}\right]}^2\end{array}$$

(78)

Then, design the intermediate controller α _n as

$$\begin{array}{ll}{\alpha }_n\hfill & =N\left({\xi }_n\right){\overline{\alpha }}_n,\stackrel{.}{{\xi }_n}=v_n^3{\overline{\alpha }}_n,\hfill \\ {\overline{\alpha }}_n\hfill & ={\lambda }_n{v}_n^{4\beta -3}\tanh \left(\frac{v_n^2}{{\kappa }_{2,n}}\right)+\frac{7}{4}{v}_n+\frac{1}{2}{v}_n^3-{\dot{x}}_{n,c}+\frac{\widehat{\theta }}{2{\varsigma }_n^2}{v}_n^3{S}_n^T\left(Z_n\right)S_n\left(Z_n\right),\hfill \end{array}$$

(79)

substituting equation (79) into (78) gets

$$L{V}_{v_n}\le -{\lambda }_n{v}_n^{4\beta }\tanh \left(\frac{v_n^2}{{\kappa }_{2,n}}\right)+\left(m^{\ast }N\left({\xi }_n\right)+1\right)\stackrel{.}{{\xi }_n}-\frac{\tilde{\theta }}{2{\varsigma }_n^2}{v}_n^6{S}_n^T\left(Z_n\right)S_n\left(Z_n\right)-\frac{1}{4}{v}_n^4+c_n+\overline{c}$$

(80)

where $m^{\ast }=\frac{h_n\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0},c_n=\frac{3}{4}{a}_n^2+\frac{{\varsigma }_n^2}{2}+\frac{1}{4}{\overline{\epsilon }}_n^4\left(Z_n\right)$ and $\overline{c}=\frac{h_n\overline{m}}{1+{\mu }_0}0.2785o+\frac{1}{2}{\left[\frac{h_n\overline{m}\left|csch\left(\mathit{∍}\right)\right|}{1+{\chi }_1\left(t\right){\mu }_0}\right]}^2+\frac{1}{4}{\overline{\mathrm{\varrho }}}^4$ .

In the end, the block diagram of the event-triggered adaptive finite-time control strategy is shown in Figure 1.

Figure  1.

Block diagram of the event-triggered adaptive finite-time control strategy.

Stability analysis

Theorem 1. Consider the non-strict feedback stochastic systems equation (17) subject to unknown time-varying control directions and input nonlinearities under Assumptions 1 and 2. Then, based on the event-triggered mechanism equation (26), the error compensation mechanism equation (33), the controller and the adaptive law equation (35) – (37), it can guarantee that the SGFSP of overall the signals in the stochastic systems, the tracking error converges to arbitrarily small residual set of the origin within in a finite time, and the Zeno behavior can be excluded.

Proof: Now, choose the Lyapunov function as

$$V={\displaystyle \sum _{i=1}^n{V}_{v_i}}+\frac{{\tilde{\theta }}^2}{2{\varsigma }_i}$$

(81)

where ς _i is a design parameter. From equation (54), (63) and (80), according to the $\tilde{\theta }=\widehat{\theta }-\theta$ , then, the time derivative of V can obtained

$$\begin{array}{l}LV\le -{\displaystyle \sum _{i=1}^n{\lambda }_i}{v}_i^{4\beta }\tanh \left(\frac{v_i^2}{{\kappa }_{2,n}}\right)+{\displaystyle \sum _{i=1}^n\left({\mathrm{\Psi }}_{i}N\left({\xi }_i\right)+1\right)}\stackrel{.}{{\xi }_i}\\ -{\displaystyle \sum _{i=1}^n\frac{\tilde{\theta }\widehat{\theta }}{{\varsigma }_i}}+{\displaystyle \sum _{i=1}^n{c}_i}+\overline{c}\end{array}$$

(82)

where Ψ _i = max{h _i , m*} with $m^{\ast }=\frac{h_i\overline{m}}{1+{\chi }_1\left(t\right){\mu }_0}$ , $c_i=\frac{3}{4}{a}_i^2+\frac{{\varsigma }_i^2}{2}+\frac{1}{4}{\overline{\epsilon }}_i^4\left(Z_i\right)$ , $\overline{c}=\frac{h_i\overline{m}}{1+{\mu }_0}0.2785o+\frac{1}{2}{\left[\frac{h_i\overline{m}\left|csch\left(\mathit{∍}\right)\right|}{1+{\chi }_1\left(t\right){\mu }_0}\right]}^2+\frac{1}{4}{\overline{\mathrm{\varrho }}}^4$ .

Then, combining the definition of θ, the following inequality holds

$$-\frac{\sigma \tilde{\theta }\widehat{\theta }}{{\varsigma }_i}=-\frac{\sigma \tilde{\theta }\left(\tilde{\theta }+\theta \right)}{{\varsigma }_i}\le -\frac{\sigma {\tilde{\theta }}^2}{{\varsigma }_i}+\frac{\sigma {\theta }^2}{{\varsigma }_i}$$

(83)

Then, substituting equation (83) into (82) yield that

$$\begin{array}{l}LV\le -{\displaystyle \sum _{i=1}^n{\lambda }_i}{v}_i^{4\beta }\tanh \left(\frac{v_i^2}{{\kappa }_{2,i}}\right)+{\displaystyle \sum _{i=1}^n\left({\mathrm{\Psi }}_{i}N\left({\xi }_i\right)+1\right)}\stackrel{.}{{\xi }_i}\\ +{\displaystyle \sum _{i=1}^n\frac{\sigma {\theta }^2}{{\varsigma }_i}}-{\displaystyle \sum _{i=1}^n{\left(\frac{\sigma {\tilde{\theta }}^2}{{\varsigma }_i}\right)}^{\beta }}+{\displaystyle \sum _{i=1}^n{\left(\frac{\sigma {\tilde{\theta }}^2}{{\varsigma }_i}\right)}^{\beta }}+{\displaystyle \sum _{i=1}^n{c}_i}+\overline{c}\end{array}$$

(84)

According to Lemma 4, by choosing $y=\psi =1,x=\frac{\sigma {\tilde{\theta }}^2}{{\varsigma }_i},c=\beta$ and d = 1 − β, the following inequality holds

$${\left(\frac{\sigma {\tilde{\theta }}^2}{{\varsigma }_i}\right)}^{\beta }\le \beta \frac{\sigma {\tilde{\theta }}^2}{{\varsigma }_i}+\left(1-\beta \right)$$

(85)

substituting equation (85) into (84), we have

$$\begin{array}{l}LV\le -{\displaystyle \sum _{i=1}^n{\lambda }_i}{v}_i^{4\beta }\tanh \left(\frac{v_i^2}{{\kappa }_{2,i}}\right)-{\displaystyle \sum _{i=1}^n{\left(\frac{\sigma {\tilde{\theta }}^2}{{\varsigma }_i}\right)}^{\beta }}+{\displaystyle \sum _{i=1}^n\left({\mathrm{\Psi }}_{i}N\left({\xi }_i\right)+1\right)}\stackrel{.}{{\xi }_i}\\ +{\displaystyle \sum _{i=1}^n\frac{\sigma {\theta }^2}{{\varsigma }_i}}+{\displaystyle \sum _{i=1}^n{c}_i}+\overline{c}+\left(1-\beta \right)\end{array}$$

(86)

According to Lemma 3, we have

$$LV\le -C_1{V}^{\beta }\left(x\right)+\left(\mathrm{\Psi }N\left(\xi \right)+1\right)\dot{\xi }+C$$

(87)

therefore, based on Lemma1, the stochastic systems are SGFSP. Where

$\begin{array}{l}\left(\mathrm{\Psi }N\left(\xi \right)+1\right)\dot{\xi }={\displaystyle {\sum }_{i=1}^n\left({\mathrm{\Psi }}_{i}N\left({\xi }_i\right)+1\right)}\stackrel{.}{{\xi }_i},\hfill \\ C_1=\min \left\{4^{\beta }{\lambda }_i\tanh \left(\frac{v_i^2}{{\kappa }_{2,i}}\right),{\left(2\sigma \right)}^{\beta },i=1,\dots ,n\right\},\hfill \\ C={\displaystyle {\sum }_{i=1}^n\frac{\sigma {\theta }^2}{{\varsigma }_i}}+{\displaystyle {\sum }_{i=1}^n{c}_i}+\overline{c}+\left(1-\beta \right)\hfill \end{array}$ Moreover, for any 0 < γ < 1, according to the Lemma 1, we can further deduce that V will converge to the set $\mathrm{\Theta }=\left\{x|E\left[V\right]\le {\left\{\frac{C}{(1-\gamma )C_2}\right\}}^{1/\beta }\right\}$ in finite-time T.

Let $\mathrm{ϵ}={\left\{\frac{C}{(1-\gamma )C_2}\right\}}^{1/\beta }$ , it’s evident that

$$\left[E\left(e_i^2\right)\right]\le {\left[E\left(e_i^4\right)\right]}^{\frac{1}{2}}\le {\left(4EV\right)}^{\frac{1}{2}}\le 2\sqrt{\epsilon },\forall t\ge T$$

(88)

and

$$\left[E\left({\tilde{\theta }}^2\right)\right]\le 2\varsigma \left(EV\right)\le 2\varsigma \mathrm{ϵ}$$

(89)

where ς is the upper bound of ς _i . That is, the closed-loop signals v _i and $\tilde{\theta }$ are bounded in sense of mean square value.

According to the definition of v _i = z _i − q _i , the tracking error z _i can converge to a small residual set of the origin in a finite time when q _i is convergent in a finite time. Next, we will prove that the compensating signal q _i is convergent within in a finite time T₁. The Lyapunov function candidate is chosen as $V_Q={\displaystyle {\sum }_{i=1}^n\frac{1}{2}}{q}_i^2$ . From the time derivative of V _Q can be obtained

$$L{V}_Q={\displaystyle \sum _{i=1}^{n-1}{q}_i}{\dot{q}}_i={\displaystyle \sum _{i=1}^{n-1}{q}_i}\left(-c_i{q}_i+h_i\left(t\right)\left(x_{i+1,c}-{\alpha }_i\right)+h_i\left(t\right)q_{i+1}-l_i\tanh \left(\frac{q_i}{{\kappa }_{1,i}}\right)\right)$$

(90)

Moreover, it can deduce that $\left(x_{i+1,c}-{\alpha }_i\right)\le {\vartheta }_i$ can be achieved in a finite time T₂ with a known constant ϑ _i . According to the Lemma 5, it can be obtained

$$h_i\left(t\right)q_i{q}_{i+1}\le \frac{1}{2{\upsilon }_i^2}{q}_i^2+\frac{{\overline{h}}^2{\upsilon }_i^2}{2}{q}_{i+1}^2$$

(91)

where $\overline{h}=\max \left\{\right|h_i|,i=1,\dots ,n\}$ . Substituting equation (91) into (90), we have

$$L{V}_Q\le -\left(c_1-\frac{1}{2{\upsilon }_1^2}\right)q_1^2-{\displaystyle \sum _{i=2}^{n-1}\left(c_i-\frac{1}{2{\upsilon }_i^2}-\frac{{\overline{h}}^2{\upsilon }_i^2}{2}\right)}\hspace{0.17em}{q}_i^2-{\displaystyle \sum _{i=1}^n{q}_i}\left(l_i-\overline{h}{\vartheta }_i\right)$$

(92)

When we choose the appropriate parameter to make $l_i\ge \overline{h}{\vartheta }_i$ hold. Then, combining Lemma 3, we can obtain

$$L{V}_Q\le -Q_1{V}_Q-Q_2{V}_Q^{\frac{1}{2}},$$

(93)

where $Q_1=\min \{2c_1-\frac{1}{{\upsilon }_1^2},2c_i-\frac{1}{{\upsilon }_i^2}-{\overline{h}}^2{\upsilon }_i^2,i=1,\dots ,n-1\}$ and $Q_2=\min \{\sqrt{2}(l_i-\overline{h}{\vartheta }_i),i=1,\dots ,n\}$ . Hence, we can know the V _Q from Lemma 2 that can converge to the origin in a finite time T₁.

From the $E\left(\right|y-y_d{|}^2)=E\left(\right|z_1{|}^2)\le 2\sqrt{\mathrm{ϵ}}$ can attained. Thus, the tracking error z₁ can converge to a small residual set of the origin in a finite time $\overline{T}=\max \{T,T_1+T_2\}$ .

Now, to demonstrate Zeno behavior will be excluded, let existing positive number $\overline{t}$ with $\{t_{k+1}-t_k\}\ge \overline{t},k\in R^{+}$ . Moreover, the $\left|e(t)\right|\hspace{0.17em}\le {\mu }_0\left|v(t)\right|+\mu Z_0+{\mu }_1|csch(v_n^3{Z}_{0}t+\mathit{∍})|,\forall t\in [t_k,t_{k+1})$ can be derived by. Then

$$\frac{d}{dt}\left|e\right|\hspace{0.17em}\le \frac{d}{dt}{\left(ee\right)}^{\frac{1}{2}}=sign\left(e\right)\dot{e}\le \dot{\varpi }$$

(94)

Combining the definition of ϖ(t) and Ito Lemma, the $\dot{\varpi }$ can be given

$$d\varpi =L\varpi +{\left(\frac{\partial \varpi }{\partial x_n}{H}_n\right)}^{T}d\omega$$

(95)

where

$$\begin{array}{l}L\varpi ={\displaystyle \sum _{j=1}^n\frac{\partial \varpi }{\partial x_j}}\left[f_i+x_{j+1}\right]+\frac{\partial \varpi }{\partial \widehat{\theta }}\stackrel{.}{\widehat{\theta }}\\ \hspace{0.17em}+{\displaystyle \sum _{j=1}^n\left(\frac{\partial \varpi }{\partial q_j}{\dot{q}}_j+\frac{\partial \varpi }{\partial x_{j,c}}{\dot{x}}_{j,c}\right)}+\frac{1}{2}{\displaystyle \sum _{j,\hslash =1}^n\frac{{\partial }^2\varpi }{\partial x_j\partial x_{\hslash }}}Tr\left\{H_j^T{H}_{\hslash }\right\}\end{array}$$

(96)

From equation (96), $\dot{\varpi }$ is continuous function of bounded variables $X,\widehat{\theta },{\overline{q}}_n,{\overline{x}}_{n,c},and{\dot{x}}_{1,c}$ , which implies $\left|\dot{\varpi }\right|$ existing a bound. Combining the Lemma 6, it can be distinctly obtained that $\left|\dot{\varpi }\right|<\ell$ and ℓ is a positive number. Furthermore, it can be acquired that ${\text{lim}}_{t\to t_{k+1}}={\mu }_0\left|v\left(t\right)\right|+\mu Z_0+{\mu }_1\left|csch(v_n^3{Z}_{0}t+\mathit{∍})\right|$ and e(t _k ) = 0.

Applying the Lagrange Mean Value Theorem, it follows that

$$e\left(t_{k+1}\right)-e\left(t_k\right)\le e\left(t_{\chi }\right)\left(t_{k+1}-t_k\right)t_{\chi }\in \left(t_k{t}_{k+1}\right)$$

(97)

According to (94), (97), ${\text{lim}}_{t\to t_{k+1}}={\mu }_0\left|v\left(t\right)\right|+\mu Z_0+{\mu }_1\left|csch(v_n^3{Z}_{0}t+\mathit{∍})\right|$ and e(t _k ) = 0 one gets

$$\mu \left(\frac{1}{{\mu }_0}\right)+{\mu }_1\csc h(∍)\le {\mu }_0\left|v\left(t\right)\right|+\mu Z_0+{\mu }_1|\csc h\left(v_n^3{Z}_{0}t+∍\right)|\le \ell \left(t_{k+1}-t_k\right)$$

(98)

From equation (98), it can be deduced that $\overline{t}=\hspace{0.17em}\hspace{0.17em}|t_{k+1}-t_k|\hspace{0.17em}\hspace{0.17em}\ge \frac{\mu (1/z_0)+{\mu }_{1csch\left(\mathit{∍}\right)}}{\ell }$ . Therefore, the Zeno behavior is avoided completely in the proposed relative threshold triggered mechanism.

Remark 3. Note that the first-order derivative of the controllers does not appear as ”singularity” in this paper due to the presence of tanh(·). In, ⁶² the derivative of the controller appears as $v_i^{4\beta -4}$ , which is obviously 4β − 4 < 0. Thus, when v _i approaches zero, the ”singularity” can be encountered in the first-order derivative of the controllers.

Simulation

In this section, two simulation examples are given to demonstrate the validity of the proposed event-triggered adaptive neural finite-time control scheme.

Example 1: Consider the following non-strict feedback stochastic nonlinear system subject to unknown time-varying control directions, and input nonlinearities

$$\{\begin{array}{l}d{x}_1=\left(h_1\left(t\right)x_2+x_1{x}_2^2\right)dt+0.1x_2\cos \left(x_1\right)d\omega \left(t\right),\hfill \\ d{x}_2=\left(h_2\left(t\right)uv\left(t\right)\right)+x_1\cos \left(x_2^2\right)dt+0.5x_2{e}^{-x_1^2}\cos \left(x_1\right)d\omega \left(t\right),\hfill \\ y=x_1,\hfill \end{array}$$

(99)

where h₁(t) = −5 + 0.1 sin(0.8t) and h₂(t) = −2 + 0.1 sin(3t) represent the unknown time-varying control direction. The desired trajectory is selected as y _d = sin(t).

The input nonlinearities of system are described below

$$u\left(v\left(t\right)\right)=\{\begin{array}{l}1.5,if1.5\le v\left(t\right),\hfill \\ v\left(t\right),if-0.5<v\left(t\right)<1.5\hfill \\ -0.5,ifv\left(t\right)\le -\mathrm{0.5.}\hfill \end{array}$$

(100)

In the simulation experiment, design the compensating signals as follow

$$\begin{array}{l}{\dot{q}}_1=-c_1{q}_1+h_1\left(t\right)\left(x_{2,c}-{\alpha }_1\right)+h_1\left(t\right)q_2-l_1\tanh \left(\frac{q_1}{{\kappa }_{\mathrm{1,1}}}\right),\hfill \\ {\dot{q}}_2=0.\hfill \end{array}$$

(101)

In addition, the virtual controllers α _i , and adaptive law $\widehat{\theta }$ are designed as follow

$${\alpha }_i=N\left({\xi }_i\right){\overline{\alpha }}_i,\stackrel{.}{{\xi }_i}=v_i^3{\overline{\alpha }}_i,i=\mathrm{1,2}$$

(102)

$$\begin{array}{lll}{\overline{\alpha }}_1\hfill & =\hfill & {\lambda }_1{v}_1^{4\beta -3}\tanh \left(\frac{v_1^2}{{\kappa }_{\mathrm{2,1}}}\right)+\frac{3}{4}{v}_1+c_1{q}_1-{\dot{x}}_{1,c}+\frac{\widehat{\theta }}{2{\varsigma }_1^2}{v}_1^3{S}_1^T\left(Z_1\right)S_1\left(Z_1\right),\hfill \\ {\overline{\alpha }}_2\hfill & =\hfill & {\lambda }_2{v}_2^{4\beta -3}\tanh \left(\frac{v_2^2}{{\kappa }_{\mathrm{2,2}}}\right)+\frac{7}{4}{v}_2+\frac{1}{2}{v}_2^3-{\dot{x}}_{2,c}+\frac{\widehat{\theta }}{2{\varsigma }_2^2}{v}_2^3{S}_2^T\left(Z_2\right)S_2\left(Z_2\right),\hfill \\ \dot{\widehat{\theta }}\hfill & =\hfill & {\displaystyle \sum _{i=1}^2\frac{e_i^6{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)}{2{\varsigma }_i}}-\sigma \widehat{\theta }.\hfill \end{array}$$

(103)

Moreover, the event-triggered controller is designed as follow

$$\begin{array}{l}\varpi \left(t\right)={\alpha }_n-\iota Z_0\tanh \left(\frac{\iota Z_0{V}_n^3}{o}\right),\hfill \\ v\left(t\right)=\varpi \left(t_k\right)\forall t\in \left(t_k,t_{k+1}\right),k\in R^{+},\hfill \\ t_{k+1}=\inf \left\{t\in R^{+}\left|\right|e\left(t\right)|\hspace{0.17em}\hspace{0.17em}\ge {\mu }_0|v\left(t\right)|+\mu Z_0+{\mu }_1|\csc h(v_n^3{Z}_{0}t+∍)|\right\}.\hfill \end{array}$$

(104)

In this simulation, the $e^{{\xi }^2/2}(k^2+2)\sin \left(\xi \right)$ function is employed as Nussbaum function. All initial values are 0. Then, select the design parameters as $c_1=5,{\kappa }_{\mathrm{1,1}}=1,{\lambda }_1=10,{\lambda }_1=6,{\kappa }_{\mathrm{2,1}}={\kappa }_{\mathrm{2,2}}=1/2000,{\varsigma }_1={\varsigma }_2=0.2,\sigma =0.1,\overline{\aleph }=250,\aleph =50,\hspace{0.17em}{\iota }_1=1/4\iota =5,o=0.0001,$ ${\mu }_0=0.2,\mu =0.05,{\mu }_1=0.2,z_0=30,$ and ϶ = 1.

The simulation results are presented by Figure 2. From Figure 2, the trajectory of the system output x₁ and the desired trajectory y _d are shown. The tracking error z₁ is illustrated in Figure 3 to converge to a small residual set of the origin within in a finite time. Output of actuator nonlinearities u(v(t)) and event-triggered control input v(t) are shown in Figure 4. Time-triggered control signal ϖ(t) and event-triggered control input v(t) are illustrated in Figure 5. The curves of ξ₂ and N(ξ₂) are presented in Figure 6. The system state x₂ and the adaptive parameter $\widehat{\theta }$ are illustrated in Figure 7 in Figure 8, respectively. Time intervals t_k+1 − t _k of event-triggered mechanism are shown in Figure 9.

Figure  2.

Trajectories of x₁ and y _d (Example 1).

Figure  3.

The tracking error z₁ (Example 1).

Figure  4.

Trajectories u(v(t)) and v(t) (Example 1).

Figure  5.

Trajectories ϖ(t) and v(t) (Example 1).

Figure  6.

Trajectories ξ₂ and N(ξ₂) (Example 1).

Figure  7.

The state x₂ (Example 1).

Figure  8.

Adaptive parameter $\widehat{\theta }$ (Example 1).

Figure  9.

Intervals t_k+1 − t _k of triggering events (Example 1).

Example 2: Consider a mass-spring-damper system in ⁶³ and, ³³ the dynamic equation is described as follow

$$\{\begin{array}{l}d{x}_1=h_1\left(t\right)x_{2}dt,\hfill \\ d{x}_2=\left(h_2\left(t\right)u\left(v\left(t\right)\right)-\frac{1}{m}f(x_1,x_2)\right)dt,-\frac{1}{m}g\left(x_1,x_2\right)d\omega \left(t\right)\hfill \\ y=x_1,\hfill \end{array}$$

(105)

where h₁(t) = −2 + 0.5 sin(5), h₂(t) = −2 + 0.1 sin(3t), $f(x_1,x_2)=2x_1^2+x_1^3\sin \left(x_1{x}_2\right)+0.2x_2^2\cos \left(x_2^2\right)$ , g(x₁, x₂) = 0.6x₁x₂,and m = 2. The desired trajectory is selected as y _d = sin(t). The input nonlinearities of system are described below

$$u\left(v\left(t\right)\right)=\{\begin{array}{l}5,if5\le v\left(t\right),\hfill \\ v\left(t\right),if-5<v\left(t\right)<5,\hfill \\ -5,ifv\left(t\right)\le -5.\hfill \end{array}$$

(106)

It is noting that the first subsystem is a linear differential equation. Then, the virtual control ${\overline{\alpha }}_1$ is described as ${\overline{\alpha }}_1={\lambda }_1{v}_1^{4\beta -3}\text{tanh}\left(\frac{v_1^2}{{\kappa }_{\mathrm{2,1}}}\right)+\frac{3}{4}{v}_1+c_1{q}_1-{\dot{x}}_{1,c}$ and other controls similar to (102). In this simulation, the $e^{{\xi }^2/2}(k^2+2)\sin \left(\xi \right)$ function is employed as Nussbaum function. The initial states are chosen as $x_1\left(0\right)=x_2\left(0\right)=0.1$ , and the other initial values are 0. Then, select the design parameters as $c_1=9,{\kappa }_{\mathrm{1,1}}=1,{\lambda }_1=14,{\lambda }_1=1,{\kappa }_{\mathrm{2,1}}={\kappa }_{\mathrm{2,2}}=1/2000,{\varsigma }_1={\varsigma }_2=0.1,\sigma =0.1,\overline{\mathbf{\aleph }}=250,\mathbf{\aleph }=50,\hspace{0.17em}{\iota }_1=1/9\iota =5,$ $o=0.0001,{\mu }_0=0.2,\mu =0.05,z_0=30,{\mu }_1=0.2,$ and ϶ = 1.

The simulation results are obtained as Figure 10. Specifically, From Figure 10, the trajectory of the system output x₁ and the desired trajectory y _d are presented. The tracking error z₁ is illustrated in Figure 11 to converge to a small residual set of the origin in a finite time. Output of actuator nonlinearities u(v(t)) and event-triggered control input v(t) are shown in Figure 12. Time-triggered control signal ϖ(t) and event-triggered control input v(t) are illustrated in Figure 13. The curves of ξ₂ and N(ξ₂) are presented in Figure 14. The system state x₂ and the adaptive parameter $\widehat{\theta }$ are exhibited in Figure 15 in Figure 16, respectively. Time intervals t_k+1 − t _k of event-triggered mechanism are shown in Figure 17.

Figure  10.

Trajectories of x₁ and y _d (Example 2).

Figure  11.

The tracking error z₁ (Example 2).

Figure  12.

Trajectories u(v(t)) and v(t) (Example 2).

Figure  13.

Trajectories ϖ(t) and v(t) (Example 2).

Figure  14.

Trajectories of ξ₂ and N(ξ₂) (Example 2).

Figure  15.

The state x₂ (Example 2).

Figure  16.

Adaptive parameter $\widehat{\theta }$ (Example 2).

Figure  17.

Intervals t_k+1 − t _k of triggering events (Example 2).

From the above simulation images, the designed event-triggered adaptive neural finite-time tracking control scheme can achieve the desirable control performance and the Zeno behavior can be avoided.

Conclusion

In this paper, a dynamic-surface-based event-triggered adaptive neural finite-time tracking control scheme has been developed for a class of non-strict feedback stochastic nonlinear systems subject to time-varying unknown control directions, and input nonlinearities. To begin with, an improved event-triggered mechanism is presented to preserve communication resources and lessen the strain of communication. Secondly, a novel criterion is proposed in Lemma 1 to provide a powerful tool for analyzing time-varying unknown control directions, and input nonlinearities of stochastic systems. Furthermore, neural networks (NNs) are utilized to cope with unknown functions.

Note that this work considers only the condition in which system states can be measurable. Supposing that system states cannot be measurable, how to design an event-triggered adaptive neural finite-time tracking controller for the systems is interesting and meaningful. Therefore, we will further tend the event-triggered adaptive finite-time control for stochastic nonlinear multi-agent systems when system states are unmeasurable.