Dynamic event-triggered tracking control for a class of p- normal switched stochastic nonlinear systems

Abstract

This paper investigates the event-triggered tracking issue for p-normal switched stochastic nonlinear systems. Unlike the existing event-triggered schemes, an event-triggering mechanism is proposed with a dynamic gain. Then, a new adaptive event-triggered controller (ETC) is designed via output feedback. The presented control scheme can render that all signals of the closed-loop system are bounded almost surely. Moreover, the Zeno phenomenon is excluded. Three examples are provided to verify the efficiency of the presented strategy.

Keywords

Event-triggered control dynamic gain stochastic nonlinear systems output feedback tracking control

Introduction

It is widely known that nonlinearities exist in many practical engineering fields, such as agronomic systems, single-link robot systems, chemical reactor systems, tunnel diode circuit systems, and complex networked systems. Recently, the investigation for nonlinear systems has drawn considerable attention (Bibi et al., 2020; Chu et al., 2021; Li et al., 2021; Wu et al., 2020a; Zahedi and Binazadeh, 2020; Zhai, 2019). Many effective control algorithms have been put forward, such as adaptive control (Lin and Qian, 2002; Niu et al., 2020; Zhai and Karimi, 2019), finite-time control (Mei et al., 2020; Min et al., 2021), and output-feedback control (Yan et al., 2017). However, stochastic disturbances exist in many practical nonlinear systems. Over the past few decades, many meaningful accomplishments have been obtained for stochastic nonlinear systems (Jiang and Zhai, 2019; Lan and Li, 2017; Li et al., 2020; Min et al., 2019; Niu and Liu, 2017; Qin and Min, 2020; Song and Zhai, 2019). To name a few, by means of an auxiliary system, an output-feedback controller was presented in Min et al. (2019) for stochastic nonlinear systems with input saturation. On account of the existence of unknown measurement sensitivity, the authors employed the low-gain linear K-filters to design a controller without using the information of the output function in Li et al. (2020). In Song and Zhai (2019), the problem of finite-time control was discussed for high-order stochastic systems whose nonlinearities were restricted by different low-triangular and upper-triangular structure.

Note that tracking control plays an important role in control theory, which can be applied in many practical engineering (Hua et al., 2020; Li et al., 2018; Sun et al., 2020; Wu et al., 2020b; Yao and Zhang, 2019; Zhang et al., 2018; Zhao et al., 2015). It is well known that tracking control is to design a controller to track a predetermined signal, which is more challenging than the conventional stabilization problem. In Zhang et al. (2018), the finite-time tracking control issue was tackled using the backstepping technique. With the help of a new full-order high gain auxiliary system, in Hua et al. (2020), the adaptive tracking control problem was tackled for stochastic nonlinear systems with input saturation. On the basis of the backstepping method, the authors presented an adaptive controller for stochastic switched systems in Yao and Zhang (2019).

To save communication and computation resources, event-triggered control is presented and has aroused widespread attention. Up to now, various results on event-triggered control have been proposed (Huang and Liu, 2019; Li and Liu, 2020; Ma et al., 2019; Mu et al., 2021; Wang and Chen, 2021; Xing et al., 2019). For instance, by employing a dynamic gain and two event-triggering mechanisms, two new adaptive tracking strategies for uncertain nonlinear systems were developed in Huang and Liu (2019), which avoided infinitely fast sampling/execution and ensured the prespecified tracking accuracy. In Xing et al. (2019), by introducing two different event-triggered strategies, the control problem of uncertain nonlinear systems was addressed via output feedback. By means of an adaptive dynamic gain and a dynamic-gain observer, Li and Liu (2020) proposed an event-triggering mechanism and further designed a new adaptive controller. In Ma et al. (2019), an event-triggered control law was proposed by combining the barrier Lyapunov functions with a reduced-order observer.

Motivated by the above discussions, we attempt to solve the adaptive tracking control issue for p-normal switched stochastic nonlinear systems by event-triggered output feedback. The contributions of this article are summarized as follows:

Distinct from the existing works (Lan and Li, 2017; Niu and Liu, 2017), uncertain parameters and constant terms simultaneously exist in the considered system.

To tackle the unknown homogeneous growth rate, a dynamic gain is introduced.

The newly proposed event-triggering mechanism is a dynamic one, which merges delicately with a dynamic gain. Then, a new event-triggered controller (ETC) is constructed via output feedback, which can ensure all signals of the system are bounded almost surely.

Notations

In this paper, $R$ is the family composed of all real numbers. $R^{n}$ is the real n-dimensional space. $R_{odd}^{+} \overset{Δ}{=} {q \in R : q > 0 and q is a ratio of odd integers}$ . $R_{odd}^{\geq 1} \overset{Δ}{=} {r \in R : r \geq 1 and r is a ratio of odd integers}$ . $Z^{+}$ stands for the set of all positive integers. $Tr {\cdot}$ is the trace of a square matrix. $C^{i}$ is the set of all functions with continuous ith partial derivatives.

Problem description and preliminaries

Stochastic stability

Consider the following system

d x = f (x) d t + g^{T} (x) d w

(1)

where $x \in R^{n}$ represents the system state, $w$ is an r-dimensional standard Wiener process defined on the complete probability space $(Ω, F, P)$ . The functions $f (\cdot)$ and $g^{T} (\cdot)$ are locally $C^{1}$ with $f (0) = 0$ and $g (0) = 0$ .

Definition 1

For a given function $V (x) \in C^{2}$ associated with system (1), the infinitesimal generator $L$ is (Krstic and Deng, 1998)

L V (x) = \frac{\partial V}{\partial x} f (x) + \frac{1}{2} T r {g (x) \frac{\partial^{2} V}{\partial x^{2}} g^{T} (x)}

(2)

Definition 2

Consider system (1), and assume there is a positive definite function $V (x) \in C^{2}$ , two functions $δ_{1} (\cdot)$ , $δ_{2} (\cdot) \in K_{\infty}$ , constants $ι_{1} > 0$ and $ι_{2} \geq 0$ such that (Gao et al., 2017)

\begin{array}{l} δ_{1} (‖ x ‖) \leq V (x) \leq δ_{2} (‖ x ‖) \\ ℒ V (x) \leq - ι_{1} V (x) + ι_{2} \end{array}

(3)

then,

there is an almost surely unique solution on $[0, \infty)$ ;

the solution process is bounded almost surely;

the solution process satisfies

E [V (x)] \leq e^{- ι_{1} t} V (x_{0}) + ι_{1}^{- 1} ι_{2}

(4)

System description

Consider the following system

\begin{matrix} d x_{i} = (x_{i + 1}^{p_{i}} + f_{i, σ (t)} ({\bar{x}}_{i})) dt + g_{i, σ (t)}^{T} ({\bar{x}}_{i}) dw, i = 1, \dots, n - 1 \\ d x_{n} = (u^{p_{n}} + f_{n, σ (t)} (x)) dt + g_{n, σ (t)}^{T} (x) dw \\ y = x_{1} - y_{r} \end{matrix}

(5)

where ${\bar{x}}_{i} = (x_{1}, \dots, x_{i})^{T} \in R^{i} (i = 1, \dots, n)$ are the system states with ${\bar{x}}_{n} = x$ , $y \in R$ denotes the output, and $y_{r} \in R$ is the reference signal. $σ (t) : [0, + \infty) \to M = {1, 2, \dots, ι}$ is the switching signal. $w$ is defined as those in equation (1). For $i = 1, \dots, n$ and $k \in M$ , $f_{i, k}$ and $g_{i, k}$ are $C^{1}$ with $f_{i, k} (0) = 0$ and $g_{i, k} (0) = 0$ . $u$ is the input signal. $p_{i} \in R_{odd}^{\geq 1}$ , $i = 1, \dots, n - 1$ , and $p_{n} = 1$ are the system powers.

Assumption 1

There exist two non-negative constants $l_{i, k}$ , $c_{i, k}$ , and two unknown constants $θ_{i, k} > 0$ , $γ_{i, k} > 0$ , such that for $i = 1, \dots, n$ and $k \in M$

\begin{matrix} | f_{i, k} | \leq θ_{i, k} \sum_{j = 1}^{i} | x_{j} |^{\frac{r_{i} + ϖ}{r_{j}}} + l_{i, k} \\ ‖ g_{i, k} ‖ \leq γ_{i, k} \sum_{j = 1}^{i} | x_{j} |^{\frac{2 r_{i} + ϖ}{2 r_{j}}} + c_{i, k} \end{matrix}

where $ϖ \geq 0$ and

r_{1} = 1, r_{i + 1} p_{i} = r_{i} + ϖ, i = 1, \dots, n

(6)

Next, we set $ϖ_{0} = \max_{1 \leq i \leq n} {r_{i}}$ . Furthermore, one of the conditions below should be met:

$r_{n} + ϖ \geq r_{i}$ , if $ϖ_{0} / r_{i} = 1$ or $ϖ_{0} / r_{i} \geq 2$ for all $i = 1, 2, \dots, n$ ;

$r_{n} + ϖ \geq 2 r_{i}$ , otherwise.

Assumption 2

The reference signal $y_{r}$ and its derivative are bounded. For a constant $D > 0$ , one has $| y_{r} | \leq D$ and $| {\overset{\cdot}{y}}_{r} | \leq D$ .

Remark 1

As a matter of fact, Assumption 1 is more general than those in Lan and Li (2017) and Niu and Liu (2017). When $p_{i} = p$ and $ϖ = p - 1$ , Assumption 1 degrades into that in Niu and Liu (2017). If $p_{i} = 1$ , $ϖ = 0$ , and $θ_{i, k}$ , $γ_{i, k}$ are known, Assumption 1 coincides with that in Lan and Li (2017). For simplicity, let $ϖ = a_{0} / {\bar{a}}_{0}$ with $a_{0}$ being an even integer and ${\bar{a}}_{0}$ being an odd integer.

Remark 2

Assumption 2 is widely used to tackle the tracking control problem, such as in Zhao et al. (2015), Yao and Zhang (2019).

Lemma 1

For $x \in R$ and $y \in R$ , if $r \geq 1$ , one has (Lin and Qian, 2000)

\begin{matrix} 1 . | x + y |^{r} \leq 2^{r - 1} | x^{r} + y^{r} | \\ 2 . {(| x | + | y |)}^{\frac{1}{r}} \leq | x |^{\frac{1}{r}} + | y |^{\frac{1}{r}} \end{matrix}

if $r \in R_{odd}^{\geq 1}$ , one gets

\begin{matrix} 3 . | x - y |^{r} \leq 2^{r - 1} | x^{r} - y^{r} | \\ 4 . | x^{\frac{1}{r}} - y^{\frac{1}{r}} | \leq 2^{1 - \frac{1}{r}} | x - y |^{\frac{1}{r}} \end{matrix};

Lemma 2

For $x \in R$ and $y \in R$ , it can be obtained that (Lin and Qian, 2000)

| x |^{ς_{1}} | y |^{ς_{2}} \leq \frac{ς_{1}}{ς_{1} + ς_{2}} ς_{3} | x |^{ς_{1} + ς_{2}} + \frac{ς_{2}}{ς_{1} + ς_{2}} ς_{3}^{- \frac{ς_{1}}{ς_{2}}} | y |^{ς_{1} + ς_{2}}

where $ς_{1}$ , $ς_{2}$ , and $ς_{3}$ are the positive constants.

Lemma 3

Assume a function $V : R^{n} \to R$ is homogeneous of degree (HOD) $ϖ$ with respect to $Δ = (r_{1}, \dots, r_{n})$ (Bacciotti and Rosier, 2001). Then,

$\partial V / \partial x_{i}$ is HOD $ϖ - r_{i}$ ;

for a constant $\bar{b}$ , one has $V (x) \leq \bar{b} ‖ x ‖_{Δ}^{ϖ}$ .

Main results

Our objective is to propose an event-triggered tracking scheme for system (5). First, we consider the following nominal system

d z_{1} = z_{2}^{p_{1}} dt, \dots, d z_{n} = v^{p_{n}} dt, y = z_{1}

(7)

where $z = (z_{1}, \dots, z_{n})^{T} \in R^{n}$ represents the state, $v \in R$ is the control input and $y \in R$ is the output. In the analysis below, several parameters are introduced. Let $λ \in R_{odd}^{+}$ satisfy $λ \geq \max_{1 \leq i \leq n} {r_{i} + ϖ, ρ}$ . Meanwhile, $ρ$ is selected in the manner:

choose $ρ = ϖ_{0}$ , if the condition (a) of Assumption 1 is satisfied;

if the condition (b) of Assumption 1 holds, then $ρ$ can be chosen as $\forall ρ \in R_{odd}^{+}$ satisfying $\max_{1 \leq i \leq n} {2 r_{i}} \leq ρ \leq r_{n} + ϖ$ .

Similar to the design process in Zhai and Du (2014), a state feedback controller is designed as shown in the following lemma.

Lemma 4

For system (7), a family of virtual controllers $z_{1}^{*}$ , $\dots$ , $z_{n}^{*}$ are defined as

\begin{matrix} z_{1}^{*} = 0, ξ_{1} = z_{1}^{\frac{ρ}{r_{1}}} - z_{1}^{* \frac{ρ}{r_{1}}}, \\ z_{i}^{*} = - β_{i - 1} ξ_{i - 1}^{\frac{r_{i}}{ρ}}, ξ_{i} = z_{i}^{\frac{ρ}{r_{i}}} - z_{i}^{* \frac{ρ}{r_{i}}}, i = 2, . . ., n \end{matrix}

(8)

with $β_{i} > 0$ $(i = 1, \dots, n - 1)$ . There is a state feedback controller

z_{n + 1}^{*} = - β_{n} {(z_{n}^{\frac{ρ}{r_{n}}} + β_{n - 1}^{\frac{ρ}{r_{n}}} (z_{n - 1}^{\frac{ρ}{r_{n - 1}}} + \dots + β_{2}^{\frac{ρ}{r_{3}}} (z_{2}^{\frac{ρ}{r_{2}}} + β_{1}^{\frac{ρ}{r_{2}}} z_{1}^{\frac{ρ}{r_{1}}}) \dots))}^{\frac{r_{n + 1}}{ρ}}

(9)

such that

\begin{matrix} L V_{n} \leq - \sum_{i = 1}^{n} ξ_{i}^{\frac{4 λ}{ρ}} + ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} (v - z_{n + 1}^{*}) \end{matrix}

(10)

with $β_{n} > 0$ and $V_{n} = \sum_{i = 1}^{n} \int_{z_{i}^{*}}^{z_{i}} {(s^{\frac{ρ}{r_{i}}} - z_{i}^{* \frac{ρ}{r_{i}}})}^{\frac{4 λ - r_{i} - ϖ}{ρ}} ds$ .

In what follows, for system (7), we construct a homogeneous observer

d η_{i} = - b_{i - 1} {\hat{z}}_{i}^{p_{i - 1}} dt, {\hat{z}}_{i} = {(η_{i} + b_{i - 1} {\hat{z}}_{i - 1})}^{\frac{r_{i}}{r_{i - 1}}}, i = 2, \dots, n

(11)

where ${\hat{z}}_{1} = z_{1}$ and $b_{i} > 0$ $(i = 1, \dots, n - 1)$ . According to equation (9), an output-feedback controller can be represented as

\bar{v} (\hat{z}) = - β_{n} {({\hat{z}}_{n}^{\frac{ρ}{r_{n}}} + β_{n - 1}^{\frac{ρ}{r_{n}}} ({\hat{z}}_{n - 1}^{\frac{ρ}{r_{n - 1}}} + \dots + β_{2}^{\frac{ρ}{r_{3}}} ({\hat{z}}_{2}^{\frac{ρ}{r_{2}}} + β_{1}^{\frac{ρ}{r_{2}}} z_{1}^{\frac{ρ}{r_{1}}}) \dots))}^{\frac{r_{n + 1}}{ρ}}

(12)

where $\hat{z} = (z_{1}, {\hat{z}}_{2}, \dots, {\hat{z}}_{n})^{T}$ .

For $i = 2, \dots, n$ , we denote

U_{i} = \int_{ς_{i}^{(4 λ - r_{i - 1} - ϖ) / r_{i - 1}}}^{z_{i}^{(4 λ - r_{i - 1} - ϖ) / r_{i}}} (s^{\frac{r_{i - 1}}{4 λ - r_{i - 1} - ϖ}} - ς_{i}) d s

(13)

where $ς_{i} = η_{i} + b_{i - 1} z_{i - 1}$ .

Denote $e_{i} = (z_{i}^{p_{i - 1}} - {\hat{z}}_{i}^{p_{i - 1}})^{\frac{ρ}{r_{i} p_{i - 1}}}$ , $i = 2, \dots, n$ . Then, one can obtain the following result

\begin{matrix} L U_{i} = \frac{4 λ - r_{i - 1} - ϖ}{r_{i}} z_{i}^{\frac{4 λ - r_{i - 1} - r_{i} - ϖ}{r_{i}}} (z_{i}^{\frac{r_{i - 1}}{r_{i}}} - ς_{i}) z_{i + 1}^{p_{i}} \\ - b_{i - 1} e_{i}^{\frac{r_{i} p_{i - 1}}{ρ}} (z_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}} - {\hat{z}}_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}}) \\ - b_{i - 1} e_{i}^{\frac{r_{i} p_{i - 1}}{ρ}} ({\hat{z}}_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}} - ς_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i - 1}}}) \end{matrix}

(14)

where $z_{n + 1} = \bar{v}$ .

Based on Lemma 1, one has

\begin{matrix} - b_{i - 1} e_{i}^{\frac{r_{i} p_{i - 1}}{ρ}} (z_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}} - {\hat{z}}_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}}) = \\ - b_{i - 1} e_{i}^{\frac{r_{i} p_{i - 1}}{ρ}} ({(z_{i}^{p_{i - 1}})}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i} p_{i - 1}}} - {({\hat{z}}_{i}^{p_{i - 1}})}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i} p_{i - 1}}}) \\ \leq - b_{i - 1} α_{i} e_{i}^{\frac{4 λ}{ρ}} \end{matrix}

(15)

where $α_{i} = 2^{1 - \frac{4 λ - ϖ - r_{i - 1}}{r_{i} p_{i - 1}}}$ .

Similar to the method in Zhai and Du (2014), the following propositions can be obtained.

Proposition 1

For $i = 2, \dots, n - 1$ , it can be obtained that

\frac{4 λ - r_{i - 1} - ϖ}{r_{i}} z_{i}^{\frac{4 λ - r_{i - 1} - r_{i} - ϖ}{r_{i}}} (z_{i}^{\frac{r_{i - 1}}{r_{i}}} - ς_{i}) z_{i + 1}^{p_{i}} \leq \frac{1}{12} \sum_{j = i - 1}^{i + 1} ξ_{j}^{\frac{4 λ}{ρ}} + d_{i} e_{i}^{\frac{4 λ}{ρ}} + {\tilde{κ}}_{i} (b_{i - 1}) e_{i - 1}^{\frac{4 λ}{ρ}}

(16)

with $d_{i} > 0$ and a continuous function ${\tilde{κ}}_{i} (\cdot) ({\tilde{κ}}_{2} (b_{1}) = 0)$ .

Proposition 2

When $i = n$ , one has

\frac{4 λ - r_{n - 1} - ϖ}{r_{n}} z_{n}^{\frac{4 λ - r_{n - 1} - r_{n} - ϖ}{r_{n}}} (z_{n}^{\frac{r_{n - 1}}{r_{n}}} - ς_{n}) {\bar{v}}^{p_{n}} (\hat{z}) \leq \frac{1}{8} \sum_{i = 1}^{n} ξ_{i}^{\frac{4 λ}{ρ}} + \bar{d} \sum_{i = 2}^{n} e_{i}^{\frac{4 λ}{ρ}} + {\tilde{κ}}_{n} (b_{n - 1}) e_{n - 1}^{\frac{4 λ}{ρ}}

(17)

where ${\tilde{κ}}_{n} (\cdot)$ is a continuous function and $\bar{d} > 0$ .

Proposition 3

For $i = 3, \dots, n$ , we can get

- b_{i - 1} e_{i}^{\frac{r_{i} p_{i - 1}}{ρ}} ({\hat{z}}_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}} - ς_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i - 1}}}) \leq \frac{1}{16} \sum_{j = i - 1}^{i} ξ_{j}^{\frac{4 λ}{ρ}} + e_{i}^{\frac{4 λ}{ρ}} + {\bar{κ}}_{i} (b_{i - 1}) e_{i - 1}^{\frac{4 λ}{ρ}}

(18)

with ${\bar{κ}}_{i} (\cdot) \geq 0$ being a continuous function of $b_{i - 1}$ .

According to equation (15) and the above propositions, one has

\begin{matrix} L U \leq \frac{1}{2} \sum_{i = 1}^{n} ξ_{i}^{\frac{4 λ}{ρ}} - (b_{1} α_{2} - d_{2} - \bar{d} - {\tilde{κ}}_{3} (b_{2}) - {\bar{κ}}_{3} (b_{2})) e_{2}^{\frac{4 λ}{ρ}} \\ - \sum_{i = 3}^{n - 1} (b_{i - 1} α_{i} - 1 - d_{i} - \bar{d} - {\tilde{κ}}_{i + 1} (b_{i}) - {\bar{κ}}_{i + 1} (b_{i})) e_{i}^{\frac{4 λ}{ρ}} \\ - (b_{n - 1} α_{n} - 1 - \bar{d}) e_{n}^{\frac{4 λ}{ρ}} \end{matrix}

(19)

where $U = \sum_{i = 2}^{n} U_{i}$ .

Motivated by Xing et al. (2019), we adopt the following event-triggering mechanism

v (t) = \tilde{ω} (t_{l}), \forall t \in [t_{l}, t_{l + 1})

(20)

t_{l + 1} = \inf {t > t_{l} | | \tilde{ω} (t) - v (t) | > \frac{γ_{1}}{L} | v (t) | + \frac{γ_{2}}{L}}

(21)

where $\tilde{ω} (t)$ is defined as

\tilde{ω} (t) = (1 + γ_{1} L^{- 1}) (\bar{v} (\hat{z}) - {\overset{ˇ}{d}}_{0} \tanh (\frac{{\overset{ˇ}{d}}_{0}}{\bar{ε}} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}}))

(22)

with ${\overset{ˇ}{d}}_{0} = γ_{2} / 1 - γ_{1}$ , $\bar{ε} > 0$ , $0 < γ_{1} < 1$ , and $γ_{2} > 0$ . $t_{l}$ , $l \in Z^{+}$ represents the update time with $t_{1} = 0$ . $L \geq 1$ is a dynamic gain and its derivative is defined as

\overset{\cdot}{L} = a_{1} L^{2} \max {a_{2} z_{1}^{ϖ} L^{- ϵ_{0}} - a_{3}, 0}, L (0) = 1

(23)

where $a_{1}$ , $a_{2}$ , and $a_{3}$ are the positive constants. $ϵ_{0}$ is a positive constant to be determined later. If equation (21) holds, the control signal $v (t)$ will be updated to $v (t) = \tilde{ω} (t_{l + 1})$ . When $t \in [t_{l + 1}, t_{l + 2})$ , $\tilde{ω} (t_{l + 1})$ will be kept. From equation (21), we get

| \tilde{ω} (t) - v (t) | \leq \frac{γ_{1}}{L} | v (t) | + \frac{γ_{2}}{L}, \forall t \in [t_{l}, t_{l + 1}), l \in Z^{+}

(24)

By equation (24), one can obtain

\tilde{ω} (t) = (1 + ζ_{1} (t) \frac{γ_{1}}{L}) v (t) + ζ_{2} (t) \frac{γ_{2}}{L}

(25)

where $| ζ_{1} (t) | \leq 1$ and $| ζ_{2} (t) | \leq 1$ are two continuous functions. Then, one holds

v (t) = \frac{\tilde{ω} (t)}{1 + ζ_{1} (t) γ_{1} L^{- 1}} - \frac{γ_{2} ζ_{2} (t) L^{- 1}}{1 + ζ_{1} (t) γ_{1} L^{- 1}}

(26)

From equations (8) and (9), it produces $ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} z_{n + 1}^{*} \leq 0$ . Then, the following result can be obtained

\begin{array}{l} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} (v - z_{n + 1}^{*}) = ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} (\frac{\tilde{ω} (t)}{1 + ζ_{1} (t) γ_{1} L^{- 1}} - \frac{γ_{2} ζ_{2} (t) L^{- 1}}{1 + ζ_{1} (t) γ_{1} L^{- 1}} - z_{n + 1}^{*}) \\ = \frac{1 + γ_{1} L^{- 1}}{1 + ζ_{1} (t) γ_{1} L^{- 1}} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} (\bar{v} (\hat{z}) - z_{n + 1}^{*}) + (\frac{1 + γ_{1} L^{- 1}}{1 + ζ_{1} (t) γ_{1} L^{- 1}} - 1) ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} z_{n + 1}^{*} \\ - \frac{γ_{2} ζ_{2} (t) L^{- 1}}{1 + ζ_{1} (t) γ_{1} L^{- 1}} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} - \frac{{\overset{ˇ}{d}}_{0}}{L} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} \tanh (\frac{{\overset{ˇ}{d}}_{0}}{\bar{ε}} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}}) \\ \leq {\overset{ˇ}{d}}_{1} | ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} (\bar{v} (\hat{z}) - z_{n + 1}^{*}) | - \frac{1}{L} {\overset{ˇ}{d}}_{0} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} \tanh (\frac{{\overset{ˇ}{d}}_{0}}{\bar{ε}} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}}) + \frac{1}{L} | {\overset{ˇ}{d}}_{0} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} | \end{array}

(27)

where ${\overset{ˇ}{d}}_{1} = 1 + γ_{1} / 1 - γ_{1}$ .

In a similar manner as Proposition 3.7 in Zhai and Du (2014), one can obtain

{\overset{ˇ}{d}}_{1} | ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} (\bar{v} (\hat{z}) - z_{n + 1}^{*}) | \leq \frac{1}{4} \sum_{i = 1}^{n} ξ_{i}^{\frac{4 λ}{ρ}} + {\overset{ˇ}{d}}_{2} \sum_{i = 2}^{n} e_{i}^{\frac{4 λ}{ρ}}

(28)

with ${\overset{ˇ}{d}}_{2} > 0$ being a constant.

Besides, as shown in Ren et al. (2009), one has

0 \leq | \overset{ˇ}{v} | - \overset{ˇ}{v} \tanh (\frac{\overset{ˇ}{v}}{\bar{ε}}) \leq 0.2785 \bar{ε}, \forall \overset{ˇ}{v} \in R

(29)

where $\bar{ε}$ is defined in equation (22).

Hence, we have

| {\overset{ˇ}{d}}_{0} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} | - {\overset{ˇ}{d}}_{0} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}} \tanh (\frac{{\overset{ˇ}{d}}_{0}}{\bar{ε}} ξ_{n}^{\frac{4 λ - r_{n} - ϖ}{ρ}}) \leq 0.2785 \bar{ε}

(30)

According to the above inequalities, it is easy to obtain

\begin{array}{l} ℒ Y \leq - \frac{1}{4} \sum_{i = 1}^{n} ξ_{i}^{\frac{4 λ}{ρ}} - (b_{1} α_{2} - d_{2} - \bar{d} - {\overset{ˇ}{d}}_{2} - {\tilde{κ}}_{3} (b_{2}) - {\bar{κ}}_{3} (b_{2})) e_{2}^{\frac{4 λ}{ρ}} \\ - \frac{1}{4} \sum_{i = 3}^{n - 1} (b_{i - 1} α_{i} - 1 - d_{i} - \bar{d} - {\overset{ˇ}{d}}_{2} - {\tilde{κ}}_{i + 1} (b_{i}) - {\bar{κ}}_{i + 1} (b_{i})) e_{i}^{\frac{4 λ}{ρ}} \\ - (b_{n - 1} α_{n} - 1 - \bar{d} - {\overset{ˇ}{d}}_{2}) e_{n}^{\frac{4 λ}{ρ}} + c_{0} \end{array}

(31)

where $Y = V_{n} + U$ and $c_{0} = 0.2785 \bar{ε} L^{- 1}$ .

Then, one can choose $b_{i} (i = 1, \dots, n - 1)$ as

\begin{array}{l} b_{n - 1} \geq \frac{(\frac{5}{4} + \bar{d} + {\overset{ˇ}{d}}_{2})}{α_{n}} \\ b_{i - 1} \geq \frac{(\frac{5}{4} + d_{i} + \bar{d} + {\overset{ˇ}{d}}_{2} + {\tilde{κ}}_{i + 1} (b_{i}) + {\bar{κ}}_{i + 1} (b_{i}))}{α_{i}}, i = 3, \dots, n - 1 \\ b_{1} \geq \frac{(\frac{1}{4} + d_{2} + \bar{d} + {\overset{ˇ}{d}}_{2} + {\tilde{κ}}_{3} (b_{2}) + {\bar{κ}}_{3} (b_{2}))}{α_{2}} \end{array}

(32)

By equations (31) and (32), the following inequality holds

L Y \leq - \frac{1}{4} \sum_{i = 1}^{n} ξ_{i}^{\frac{4 λ}{ρ}} - \frac{1}{4} \sum_{i = 1}^{n} e_{i}^{\frac{4 λ}{ρ}} + c_{0}

(33)

Based on the above analysis, it is easy to obtain that $Y$ is the positive definite and proper with respect to $Z \overset{Δ}{=} (z_{1}, \dots, z_{n}, η_{2}, \dots, η_{n})^{T}$ . From equations (7) and (11), one has

dZ = E (Z) dt = {(z_{2}^{p_{1}}, \dots, z_{n}^{p_{n - 1}}, v^{p_{n}}, - b_{1} {\hat{z}}_{2}^{p_{1}}, \dots, - b_{n - 1} {\hat{z}}_{n}^{p_{n - 1}})}^{T} dt

(34)

By introducing

Δ = (r_{1}, \dots, r_{n}, r_{1}, \dots, r_{n - 1})

(35)

it is easy to prove that $Y$ is HOD $4 λ - ϖ$ . According to Lemma 3, one has

L Y \leq - ϱ_{1} ‖ Z ‖_{Δ}^{4 λ} + c_{0}

(36)

where $ϱ_{1} > 0$ .

Remark 3

From the construction of $Y$ , it follows that

\begin{array}{l} Y (Δ_{ε} (Z)) = \sum_{i = 1}^{n} \int_{ε^{r_{i}} z_{i}^{*}}^{ε^{r_{i}} z_{i}} {(s^{\frac{ρ}{r_{i}}} - ε^{ρ} z_{i}^{* \frac{ρ}{r_{i}}})}^{\frac{4 λ - r_{i} - ϖ}{ρ}} d s \\ + \sum_{i = 2}^{n} \int_{{(ε^{r_{i - 1}} ς_{i})}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i - 1}}}}^{{(ε^{r_{i}} z_{i})}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}}} (s^{\frac{r_{i - 1}}{4 λ - r_{i - 1} - ϖ}} - ε^{r_{i - 1}} ς_{i}) d s \\ = \sum_{i = 1}^{n} \int_{z_{i}^{*}}^{z_{i}} {(ε^{ρ} q_{1}^{\frac{ρ}{r_{i}}} - ε^{ρ} z_{i}^{* \frac{ρ}{r_{i}}})}^{\frac{4 λ - r_{i} - ϖ}{ρ}} ε^{r_{i}} d q_{1} \\ + \sum_{i = 2}^{n} \int_{ς_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i - 1}}}}^{z_{i}^{\frac{4 λ - r_{i - 1} - ϖ}{r_{i}}}} (ε^{r_{i - 1}} q_{2}^{\frac{r_{i - 1}}{4 λ - r_{i - 1} - ϖ}} - ε^{r_{i - 1}} ς_{i}) ε^{4 λ - r_{i - 1} - ϖ} d q_{2} \\ = ε^{4 λ - ϖ} Y (Z) \end{array}

with $q_{1} = ε^{- r_{i}} s$ and $q_{2} = ε^{r_{i - 1} + ϖ - 4 λ} s$ , which means that $Y (Z)$ is HOD $4 λ - ϖ$ .

Remark 4

In this paper, a new event-triggered control strategy is established by coupling with a dynamic gain. The triggering threshold in event-triggered condition (21) can be adaptively adjusted in line with $L$ , which is more flexible. Besides, when $L = 1$ and $γ_{1} = 0$ in equation (21), the triggering mechanism coincides with the fixed threshold strategy in Xing et al. (2019); if $L = 1$ in equation (21), the mechanism degrades into the relative threshold strategy in Ma et al. (2019) and Xing et al. (2019).

Thus, according to the above analysis, we can summarize the theorem as follows.

Theorem 1

For system (5) with Assumptions 1 and 2, under the event-triggering rulers (20)–(22), the tracking problem can be solved and the Zeno phenomenon is avoided.

Proof

For system (5), choose the transformation of coordinates

z_{1} = \frac{x_{1} - y_{r}}{L^{m_{1}}}, z_{i} = \frac{x_{i}}{L^{m_{i}}}, i = 2, \dots, n, v = \frac{u}{L^{m_{n + 1}}}

(37)

where $m_{1} = 0$ , $m_{i + 1} p_{i} = m_{i} + 1$ , $i = 1, \dots, n$ , and $L \geq 1$ is a dynamic gain. With the help of the new z-coordinate, we can rewrite system (5) as

\begin{matrix} d z_{i} = ({Lz}_{i + 1}^{p_{i}} - \frac{\overset{\cdot}{L}}{L} m_{i} z_{i} + {\tilde{f}}_{i, k} (\cdot)) dt + {\tilde{g}}_{i, k}^{T} (\cdot) dw, i = 1, \dots, n - 1 \\ d z_{n} = (L v^{p_{n}} - \frac{\overset{\cdot}{L}}{L} m_{n} z_{n} + {\tilde{f}}_{n, k} (\cdot)) dt + {\tilde{g}}_{n, k}^{T} (\cdot) dw \\ y = z_{1} \end{matrix}

(38)

where ${\tilde{f}}_{1, k} (\cdot) = f_{1, k} (\cdot) - {\overset{\cdot}{y}}_{r}$ , ${\tilde{f}}_{i, k} (\cdot) = f_{i, k} (\cdot) / L^{m_{i}}$ , $i = 2, \dots, n$ , and ${\tilde{g}}_{i, k} (\cdot) = g_{i, k} (\cdot) / L^{m_{i}}$ , $i = 1, \dots, n$ .

Then, an observer is designed as

d η_{i} = - L b_{i - 1} {\hat{z}}_{i}^{p_{i - 1}} dt, {\hat{z}}_{i} = {(η_{i} + b_{i - 1} {\hat{z}}_{i - 1})}^{\frac{r_{i}}{r_{i - 1}}}, i = 2, \dots, n

(39)

where $b_{i - 1} > 0$ are chosen in equation (32).

Using the definition of $Z$ yields

dZ = (LE (Z) - \frac{\overset{\cdot}{L}}{L} {(m_{1} z_{1}, \dots, m_{n} z_{n}, 0, \dots, 0)}^{T} + F (Z)) dt + G^{T} (Z) dw

(40)

where $F (Z) = ({\tilde{f}}_{1, k} (\cdot), \dots, {\tilde{f}}_{n, k} (\cdot), 0, \dots, 0)^{T}$ and $G (Z) = ({\tilde{g}}_{1, k} (\cdot), \dots, {\tilde{g}}_{n, k} (\cdot), 0, \dots, 0)$ .

In the light of equations (36) and (40), one has

\begin{matrix} L Y \leq - ϱ_{1} L ‖ Z ‖_{Δ}^{4 λ} + \frac{\partial Y}{\partial Z} F (Z) - \frac{\overset{\cdot}{L}}{L} \frac{\partial Y}{\partial Z} {(m_{1} z_{1}, \dots, m_{n} z_{n}, 0, \dots, 0)}^{T} \\ + \frac{1}{2} Tr {G (Z) \frac{\partial^{2} Y}{\partial Z^{2}} G^{T} (Z)} + L c_{0} \\ \leq - ϱ_{1} L ‖ Z ‖_{Δ}^{4 λ} + | \frac{\partial Y}{\partial Z} F (Z) | + \frac{\overset{\cdot}{L}}{L} \sum_{i = 1}^{n} | \frac{\partial Y}{\partial Z_{i}} m_{i} z_{i} | \\ + \frac{1}{2} Tr {G (Z) \frac{\partial^{2} Y}{\partial Z^{2}} G^{T} (Z)}} + c_{1} \end{matrix}

(41)

where $c_{1} = 0.2785 \bar{ε}$ .

According to Assumption 1 and equation (37), one has

\begin{matrix} {\tilde{f}}_{i, k} \leq \frac{θ_{i, k}}{L^{m_{i}}} (| x_{1} |^{\frac{r_{i} + ϖ}{r_{1}}} + \dots + | x_{i} |^{\frac{r_{i} + ϖ}{r_{i}}}) + \frac{l_{i, k} + | {\overset{\cdot}{y}}_{r} |}{L^{m_{i}}} \\ \leq \frac{θ_{i, k}}{L^{m_{i}}} (| z_{1} + y_{r} |^{\frac{r_{i} + ϖ}{r_{1}}} + \dots + | L^{m_{i}} z_{i} |^{\frac{r_{i} + ϖ}{r_{i}}}) + \frac{l_{i, k} + | {\overset{\cdot}{y}}_{r} |}{L^{m_{i}}} \\ \leq {\bar{θ}}_{i} L^{1 - ϵ_{1}} (| z_{1} |^{\frac{r_{i} + ϖ}{r_{1}}} + \dots + | z_{i} |^{\frac{r_{i} + ϖ}{r_{i}}}) + \frac{l_{i, k} + | {\overset{\cdot}{y}}_{r} | + {\bar{θ}}_{i} | y_{r} |^{r_{i} + ϖ}}{L^{m_{i}}} \\ \leq \bar{θ} L^{1 - ϵ_{1}} (| z_{1} |^{\frac{r_{i} + ϖ}{r_{1}}} + \dots + | z_{i} |^{\frac{r_{i} + ϖ}{r_{i}}}) + h_{1} \end{matrix}

(42)

with ${\bar{θ}}_{i}$ and $\bar{θ}$ being the unknown constants, $h_{1} > 0$ and $ϵ_{1} \in (0, 1]$ .

Similar to equation (42), it holds

‖ {\tilde{g}}_{i, k} ‖ \leq \bar{γ} L^{\frac{1}{2} - ϵ_{2}} (| z_{1} |^{\frac{2 r_{i} + ϖ}{2 r_{1}}} + \dots + | z_{i} |^{\frac{2 r_{i} + ϖ}{2 r_{i}}}) + h_{2}

(43)

with $\bar{γ}$ being an unknown positive constant, $0 < ϵ_{2} \leq 1 / 2$ and $h_{2} > 0$ .

Since $\partial Y / \partial Z_{i}$ is HOD $4 λ - ϖ - r_{i}$ , one has

| \frac{\partial Y}{\partial Z} F (Z) | \leq \sum_{i = 1}^{n} | \frac{\partial Y}{\partial Z_{i}} | | {\tilde{f}}_{i, k} | \leq \tilde{θ} L^{1 - ϵ_{1}} ‖ Z ‖_{Δ}^{4 λ} + c_{2}

(44)

with $\tilde{θ}$ being an unknown constant and $c_{2} > 0$ .

According to Lemma 3, one obtains that $\partial^{2} Y / \partial Z_{i} \partial Z_{j}$ is HOD $4 λ - r_{i} - r_{j} - ϖ$ . Hence, for an unknown positive constant $\tilde{γ}$ , one has

\frac{1}{2} Tr {G (Z) \frac{\partial^{2} Y}{\partial Z^{2}} G^{T} (Z)} \leq \tilde{γ} L^{1 - ϵ_{3}} ‖ Z ‖_{Δ}^{4 λ} + c_{3}

(45)

where $0 < ϵ_{3} \leq 1$ and $c_{3} > 0$ .

By equations (44) and (45), it can be obtained that

\begin{matrix} | \frac{\partial Y}{\partial Z} F (Z) | + \frac{1}{2} Tr {G (Z) \frac{\partial^{2} Y}{\partial Z^{2}} G^{T} (Z)} \leq \\ (\tilde{θ} + \tilde{γ}) L^{1 - ϵ_{0}} ‖ Z ‖_{Δ}^{4 λ} + c_{2} + c_{3} \end{matrix}

(46)

where $ϵ_{0} = min {ϵ_{1}, ϵ_{3}}$ .

Based on the homogeneous theory, we can get

| z_{i} | \leq {(\sum_{i = 1}^{n} z_{i}^{\frac{2}{r_{i}}} + \sum_{i = 2}^{n} η_{i}^{\frac{2}{r_{i - 1}}})}^{\frac{r_{i}}{2}} = ‖ Z ‖_{Δ}^{r_{i}}

(47)

According to equations (23) and (47), one has

\frac{\overset{\cdot}{L}}{L} \sum_{i = 1}^{n} | \frac{\partial Y}{\partial Z_{i}} m_{i} z_{i} | \leq ϱ_{2} L^{1 - ϵ_{0}} ‖ Z ‖_{Δ}^{4 λ}

(48)

where $ϱ_{2} > 0$ .

Then, substituting equations (44)–(48) into equation (41) yields

\begin{matrix} L Y \leq - ϱ_{1} L ‖ Z ‖_{Δ}^{4 λ} + (\tilde{θ} + \tilde{γ} + ϱ_{2}) L^{1 - ϵ_{0}} ‖ Z ‖_{Δ}^{4 λ} + {\bar{c}}_{1} \\ \leq - L (ϱ_{1} - \frac{\tilde{θ} + \tilde{γ} + ϱ_{2}}{L^{ϵ_{0}}}) ‖ Z ‖_{Δ}^{4 λ} + {\bar{c}}_{1} \end{matrix}

(49)

where ${\bar{c}}_{1} = \sum_{i = 1}^{3} c_{i}$ .

From equation (12), one has

\frac{\partial {\bar{v}}^{p_{n}} (\hat{z})}{\partial {\hat{z}}_{i}} = - Λ_{i} {\hat{z}}_{i}^{\frac{ρ - r_{i}}{r_{i}}} {({\hat{z}}_{n}^{\frac{ρ}{r_{n}}} + β_{n - 1}^{\frac{ρ}{r_{n}}} {\hat{z}}_{n - 1}^{\frac{ρ}{r_{n - 1}}} + \dots + β_{n - 1}^{\frac{ρ}{r_{n}}} β_{n - 2}^{\frac{ρ}{r_{n - 1}}} \dots β_{1}^{\frac{ρ}{r_{2}}} z_{1}^{\frac{ρ}{r_{1}}})}^{\frac{r_{n} + ϖ - ρ}{ρ}}

(50)

where $Λ_{i} = (r_{n} + ϖ / r_{i}) β_{n}^{p_{n}} β_{n - 1}^{\frac{ρ}{r_{n}}} \dots β_{i}^{\frac{ρ}{r_{i + 1}}}$ , $i = 1, \dots, n$ .

With the definition of $ρ$ in mind, one has $r_{n} + ϖ \geq ρ \geq \max_{1 \leq i \leq n} {r_{i}}$ , which means that

\frac{r_{n} + ϖ - ρ}{ρ} \geq 0, \frac{ρ - r_{i}}{r_{i}} \geq 0, \frac{ρ}{r_{i}} \geq 1

(51)

Hence, it follows from equation (50) that $\partial {\bar{v}}^{p_{n}} (\hat{z}) / \partial {\hat{z}}_{i}$ is continuous and ${\bar{v}}^{p_{n}}$ is $C^{1}$ . Meanwhile, based on the construction of $\overset{\cdot}{L}$ , it is easily obtained that $\overset{\cdot}{L} \geq 0$ and $L (t) \geq L (0) = 1$ . Based on the above analysis, it is obvious that the closed-loop system satisfies the locally Lipschitz condition. In the light of Definition 2, for $\forall (x (0), η (0), L (0))$ , there exists a unique solution $(x (t), η (t), L (t))$ .

We first prove L(t) is bounded on $[0, t_{f})$ . Due to $\overset{\cdot}{L} (t) \geq 0$ , $L (t)$ is a monotone non-decreasing function. There is a time $t_{0} \in [0, t_{f})$ such that

L (t) \geq {(\frac{2 (\tilde{θ} + \tilde{γ} + ϱ_{2})}{ϱ_{1}})}^{\frac{1}{ϵ_{0}}}, \forall t \in [t_{0}, t_{f})

(52)

Then, one can deduce that

L Y \leq - \frac{ϱ_{1}}{2} L ‖ Z ‖_{Δ}^{4 λ} + {\bar{c}}_{1}

(53)

By virtue of Lemma 3, for two positive constants $\bar{o}$ and $\underline{o}$ , one has

\underline{o} ‖ Z ‖_{Δ}^{4 λ - ϖ} \leq Y (Z) \leq \bar{o} ‖ Z ‖_{Δ}^{4 λ - ϖ}

(54)

By Lemma 2, one has

\bar{o} ‖ Z ‖_{Δ}^{4 λ - ϖ} \leq L ‖ Z ‖_{Δ}^{4 λ} + \frac{ϖ {(4 λ - ϖ)}^{\frac{4 λ - ϖ}{ϖ}} {\bar{o}}^{\frac{4 λ}{ϖ}}}{L^{\frac{4 λ - ϖ}{ϖ}} {(4 λ)}^{\frac{4 λ}{ϖ}}}

(55)

which yields

- L ‖ Z ‖_{Δ}^{4 λ} \leq - \bar{o} ‖ Z ‖_{Δ}^{4 λ - ϖ} + \frac{ϖ {(4 λ - ϖ)}^{\frac{4 λ - ϖ}{ϖ}} {\bar{o}}^{\frac{4 λ}{ϖ}}}{L^{\frac{4 λ - ϖ}{ϖ}} {(4 λ)}^{\frac{4 λ}{ϖ}}}

(56)

Substituting equation (56) into equation (53), we have

L Y \leq - \frac{ϱ_{1} \bar{o}}{2} ‖ Z ‖_{Δ}^{4 λ - ϖ} + {\bar{c}}_{2} \leq - \frac{ϱ_{1}}{2} Y + {\bar{c}}_{2}

(57)

where ${\bar{c}}_{2} = (ϱ_{1} ϖ / 2) (4 λ - ϖ)^{\frac{4 λ - ϖ}{ϖ}} {\bar{o}}^{\frac{4 λ}{ϖ}} / (4 λ)^{\frac{4 λ}{ϖ}} + {\bar{c}}_{1}$ .

From equation (57) and Definition 2, we know that $z (t)$ and $η (t)$ are also bounded on $[0, t_{f})$ , which implies $z_{1} (t)$ is bounded on $[0, t_{f})$ . Assume that $li m_{t \to t_{f}} L (t) = + \infty$ . According to equation (23), it holds that $\overset{\cdot}{L} = a_{1} L^{2} \max {- a_{3}, 0} = 0$ , which contradicts with the assumption $li m_{t \to t_{f}} L (t) = + \infty$ . Hence, $L (t)$ is bounded on $[0, t_{f})$ .

Due to the continuity of the solution, $t_{f}$ can be infinite. Hence, we know that $L (t)$ , $z (t)$ , and $η (t)$ are bounded on $[0, + \infty)$ . Based on equation (37) and the boundedness of $L (t)$ , we can infer that $x_{2}, \dots, x_{n}$ are bounded almost surely. By virtue of equation (57) and Definition 2, there holds

E (Y (t)) \leq e^{- \frac{ϱ_{1}}{2} t} Y (0) + \frac{2}{ϱ_{1}} {\bar{c}}_{2}

(58)

Thus, one has

E (| z_{1} (t) |^{4 λ - ϖ}) \leq (4 λ - ϖ) e^{- \frac{ϱ_{1}}{2} t} Y (0) + \frac{2 {\bar{c}}_{2} (4 λ - ϖ)}{ϱ_{1}}

(59)

Noting that $z_{1} = x_{1} - y_{r}$ , from equation (59), one has

\lim_{t \to + \infty} E | x_{1} (t) - y_{r} (t) | \leq {\bar{c}}_{3}

(60)

where ${\bar{c}}_{3} = (2 c (4 λ - ϖ) / ϱ_{1})^{\frac{1}{4 λ - ϖ}}$ . Following the boundedness of $y_{r}$ , one holds that $x_{1}$ is bounded almost surely. Therefore, all signals of the closed-loop system are bounded almost surely.

Next, we will show that Zeno behavior does not happen. From equation (12), equation (22), and the definition of $ρ$ , one obtains that $\tilde{ω} (t)$ is a $C^{1}$ function. According to the boundedness of $η (t)$ and $\hat{z} (t)$ , one has $| d \tilde{ω} (t) / dt |$ is bounded.

Define $Λ (t) = (\tilde{ω} (t) - v (t))^{2}$ . From equation (20), equation (22), and the boundedness of $v (t)$ , it gives

| \frac{d Λ (t)}{dt} | \leq 2 | \tilde{ω} (t) - v (t) | | \frac{d \tilde{ω} (t)}{dt} | \leq λ_{1}, t \in [t_{l}, t_{l + 1})

(61)

where $λ_{1}$ is a positive constant.

Due to $(\tilde{ω} (t_{l}) - v (t_{l}))^{2} = 0$ and $(\tilde{ω} (t_{l + 1}^{-}) - v (t_{l + 1}^{-}))^{2} = (γ_{1} / L | v (t_{l}) | + γ_{2} / L)^{2}$ , it holds

\begin{array}{l} {(\frac{γ_{1}}{L} | v (t_{l}) | + \frac{γ_{2}}{L})}^{2} = {(\tilde{ω} (t_{l + 1}^{-}) - v (t_{l + 1}^{-}))}^{2} - {(\tilde{ω} (t_{l}) - v (t_{l}))}^{2} \\ \leq \int_{t_{l}}^{t_{l + 1}^{-}} λ_{1} d t \leq λ_{1} (t_{l + 1} - t_{l}) \end{array}

(62)

which leads to $t_{l + 1} - t_{l} \geq (γ_{1} / L | v (t_{l}) | + γ_{2} / L)^{2} / λ_{1} > 0$ . Therefore, the Zeno behavior does not occur.

Remark 5

It should be mentioned that the condition imposed on nonlinear drift and diffusion terms is relaxed, which is more widely used. In this paper, one main challenge is to tackle the issue of the unknown homogeneous growth rate. To overcome this difficulty, a dynamic gain $L$ is introduced instead of the scaling gain in Li et al. (2018).

Remark 6

These variables/parameters used in controller design and analysis are described in Table 1.

Table 1.

List of the variables/parameters.

Variable/parameter	Implication
$β_{i} (i = 1, \dots, n)$ , $d_{i} (i = 2, \dots, n - 1)$ , $α_{i} (i = 2, \dots, n)$ , $b_{i} (i = 1, \dots, n - 1)$ , $a_{i} (i = 1, 2, 3)$ , $c_{i} (i = 1, 2, 3)$ , ${\bar{c}}_{i} (i = 1, 2, 3)$ , $ϵ_{i} (i = 0, \dots, 3)$ , ${\overset{ˇ}{d}}_{i} (i = 0, 1, 2)$ , $\bar{d}$ , $γ_{1}$ , $γ_{2}$ , $\bar{ε}$ , $ϱ_{1}$ , $ϱ_{2}$ , $h_{1}$ , $h_{2}$ , $\bar{o}$ , $\underline{o}$ , $λ_{1}$	Known positive constants
$\bar{θ}$ , $\bar{γ}$ , $\tilde{θ}$ , $\tilde{γ}$	Unknown positive constants
${\tilde{κ}}_{i} (b_{i - 1}) (i = 2, \dots, n)$ , ${\bar{κ}}_{i} (b_{i - 1}) (i = 3, \dots, n)$	Continuous functions $b_{i - 1}$ with $({\tilde{κ}}_{2} (\cdot) = 0)$

Simulation examples

Example 1

Consider the following switched system

\begin{matrix} d x_{1} = (x_{2}^{p_{1}} + f_{1, σ (t)} (x_{1})) dt + g_{1, σ (t)}^{T} (x_{1}) dw \\ d x_{2} = (u + f_{2, σ (t)} (x)) dt + g_{2, σ (t)}^{T} (x) dw \end{matrix}

(63)

y = x_{1} - y_{r}

where $p_{1} = 13 / 11$ , $σ (t) : [0, + \infty) \to M = {1, 2}$ . In order to verify the robustness of the proposed method, nonlinear drift and diffusion terms are given by the two cases as follows:

Case 1: $f_{1, 1} = 0.1 x_{1}^{\frac{13}{11}}$ , $f_{2, 1} = 0.4 \sin (x_{1}) x_{2}$ , $f_{1, 2} = 0.2 x_{1}$ , $f_{2, 2} = 0.1 x_{1} x_{2}$ , $g_{1, 1}^{T} = 0$ , $g_{2, 1}^{T} = 0.3 x_{2}^{\frac{12}{11}}$ , $g_{1, 2}^{T} = 0.2 x_{1}^{\frac{12}{11}}$ , and $g_{2, 2}^{T} = 0.5 x_{1} x_{2}$ .

Case 2: $f_{1, 1} = 0.3 \cos (x_{1}) x_{1}^{\frac{13}{11}}$ , $f_{2, 1} = 0.5 \cos (x_{1}) x_{2}$ , $f_{1, 2} = 0.4 \sin (t) x_{1}$ , $f_{2, 2} = 0.2 \sin (x_{1}) x_{2}$ , $g_{1, 1}^{T} = 0.1 \sin (x_{1}) x_{1}^{\frac{12}{11}}$ , $g_{2, 1}^{T} = 0.4 \cos (t) x_{2}^{\frac{12}{11}}$ , $g_{1, 2}^{T} = 0.4 x_{1}^{\frac{12}{11}}$ , and $g_{2, 2}^{T} = 0.6 \cos (x_{1}) x_{2}$ .

By choosing $ϖ = 2 / 11$ , one obtains $r_{1} = r_{2} = 1$ , $m_{1} = 0$ , $m_{2} = 11 / 13$ , and $m_{3} = 24 / 13$ . Obviously, Assumption 1 holds. By Theorem 1, the observer and controller are given by

\begin{array}{l} d η_{2} = - L b_{1} {\hat{z}}_{2}^{\frac{13}{11}} d t, {\hat{z}}_{2} = η_{2} + b_{1} z_{1} \\ \dot{L} = a_{1} L^{2} \max {a_{2} z_{1}^{\frac{2}{11}} L^{- ϵ_{0}} - a_{3}, 0}, L (0) = 1 \\ u (t) = L^{\frac{24}{13}} \tilde{ω} (t_{l}), t \in [t_{l}, t_{l + 1}), l \in Z^{+} \\ \tilde{ω} (t) = - (1 + \frac{γ_{1}}{L}) (β_{2} {({\hat{z}}_{2} + β_{1} z_{1})}^{\frac{13}{11}} + {\overset{ˇ}{d}}_{0} \tanh (\frac{{\overset{ˇ}{d}}_{0}}{\bar{ε}} ξ_{2}^{\frac{39}{11}})) \end{array}

In simulation, the reference signal is given by $y_{r} = 2 \sin (2 t)$ . The initial value is selected as $(x_{1} (0), x_{2} (0), η_{2} (0)) = (0.2, - 0.2, 0)$ . For Case 1, the parameters are adopted as $γ_{1} = 0.3$ , $γ_{2} = 3$ , $β_{1} = 6$ , $β_{2} = 9$ , $b_{1} = 5$ , $a_{1} = 9$ , $a_{2} = 1$ , $ϵ_{0} = 0.8$ , $a_{3} = 0.45$ , and $\bar{ε} = 0.8$ . For Case 2, the parameters are set to $γ_{1} = 0.2$ , $γ_{2} = 2.5$ , $β_{1} = 3$ , $β_{2} = 11$ , $b_{1} = 15$ , $a_{1} = 6$ , $a_{2} = 1$ , $ϵ_{0} = 0.8$ , $a_{3} = 0.45$ , and $\bar{ε} = 0.2$ . Figures 1 –5 depict all simulation results, which verify the effectiveness of the developed control method.

Figure 1.

System state $x_{1}$ and reference signal $y_{r}$ in Example 1.

Figure 2.

Dynamic gain $L$ in Example 1.

Figure 3.

Control input $u$ in Example 1.

Figure 4.

Switching signal $σ (t)$ in Example 1.

Figure 5.

The time interval of triggering events in Example 1.

Example 2

As is shown in Hou et al. (2013), the continuous stirred tank reactor (CSTR) with two modes feed stream is described by

\begin{matrix} d x_{1} = (x_{2} + f_{1, σ (t)} (x_{1})) dt + \frac{1}{4} f_{1, σ (t)} (x_{1}) dw \\ d x_{2} = udt \\ y = x_{1} - y_{r} \end{matrix}

(64)

where $σ (t) : [0, + \infty) \to M = {1, 2}$ , $f_{1, 1} = 0.5 x_{1}$ , and $f_{1, 2} = 2 x_{1}$ . By choosing $ϖ = 0$ and $r_{1} = r_{2} = r_{3} = 1$ , it can be verified that Assumption 1 is met.

In the practical simulation, we choose $y_{r} = \sin (2 t) + \sin (t)$ . The initial value is $(x_{1} (0), x_{2} (0), η_{2} (0)) = (0.4, - 0.1, 0)$ . In order to analyze the influence of control parameters on the closed-loop system, two groups of the parameters are selected as follows:

Case I: $γ_{1} = 0.2$ , $γ_{2} = 1.7$ , $β_{1} = 7$ , $β_{2} = 9$ , $b_{1} = 12$ , $a_{1} = 3$ , $a_{2} = 1$ , $ϵ_{0} = 1$ , $a_{3} = 0.6$ , and $\bar{ε} = 0.02$ .

Case II: $γ_{1} = 0.2$ , $γ_{2} = 1.7$ , $β_{1} = 8$ , $β_{2} = 21$ , $b_{1} = 15$ , $a_{1} = 3$ , $a_{2} = 1$ , $ϵ_{0} = 1$ , $a_{3} = 0.5$ , and $\bar{ε} = 0.03$ .

Simulation results are demonstrated in Figures 6 –10, which can indicate the efficiency of the presented scheme. From these figures, it can be seen that the selection of control parameters affects the system output tracking performance.

Figure 6.

System state $x_{1}$ and reference signal $y_{r}$ in Example 2.

Figure 7.

Dynamic gain $L$ in Example 2.

Figure 8.

Control input $u$ in Example 2.

Figure 9.

Switching signal $σ (t)$ in Example 2.

Figure 10.

The time interval of triggering events in Example 2.

Example 3

The single-link robot arm borrowed from Min et al. (2021) is given to compare the effects of the control scheme in this paper with that in Li et al. (2018). The corresponding dynamic system without considering switching is given by

J_{s} \overset{\cdot\cdot}{q} = M_{p} g l_{0} \sin q - B_{s} \overset{\cdot}{q} + u

(65)

where $q$ and $u$ are the link angular displacement and actual input, respectively. $B_{s} \overset{\cdot}{q}$ is the viscous damping with the damping coefficient $B_{s}$ . $g$ , $J_{s}$ , $M_{p}$ , and $l_{0}$ represent the acceleration of gravity, the moment of inertia, the mass, and length of the link, respectively.

Letting $B_{s} = B_{0} + Δ B$ with $B_{0} = 0.5 N \cdot m \cdot s$ as the uncertain variable and defining $x_{1} = \overset{\cdot}{q}$ and $x_{2} = \overset{\cdot}{q}$ , system (65) can be represented as

\begin{matrix} d x_{1} = x_{2} dt \\ d x_{2} = \frac{1}{J_{s}} udt + f_{2} (x_{1}, x_{2}) dt + g_{2}^{T} (x_{1}, x_{2}) dw \end{matrix}

where $f_{2} = - 1 / J_{s} M_{p} g \sin (x_{1}) - B_{0} l_{0} / J_{s} x_{2}$ and $g_{2}^{T} = - D_{s} / J_{s} x_{2}$ with $D_{s}^{2}$ being the covariance of stochastic noise. Choose the parameters: $M_{p} = 1 kg$ , $g = 9.8 m / s^{2}$ , $l_{0} = 1 m$ , $J_{s} = 1 kg \cdot m^{2}$ , and $D_{s} = 0.1$ .

In the practical simulation, choose $y_{r} = \cos t$ . In this paper, the design parameters are given by $b_{1} = 17$ , $β_{1} = 5$ , $β_{2} = 12$ , $γ_{1} = 0.1$ , $γ_{2} = 0.5$ , $a_{1} = 4$ , $a_{2} = 0.7$ , $ϵ_{0} = 0.7$ , $a_{3} = 0.1$ , and $\bar{ε} = 0.4$ . The parameters in Li et al. (2018) are chosen as $l_{1} = 17$ , $β_{1} = 5$ , $β_{2} = 12$ , and $L = 1.4$ . Under the same initial values $(x_{1} (0)$ , $x_{2} (0)$ , $η_{2} (0)) = (- 1, 0.6, 0)$ , the simulation results of the presented strategy and Li et al. (2018) are demonstrated in Figures 11 and 12. These figures show that the output tracking performance in this paper is better than that in Li et al. (2018).

Figure 11.

System state $x_{1}$ and reference signal $y_{r}$ in Example 3.

Figure 12.

Tracking error $y$ in Example 3.

Conclusion

In this paper, the adaptive tracking control problem has been considered for p-normal switched stochastic nonlinear systems under dynamic event-triggering mechanism. The growth condition imposed on nonlinear drifts/diffusions is further relaxed and a new dynamic ETC is designed. Simulation results indicate the validity of the presented scheme. Our future work is to explore how to solve the event-triggered tracking control problem of constrained nonlinear systems with actuator faults.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by the National Natural Science Foundation of China (grant no. 61873061) and the Natural Science Foundation of Jiangsu Province (BK20211162).

ORCID iDs

Manman Yuan

Junyong Zhai

Hui Ye

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