Event-triggered finite-time neural control for uncertain nonlinear systems with unknown disturbances and its application in SVC

Abstract

In this article, an event-triggered finite-time neural control strategy is proposed for nonlinear power systems with unknown disturbances and static var compensator (SVC). We first transform the power system with SVC into a three-dimensional uncertain nonlinear system and then extend it to an $n$ -dimensional uncertain nonlinear system. The disturbance observer is established to estimate external disturbances and the unknown nonlinear terms are approximated by the radial basis function neural networks. Moreover, to avoid the complexity explosion problem in the traditional backstepping method, the command filtering technique is adopted, and the error caused by the command filters is compensated. The adaptive event-triggered finite-time controller ensures that all signals are bounded in finite time and excludes Zeno phenomena. In the end, the simulation for the two-area interconnected power system with SVC is presented to verify the availability and feasibility of the proposed approach.

Keywords

Finite-time control command filtering technique neural networks disturbance observer event-triggered control

Introduction

With the increasing demand for electric energy and power quality, the power system has been expanded and developed rapidly, how to maintain the stability of the increasingly complex power systems has become an urgent and non-negligible issue Alireza et al. (2019), Lai et al. (2021), Zhao et al. (2019) and Sabo et al. (2021). Static var compensator (SVC) can quickly adjust the reactive power by changing the reactance and making reactive power compensation for the system at any time, so as to stabilize the system voltage at a constant state and improve the voltage stability of long-distance transmission lines Amirreza et al. (2020). Thus, SVC has been widely used in power systems, and numerous scholars have carried out studies on the SVC Roselyn and Devaraj (2018), Hamid and Hossein (2019), and Thierry et al. (2020). In Wan et al. (2018), nonlinear models and nonlinear control of single-machine infinite systems with SVC have been studied. However, in the actual working environment, there are commonly multiple generators working at the same time. Therefore, it is a really valuable topic to study the multi-machine power systems with SVC.

In recent years, many achievements have been made in the finite-time Lyapunov theory, and a finite-time controller scheme was designed by Du et al. (2018) to make the proposed nonlinear system reach a stable state in finite time. However, the complexity explosion problem was caused by backstepping during the design process. To solve the complexity explosion problem, the command filtering control technology was introduced by Wang et al. (2019) to simplify the calculation of controller design and compensate for filtering errors. For unknown nonlinear functions in nonlinear systems, the approximation properties of neural networks (NNs) or fuzzy logic systems provide an effective way to estimate them. Then, an adaptive controller combining barrier Lyapunov function and NNs was proposed by Liu et al. (2019). which could not only ensure the accuracy of the estimation of unknown nonlinear terms but also achieve the global boundedness of the closed-loop system. In Fan and Li (2018), a kind of stochastic system was discussed, and a tracking control method based on backstepping control technology was proposed by combining adaptive control and NNs. This method could ensure that the tracking error was finally constrained in the adjustable neighborhood of the origin. In Liu et al. (2018), an adaptive fault-tolerant control (FTC) method based on NNs was designed to identify the unknown internal dynamics in the switched system. Then, a new FTC scheme was proposed by Su and Che (2021) by using radial basis function neural networks (RBFNNs), which could ensure that the output tracking error is bounded in the sense of mean square under denial-of-service attacks.

On the other hand, event-triggered control is also a research hotspot in recent years. Unlike the traditional time-triggered control, which maintains the starting state at all times, the mechanism of event-triggered control is that it is triggered only when the system needs it. Thus, event-triggered control can not only reduce the communication burden of the system but also save communication resources, which has great application value and some remarkable research results have been achieved in recent years Tabuada (2007), Girard (2014), Postoyan et al. (2014), Xing et al. (2016, 2017), Ma et al. (2019a, 2019b), Zhang and Yang (2019), Huang et al. (2019), and Wu et al. (2023). In Tabuada (2007), static rules of system state were proposed. In Girard (2014) and Postoyan et al. (2014), an internal dynamic variable was added, on which a dynamic controller triggering event was designed. However, the above control design schemes were all based on the assumption that the closed-loop system had input-state stability (ISS) in terms of measurement errors, which was rarely met in practical control projects. Therefore, the controllers designed by Xing et al. (2016, 2017), Ma et al. (2019a, 2019b), and Zhang and Yang (2019) combined with the event-triggering mechanism avoided the ISS assumption of measurement errors. In Ma et al. (2019b), an event-triggered control scheme based on an observer was established by using NNs to identify the unknown nonlinear terms in the controlled objects. Although some research results have been achieved on event-triggered control, there are few scientific achievements on the finite-time control based on event-triggered control for uncertain nonlinear systems in the existing references, which is worthy of further research by the majority of researchers.

Motivated by the above references, for the uncertain nonlinear systems considering external disturbances, an event-triggered finite-time controller is proposed in this paper, which is applied to the two-area interconnected power system with SVC. The main contributions are as follows:

An event-triggered finite-time control method is proposed for the multi-machine power systems with SVC; compared with the existing references, the designed controller not only has a faster convergence speed than the control methods in Zhang and Yang (2019) but also solves the complexity explosion problem of the traditional backstepping method in Liu et al. (2018).

In the design of the event-triggered control scheme in this paper, compared with Huang et al. (2019), the event-triggered control signal is embedded into the adaptive control laws by introducing the intermediate variable to further effectively eliminate the redundant. At the same time, the system proposed in this paper considers the external disturbances, which have better adaptability and greater robustness.

Compared with the system in Li (2019) and Nai et al. (2020), the nonlinear system proposed in this paper considers the unknown external disturbances and uncertain nonlinear terms simultaneously. In addition, the adaptive finite-time anti-disturbance control method in this paper uses the NNs to approximate the uncertain nonlinear functions of the $n$ -dimensional uncertain nonlinear systems; the case of the third-order SVC system is contained in these systems.

Problem statement and preliminaries

Description of the system

Considering the two-area interconnected power system with SVC, the system structure is shown in Figure 1. Figure 2 shows that the two-area interconnected power system is equivalent to a two-machine system, in which $G_{1}$ and $G_{2}$ are equivalent generators Gu et al. (2015). The mathematical model of the system with SVC can be established as follows:

The $(δ, ω)$ second-order model is selected to establish the mathematical equation, assuming that the input mechanical power $(P_{m 1}, P_{m 2})$ and the transient potential $(E_{1}, E_{2})$ are constants. The damping coefficient of the system cannot be accurately measured. Therefore, the mathematical model of the system without SVC is

{\begin{matrix} {\overset{\cdot}{δ}}_{1} = ω_{1} - ω_{10} \\ {\overset{\cdot}{δ}}_{2} = ω_{2} - ω_{20} \\ {\overset{\cdot}{ω}}_{1} = \frac{ω_{10}}{H_{1}} (P_{m 1} - P_{e 1}) - \frac{D_{1}}{H_{1}} (ω_{1} - ω_{10}) \\ {\overset{\cdot}{ω}}_{2} = \frac{ω_{20}}{H_{2}} (P_{m 2} - P_{e 2}) - \frac{D_{2}}{H_{2}} (ω_{2} - ω_{20}) \\ y = δ_{1} - δ_{2} \end{matrix}

(1)

where the parameters of equivalent generators $G_{1}$ and $G_{2}$ are as follows: $δ_{1}$ and $δ_{2}$ are the rotor operating angles of the two generators; $ω_{1}$ and $ω_{2}$ are the rotor angular velocities of the two generators; $ω_{10}$ and $ω_{20}$ are the rated rotor angular velocity of the two generators, respectively. $H_{1}$ and $H_{2}$ are the rotational inertia of the two generators, respectively. $D_{1}$ and $D_{2}$ are the damping coefficients of the two generators, respectively. $P_{m 1}$ and $P_{m 2}$ are, respectively, the input mechanical power of the two generators; $P_{e 1}$ and $P_{e 2}$ are the electromagnetic power of the two generators, respectively.

Assuming that the load power and line loss are not considered, the electromagnetic power of two equivalent generators $G_{1}$ and $G_{2}$ is

P_{e 1} = - P_{e 2} = \frac{E_{1} E_{2} \sin (δ_{1} - δ_{2})}{X_{Σ}}

(2)

where $X_{Σ}$ is the total reactance in the contact line, $X_{Σ} = X_{a} + X_{b} + X_{a} X_{b} B_{SVC}$ , $B_{SVC}$ is the susceptance of SVC.

In the study of power system stability, SVC is generally equivalent to a first-order inertia link as follows

{\overset{\cdot}{B}}_{SVC} = \frac{1}{T} (- B_{SVC} + B_{SVC 0} + u)

(3)

where $T$ , $u$ , and $B_{SVC 0}$ are the inertia time constant, control quantity, and initial value of susceptance of SVC, respectively.

Let $Δ δ_{12} = δ_{1} - δ_{2} - δ_{0}$ (where $δ_{0}$ is the initial value of the power angle difference between the two generators), and $Δ ω_{12} = ω_{1} - ω_{2}$ . Then, system (1) can be written as

{\begin{matrix} \dot{Δ} δ_{12} = ω_{1} - ω_{2} \\ \begin{array}{l} \dot{Δ} ω_{12} = (\frac{ω_{10}}{H_{1}} P_{m 1} \frac{ω_{20}}{H_{2}} P_{m 2}) - (\frac{ω_{10}}{H_{1}} + \frac{ω_{20}}{H_{2}}) E_{1} E_{2} u \sin (δ_{1} - δ_{2}) \\ - \frac{D_{1}}{H_{1}} (ω_{1} - ω_{10}) + \frac{D_{2}}{H_{2}} (ω_{2} - ω_{20}) \end{array} \\ y = δ_{1} - δ_{2} \end{matrix}

(4)

Set the state variables as $[x_{1}, x_{2}, x_{3}] = [Δ δ_{12}, Δ ω_{12}, \overset{\cdot}{Δ} ω_{12}]$ , assuming the damping of the system is uniform, then let $β = - \frac{D_{1}}{H_{1}} = - \frac{D_{2}}{H_{2}}$ . Considering the influence of external interference, unknown interference $d_{2} (t)$ and $d_{3} (t)$ are added to the system. Then, system (4) can be expressed as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} + d_{2} (t) \\ {\overset{\cdot}{x}}_{3} = β x_{3} + f (x_{1}, x_{2}) + h_{3} \sin (x_{1} + δ_{0}) u + d_{3} (t) \\ y = x_{1} + δ_{0} \end{matrix}

(5)

where

a_{1} = - (\frac{ω_{10}}{H_{1}} + \frac{ω_{20}}{H_{2}}) E_{1} E_{2}

(6)

h_{1} = \frac{a_{1}}{X_{Σ}}

(7)

h_{2} = h_{3} (B_{SVC 0} - B_{SVC})

(8)

h_{3} = - \frac{a_{1} X_{a} X_{b}}{{X_{Σ}}^{2} T}

(9)

f (x_{1}, x_{2}) = h_{1} x_{2} \cos (x_{1} + δ_{0}) + h_{2} \sin (x_{1} + δ_{0})

(10)

The control objective of this paper is to construct a finite-time event-triggered control method based on NNs for the equivalent two-machine system with SVC. Considering the influence of unknown disturbances, the system can still be stable in a finite time.

Remark 1. For a given positive constant $k_{b}$ , if the initial value of $x_{1}$ is less than it, then the designed controller can make the output of the power angle difference $y$ of the equivalent two-machine system within the range of $(δ_{0} - k_{b}, δ_{0} + k_{b})$ , that is, the output always within the limit of the constraint.

Problem formulation for $n$ -dimensional nonlinear systems

To make the designed control scheme more general, the third-order nonlinear power system (5) will be extended to a class of $n$ -dimensional nonlinear systems. The nonlinear system with uncertain nonlinear terms and external disturbances can be described by

{\begin{matrix} {\overset{\cdot}{x}}_{i} = g_{i} ({\bar{x}}_{i}) x_{i + 1} + f_{i} ({\bar{x}}_{i}) + d_{i} (t), i = 1, 2, \dots, n - 1 \\ {\overset{\cdot}{x}}_{n} = g_{n} (x) u (t) + f_{n} (x) + d_{n} (t) \end{matrix}

(11)

where ${\bar{x}}_{i} = [x_{1}, x_{2}, \dots, x_{i}]^{T} \in R^{i}$ and $x = {\bar{x}}_{n}$ is the system state vector. $u (t)$ is the control input, $d_{i} (t)$ is the unknown external disturbance, respectively. $f_{i} (\cdot)$ is an unknown nonlinear function. $g_{i} (\cdot)$ is a known smooth function satisfying $g_{i} (\cdot) \neq 0$ , and $| g_{i} (\cdot) | \leq {\bar{g}}_{i}$ , where ${\bar{g}}_{i}$ is a positive constant.

Assumption 1. The derivative of external disturbance $d_{i}$ in system (11) is bounded, satisfying $lim_{t \to \infty} {\overset{\cdot}{d}}_{i} = 0$ .

RBFNNs and preliminary knowledge

To approximate the unknown nonlinear function $f_{i} (\cdot)$ , the RBFNNs are introduced in this paper, the definitions of which are as follows.

If there is an unknown continuous function $F (χ)$ $(χ \in Ω_{χ}, Ω_{χ} \subset R^{n})$ and $\forall ϵ > 0$ , there’s going to be an integer $ϑ^{* T} S (χ)$ makes the following inequation exists

sup_{χ \in Ω_{χ}} | ϑ^{* T} S (χ) - F (χ) | \leq ϵ

(12)

where $χ \in R^{n}$ is the RBFNNs’ input vector, $ϑ^{*} \in R^{τ}$ is the ideal weight vector, $S (χ) = [S_{1} (χ), . . ., S_{Ψ} (χ)]^{T}$ is the smooth basis vector and $Ψ$ is the number of node. Thus, combining with the Gaussian functions, $S_{ϱ} (χ), ϱ = 1, . . ., Ψ$ can be defined as

S_{ϱ} (χ) = \exp (- \frac{{(χ - κ_{ϱ})}^{T} (χ - κ_{ϱ})}{v_{ϱ}^{2}}), ϱ = 1, \dots, Ψ

(13)

where $κ_{ϱ} = [κ_{ϱ 1}, . . ., κ_{ϱ n}]$ are the centers of RBFNNs, $v_{ϱ}$ are the width of RBFNNs.

According to the monotonicity of the exponential function, there exists $0 < S_{ρ} (\cdot) < 1$ , which means $S_{ρ}^{2} (\cdot) < 1$ . Thus, we can get the following inequation

S^{T} (χ) S (χ) \leq Ψ

(14)

Thus, for an unknown function $F (χ)$ , it can be approximated by RBFNNs as

F (χ) = ϑ^{* T} S (χ) + ϵ (χ)

(15)

where $ϵ (χ)$ is the smallest approximation error. Assuming that $ϵ (χ) \leq \bar{ϵ}$ , $\bar{ϵ}$ is the boundary of $ϵ (χ)$ .

Definition 1. Zhu et al. (2011) For a nonlinear system $\overset{\cdot}{χ} = f (χ, u)$ , where $χ$ is the state vector, $u$ is the input vector. The solution is practical finite-time stable (PFS) if, for all $χ (t_{0}) = χ_{0}$ , there exists $δ > 0$ and $T (δ, χ_{0}) < \infty$ , such that $χ (t) < δ$ , for all $t \geq t_{0} + T$ .

Lemma 1. Zhu et al. (2011) and Bhat and Bernstein (2000) Suppose $V (χ)$ is a $C^{1}$ smooth positive-definite function (defined on $U \subset R^{n}$ ) and $\overset{\cdot}{V} (χ) + F^{'} V^{α} (χ)$ is a negative semi-definite function on $U \subset R^{n}$ for $α \in (0, 1)$ and $F^{'} \in R^{+}$ , then there exists an area $U_{0} \subset R^{n}$ such that any $V (χ)$ which starts from $U_{0} \subset R^{n}$ can reach $V (χ) \equiv 0$ in finite time. Moreover, if $T_{reach}$ is the time needed to reach $V (χ) \equiv 0$ , then

T_{reach} \leq \frac{V^{1 - α} (χ_{0})}{F^{'} (1 - α)}

(16)

where $V (χ_{0})$ is the initial value of $V (χ)$ .

Lemma 2. Cai et al. (2016) For a nonlinear system $\overset{\cdot}{χ} = f (χ, u)$ . Assuming that there exists a continuous function $V (χ)$ and parameters $F^{'} > 0$ , $0 < α < 1$ , $0 < η < \infty$ such that the following formula holds

\overset{\cdot}{V} (χ) \leq - F^{'} V^{α} (χ) + η

(17)

when the above conditions are met, the system $\overset{\cdot}{χ} = f (χ, u)$ is PFS.

Lemma 3. Xing et al. (2017) and Li (2019)For $\forall χ \in R$ and $\forall ϑ > 0$ , there exists the following inequality

0 \leq | χ | - χ \tanh (χ / ϑ) \leq 0.2785 ϑ = ϑ^{'}

(18)

Lemma 4. Xing et al. (2017) For $\forall s_{i} \in R, i = 1, 2, \dots, n$ and $0 < ς \leq 1$ , there exists the following inequality

{(\sum_{i = 1}^{n} | s_{i} |)}^{ς} \leq \sum_{i = 1}^{n} | s_{i} |^{ς} \leq n^{1 - ς} {(\sum_{i = 1}^{n} | s_{i} |)}^{ς}

(19)

Lemma 5. Nai et al. (2020) For the system $\overset{\cdot}{ξ} = f (ξ)$ , if there exist positive definite function $V (ξ)$ and parameters $a > 0$ , $b > 0$ , $0 < c < \infty$ , and $0 < ρ < 1$ making the following inequality true

\overset{\cdot}{V} (ξ) \leq - aV (ξ) - bV (ξ)^{ρ} + c

(20)

then, the finite-time stability of the system $\overset{\cdot}{ξ} = f (ξ)$ is guaranteed. The setting time $T_{r}$ satisfies that $T_{r} \leq \max {t_{0} + \frac{1}{τ a (1 - ρ)} \ln \frac{τ a V^{1 - ρ} (t_{0}) + b}{b}, t_{0} + \frac{1}{a (1 - ρ)} \ln \frac{a V^{1 - ρ} (t_{0}) + τ b}{τ b}}$ , where $0 < τ < 1$ .

Control scheme design and stability analysis

In this section, the disturbance observer is designed to estimate the unknown external disturbances. Then, the event-triggered controller based on NNs and a command filter are proposed to ensure the system is stable in finite time. In the end, we analyze the stability of the designed control method.

Disturbance observer design

To compensate for the effect of external disturbances, a disturbance observer is designed. The specific processes are as follows.

Rewrite equation (11) as

{\begin{matrix} \overset{\cdot}{X} = AX + F (X) + G (X) u + Ed (t) \\ \overset{\cdot}{y} = x_{1} + δ_{0} \end{matrix}

(21)

where

X = [x_{1}, x_{2}, . . ., x_{n}]^{T}

(22)

A = [\begin{matrix} 0 & g_{1} ({\bar{x}}_{1}) & 0 & . . . & 0 \\ 0 & 0 & g_{2} ({\bar{x}}_{2}) & . . . & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & . . . & g_{n - 1} ({\bar{x}}_{n - 1}) \\ 0 & 0 & 0 & . . . & 0 \end{matrix}]

(23)

F (X) = [f_{1} ({\bar{x}}_{1}), f_{2} ({\bar{x}}_{2}), . . ., f_{n} ({\bar{x}}_{n})]^{T}

(24)

G (X) = [0, 0, . . ., g_{n} ({\bar{x}}_{n})]^{T}

(25)

E = I^{n}

(26)

d (t) = [d_{1} (t), d_{2} (t), . . ., d_{n} (t)]^{T}

(27)

Then, the following disturbance observer is established to estimate the unknown disturbances of the system

{\begin{matrix} \overset{\cdot}{p} = - W (X) Ep - W (X) [Em (X) + AX + F (X) + G (X) u] \\ \hat{d} (t) = p + m (X) \end{matrix}

(28)

where $p$ is the internal state of the nonlinear disturbance observer, $\hat{d} (t)$ is the estimate of $d (t)$ , $m (X)$ is the nonlinear vector function to be designed. The gain matrix $W (X)$ of the nonlinear disturbance observer is defined as

W (X) = \frac{\partial m (X)}{\partial X}

(29)

The observation error of nonlinear disturbance observer $e_{d} (t)$ is defined as

e_{d} (t) = d (t) - \hat{d} (t)

(30)

The error dynamics can be derived as

\begin{matrix} {\overset{\cdot}{e}}_{d} (t) = \overset{\cdot}{d} (t) - \overset{\cdot}{\hat{d}} (t) \\ = \overset{\cdot}{d} (t) - \overset{\cdot}{p} - W (X) [AX + F (X) + G (X) u + Ed (t)] \end{matrix}

(31)

Substituting (28) into (31) yields

{\overset{\cdot}{e}}_{d} (t) = \overset{\cdot}{d} (t) - W (X) E e_{d} (t)

(32)

Considering that the disturbance estimation of the output of the disturbance observer can gradually track the disturbance in the system (21), the observer gains $W (X)$ is selected to make $W (X) E > 0$ valid, which means that ${\overset{\cdot}{e}}_{d} (t) + W (X) E e_{d} (t) = 0$ , the disturbance estimation error converges to 0.

Remark 2. Compared with the linear disturbance observer proposed by Yao et al. (2018), the nonlinear disturbance observer proposed in this paper has stronger robustness due to its feedback control mechanism and the introduction of higher-order terms to compensate for the system’s nonlinearity. In addition, the disturbance observer does not need to know exactly the mathematical model of the control system and the precise dynamic characteristics of the system, which makes the application of the disturbance observer wider. Compared with the linear reduced-order observer proposed by Chen et al. (2019), the higher-order terms introduced in the nonlinear disturbance observer make it have higher estimation accuracy.

Adaptive event-triggered finite-time controller design

Combined with the backstepping technology, coordinate transformation is applied to state variables as

ψ_{1} = x_{1}

ψ_{i} = x_{i} - ζ_{i}, i = 2, \dots, n

(33)

where $ζ_{i}$ is the output of the command filter, which is introduced as follows

a_{i} {\overset{\cdot}{ζ}}_{i} + ζ_{i} = α_{i}, ζ_{i} (0) = α_{i} (0), i = 2, \dots, n

(34)

where $a_{i} > 0$ is the design parameter, $α_{i}$ is the virtual controllers to be designed, and the input of the command filter.

To compensate for filter errors, the auxiliary system can be designed as follows

\begin{matrix} {\overset{\cdot}{ξ}}_{1} = - ν_{1} ξ_{1} + g_{1} (ζ_{2} - α_{2}) + g_{1} ξ_{2} - ϰ_{1} sign (ξ_{1}) \\ {\overset{\cdot}{ξ}}_{i} = - ν_{i} ξ_{i} + g_{i} (ζ_{i + 1} - α_{i + 1}) - g_{i - 1} ξ_{i - 1} + g_{i} ξ_{i + 1} - ϰ_{i} sign (ξ_{i}), \\ i = 2, \dots, n - 1 \\ {\overset{\cdot}{ξ}}_{n} = - ν_{n} ξ_{n} - g_{n - 1} ξ_{n - 1} - ϰ_{n} sign (ξ_{n}) \end{matrix}

(35)

where $ν_{i} > 0$ , $ϰ_{i} > 0$ are design parameters, respectively.

Remark 3. In the process of using the traditional backstepping method, with the improvement of the system order, it is necessary to design a higher-order feedback controller, and the computational complexity increases exponentially, which makes the control system difficult to realize. Therefore, we introduced the command filtering technology. By filtering the input commands of the controller, the redundant components brought by the high-order controller could be eliminated in the control system, so as to reduce the order of the feedback controller and reduce the calculation load.

The definition of the compensated error is

ϖ_{i} = ψ_{i} - ξ_{i}

(36)

Step 1: Taking the derivative of $ϖ_{1}$ , we have

\begin{matrix} {\overset{\cdot}{ϖ}}_{1} = {\overset{\cdot}{ψ}}_{1} - {\overset{\cdot}{ξ}}_{1} = {\overset{\cdot}{x}}_{1} + ν_{1} ξ_{1} - g_{1} ζ_{2} + g_{1} α_{2} - g_{1} ξ_{2} + ϰ_{1} sign (ξ_{1}) \\ = g_{1} x_{2} + f_{1} (x_{1}) + d_{1} + ν_{1} ξ_{1} - g_{1} ζ_{2} + g_{1} α_{2} \\ - g_{1} ξ_{2} + ϰ_{1} sign (ξ_{1}) \\ = g_{1} ϖ_{2} + g_{1} α_{2} + f_{1} (x_{1}) + d_{1} + ν_{1} ξ_{1} + ϰ_{1} sign (ξ_{1}) \end{matrix}

(37)

Select the following Lyapunov function

V_{1} = \frac{1}{2} ϖ_{1}^{2} + \frac{1}{2} {\tilde{ϑ}}_{1}^{2}

(38)

where ${\tilde{ϑ}}_{1} = ϑ_{1} - {\hat{ϑ}}_{1}$ is the estimation error, ${\hat{ϑ}}_{1}$ is the estimation of $ϑ_{1}$ .

Taking the derivative of $V_{1}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = ϖ_{1} {\overset{\cdot}{ϖ}}_{1} - {\tilde{ϑ}}_{1} {\overset{\cdot}{\hat{ϑ}}}_{1} \\ = ϖ_{1} [g_{1} (ϖ_{2} + α_{2}) + f_{1} (x_{1}) + d_{1} + ν_{1} ξ_{1} + ϰ_{1} sign (ξ_{1})] - {\tilde{ϑ}}_{1} {\overset{\cdot}{\hat{ϑ}}}_{1} \\ = ϖ_{1} [g_{1} (ϖ_{2} + α_{2}) + ϑ_{1}^{* ⊤} S_{1} (z_{1}) + ϵ_{1}^{*} (z_{1}) + d_{1} + ν_{1} ξ_{1} \\ + ϰ_{1} sign (ξ_{1})] - {\tilde{ϑ}}_{1} {\overset{\cdot}{\hat{ϑ}}}_{1} \end{matrix}

(39)

where $ϑ_{1}^{*}$ is the NNs’ weight vector, $S_{1} (z_{1})$ is the radial basis function, $z_{1} = x_{1}$ is the input vector of RBFNNs, $ϵ_{1}^{*}$ is the approximation error.

Using the Young’s inequality, we have

\begin{matrix} ϖ_{1} ϑ_{1}^{* T} S_{1} (z_{1}) \leq \frac{1}{2} + \frac{1}{2} ϖ_{1}^{2} ϑ_{1} S_{1}^{⊤} (z_{1}) S_{1} (z_{1}) \\ ϖ_{1} ϵ_{1}^{*} (z_{1}) \leq \frac{1}{2} ϖ_{1}^{2} + \frac{1}{2} {\bar{ϵ}}_{1}^{2} \end{matrix}

(40)

Substituting (40) into (39), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq ϖ_{1} [g_{1} (ϖ_{2} + α_{2}) + \frac{1}{2} ϖ_{1} ϑ_{1} S_{1} {(z_{1})}^{⊤} S_{1} (z_{1}) + \frac{1}{2} ϖ_{1} + d_{1} \\ + ν_{1} ξ_{1} + ϰ_{1} sign (ξ_{1})] - {\tilde{ϑ}}_{1} {\overset{\cdot}{\hat{ϑ}}}_{1} + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{1}^{2} \end{matrix}

(41)

The finite-time virtual controller $α_{2}$ is designed as

α_{2} = \frac{1}{g_{1}} (- ν_{1} ψ_{1} - \frac{1}{2} ϖ_{1} {\hat{ϑ}}_{1} S_{1}^{T} (z_{1}) S_{1} (z_{1}) - {\hat{d}}_{1} - | ϖ_{1} |^{c} sign (ϖ_{1}))

(42)

where $0 < c < 1$ is a constant.

The parameter adaptive law $\overset{\cdot}{{\hat{ϑ}}_{1}}$ is designed as

\overset{\cdot}{{\hat{ϑ}}_{1}} = - γ_{1} \hat{ϑ_{1}} + \frac{1}{2} ϖ_{1}^{2} S_{1}^{⊤} (z_{1}) S_{1} (z_{1})

(43)

Combining equations (41)–(43), we can conclude that

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq (\frac{1}{2} - ν_{1}) ϖ_{1}^{2} + g_{1} ϖ_{1} ϖ_{2} + ϖ_{1} ϰ_{1} sign (ξ_{1}) - ϖ_{1} | ϖ_{1} |^{c} sign (ϖ_{1}) \\ + γ_{1} \hat{ϑ_{1}} {\tilde{ϑ}}_{1} + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{1}^{2} \end{matrix}

(44)

Consider the following fact

γ_{1} {\hat{ϑ}}_{1} {\tilde{ϑ}}_{1} \leq \frac{γ_{1}}{2} ϑ_{1}^{2} - \frac{γ_{1}}{2} {\tilde{ϑ}}_{1}^{2}

(45)

Then, one yields

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - (ν_{1} - \frac{1}{2}) ϖ_{1}^{2} - ϖ_{1} {| ϖ_{1} |}^{c} sign (ϖ_{1}) - \frac{γ_{1}}{2} {\tilde{ϑ}}_{1}^{2} \\ + g_{1} ϖ_{1} ϖ_{2} + ϖ_{1} ϰ_{1} sign (ξ_{1}) + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{1}^{2} + \frac{γ_{1}}{2} ϑ_{1}^{2} \end{matrix}

(46)

Step 2: Taking the derivative of $ϖ_{2}$ , we have

\begin{matrix} {\overset{\cdot}{ϖ}}_{2} = {\overset{\cdot}{ψ}}_{2} - {\overset{\cdot}{ξ}}_{2} \\ = g_{2} x_{3} + f_{2} ({\bar{x}}_{2}) + d_{2} + ν_{2} ξ_{2} - g_{2} ζ_{3} + g_{2} α_{3} - g_{2} ξ_{3} \\ + g_{1} ξ_{1} + ϰ_{2} sign (ξ_{2}) - {\overset{\cdot}{ζ}}_{2} \\ = g_{2} ϖ_{3} + g_{2} α_{3} + f_{2} ({\bar{x}}_{2}) + d_{2} + ν_{2} ξ_{2} + ϰ_{2} sign (ξ_{2}) + g_{1} ξ_{1} - {\overset{\cdot}{ζ}}_{2} \end{matrix}

(47)

Select the following Lyapunov function

V_{2} = V_{1} + \frac{1}{2} ϖ_{2}^{2} + \frac{1}{2} {\tilde{ϑ}}_{2}^{2}

(48)

where ${\tilde{ϑ}}_{2} = ϑ_{2} - {\hat{ϑ}}_{2}$ is the estimation error, ${\hat{ϑ}}_{2}$ is the estimation of $ϑ_{2}$ .

Taking the derivative of $V_{2}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + ϖ_{2} {\overset{\cdot}{ϖ}}_{2} - {\tilde{ϑ}}_{2} {\overset{\cdot}{\hat{ϑ}}}_{2} \\ = {\overset{\cdot}{V}}_{1} + ϖ_{2} [g_{2} ϖ_{3} + g_{2} α_{3} + ϑ_{2}^{* ⊤} S_{2} (z_{2}) + ϵ_{2}^{*} (z_{2}) + d_{2} + ν_{2} ξ_{2} \\ + ϰ_{2} sign (ξ_{2}) + g_{1} ξ_{1} - {\overset{\cdot}{ζ}}_{2}] - {\tilde{ϑ}}_{2} {\overset{\cdot}{\hat{ϑ}}}_{2} \end{matrix}

(49)

where $z_{2} = [x_{1}, x_{2}]^{T}$ is the input vector of NNs.

Using the Young’s inequality, we have

\begin{matrix} ϖ_{2} ϑ_{2}^{* ⊤} S_{2} (z_{2}) \leq \frac{1}{2} + \frac{1}{2} ϖ_{2}^{2} ϑ_{2} S_{2}^{⊤} (z_{2}) S_{2} (z_{2}) \\ ϖ_{2} ϵ_{2}^{*} (z_{2}) \leq \frac{1}{2} ϖ_{2}^{2} + \frac{1}{2} {\bar{ϵ}}_{2}^{2} \end{matrix}

(50)

Substituting (50) into (49), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq {\overset{\cdot}{V}}_{1} + ϖ_{2} [g_{2} (ϖ_{3} + α_{3}) + \frac{1}{2} ϖ_{2} ϑ_{2} S_{2} {(z_{2})}^{⊤} S_{2} (z_{2}) + \frac{1}{2} ϖ_{2} + d_{2} . \end{matrix} + ν_{2} ξ_{2} + ϰ_{2} sign (ξ_{2}) + g_{1} ξ_{1} - {\overset{\cdot}{ζ}}_{2}] - {\tilde{ϑ}}_{2} {\overset{\cdot}{\hat{ϑ}}}_{2} + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{2}^{2}

(51)

The finite-time virtual controller $α_{2}$ is designed as

α_{3} = \frac{1}{g_{2}} (- g_{1} ψ_{1} - ν_{2} ψ_{2} - \frac{1}{2} ϖ_{2} {\hat{ϑ}}_{2} S_{2}^{⊤} (z_{2}) S_{2} (z_{2}) - {\hat{d}}_{2} - | ϖ_{2} |^{c} sign (ϖ_{2}) + {\overset{\cdot}{ζ}}_{2})

(52)

The parameter adaptive law $\overset{\cdot}{{\hat{ϑ}}_{2}}$ is designed as

\overset{\cdot}{{\hat{ϑ}}_{2}} = - γ_{2} {\hat{ϑ}}_{2} + \frac{1}{2} ϖ_{2}^{2} S_{2}^{⊤} (z_{2}) S_{2} (z_{2})

(53)

Combining equations (51)–(53), we can conclude that

\begin{array}{l} {\dot{V}}_{2} \leq - (ν_{1} - \frac{1}{2}) ϖ_{1}^{2} - ϖ_{1} | ϖ_{1} |^{c} sign (ϖ_{1}) - \frac{γ_{1}}{2} {\tilde{ϑ}}_{1}^{2} \\ + ϖ_{1} ϰ_{1} sign (ξ_{1}) + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{1}^{2} + \frac{γ_{1}}{2} ϑ_{1}^{2} \\ + g_{2} ϖ_{2} ϖ_{3} - (ν_{2} - \frac{1}{2}) ϖ_{2}^{2} - ϖ_{2} | ϖ_{2} |^{2} sign (ϖ_{2}) + \frac{1}{2} + \frac{1}{2} ϵ_{2}^{2} \\ + γ_{2} {\hat{ϑ}}_{2} {\tilde{ϑ}}_{2} + ϖ_{2} ϰ_{2} sign (ξ_{2}) \end{array}

(54)

Consider the following fact

γ_{2} \hat{ϑ_{2}} {\tilde{ϑ}}_{2} \leq \frac{γ_{2}}{2} ϑ_{2}^{2} - \frac{γ_{2}}{2} {\tilde{ϑ}}_{2}^{2}

(55)

Then, one yields

\begin{array}{l} {\dot{V}}_{2} \leq - (ν_{1} - \frac{1}{2}) ϖ_{1}^{2} - ϖ_{1} | ϖ_{1} |^{c} sign (ϖ_{1}) - \frac{γ_{1}}{2} {\tilde{ϑ}}_{1}^{2} \\ + ϖ_{1} ϰ_{1} sign (ξ_{1}) + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{1}^{2} + \frac{γ_{1}}{2} ϑ_{1}^{2} \\ + g_{2} ϖ_{2} ϖ_{3} - (ν_{2} - \frac{1}{2}) ϖ_{2}^{2} - ϖ_{2} | ϖ_{2} |^{c} sign (ϖ_{2}) - \frac{γ_{2}}{2} {\tilde{ϑ}}_{2}^{2} \\ + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{2}^{2} + \frac{γ_{2}}{2} ϑ_{2}^{2} + ϖ_{2} ϰ_{2} sign (ξ_{2}) \end{array}

(56)

Step i $(i = 2, \dots, n - 1)$ : Taking the derivative of $ϖ_{i}$ , one has

\begin{matrix} {\overset{\cdot}{ϖ}}_{i} = {\overset{\cdot}{ψ}}_{i} - {\overset{\cdot}{ξ}}_{i} \\ = g_{i} x_{i + 1} + f_{i} ({\bar{x}}_{i}) + d_{i} + ν_{i} ξ_{i} - g_{i} ζ_{i + 1} + g_{i} α_{i + 1} - g_{i} ξ_{i + 1} \\ + g_{i - 1} ξ_{i - 1} + ϰ_{i} sign (ξ_{i}) - {\overset{\cdot}{ζ}}_{i} \\ = g_{i} ϖ_{i + 1} + g_{i} α_{i + 1} + f_{i} ({\bar{x}}_{i}) + d_{i} + ν_{i} ξ_{i} + ϰ_{i} sign (ξ_{i}) \\ + g_{i - 1} ξ_{i - 1} - {\overset{\cdot}{ζ}}_{i} \end{matrix}

(57)

Select the following Lyapunov function

V_{i} = V_{i - 1} + \frac{1}{2} ϖ_{i}^{2} + \frac{1}{2} {\tilde{ϑ}}_{i}^{2}

(58)

where ${\tilde{ϑ}}_{i} = ϑ_{i} - {\hat{ϑ}}_{i}$ is the estimation error, ${\hat{ϑ}}_{i}$ is the estimation of $ϑ_{i}$ .

Then, taking the derivative of $V_{i}$ with time, we have

\begin{matrix} {\overset{\cdot}{V}}_{i} = {\overset{\cdot}{V}}_{i - 1} + ϖ_{i} {\overset{\cdot}{ϖ}}_{i} - {\tilde{ϑ}}_{i} {\overset{\cdot}{\hat{ϑ}}}_{i} \\ = {\overset{\cdot}{V}}_{i - 1} + ϖ_{i} [g_{i} ϖ_{i + 1} + g_{i} α_{i + 1} + ϑ_{i}^{* T} S_{i} (z_{i}) + ϵ_{i}^{*} (z_{i}) + d_{i} + ν_{i} ξ_{i} \\ + ϰ_{i} sign (ξ_{i}) + g_{i - 1} ξ_{i - 1} - {\overset{\cdot}{ζ}}_{i}] - {\tilde{ϑ}}_{i} {\overset{\cdot}{\hat{ϑ}}}_{i} \end{matrix}

(59)

where $z_{i} = [x_{1}, \dots, x_{i}]^{T}$ is the input vector of RBFNNs.

The finite-time virtual controller $α_{i + 1}$ is designed as

\begin{matrix} α_{i + 1} = \frac{1}{g_{i}} \\ (- g_{i - 1} ψ_{i - 1} - ν_{i} ψ_{i} - \frac{1}{2} ϖ_{i} {\hat{ϑ}}_{i} S_{i}^{⊤} (z_{i}) S_{i} (z_{i}) - {\hat{d}}_{i} - | ϖ_{i} |^{c} sign (ϖ_{i}) + {\overset{\cdot}{ζ}}_{i}) \end{matrix}

(60)

The parameter adaptive law $\overset{\cdot}{{\hat{ϑ}}_{i}}$ is designed as

\overset{\cdot}{{\hat{ϑ}}_{i}} = - γ_{i} {\hat{ϑ}}_{i} + \frac{1}{2} ϖ_{i}^{2} S_{i}^{⊤} (z_{i}) S_{i} (z_{i})

(61)

Consider the following fact

γ_{i} {\hat{ϑ}}_{i} {\tilde{ϑ}}_{i} \leq \frac{γ_{i}}{2} ϑ_{i}^{2} - \frac{γ_{i}}{2} {\tilde{ϑ}}_{i}^{2}

(62)

Then, one yields

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq \sum_{j = 1}^{i} - (ν_{j} - \frac{1}{2}) ϖ_{j}^{2} - \sum_{j = 1}^{i} ϖ_{i} | ϖ_{i} |^{c} sign (ϖ_{i}) - \sum_{j = 1}^{i} \frac{γ_{j}}{2} {\tilde{ϑ}}_{j}^{2} \\ + \sum_{j = 1}^{i} \frac{γ_{j}}{2} ϑ_{j}^{2} + \frac{i}{2} + \sum_{j = 1}^{i} \frac{1}{2} {\bar{ϵ}}_{j}^{2} \\ + \sum_{j = 1}^{i} ϖ_{i} ϰ_{i} sign (ξ_{i}) + g_{i} ϖ_{i} ϖ_{i + 1} \end{matrix}

(63)

Step n: The finite-time virtual controller $α_{n}$ is designed as

\begin{matrix} α_{n} = \frac{1}{g_{n}} (- g_{n - 1} φ_{n - 1} - ν_{n} φ_{n} - | ϖ_{n} |^{c} sign (ϖ_{n}) \\ - \frac{1}{2} ϖ_{n} {\hat{ϑ}}_{n} S_{n}^{T} (z_{n}) S_{n} (z_{n}) - {\hat{d}}_{n} (t) + {\overset{\cdot}{ζ}}_{n}) \end{matrix}

(64)

The parameter adaptive law $\overset{\cdot}{{\hat{ϑ}}_{n}}$ is designed as

\begin{matrix} {\overset{\cdot}{\hat{ϑ}}}_{n} = - γ_{n} {\hat{ϑ}}_{n} + \frac{1}{2} ϖ_{n}^{2} S_{n}^{T} (z_{n}) S_{n} (z_{n}) \end{matrix}

(65)

Then, we set up the following finite-time event-triggered controller

\begin{matrix} Φ (t) = - (1 + Γ) (α_{n} \tanh (\frac{ϖ_{n} α_{n}}{σ}) + \bar{a} \tanh (\frac{ϖ_{n} \bar{a}}{σ})) \\ u (t) = Φ (t_{q}), \forall t \in [t_{q}, t_{q + 1}) \end{matrix}

(66)

Select the following event-triggering mechanism as

t_{q + 1} = inf {t \in R | | π (t) | \geq Γ | u (t) | + a}

(67)

where $π (t) = Φ (t) - u (t)$ is measurement error, $σ > 0, a > 0, 0 < Γ < 1$ , and $\bar{a} > \frac{a}{(1 - Γ)}$ are positive design parameters. $t_{q} (q \in z^{+})$ is the controller update time, that is, whenever (67) is triggered, the triggering time is marked as $t_{q + 1}$ and the control value $u (t_{q + 1})$ is applied to the controlled system. During $t \in [t_{q}, t_{q + 1})$ , the control value is a constant, $Φ (t_{q})$ .

According to equation (66), the following equation can be established

Φ (t) = (1 + e_{1} (t) Γ) u (t) + e_{2} (t) a, t \in [t_{q}, t_{q + 1})

(68)

where $| e_{1} (t) | \leq 1$ and $| e_{2} (t) | \leq 1$ are time-varying parameters. Then, the above equation can be rewritten as

u (t) = \frac{Φ (t) - e_{2} (t) a}{1 + e_{1} (t) Γ}

(69)

Select the following Lyapunov function

V_{n} = V_{n - 1} + \frac{1}{2} ϖ_{n}^{2} + \frac{1}{2} {\tilde{ϑ}}_{n}^{2}

(70)

where ${\tilde{ϑ}}_{n} = ϑ_{n} - {\hat{ϑ}}_{n}$ is the estimation error, ${\hat{ϑ}}_{n}$ is the estimation of $ϑ_{n}$ .

Taking derivative of $V_{n}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + ϖ_{n} {\overset{\cdot}{ϖ}}_{n} - {\tilde{ϑ}}_{n} {\overset{\cdot}{\hat{ϑ}}}_{n} \\ = {\overset{\cdot}{V}}_{n - 1} + ϖ_{n} [g_{n} \frac{Φ (t) - e_{2} (t) a}{1 + e_{1} (t) Γ} + ϑ_{n}^{* T} S_{n} (z_{n}) \\ ϵ_{n}^{*} (z_{n}) + d_{n} (t) - {\overset{\cdot}{ζ}}_{n} + ν_{n} ξ_{n} \\ + g_{n - 1} ξ_{n - 1} ϰ_{n} sign (ξ_{n})] - {\tilde{ϑ}}_{n} {\overset{\cdot}{\hat{ϑ}}}_{n} \end{matrix}

(71)

where $z_{n} = [x_{1}, \dots, x_{n}]^{T}$ is the input vector of RBFNNs.

Using the Young’s inequality, we have

\begin{array}{l} ϖ_{n} ϑ_{n}^{* T} S_{n} (z_{n}) \leq \frac{1}{2} + \frac{1}{2} ϖ_{n}^{2} ϑ_{n} S_{n}^{T} (z_{n}) S_{n} (z_{n}) \\ ϖ_{n} ϵ_{n}^{*} (z_{n}) \leq \frac{1}{2} ϖ_{n}^{2} + \frac{1}{2} {\bar{ϵ}}_{n}^{2} \end{array}

(72)

From $e_{1} (t) \in [- 1, 1]$ and $e_{2} (t) \in [- 1, 1]$ , we can get the following equation

\frac{ϖ_{n} Φ (t)}{1 + e_{1} (t) Γ} \leq \frac{ϖ_{n} Φ (t)}{1 + Γ} | \frac{e_{2} (t) a}{1 + e_{1} (t)} | \leq \frac{a}{1 - Γ}

(73)

Combining equations (66) and (71)–(73), we can conclude that

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq {\overset{\cdot}{V}}_{n - 1} - ϖ_{n} g_{n} (α_{n} \tanh (\frac{ϖ_{n} α_{n}}{σ}) + \bar{a} \tanh (\frac{ϖ_{n} \bar{a}}{σ})) \\ + | \frac{g_{n} ϖ_{n} a}{1 - Γ} | + ϖ_{n} [\frac{1}{2} ϖ_{n} ϑ_{n} S_{n}^{⊤} (z_{n}) S_{n} (z_{n}) \frac{1}{2} ϖ_{n} + d_{n} (t) - {\overset{\cdot}{ζ}}_{n} + ν_{n} ξ_{n} \\ + g_{n - 1} ξ_{n - 1} + ϰ_{n} sign (ξ_{n})] + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{n}^{2} - {\tilde{ϑ}}_{n} {\overset{\cdot}{\hat{ϑ}}}_{n} \end{matrix}

(74)

According to Lemma 3, equation (74) can be redescribed as

\begin{array}{l} {\dot{V}}_{n} \leq {\dot{V}}_{n - 1} + g_{n} ϖ_{n} α_{n} + ϖ_{n} \\ [\frac{1}{2} ϖ_{n} ϑ_{n} S_{n}^{⊤} (z_{n}) S_{n} (z_{n}) + \frac{1}{2} ϖ_{n} + d_{n} (t) - {\dot{ζ}}_{n} + ν_{n} ξ_{n} \\ + g_{n - 1} ξ_{n - 1} + ϰ_{n} sign (ξ_{n})] + \frac{1}{2} + \frac{1}{2} {\bar{ϵ}}_{n}^{2} - | ϖ_{n} g_{n} \bar{a} | + | \frac{ϖ_{n} g_{n} a}{1 - Γ} | \\ + 0.557 g_{n} σ - {\tilde{ϑ}}_{n} {\dot{\hat{ϑ}}}_{n} \end{array}

(75)

Substituting (63) into (75) yields

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq \sum_{j = 1}^{n} - (ν_{j} - \frac{1}{2}) ϖ_{j}^{2} - \sum_{j = 1}^{n} ϖ_{j} {| ϖ_{j} |}^{c} sign (ϖ_{j}) - \sum_{j = 1}^{n} \frac{γ_{j}}{2} {\tilde{ϑ}}_{j}^{2} \\ + \sum_{j = 1}^{n} \frac{γ_{j}}{2} ϑ_{j}^{2} \\ + \frac{n}{2} + \frac{1}{2} \sum_{j = 1}^{n} {\bar{ϵ}}_{j}^{2} + \sum_{j = 1}^{n} ϖ_{j} ϰ_{j} sign (ϵ_{j}) + 0.557 {\bar{g}}_{n} σ + γ_{n} {\hat{ϑ}}_{n} {\tilde{ϑ}}_{n} \end{matrix}

(76)

Using the Young’s inequality, we have

ϖ_{j} ϰ_{j} sign (ϵ_{j}) \leq \frac{1}{2} ϖ_{j}^{2} + \frac{1}{2} ϰ_{j}^{2}

(77)

Consider the following fact

γ_{n} {\hat{ϑ}}_{n} {\tilde{ϑ}}_{n} \leq \frac{γ_{n}}{2} ϑ_{n}^{2} - \frac{γ_{n}}{2} {\tilde{ϑ}}_{n}^{2}

(78)

Then, one yields

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq \sum_{j = 1}^{n} - (ν_{j} - 1) ϖ_{j}^{2} - \sum_{j = 1}^{n} | ϖ_{j} |^{c + 1} - \sum_{j = 1}^{n} \frac{γ_{j}}{2} {\tilde{ϑ}}_{j}^{2} + \sum_{j = 1}^{n} \frac{γ_{j}}{2} ϑ_{j}^{2} \\ + \frac{n}{2} + \sum_{j = 1}^{n} \frac{1}{2} {\bar{ϵ}}_{j}^{2} + \sum_{j = 1}^{n} \frac{1}{2} ϰ_{j}^{2} + 0.557 {\bar{g}}_{n} σ \end{matrix}

(79)

Stability analysis

Theorem 6. Consider the nonlinear system (11) with external disturbances and unknown nonlinear functions. Under Assumption 2.1, the virtual controller (42), (52), (60), (64), the event-triggered controller (66), and the adaptive laws (43), (53), (61), (65), the following conclusions can be obtained:

All the variables in the closed-loop system are bounded;

The output $y (t)$ can track the given signal $y_{d}$ in the sense of finite time stability;

There exists a time $t^{'} > 0$ that satisfies $t_{q + 1} - t_{q} \geq t^{'} (\forall q \in z^{+})$ , which means that the Zeno phenomenon is effectively avoided.

Proof. Select the following common Lyapunov function

V = V_{n} = \sum_{i = 1}^{n} \frac{1}{2} ϖ_{j}^{2} + \sum_{i = 1}^{n} \frac{1}{2} {\tilde{ϑ}}_{j}^{2}

(80)

From (79), if $ν_{j} > 1$ , we have

\overset{\cdot}{V} \leq - a^{'} V + b^{'}

(81)

where

a^{'} = min {2 (ν_{i} - 1), γ_{i}, i = 1, \dots, n}

(82)

b^{'} = \frac{n}{2} + \sum_{j = 1}^{n} \frac{γ_{j}}{2} ϑ_{j}^{2} + \sum_{j = 1}^{n} \frac{1}{2} {\bar{ϵ}}_{j}^{2} + 0.557 {\bar{g}}_{n} σ + \sum_{j = 1}^{n} \frac{1}{2} ϰ_{j}^{2}

(83)

The integral from time 0 to $t$ of the inequality (81) can be expressed as

\begin{matrix} V (t) \leq ϖ^{- a^{'} t} V (0) + \frac{b^{'}}{F} (1 - ϖ^{- F t}) \\ \leq ϖ^{- a^{'} t} V (0) + \frac{b^{'}}{F} \end{matrix}

(84)

From the above analysis, we can conclude that $ϖ$ and ${\tilde{ϑ}}_{i}, i = 1, 2, . . ., n$ are bounded for bounded initial conditions. Furthermore, based on ${\tilde{ϑ}}_{i} = ϑ_{i} - {\hat{ϑ}}_{i}$ , ${\hat{ϑ}}_{i}$ are bounded because of the boundedness of $ϑ_{i}$

Form the above analysis, ${\tilde{ϑ}}_{i}$ is bounded. Then, we denote $ϑ_{iM}$ as the bounds of $\tilde{ϑ}$ . This means that $| {\tilde{ϑ}}_{i} | \leq ϑ_{iM}$ .

Combining equations (37), (42), (57), (60), (61), (64), and (79), and considering $\bar{a} > \frac{a}{(1 - Γ)}$ , we have:

\begin{matrix} {\overset{\cdot}{V}}_{ϖ n} \leq - \sum_{i = 1}^{n} (ν_{i} - 1 - \frac{1}{2} τ_{i} ϑ_{iM}) ϖ_{i}^{2} - \sum_{i = 1}^{n} | ϖ_{i} |^{c + 1} + \sum_{i = 1}^{n} \frac{1}{2} ϰ_{i}^{2} \\ + 0.557 {\bar{g}}_{n} σ + \frac{n}{2} + \sum_{i = 1}^{n} \frac{1}{2} {\bar{ϵ}}_{i}^{2} \end{matrix}

(85)

If ν_{i} > 1 + \frac{1}{2} τ_{i} ϑ_{iM}, i = 1, 2, . . ., n

\begin{matrix} {\overset{\cdot}{V}}_{ϖ n} \leq - \sum_{i = 1}^{n} | ϖ_{i} |^{c + 1} + \sum_{i = 1}^{n} \frac{1}{2} ϰ_{i}^{2} + 0.557 {\bar{g}}_{n} σ + \frac{n}{2} + \sum_{i = 1}^{n} \frac{1}{2} {\bar{ϵ}}_{i}^{2} \\ \leq - V_{ϖ n}^{c'} + b^{''} \end{matrix}

(86)

where $c^{'} = \frac{c + 1}{2}$ , $b^{''} = \sum_{i = 1}^{n} \frac{1}{2} ϰ_{i}^{2} + 0.557 {\bar{g}}_{n} σ + \frac{n}{2} + \sum_{i = 1}^{n} \frac{1}{2} {\bar{ϵ}}_{i}^{2}$ .

From the above analysis procedure, we choose the following common Lyapunov equation

V_{ϖ} = V_{ϖ n} = \sum_{i = 1}^{n} \frac{1}{2} ϖ_{i}^{2}

(87)

then, according to (86), we have

\begin{matrix} {\overset{\cdot}{V}}_{ϖ} \leq - V_{ϖ n}^{c'} + b^{''} \end{matrix}

(88)

Based on Lemma 2, we can know that $ϖ_{i} (i = 1, 2, . . ., n)$ are PFS. Moreover, there exists a positive constant $ϒ \in (0, 1]$ such that (86) can be represented as

{\overset{\cdot}{V}}_{ϖ} \leq - ϒ V_{ϖ}^{c^{'}} - (1 - ϒ) V_{ϖ}^{c^{'}} + b^{''}

(89)

then, if $V_{ϖ}^{c^{'}} > b^{''} / (1 - ϒ)$ , one has ${\overset{\cdot}{V}}_{ϖ} \leq - ϒ V_{ϖ}^{c^{'}}$ . According to Lemma 1, we can conclude that $ϖ_{i} (i = 1, 2, . . ., n)$ can achieve $V_{ϖ}^{c^{'}} > b^{''} / (1 - ϒ)$ in finite time

lim_{ϒ \to ϒ_{0}} ϖ \in (V_{ϖ}^{c^{'}} \leq \frac{b^{''}}{(1 - ϒ)})

(90)

where $ϒ_{0} \in (0, 1]$ , and the reach time of (88) is bounded by

T_{1} \leq \frac{V_{ϖ}^{1 - c^{'}} (0)}{ϒ_{0} (1 - c^{'})}

(91)

where $V_{ϖ} (0)$ is the initial value of $V_{ϖ}$ .

Further, it can be seen from the definition $ϖ_{i} = ψ_{i} - ξ_{i}$ that if the finite time stability of $ξ_{i}$ is guaranteed, then the finite time stability of $ψ_{i}$ can also be guaranteed. Therefore, the finite time stability of $ψ_{i}$ is proved next.

Select the following Lyapunov function

V_{ξ} = \sum_{i = 1}^{n} \frac{1}{2} ξ_{i}^{2}

(92)

The derivative of (92) is calculated as

\begin{matrix} {\overset{\cdot}{V}}_{ξ} = - ν_{1} ξ_{1}^{2} + g_{1} ξ_{1} (ζ_{2} - α_{2}) + g_{1} ξ_{1} ξ_{2} - ξ_{1} ϰ_{1} sign (ξ_{1}) + \dots \\ - ν_{i} ξ_{i}^{2} + g_{i} ξ_{i} (ζ_{i + 1} - α_{i + 1}) - g_{i - 1} ξ_{i} ξ_{i - 1} + g_{i} ξ_{i} ξ_{i + 1} \\ - ξ_{i} ϰ_{i} sign (ξ_{i}) + \dots - ν_{n} ξ_{n}^{2} - g_{n - 1} ξ_{n} ξ_{n - 1} - ξ_{n} ϰ_{n} sign (ξ_{n}) \\ = - \sum_{i = 1}^{n} ν_{i} ξ_{i}^{2} - \sum_{i = 1}^{n} ϰ_{i} | ξ_{i} | + \sum_{i = 1}^{n - 1} g_{i} ξ_{i} (ζ_{i + 1} - α_{i + 1}) \end{matrix}

(93)

According to Lemma and Li (2019), the output $ζ_{i + 1}$ satisfies $| ζ_{i + 1} - α_{i + 1} | \leq Λ_{i}$ in finite time $T_{2}$ , and $Λ_{i} > 0$ is a known constant. Therefore, combining with Lemma 4, we can obtain the following inequation

\begin{matrix} {\overset{\cdot}{V}}_{ξ} \leq - \sum_{i = 1}^{n} ν_{i} ξ_{i}^{2} - \sum_{i = 1}^{n} ϰ_{i} | ξ_{i} | + \sum_{i = 1}^{n - 1} | g_{i} | | ξ_{i} | | ζ_{i + 1} - α_{i + 1} | \\ + | g_{n} | | ξ_{n} | | Λ_{n} | \\ \leq - \sum_{i = 1}^{n} ν_{i} ξ_{i}^{2} - \sum_{i = 1}^{n} (ϰ_{i} - {\bar{g}}_{i} ε_{i}) | ξ_{i} | \\ \leq - ν^{'} V_{ξ} - d^{'} V_{ξ}^{\frac{1}{2}} \end{matrix}

(94)

where $ν^{'} = min {2 ν_{i}}$ , $d^{'} = min {\sqrt{2} (ϰ_{i} - {\bar{g}}_{i} Λ_{i})}$ , and $ϰ_{i} > {\bar{g}}_{i} Λ_{i}$ .

According to Lemma 5, $ξ_{i}$ is bounded in finite time $T_{3}$ . Thus, system (5) will be stable in finite time $T \leq T_{1} + T_{2} + T_{3}$ .

For $\forall q \in z^{+}$ , if there exists $t^{'} > 0$ satisfying the inter sampling intervals $t_{q + 1} - t_{q} \geq t^{'}$ in terms of $π (t) = Φ (t) - u (t)$ such that $\frac{d}{dt} | π | = \frac{d}{dt} (π \times π)^{\frac{1}{2}} = sign (π) \overset{\cdot}{π} \leq | \overset{\cdot}{Φ} |$ . According to (64) and (66), it is obvious that $Φ$ is derivable and $| \overset{\cdot}{Φ} | \leq ι$ (where $ι > 0$ is a constant). Based on the fact that $π (t_{q}) = 0$ and $lim_{t \to t_{q + 1}} | π (t) | = Γ | u (t) | + a$ , there exists $t^{'} \geq (Γ | u (t) | + a) / ι$ . Thus, the Zeno phenomenon can be avoided.

This completes the proof.

Simulation results

In this section, according to the designed controller, the transient stability of the two-area interconnected power system with SVC in Figure 3 is simulated. Because the parameters of generators $G_{1}$ and $G_{3}$ , $G_{2}$ and $G_{4}$ are the same, respectively, it is only necessary to select generators $G_{1}$ and $G_{2}$ in the two areas for study Kundur and Malik (2022). Then, system (5) can be represented as follows:

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} + d_{2} (t) \\ {\overset{\cdot}{x}}_{3} = g_{3} (x) u (t) + f_{3} (x) + d_{3} (t) \end{matrix}

(95)

where $g_{3} (x) = h_{3} \sin (x_{1} + δ_{0})$ , $f_{3} (x) = ζ x_{3} + f (x_{1}, x_{2})$

Select the system parameters in system (5) as $δ_{0} = 15 °$ , $ω_{10} = ω_{20} = 314.159 rad / s$ , $H_{1} = 6.5$ , $H_{2} = 6.175$ , $D_{1} = 1$ , $E_{1} = E_{2} = 1$ , $X_{a} = X_{b} = 0.5$ , $B_{SVC} = 0.4$ , $T = 0.02$ , $k_{b} = 10 °$ . The external disturbances are designed as $d_{2} (t) = 0.1 * \sin (5 t)$ , $d_{3} (t) = \cos (5 t)$ . The gain matrix of the disturbance observer is $W (X) = diag (10, 10, 10)$ .

Set the initial values of the system to $x (0) = [0.05, 0.01, 0.05]^{T}$ . The input vectors of RBFNNs are $z_{1} = x_{1}$ , $z_{2} = [x_{1}, x_{2}]^{T}$ , and $z_{3} = [x_{1}, x_{2}, x_{3}]$ , respectively. The network structure is selected as $3 - 9 - 1$ , and the center point vector value of the Gaussian function is selected according to the effective mapping range of the input values. The values of control parameters are shown in Table 1.

Table 1.

The values of control parameters.

Parameters	Value	Parameters	Value	Parameters	Value	Parameters	Value
$ν_{1}$	10	$γ_{1}$	0.5	$ϰ_{1}$	1	$a_{2}$	0.05
$ν_{2}$	100	$γ_{2}$	0.5	$ϰ_{2}$	1	$a_{3}$	0.01
$ν_{3}$	100	$γ_{3}$	0.5	$ϰ_{3}$	1	$\bar{a}$	3
$c$	0.9	$Γ$	0.5	$σ$	10	$Ψ$	9

Remark 4. The network structure of RBFNNs is divided into three layers, which are the input layer, hidden layer, and output layer. In this paper, the input of the neural network is three, so the input layer is three. The vector value of the center point of the Gaussian function is selected according to the effective mapping range of the input value, and the center points $κ_{ρ}$ are selected as [−2 −1.5 1 −0.5 0 0.5 1 1.5 2] according to the value range of the actual input, so there are nine hidden layers. The diameter basis width of the radial basis function determines the rate at which RBFNNs drop. In order to make RBFNNs drop faster, the diameter basis width $v_{ρ}$ is selected as 9. Since the system proposed in this paper contains only one nonlinear term, the output layer is one layer. Finally, the structure of the RBFNNs network is determined as $3 - 9 - 1$ .

Remark 5. By reasonably selecting bounded constant $ϵ$ , external disturbance $d_{i} (t)$ , disturbance observer gains $W (X)$ , controller parameters $ν_{i}, ϰ_{i}, c$ and event-triggered input $Φ (t)$ , the control performance and stability of the system can be effectively guaranteed. At the same time, we can increase $γ_{i}$ , so that the compensation error $ϖ_{1}$ can be maintained in a narrow margin.

The simulation results are shown in Figures 4 –9. Figure 4 exhibits the response curve of rotational speed difference $ω$ under the designed controller of this paper. Figure 5 shows the response curve of power angle difference $δ$ under the controller of this paper. Figures 6–7 display the tracking trajectory of external disturbances $d_{2}$ , $d_{3}$ and their estimated values ${\hat{d}}_{2}$ , ${\hat{d}}_{3}$ . We can find that the designed disturbance observer has a good effect on the estimation of external disturbances. Therefore, the designed controller can well compensate for the influence of external disturbances.

In addition, Figure 8 shows the trajectory of $u (t)$ , and Figure 9 shows the trigger time interval $t_{q + 1} - t_{q}$ . It can be seen from Figures 8 and 9 that compared with the traditional time-triggered control, the event-triggered control scheme can not only reduce the communication burden but also save communication resources.

Conclusion

Considering the influence of unknown nonlinear terms and external disturbances, an adaptive event-triggered finite-time control method based on NNs and disturbance observer was proposed for n-dimensional uncertain nonlinear systems. The disturbance observer was used to estimate external disturbances, and RBFNNs were used to approximate the unknown nonlinear terms of the system. At the same time, the command filtering technique was used to avoid the complexity explosion in the traditional backstepping method, and the error caused by it was compensated. The controller ensured that all signals could be bounded in finite time and considerably reduced the communication burden. Finally, the simulation results of the two-area interconnected power system with SVC verified the availability of the proposed method. Further, we will study the control of nonlinear power systems with SVC with time delay and input quantization problems.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Wenhui Liu

References

Alireza

Meysam

Farrokh

(2019) Power system flexibility: An overview of emergence to evolution. Journal of Modern Power Systems and Clean Energy 7(5): 987–1007.

Amirreza

Zulkurnain

Heidari

, et al. (2020) Optimal designing of static var compensator to improve voltage profile of power system using fuzzy logic control. Energy 192: 116665.

Bhat

Bernstein

(2000) Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization 38(3): 751–766.

Cai

Xiang

Guo

(2016) Adaptive finite-time control for uncertain nonlinear systems with application to mechanical systems. Nonlinear Dynamics 84(2): 943–958.

Chen

Xiong

(2019) Tracking flight control of quadrotor based on disturbance observer. IEEE Transactions on Systems, Man, and Cybernetics: Systems 51(3): 1414–1423.

Zhai

Chen

, et al. (2018) Robustness analysis of a continuous higher order finite-time control system under sampled-data control. IEEE Transactions on Automatic Control 64(6): 2488–2494.

Fan

(2018) Adaptive neural network prescribed performance matrix projection synchronization for unknown complex dynamical networks with different dimensions. Neurocomputing 281: 55–66.

Girard

(2014) Dynamic triggering mechanisms for event-triggered control. IEEE Transactions on Automatic Control 60(7): 1992–1997.

Tang

Sun

, et al. (2015) The effect of the angle swing of weak-connection generator groups in a regional grid to the angle stability of the interconnected power system. Proceedings of the Chinese Society of Electrical Engineering 35(6): 1336–1343.

10.

Hamid

Hossein

(2019) Enhancing voltage stability and LVRT capability of a wind-integrated power system using a fuzzy-based SVC. Engineering Science and Technology, an International Journal 22(3): 827–839.

11.

Huang

Wang

Wen

, et al. (2019) Adaptive event-triggered control of nonlinear systems with controller and parameter estimator triggering. IEEE Transactions on Automatic Control 65(1): 318–324.

12.

Kundur

Malik

(2022) Power System Stability and Control. New York: McGraw Hill Education.

13.

Lai

Giorgio

Andrew

, et al. (2021) A review on long-term electrical power system modeling with energy storage. Journal of Cleaner Production 280: 124298.

14.

(2019) Finite time command filtered adaptive fault tolerant control for a class of uncertain nonlinear systems. Automatica 106: 117–123.

15.

Liu

Tong

(2018) Neural networks-based adaptive finite-time fault-tolerant control for a class of strict-feedback switched nonlinear systems. IEEE Transactions on Cybernetics 49(7): 2536–2545.

16.

Liu

, et al. (2019) Barrier function-based adaptive control for uncertain strict-feedback systems within predefined neural network approximation sets. IEEE Transactions on Neural Networks and Learning Systems 31(8): 2942–2954.

17.

Liang

, et al. (2019a) Adaptive fuzzy event-triggered control for stochastic nonlinear systems with full state constraints and actuator faults. IEEE Transactions on Fuzzy Systems 27(11): 2242–2254.

18.

, et al. (2019b) Event-triggered adaptive neural network control for nonstrict-feedback nonlinear time-delay systems with unknown control directions. IEEE Transactions on Neural Networks and Learning Systems 31(10): 4196–4205.

19.

Nai

Yang

Zhang

(2020) Adaptive neural output feedback compensation control for intermittent actuator faults using command-filtered backstepping. IEEE Transactions on Neural Networks and Learning Systems 31(9): 3497–3511.

20.

Postoyan

Tabuada

Nešić

, et al. (2014) A framework for the event-triggered stabilization of nonlinear systems. IEEE Transactions on Automatic Control 60(4): 982–996.

21.

Roselyn

Devaraj

(2018) Adaptive multi objective differential evolution with fuzzy decision making in preventive and corrective control approaches for voltage security enhancement. Journal of the Franklin Institute 355(11): 4553–4582.

22.

Sabo

Abdul Wahab

Othman

, et al. (2021) Application of a neuro-fuzzy controller for single machine infinite bus power system to damp low-frequency oscillations. Transactions of the Institute of Measurement and Control 43(16): 3633–3646.

23.

Che

(2021) Fault-tolerant control for model-free networked control systems under DoS attacks. Journal of the Franklin Institute 358(17): 9023–9033.

24.

Tabuada

(2007) Event-triggered real-time scheduling of stabilizing control tasks. IEEE Transactions on Automatic Control 52(9): 1680–1685.

25.

Thierry

Mevludin

William

, et al. (2020) Test systems for voltage stability studies. IEEE Transactions on Power Systems 35(5): 4078–4087.

26.

Wan

Murad

Liu

, et al. (2018) Voltage frequency control using SVC devices coupled with voltage dependent loads. IEEE Transactions on Power Systems 34(2): 1589–1597.

27.

Wang

Kang

Feng

(2019) Finite-time adaptive fuzzy command filtered backstepping control for a class of nonlinear systems. International Journal of Fuzzy Systems 21(8): 2575–2587.

28.

Liu

(2023) Observer-based event-triggered adaptive neural control for time-delay nonlinear systems with input saturation and external disturbances. Transactions of the Institute of Measurement and Control 45: 2489–2500.

29.

Xing

Wen

Liu

, et al. (2016) Event-triggered adaptive control for a class of uncertain nonlinear systems. IEEE Transactions on Automatic Control 62(4): 2071–2076.

30.

Xing

Wen

Liu

, et al. (2017) Adaptive compensation for actuator failures with event-triggered input. Automatica 85: 129–136.

31.

Yao

Park

, et al. (2018) Disturbance-observer-based composite hierarchical antidisturbance control for singular markovian jump systems. IEEE Transactions on Automatic Control 64(7): 2875–2882.

32.

Zhang

Yang

(2019) Event-triggered global finite-time control for a class of uncertain nonlinear systems. IEEE Transactions on Automatic Control 65(3): 1340–1347.

33.

Zhao

Antonio

Marcos

, et al. (2019) Power system dynamic state estimation: Motivations, definitions, methodologies, and future work. IEEE Transactions on Power Systems 34(4): 3188–3198.

34.

Zhu

Xia

(2011) Attitude stabilization of rigid spacecraft with finite-time convergence. International Journal of Robust and Nonlinear Control 21(6): 686–702.

Event-triggered finite-time neural control for uncertain nonlinear systems with unknown disturbances and its application in SVC

Abstract

Keywords

Introduction

Problem statement and preliminaries

Description of the system

Problem formulation for n -dimensional nonlinear systems

RBFNNs and preliminary knowledge

Control scheme design and stability analysis

Disturbance observer design

Adaptive event-triggered finite-time controller design

Stability analysis

Simulation results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Problem formulation for $n$ -dimensional nonlinear systems