Event-Triggered Adaptive Neural Prescribed Performance Tracking Control for Nonlinear Cyber–Physical Systems against Deception Attacks

Li, Chunyan; Li, Yinguang; Zhang, Jianhua; Li, Yang

doi:10.3390/math12121838

Open AccessArticle

Event-Triggered Adaptive Neural Prescribed Performance Tracking Control for Nonlinear Cyber–Physical Systems against Deception Attacks

¹

School of Information Technology, Zhejiang Financial College, Hangzhou 310018, China

²

School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266525, China

^*

Author to whom correspondence should be addressed.

Mathematics 2024, 12(12), 1838; https://doi.org/10.3390/math12121838

Submission received: 14 May 2024 / Revised: 7 June 2024 / Accepted: 8 June 2024 / Published: 13 June 2024

(This article belongs to the Special Issue Nonlinear Dynamics and Control: Challenges and Innovations)

Download

Browse Figures

Versions Notes

Abstract

:

This paper investigates the problem of the adaptive neural network tracking control of nonlinear cyber–physical systems (CPSs) subject to unknown deception attacks with prescribed performance. The considered system is under the influence of unknown deception attacks on both actuator and sensor networks, making the research problem challenging. The outstanding contribution of this paper is that a new anti-deception attack-prescribed performance tracking control scheme is proposed through a special coordinate transformation and funnel function, combined with backstepping and bounded estimation methods. The transient performance of the system can be ensured by the prescribed performance control scheme, which makes the indicators of the controlled system, such as settling time and tracking accuracy, able to be pre-assigned offline according to the task needs, and the applicability of the prescribed performance is tested by selecting different values of the settling time (0.5 s, 1 s, 1.5 s, 2 s, 2.5 s, and 3 s). In addition, to save the computational and communication resources of the CPS, this paper uses a finite-time differentiator to approximate the virtual control law differentiation to avoid “complexity explosion” and a switching threshold event triggering mechanism to save the communication resources for data transmission. Finally, the effectiveness of the proposed control strategy is further verified by an electromechanical system simulation example.

Keywords:

nonlinear cyber–physical systems; adaptive; deception attacks; prescribed performance; event triggering mechanism (ETM)

MSC:

93C10; 68T05; 93-10; 93C40; 93D40; 93D21

1. Introduction

The CPS is a type of control system that combines physical components with network communications and computers, and has strong advantages in realizing dynamic control, real-time sensing and information services for large-scale engineering systems, which are widely used in industrial control systems, distributed power systems, intelligent transport systems and other key infrastructure fields [1,2,3]. Through the combination of adaptive techniques and backstepping, the control tracking theory of nonlinear systems has been widely developed [4,5,6], and many excellent research studies have emerged in the field of the tracking control of CPSs [7,8,9].

Transmitting data over shared network communication channels provides some benefits to traditional control systems, such as scalability, flexibility and efficiency. However, this also makes the system vulnerable to adversarial attacks such as denial-of-service (Dos) attacks, deception attacks [10,11] and replay attacks. The Dos attack is a type of attack that disables a control system by blocking the communication channel so that the control system is missing real-time data [12]. Replay attacks damage system stability by maliciously transmitting the same data repeatedly [13]. Among them, the deception attack is a type of attack in which an attacker maliciously injects false data into the network system [14]. Without addressing these cyber-attacks, system performance will deteriorate and even lead to disaster. Therefore, it is crucial to design effective security control algorithms for CPSs subjected to cyber-attacks to ensure overall stability, and deception attacks are one of the important attacks that need to be further investigated. So far, many studies on CPSs with deception attacks have existed [15,16,17,18]. Specifically, the literature [15] addresses the problem of sensor deception attacks on uncertain linear systems by designing an adaptive scheme. A new terminal integral adaptive sliding film control method is proposed in the literature [16] for linear systems subject to injection spoofing attacks. The adaptive control of nonlinear CPSs subjected to disinformation injection attacks has been studied in the literature [17] and the unknown time-varying gain problem has been dealt with using the Nussbaum function. The problem of adaptive security control for nonlinear CPSs subject to unknown deception attacks has been studied in the literature [18]. However, results on the tracking control of CPSs subjected to double deception attacks on sensor and actuator networks are scarce.

Cyber–physical systems typically have multiple control loops sharing available computing and communication resources. Therefore, efficient use of the only resources available is a central issue in the control design of CPSs. The use of time-triggered conventional digital control techniques may lead to a huge waste of communication resources and unnecessarily high workloads when computational and communication resources are allocated to other tasks [19]. In contrast, in systems that use an event-triggered mechanism (ETM), control inputs are updated only irregularly at certain moments, reducing the computational and communication burden [20,21,22]. A lot of research has emerged that addresses saving communication resources [23,24,25,26]. A fixed threshold ETM control strategy is proposed in reference [23]. A relative threshold ETM in which the threshold value varies with the control input signal has been designed in reference [25]. Although adopting the relative threshold ETM strategy can further save communication resources while obtaining better control accuracy, the trigger threshold is directly related to the control input; once the control input suddenly becomes larger, it is easy to cause a sudden jump of the control input, resulting in the controlled system being affected by a strong pulse. Therefore, in this paper, the switching threshold ETM strategy is selected to save the computational and communication resources of CPSs.

On the other hand, in the control of many practical complex systems, such as process control, there are not only the necessary stability requirements for the controlled system but also very high requirements for transient performance in terms of the quality of task completion or operation [27,28]. Therefore, control design to achieve pre-specified performance metrics, such as tracking error convergence rate, settling time, etc., is more in line with the current concepts of intelligent control system design. In order to solve the above problems, scholars have conducted an in-depth study of prescribed performance control in [29,30,31]. It is worth noting that the use of backstepping in the design of control strategies will lead to a “explosion of complexity” due to the repeated differentiation of the virtual control law [32]. Therefore, it would be interesting to avoid the “complexity explosion” phenomenon while achieving the prescribed performance control.

Inspired by the above discussion, this paper proposes an event-triggered adaptive prescribed performance tracking control for nonlinear CPSs of sensor and actuator networks subject to deception attacks. Compared with the existing results, the contributions of this article are summarized below:

Compared to reference [17,18], which only considers the case where sensors or actuators are subject to deception attacks, this paper investigates the problem of adaptive security control for CPSs where both sensor and actuator networks are subject to unknown deception attacks.
By introducing a performance funnel function in the control design, the adaptive prescribed performance control scheme proposed in this paper ensures that the system is stable under deception attacks for a prescribed finite period of time and the tracking error converges to a prescribed region.
Finally, while using the ETM strategy to save CPSs communication resources, computational resources are saved by avoiding “complexity explosion” by approximating the virtual control law using finite-time differentiators.

The rest of this paper is described below. Section 2 provides an introduction to the problem description and preparatory knowledge. Section 3 presents the controller design and stability analysis. Section 4 gives the simulations and the result analysis. Finally, Section 5 analyzes and concludes the paper.

2. Preliminaries

The main symbols used in this paper are shown in Table 1.

Consider nonlinear CPSs modeled as

\{\begin{matrix} {\dot{x}}_{i} = g_{i} ({\bar{x}}_{i}) x_{i + 1} + f_{i} ({\bar{x}}_{i}) + d_{i} (t) \\ {\dot{x}}_{n} = g_{n} ({\bar{x}}_{n}) u + f_{n} ({\bar{x}}_{n}) + d_{n} (t) \end{matrix}

(1)

where

{\bar{x}}_{i} = {[x_{1}, x_{2}, \dots, x_{i}]}^{T} \in R^{i}

and

x_{i} (t)

are state variables of the system, and

g_{i} (\cdot), f_{i} (\cdot)

represent the unknown smooth functions that are nonlinear.

d_{i} (t) \in R^{m}

is the external disturbance vector.

u \in R

and

y = x_{1}

mean the input and output of the system, respectively.

In this work, we assume that the controlled system is subject to a deception attack through the sensor network and actuator networks, which is shown in Figure 1. The state measurements after the deception attack are denoted as

{\tilde{x}}_{i} = x_{i} + o_{i} (t) x_{i} = λ_{i} (t) x_{i}

, where

λ_{i} (t) = 1 + o_{i} (t), i = 1, 2, \dots, n

.

o_{i} (t)

represents the uncertain time-varying attack signals, and

{\tilde{x}}_{i}

is available.

On the other hand, it is assumed that the control signal of the controlled system is subject to a deception attack at an unknown moment when the control signal is transmitted in the network channel. The model of deception attack is expressed as

\tilde{u} = \{\begin{matrix} u, t \in [0, T_{d}) \\ β (t) u + w (t) ζ (x), t \in [T_{d}, \infty) \end{matrix}

(2)

where

\tilde{u}

is the control signal value after the deception attack and u is the true control signal generated by the controller.

T_{d}

represents the moment when the actuator network is subjected to a deception attack.

β (t)

and

w (t)

are time-varying functions,

ζ (x)

is the nonlinear function. Likewise, there are four positive constants

β_{m}

,

β_{M}

,

\bar{w}

and

\bar{ζ}

satisfying

0 < β_{m} \leq β (t) \leq β_{M}

,

|\dot{w} (t)| \leq \bar{w}

and

|ζ (x)| \leq \bar{ζ}

.

Remark 1.

In CPSs, when an attacker can monitor state information through a network channel, they are able to inject malicious data into real information, which is called a deception attack. After a deception attack, the original state information

x_{i}

is no longer directly usable, and the controller u has changed to

\tilde{u}

due to the injection of false data. Moreover, the injection of false information introduces time-varying gain parameters to the system state and control inputs, and these can damage system performance or even lead to instability. Therefore, dealing with these time-varying gain parameters and spuriously injected information is a challenging task.

Assumption 1

([33]). For the attack gain

λ_{i} (t)

, we assume that

λ_{i} (t)

must be positive. There exist some known positive constants

\bar{λ}

and

λ_{0}

satisfying

|λ_{i} (t)| \leq \bar{λ}, |{\dot{λ}}_{i} λ_{i}^{- 1}| \leq λ_{0}

.

Assumption 2.

The reference trajectory

y_{d} \in R

and its first- and second-order derivatives are available.

Assumption 3.

Functions

g_{i} (t)

are bounded. Supposing that there are constants

a_{i}

, there exists

0 < g_{i}^{m} \leq g_{i} ({\bar{x}}_{i}) < \infty, i = 1, 2, \dots n

.

Remark 2.

From Assumption 2, it is obtained that the following property exists with respect to the attack gain

λ_{i} (t)

: there exist two positive constants

λ_{m}

and

λ_{M}

, satisfying

λ_{m} \leq λ_{i} λ_{i + 1}^{- 1} \leq λ_{M}

.

Definition 1

([34,35]). For given accuracy bound

μ_{T_{s}}

and convergence time

T_{s} > 0

, a time-varying smooth function

μ (μ_{T_{S}}, T_{s}, t)

defined for

t \in R_{\geq 0}

and abbreviated as

μ (t)

, is called a finite-time performance funnel function if the following hold:

$μ (t) > 0$ for all $t \geq 0$ ;
$μ (t)$ belong to $W^{1, \infty}$ , where $W^{1, \infty}$ represents the class of bounded functions with bounded derivatives;
${lim}_{t \to T_{s}} μ (t) = μ_{T_{s}}$ and $μ (t) = μ_{T_{s}}$ holds for all $t \geq T_{s}$ .

By building a controller such that the error

e (t)

always evolves within the funnel

μ (t)

, the desired transient performance can be achieved in a prescribed time. The evolution of the tracking error of the controlled system in the prescribed performance control with the performance funnel is shown schematically in Figure 2. In this paper, the following performance functions

μ (t)

will be used to implement the prescribed performance control tasks:

μ (t) = \{\begin{matrix} {[μ_{0}^{a} - a κ t]}^{\frac{1}{a}} + μ_{T_{s}}, t \in [0, T_{s}) \\ μ_{T_{s}}, t \in [T_{s}, + \infty) \end{matrix}

(3)

where

μ_{T_{s}}, μ_{0}^{a}, a

are positive design parameters and

a = (m / n)

with

n \geq m

,

n, m

are positive odd and even integers, respectively. In addition,

μ_{0}^{a} + μ_{T_{s}} = μ (0)

is the initial value,

T_{s} = \frac{a}{b}

is the settling time and

μ_{T_{S}}

represents the predefined tracking accuracy.

Lemma 1

([36]). Let

h (Z)

be an unknown continuous function over a compact set

Ω_{Z} \in R^{m}

for

\forall ε > 0

, and

l > 0

being the NN node number, there exists a radial basis function(RBF) NN, satisfying the following:

h (Z) = ϕ^{T} R (Z) + δ (Z), ∥δ (Z)∥ \leq ε

(4)

where

ϕ^{T} = arg {min}_{ϕ \in R^{n}} \{sup_{Z \in Ω_{Z}} |h (Z) - ϕ^{T} R (Z)|\}

,

Z = [Z_{1}, \dots, Z_{l}] \in R^{l}

denotes the input vector and

ϕ = {[ϕ_{1}, ϕ_{2}, \dots, ϕ_{l}]}^{T}

represents the ideal weight vectors.

δ (Z)

indicates the approaching error and

R (Z) = {[R_{1} (Z), \dots, R_{l} (Z)]}^{T}

represents the vectors of RBF basis functions, which can be shown as follows:

R_{i} (Z) = exp [- \frac{{∥Z - c_{r, i}∥}^{2}}{σ_{N N, i}^{2}}]

(5)

in which

c_{i} = {[c_{i 1}, c_{i 2}, \dots, c_{i l}]}^{T}

and

σ_{N N, i}

are the center and breadth of the RBF NN radial basis function, respectively. The structure of NN used in this paper is depicted in Figure 3.

Lemma 2

([37]). For any scalar positive function

σ (t)

and any variable

z \in R

, the following inequality holds:

0 \leq |z| - \frac{z^{2}}{\sqrt{z^{2} + σ^{2} (t)}} \leq σ (t)

(6)

Lemma 3

(Young’s Inequality): If there exist

ℏ > 0, ℘ > 0

and some positive constants

p, m_{1}, m_{2}

and

(m_{1} - 1) (m_{2} - 1) = 1

, then

ℏ ℘ \leq \frac{p^{m_{1}}}{m_{1}} {∥ℏ∥}^{m_{1}} + \frac{1}{m_{2} p^{m_{2}}} {∥℘∥}^{m_{2}}, \forall ℘, ℏ \in R^{n}

(7)

Lemma 4

([38]). For any real variables

ζ \in R

and a positive constant τ, the following inequality holds:

0 \leq |ζ| - ζ tanh (\frac{ζ}{τ}) \leq 0.2785 τ

(8)

3. Controller Design and Stability Analysis

This section is devoted to the design of a dynamic event-triggered adaptive control algorithm that ensures that the system tracking error

e (t) = y - y_{d}

remains within a predefined funnel and converges asymptotically. In order to obtain the ideal control performance, we define the following state transformation as:

s_{1} (t) = tan (\frac{π e (t)}{2 μ (t)})

(9)

Thus, we have

e (t) = \frac{2}{π} μ (t) arctan (s_{1} (t))

(10)

From (10), we have

\dot{e} (t) = \frac{2}{π} \dot{μ} (t) arctan (s_{1} (t)) + \frac{2}{π} μ (t) \frac{{\dot{s}}_{1} (t)}{1 + s_{1}^{2} (t)}

(11)

We can further obtain

{\dot{s}}_{1} (t) = h_{1} \dot{e} (t) - \frac{2}{π} \dot{μ} (t) arctan (s_{1} (t))

(12)

where

h_{1} = (π (1 + e_{1}^{2} (t)) / 2 μ (t)) > 0

is a known smooth function. From (9), it can be concluded that the tracking error can be within the range

- μ (t) < e (t) < μ (t)

. Before realizing the desired control task, the following coordinate transformations are defined:

\{\begin{matrix} z_{1} = s_{1} (t) \\ z_{i} = {\tilde{x}}_{i} - α_{i - 1} \end{matrix}

(13)

3.1. Controller Design

Step 1: Consider the presence of a deception attack, where

{\tilde{x}}_{1} = λ_{1} (t) x_{1}

is the only state information available, and hence

e (t) = {\tilde{x}}_{1} - y_{d}

. By (1), (12) and (13), one has

\begin{matrix} {\dot{z}}_{1} = h_{1} ({\dot{λ}}_{1} x_{1} + λ_{1} {\dot{x}}_{1} - {\dot{y}}_{d}) - \frac{2}{π} h_{1} \dot{μ} (t) arctan (e_{1} (t)) \\ = h_{1} ({\dot{λ}}_{1} λ_{1}^{- 1} {\tilde{x}}_{1} + λ_{1} (g_{1} x_{2} + f_{1} ({\bar{x}}_{1}) + d_{1} (t)) - {\dot{y}}_{d}) - \frac{2}{π} h_{1} \dot{μ} (t) arctan (e_{1} (t)) \\ = h_{1} ({\dot{λ}}_{1} λ_{1}^{- 1} {\tilde{x}}_{1} + λ_{1} λ_{2}^{- 1} g_{1} (z_{2} + α_{1}) + λ_{1} f_{1} ({\bar{x}}_{1}) + λ_{1} d_{1} (t) - {\dot{y}}_{d}) \\ - \frac{2}{π} h_{1} \dot{μ} (t) arctan (e_{1} (t)) \end{matrix}

(14)

Construct the first Lyapunov function candidate as

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

(15)

It can be verified that

V_{1}

is positive definite and continuously differentiable and thus a valid Lyapunov function candidate. The time derivative of

V_{1}

along (14) is

\begin{matrix} {\dot{V}}_{1} = λ_{1} λ_{2}^{- 1} g_{1} z_{1} h_{1} (z_{2} + α_{1}) + z_{1} h_{1} [h (Z) + λ_{1} d_{1} (t)] \\ - \frac{2}{π} z_{1} h_{1} \dot{μ} (t) arctan (e_{1} (t)) - \frac{1}{γ_{1}} {\tilde{θ}}_{1} {\dot{\hat{θ}}}_{1} \end{matrix}

(16)

where

h_{1} (Z) = \dot{λ} λ^{- 1} {\tilde{x}}_{1} + λ_{1} f_{1} ({\bar{x}}_{1}) - {\dot{y}}_{d}

. Since the unknown functions appear in

h (Z)

, they are approached by an RBF NN as follows:

h_{1} (Z) = {ϕ_{1}}^{T} R_{1} (Z) + δ_{1} (Z), ∥δ_{1} (Z)∥ \leq ε_{1}

(17)

To deal with the unknown external interference terms of the controlled system, define

θ_{1} = {sup}_{t \geq 0} ∥Θ_{1} (t)∥

with

Θ_{1} = {[{ϕ_{1}}^{T}, λ_{1} {\bar{d}}_{1} + ε_{1}]}^{T}

. Moreover, let

ξ_{1} = {[R_{1} (Z), 1]}^{T} \in R^{l + 1}

.

Then, by invoking Lemma 4,

\begin{matrix} z_{1} h_{1} (ϕ^{T} R (Z) + λ_{1} d_{1} (t) + δ (Z)) \\ = z_{1} h_{1} Θ_{1}^{T} ξ_{1} \leq |z_{1}| h_{1} θ_{1} ∥ξ_{1}∥ \\ \leq z_{1} θ_{1} h_{1} ∥ξ_{1}∥ tanh (\frac{z_{1} h_{1} ∥ξ_{1}∥}{τ_{1}}) + 0.2785 θ_{1} τ_{1} \end{matrix}

(18)

where

∥ξ_{1}∥

is the Euclidean norm of

ξ_{1}

.

Adding and subtracting

z_{1} h_{1} {\bar{α}}_{1}

on the right-hand side of (16), and substituting (17) and (18) into (16) leads to

\begin{matrix} {\dot{V}}_{1} \leq λ_{1} λ_{2}^{- 1} g_{1} z_{1} h_{1} (z_{2} + α_{1}) - \frac{2}{π} z_{1} h_{1} \dot{μ} (t) arctan (e_{1} (t)) - \frac{1}{γ_{1}} {\tilde{θ}}_{1} {\dot{\hat{θ}}}_{1} \\ + z_{1} θ_{1} h_{1} ∥ξ_{1}∥ tanh (\frac{z_{1} h_{1} ∥ξ_{1}∥}{τ_{1}}) + 0.2785 θ_{1} τ_{1} + z_{1} h_{1} {\bar{α}}_{1} - z_{1} h_{1} {\bar{α}}_{1} \end{matrix}

(19)

where

{\bar{α}}_{1} = k_{1} z_{1} + {\hat{θ}}_{1} ∥ξ_{1}∥ tanh (\frac{z_{1} h_{1} ∥ξ_{1}∥}{τ_{1}}) - \frac{2}{π} \dot{μ} (t) arctan (e_{1} (t))

,

{\hat{θ}}_{1}

denotes the estimate of the uncertain parameter

θ_{1}

and

{\tilde{θ}}_{1} = θ_{1} - {\hat{θ}}_{1}

. Then, one can obtain

\begin{matrix} {\dot{V}}_{1} \leq - k_{1} z_{1}^{2} χ^{2} - \frac{1}{γ_{1}} {\tilde{θ}}_{1} ({\dot{\hat{θ}}}_{1} - γ_{1} z_{1} h_{1} ∥ξ_{1}∥ tanh (\frac{z_{1} h_{1} ∥ξ_{1}∥}{τ_{1}})) + λ_{1} λ_{2}^{- 1} g_{1} z_{1} h_{1} z_{2} \\ + z_{1} h_{1} {\bar{α}}_{1} + λ_{1} λ_{2}^{- 1} g_{1} z_{1} h_{1} α_{1} + 0.2785 θ_{1} τ_{1} \end{matrix}

(20)

Choose the first virtual control law

α_{1}

and the adaptive update law for

{\hat{θ}}_{1}

as

α_{1} = - \frac{z_{1} h_{1} {\bar{α}}_{1}^{2}}{g_{m 1} \sqrt{{(z_{1} h_{1} {\bar{α}}_{1})}^{2} + σ_{1}^{2}}}

(21)

{\dot{\hat{θ}}}_{1} = γ_{1} z_{1} h_{1} ∥ξ_{1}∥ tanh (\frac{z_{1} h_{1} ∥ξ_{1}∥}{τ_{1}}) - γ_{11} {\hat{θ}}_{1}

(22)

where

g_{m 1} = min \{g_{1}^{m} λ_{m}\}

with

λ_{m}

is sourced from Remark 2, and

γ_{11}

are positive design constants. According to Lemma 2, one has

z_{1} h_{1} {\bar{α}}_{1} + λ_{1} λ_{2}^{- 1} g_{1} z_{1} h_{1} α_{1} \leq z_{1} h_{1} {\bar{α}}_{1} - z_{1} h_{1} \frac{z_{1} h_{1} {\bar{α}}_{1}^{2}}{\sqrt{{(z_{1} h_{1} {\bar{α}}_{1})}^{2} + σ_{1}^{2}}} \leq σ_{1}

(23)

Substituting (21)–(23) into (20), it follows that

{\dot{V}}_{1} \leq - k_{1} z_{1}^{2} + \frac{γ_{11}}{γ_{1}} {\tilde{θ}}_{1} {\hat{θ}}_{1} + λ_{1} λ_{2}^{- 1} g_{1} z_{1} h_{1} z_{2} + σ_{1} + 0.2785 θ_{1} τ_{1}

(24)

Step

i (2 \leq i \leq n - 1)

: The finite-time differentiator that is used to approximate the first-order derivative of

α_{i - 1}

is constructed as follows:

\{\begin{matrix} {\dot{ω}}_{(i - 1), 1} = ω_{(i - 1), 2} - κ_{1} s i g {(ω_{(i - 1), 1} - α_{i - 1})}^{\frac{1}{2}} \\ {\dot{ω}}_{(i - 1), 2} = - κ_{2} s i g n (ω_{(i - 1), 1} - α_{i - 1}) \end{matrix}

(25)

where

κ_{1}, κ_{2}

are the design positive parameter of the finite-time differentiator. According to literature [39], if the initial deviations

ω_{(i - 1), 1} (0) - α_{i - 1} (0)

and

ω_{(i - 1), 2} (0) - {\dot{α}}_{i - 1} (0)

are bounded, the differentiator can give

α_{i - 1}

with any precision. Consequently, we have

ω_{(i - 1), 2} (t) - {\dot{α}}_{i - 1} = ε_{(i - 1), α}

with bounded estimation error

ε_{(i - 1), α}

, i.e., we can find

ε_{(i - 1) M}

such that

|ε_{(i - 1), α}| \leq ε_{(i - 1) M}

.

By (1), (13) and

{\tilde{x}}_{i} = λ_{i} (t) x_{i}

, the dynamics of

z_{i}

can be expressed as

\begin{matrix} {\dot{z}}_{i} = {\dot{λ}}_{i} x_{i} + λ_{i} (g_{i} λ_{i + 1}^{- 1} {\tilde{x}}_{i + 1} + f_{i} ({\bar{x}}_{i}) + d_{i} (t)) - {\dot{α}}_{i - 1} \\ = {\dot{λ}}_{i} λ_{i}^{- 1} {\tilde{x}}_{i} + λ_{i} λ_{i + 1}^{- 1} g_{i} (z_{i + 1} + α_{i}) + λ_{i} f_{i} ({\bar{x}}_{i}) + λ_{i} d_{i} (t) - {\dot{α}}_{i - 1} \end{matrix}

(26)

To address the impact of the unknown function

f_{i} ({\bar{x}}_{i})

in

z_{i}

on the design of the adaptive controller, define

h_{i} (Z) = {\dot{λ}}_{i} λ_{i}^{- 1} {\tilde{x}}_{i} + λ_{i} f_{i} ({\bar{x}}_{i})

. Then, by Lemma 1, it follows that

h_{i} (Z) = {ϕ_{i}}^{T} R_{i} (Z) + δ_{i} (Z), ∥δ_{i} (Z)∥ \leq ε_{i}

(27)

With respect to the time-varying uncertainties, for

i \geq 3

, define

Θ_{i} = {[{ϕ_{i}}^{T}, λ_{i} {\bar{d}}_{i} + ε_{i}, ε_{(i - 1) M}, λ_{M} g_{i - 1}]}^{T}, ξ_{i} = {[R_{i} (Z), 1, 1, z_{i - 1}]}^{T}

(28)

where

λ_{M}

is sourced from Remark 2. Choose the ith Lyapunov function candidate as

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

(29)

With the help of (28), the time derivative of

V_{i}

along (26) satisfies

{\dot{V}}_{i} \leq {\dot{V}}_{i - 1} + λ_{i} λ_{i + 1}^{- 1} g_{i} z_{i} (z_{i + 1} + α_{i}) - λ_{i - 1} λ_{i}^{- 1} g_{i - 1} z_{i - 1} z_{i} - z_{i} ω_{(i - 1), 2} + z_{i} Θ_{i} ξ_{i} - \frac{1}{γ_{i}} \tilde{θ} \dot{\hat{θ}}

(30)

where, for

i = 2

, the third term on the right-hand side of (30) is

λ_{1} λ_{2}^{- 1} g_{1} z_{1} h_{1} z_{2}

and the bound estimation exists in the following form:

Θ_{2} = {[{ϕ_{2}}^{T}, λ_{2} {\bar{d}}_{2} + ε_{2}, ε_{(1) M}, λ_{M} g_{1}]}^{T}, ξ_{2} = {[R_{2} (Z), 1, 1, z_{1} h_{1}]}^{T}

(31)

Note that

Θ_{i}

is bounded for all

t \geq 0

. To deal with the bounded time-varying uncertainties, define

θ_{i} = {sup}_{t \geq 0} ∥Θ_{i}∥

. By Lemma 4, it can be deduced that

z_{i} Θ_{i} ξ_{i} \leq |z_{i}| θ_{i} |ξ_{i}| \leq z_{i} θ_{i} ∥ξ_{i}∥ tanh (\frac{z_{i} ∥ξ_{i}∥}{τ_{i}}) + 0.2785 θ_{i} τ_{i}

(32)

Adding and subtracting

z_{i} {\bar{α}}_{i}

on the right-hand side of (30) and substituting (32) into (30), we arrive at

\begin{matrix} {\dot{V}}_{i} \leq {\dot{V}}_{i - 1} + λ_{i} λ_{i + 1}^{- 1} g_{i} z_{i} (z_{i + 1} + α_{i}) - λ_{i - 1} λ_{i}^{- 1} g_{i - 1} z_{i - 1} z_{i} - z_{i} ω_{(i - 1), 2} \\ + z_{i} θ_{i} ∥ξ_{i}∥ tanh (\frac{z_{i} ∥ξ_{i}∥}{τ_{i}}) + 0.2785 θ_{i} τ_{i} - \frac{1}{γ_{i}} \tilde{θ} \dot{\hat{θ}} + z_{i} {\bar{α}}_{i} - z_{i} {\bar{α}}_{i} \end{matrix}

(33)

where

{\bar{α}}_{i} = k_{i} z_{i} + {\hat{θ}}_{i} ∥ξ_{i}∥ tanh (\frac{z_{i} ∥ξ_{i}∥}{τ_{i}}) - ω_{(i - 1), 2}

. Considering the definition

{\tilde{θ}}_{i} = θ_{i} - {\hat{θ}}_{i}

, one obtains

\begin{matrix} {\dot{V}}_{i} \leq {\dot{V}}_{i - 1} - k_{i} z_{i}^{2} + λ_{i} λ_{i + 1}^{- 1} g_{i} z_{i} (z_{i + 1} + α_{i}) + z_{i} {\bar{α}}_{i} - λ_{i - 1} λ_{i}^{- 1} g_{i - 1} z_{i - 1} z_{i} \\ - \frac{1}{γ_{i}} \tilde{θ} (\dot{\hat{θ}} - γ_{i} z_{i} ∥ξ_{i}∥ tanh (\frac{z_{i} ∥ξ_{i}∥}{τ_{i}})) + 0.2785 θ_{i} τ_{i} \end{matrix}

(34)

Let the ith

α_{i}

and the update law for

{\hat{θ}}_{i}

be designed as

α_{i} = - \frac{z_{i} {\bar{α}}_{i}^{2}}{g_{m i} \sqrt{{(z_{i} {\bar{α}}_{i})}^{2} + σ_{i}^{2}}}

(35)

{\dot{\hat{θ}}}_{i} = γ_{i} z_{i} ∥ξ_{i}∥ tanh (\frac{z_{i} ∥ξ_{i}∥}{τ_{i}}) - γ_{i 1} {\hat{θ}}_{i}

(36)

where

g_{m i} = min \{g_{i}^{m} λ_{m}\}

and

γ_{i 1}

are positive design constants.

By Lemma 4, it can be deduced that

z_{i} {\bar{α}}_{i} + λ_{i} λ_{i + 1}^{- 1} g_{i} z_{i} α_{i} \leq z_{i} {\bar{α}}_{i} - z_{i} \frac{z_{i} {\bar{α}}_{i}^{2}}{\sqrt{{(z_{i} {\bar{α}}_{i})}^{2} + σ_{i}^{2}}} \leq σ_{i}

(37)

Substituting (35)–(37) into (34), we can obtain

{\dot{V}}_{i} \leq - \sum_{j = 1}^{i} k_{j} z_{j}^{2} + \sum_{j = 1}^{i} \frac{γ_{j 1}}{γ_{j}} {\tilde{θ}}_{j} {\hat{θ}}_{j} + λ_{i} λ_{i + 1}^{- 1} g_{i} z_{i} z_{i + 1} + \sum_{j = 1}^{i} [σ_{j} + 0.2785 θ_{j} τ_{j}]

(38)

Step n: Similar to step i, we construct the following differentiator to estimate

{\dot{α}}_{n - 1}

:

\{\begin{matrix} {\dot{ω}}_{(n - 1), 1} = ω_{(n - 1), 2} - κ_{1} s i g {(ω_{(n - 1), 1} - α_{n - 1})}^{\frac{1}{2}} \\ {\dot{ω}}_{(n - 1), 2} = - κ_{2} s i g n (ω_{(n - 1), 1} - α_{n - 1}) \end{matrix}

(39)

where

κ_{1}, κ_{2}

are the design positive parameters of the finite-time differentiator. With similar reasoning to step i, we have

ω_{(n - 1), 2} (t) - {\dot{α}}_{i - 1} = ε_{(n - 1), α}

with bounded estimation error

ε_{(n - 1), α}

, i.e., we can find

ε_{(n - 1) M}

such that

|ε_{(n - 1), α}| \leq ε_{(n - 1) M}

.

Taking (1), (9) and (13) into consideration, and using the similar definitions of

Θ_{n}

and

ξ_{n}

in (28) for

i = n

, the dynamics of

z_{n}

can be expressed as

{\dot{z}}_{n} = λ_{n} g_{n} β (t) u + h_{n} (Z) + λ_{n} d_{n} (t) - {\dot{α}}_{n - 1}

(40)

To address the impact of the unknown function, define the unknown nonlinear function

h_{n} (Z)

as

h_{n} (Z) = {\dot{λ}}_{n} λ_{n}^{- 1} {\tilde{x}}_{n} + λ_{n} f_{n} ({\bar{x}}_{n}) + w (t) ζ (x)

, which can be approximated by RBF NNs as

h_{n} (Z) = {ϕ_{n}}^{T} R_{n} (Z) + δ_{n} (Z)

with

∥δ_{n} (Z)∥ \leq ε_{n}

.

To overcome the problem of limited network resources, the following switching threshold event triggering control strategy is used to save communication resources. Switching threshold ETM can switch the threshold according to the size of the control input signal to ensure better control performance. Then, the switching threshold ETM trigger mechanism is defined as

u (t) = u_{r} (t_{k}), t_{k} \leq t < t_{k + 1}

(41)

t_{k + 1} = \{\begin{matrix} inf \{t \in t_{k} ||E_{u} (t)| \geq δ |u (t)| + m_{k, 1}\}, |u (t)| \leq D \\ inf \{t \in t_{k} ||E_{u} (t)| \geq m_{k, 2}\}, |u (t)| > D \end{matrix}

(42)

where

E_{u} (t) = u (t) - u_{r} (t)

represents the measurement error. D,

m_{k}

and

δ \in (0, 1)

is the designed positive parameter. When the above triggering rule (42) is satisfied, the time will be marked as

t_{k + 1}

and the control input

u (t_{k + 1})

will be applied to the controlled system. During the time interval

t \in [t_{k}, t_{k + 1})

, the control signal will hold as a constant, i.e.,

u_{r} (t_{k})

.

Remark 3.

The switching threshold event triggering policy used in this paper provides better flexibility in balancing communication resource effectiveness and system performance compared to the fixed threshold strategy [26] and relative threshold strategy [24]. When the magnitude of the system control input is small, e.g.,

|u (t)| \leq D

, using a relative thresholding strategy results in higher control accuracy and better system performance while saving communication resources. However, when the control input

u (t)

becomes large, the measurement error

E_{u} (t)

also becomes large, which can easily cause a sudden jump in the control input, resulting in the controlled system being subjected to a strong pulse input. Since the threshold value of the fixed threshold strategy is a constant, its measurement error is constantly limited to a constant value, which prevents sudden pulse jumps. Therefore, to ensure the system performance, the fixed threshold strategy is used when the amplitude of the input signal is

|u (t)| > D

.

Then, consider the last Lyapunov function candidate as

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

(43)

Then, the derivative of

V_{n}

satisfies

\begin{matrix} {\dot{V}}_{n} \leq {\dot{V}}_{n - 1} + λ_{n} g_{n} z_{n} β (t) u - z_{n} ω_{(n - 1), 2} + z_{n} Θ_{n} ξ_{n} \\ - λ_{n - 1} λ_{n}^{- 1} g_{n - 1} z_{n - 1} z_{n} - \frac{1}{γ_{n}} {\tilde{θ}}_{n} {\dot{\hat{θ}}}_{n} \end{matrix}

(44)

where

Θ_{n}

and

ξ_{n}

are expressed as

Θ_{n} = {[{ϕ_{n}}^{T}, λ_{n} {\bar{d}}_{n} + ε_{n}, ε_{(n - 1) M}, λ_{M} g_{n - 1}]}^{T}, ξ_{n} = {[R_{n} (Z), 1, 1, z_{n - 1}]}^{T}

(45)

Define

θ_{n} = {sup}_{t \geq 0} ∥Θ_{n}∥

; by Lemma 4, it can be deduced that

z_{n} Θ_{n} ξ_{n} \leq |z_{n}| θ_{n} |ξ_{n}| \leq z_{n} θ_{n} ∥ξ_{n}∥ tanh (\frac{z_{n} ∥ξ_{n}∥}{τ_{n}}) + 0.2785 θ_{n} τ_{n}

(46)

Design the following intermediate control command

α_{n}

and the adaptive law for the closed-loop control system

α_{n} = \frac{z_{n} {\bar{α}}_{n}^{2}}{\sqrt{{(z_{n} {\bar{α}}_{n})}^{2} + σ_{n}^{2}}}

(47)

{\dot{\hat{θ}}}_{n} = γ_{n} z_{n} ∥ξ_{n}∥ tanh (\frac{z_{n} ∥ξ_{n}∥}{τ_{n}}) - γ_{n 1} {\hat{θ}}_{n}

(48)

Now, design the switching threshold event trigger controller as follows:

u_{r} = - (1 + δ) (\frac{α_{n}}{g_{m n}} tanh (\frac{z_{n} α_{n}}{τ}) + {\bar{m}}_{k} tanh (\frac{z_{n} {\bar{m}}_{k}}{τ}))

(49)

where

g_{m n} = min (g_{n}^{m} λ_{m}) β_{m}

and

{\bar{m}}_{k} \geq m_{k} / (1 - δ_{k})

Add and subtract

z_{n} {\bar{α}}_{n}

with

{\bar{α}}_{n} = k_{n} z_{n} + {\hat{θ}}_{n} ∥ξ_{n}∥ tanh (\frac{z_{n} ∥ξ_{n}∥}{τ_{n}}) - ω_{(n - 1), 2}

on the right-hand side of (44) and substitute (46) and (48) into (44). Then, by invoking Lemma 2, one has

\begin{matrix} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} z_{j}^{2} + \sum_{j = 1}^{n} \frac{γ_{j 1}}{γ_{j}} {\tilde{θ}}_{j} {\hat{θ}}_{j} + z_{n} {\bar{α}}_{n} + λ_{n} g_{n} z_{n} β (t) u \\ + \sum_{j = 1}^{n - 1} (0.2785 θ_{j} τ_{j} + σ_{j}) + 0.2785 θ_{n} τ_{n} \end{matrix}

(50)

3.2. Stability Analysis

The effect of the proposed event-triggered adaptive neural prescribed performance control scheme on the stability of the closed-loop control system is given by the following theorem.

Theorem 1.

Consider nonlinear CPSs with unknown deception attacks and the dynamic event-based adaptive neural ETM control solution (49), together with the intermediate control laws (35) and (21), adaptive laws (22), (36) and (48). If Assumptions 1–3 are satisfied and the function

μ (t)

is appropriately selected to satisfy Definition 1, with the assistance of the dynamic event triggering mechanism and the finite-time differentiator, the following statements hold:

All signals in the closed-loop control system are bounded.
The Zeno behavior caused by the DETM can be avoided.
The prescribed tracking performance of the controlled system is guaranteed, i.e., the tracking error can be controlled within a predefined accuracy range $Ω_{e} = \{e (t) \in R ||e (t)| < μ_{T_{s}}\}$ in a predefined time $T_{s}$ , where $μ_{T_{s}}$ and $T_{s}$ are both customizable by the user.

Proof.

Next, we define the following parameters:

δ_{k} = \{\begin{matrix} δ_{k, 1}, |u (t)| \leq D \\ 0, |u (t)| > D \end{matrix}, m_{k} = \{\begin{matrix} m_{k, 1}, |u (t)| \leq D \\ m_{k, 1}, |u (t)| > D \end{matrix}

(51)

Noting that there exist two time-varying functions satisfying

|ψ_{k, i}| \leq 1, i = 1, 2

, the following holds

u_{r} (t) = (1 + ψ_{k, 1} δ_{k}) u (t) + ψ_{k, 2} m_{k}, \forall t \in [t_{k}, t_{k + 1})

(52)

Then, (52) can be rewritten as follows:

u (t) = \frac{u_{r} (t)}{1 + ψ_{k, 1} δ_{k}} - \frac{ψ_{k, 2} m_{k}}{1 + ψ_{k, 1} δ_{k}}

(53)

Substituting (53) into (50), one can obtain

\begin{matrix} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} z_{j}^{2} + \sum_{j = 1}^{n} \frac{γ_{j 1}}{γ_{j}} {\tilde{θ}}_{j} {\hat{θ}}_{j} + z_{n} {\bar{α}}_{n} + \sum_{j = 1}^{n - 1} (0.2785 θ_{j} τ_{j} + σ_{j}) \\ + λ_{n} g_{n} z_{n} β (t) [\frac{u_{r} (t)}{1 + ψ_{k, 1} δ_{k}} - \frac{ψ_{k, 2} m_{k}}{1 + ψ_{k, 1} δ_{k}}] + 0.2785 θ_{n} τ_{n} \end{matrix}

(54)

According to (49) and

|ψ_{k, i}| \leq 1, i = 1, 2

, one obtains

λ_{n} g_{n} z_{n} β (t) \frac{u_{r} (t)}{1 + ψ_{k, 1} δ_{k}} \leq λ_{n} g_{n} z_{n} β (t) \frac{u_{r} (t)}{1 + δ_{k}}

(55)

- λ_{n} g_{n} z_{n} β (t) \frac{ψ_{k, 2} m_{k}}{1 + ψ_{k, 1} δ_{k}} \leq |λ_{n} g_{n} z_{n} β (t) \frac{m_{k}}{1 - δ_{k}}|

(56)

With the help of (49), (55), (56), and Lemma 4, one obtains

\begin{matrix} λ_{n} g_{n} z_{n} β (t) (\frac{u_{r} (t)}{1 + ψ_{k, 1} δ_{k}} - \frac{ψ_{k, 2} m_{k}}{1 + ψ_{k, 1} δ_{k}}) \leq - |z_{n} α_{n}| - |λ_{n} g_{n} z_{n} β (t) {\bar{m}}_{k}| \\ + |λ_{n} g_{n} z_{n} β (t) \frac{m_{k}}{1 - δ_{k}}| + 0.557 τ \end{matrix}

(57)

Substituting (57) into (54) and considering that

{\bar{m}}_{k} \geq m_{k} / (1 - δ_{k})

, one has

\begin{matrix} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} z_{j}^{2} + \sum_{j = 1}^{n} \frac{γ_{j 1}}{γ_{j}} {\tilde{θ}}_{j} {\hat{θ}}_{j} + z_{n} {\bar{α}}_{n} - z_{n} \frac{z_{n} {\bar{α}}_{n}^{2}}{\sqrt{{(z_{n} {\bar{α}}_{n})}^{2} + σ_{n}^{2}}} \\ + \sum_{j = 1}^{n - 1} (0.2785 θ_{j} τ_{j} + σ_{j}) + 0.2785 θ_{n} τ_{n} + 0.557 τ \\ \leq - \sum_{j = 1}^{n} k_{j} z_{j}^{2} + \sum_{j = 1}^{n} \frac{γ_{j 1}}{γ_{j}} {\tilde{θ}}_{j} {\hat{θ}}_{j} + Λ \end{matrix}

(58)

where

Λ = \sum_{j = 1}^{n} (0.2785 θ_{j} τ_{j} + σ_{j}) + 0.557 τ

.

Via Lemma 3 (Young’s inequality), we obtain

\sum_{j = 1}^{n} \frac{γ_{j 1}}{γ_{j}} {\tilde{θ}}_{j} {\hat{θ}}_{j} \leq - \sum_{j = 1}^{n} \frac{γ_{j 1}}{2 γ_{j}} {\tilde{θ}}_{j}^{2} + \sum_{j = 1}^{n} \frac{γ_{j 1}}{2 γ_{j}} θ_{j}^{2}

(59)

Substituting (59) into (58) and invoking Lemma 2 yields

\begin{matrix} {\dot{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} z_{j}^{2} - \sum_{j = 1}^{n} \frac{γ_{j 1}}{γ_{j}} {\tilde{θ}}_{j}^{2} + b \\ \leq - C V_{n} + b \end{matrix}

(60)

where

b = \sum_{j = 1}^{n} \frac{γ_{j 1}}{2 γ_{j}} θ_{j}^{2} + Λ

and

C = min \{2 k_{i}, γ_{i 1}\}

.

Integrating both sides of (60) in regard to time yields

V_{n} \leq (V_{n} (0) - \frac{b}{C}) exp (- C t) + \frac{b}{C}

(61)

According to (61), one can obtain that

V_{n} \leq \frac{b}{C}

as

t \to \infty

and

V_{n}

is bounded. Additionally, in accordance with (43), it can be further obtained that

z_{i}, {\tilde{θ}}_{i}

is bounded. Because

{\hat{θ}}_{i}

denotes the estimate of the uncertain parameter

θ_{i}

, and

\tilde{θ} = θ - \hat{θ}

, one has that

{\hat{θ}}_{i}

is bounded. From (21), (35) and (47), it follows that

α_{i}

are bounded because they are composed of

z_{i}

and

{\hat{θ}}_{i}

. Since the desired trajectory

y_{d}

is a continuous bounded function, and

{\tilde{x}}_{i} = λ_{i} (t) x_{i}

, the boundedness of

{\tilde{x}}_{i}

can be inferred from the boundedness of

z_{i}

. The boundedness of the controller

u_{r} (t)

from (49) can be deduced, too. Thus, we prove the boundedness of all closed-loop signals of the system.

Second, we will show that the Zeno behavior does not occur. According to DETP (41) and (42), for

\forall t \in [t_{k}, t_{k + 1})

, the control command

u (t)

is in the holding phase, i.e.,

u (t)

is a constant. Accordingly, taking the derivative of measurements error

E_{u} (t)

produces

\frac{d |E_{u} (t)|}{d t} = \frac{d}{d t} {(E_{u} (t) \cdot E_{u} (t))}^{\frac{1}{2}} = |\dot{u} (t) - {\dot{u}}_{r} (t)| \leq |{\dot{u}}_{r} (t)|

(62)

Since

{\dot{u}}_{r} (t)

is a function containing

{\dot{x}}_{i} \in L_{\infty}

,

{\dot{z}}_{i} \in L_{\infty}

and

{\dot{\hat{θ}}}_{i} \in L_{\infty}

, it follows that

{\dot{u}}_{r} (t) \in L_{\infty}

. There is a positive constant

I_{0}

satisfying

{\dot{u}}_{r} (t) \leq I_{0}

. Then, it is easily verified that

|E_{u} (t)| = \int_{t_{k}}^{t_{k + 1}} |{\dot{E}}_{u} (t)| d t \leq \int_{t_{k}}^{t_{k + 1}} I_{0} d t \leq I_{0} (t_{k + 1} - t_{k})

(63)

From the ETM designed at (42), we have

t_{k + 1} - t_{k} \geq \frac{max \{δ |u (t)| + m_{k, 1}, m_{k, 2}\}}{I_{0}}

(64)

According to inequality (64), it can be found that there exists a lower bound for the nonzero execution interval. Therefore, there is not the Zeno phenomenon under the proposed control solution, i.e., the Zeno behavior can be avoided.

Finally, we demonstrate that the prescribed tracking performance of the controlled system can be guaranteed. Earlier, we established that

z_{1}

is always bounded, which means that

e_{1} (t)

is bounded for

\forall t \in [t_{0}, \infty)

. According to (9), it can be shown that the output tracking error

e (t) = y (t) - y_{d} (t)

always evolves within the prescribed performance funnel (3), i.e.,

|e (t)| \leq μ (t)

. By recalling the definition of

μ (t)

, one can obtain

|e (t)| \leq μ_{T_{s}}

,

\forall t \geq T_{s}

. Thus, with the performance funnel function

μ (t)

, the tracking error

e (t)

can be constrained to a prescribed precision region

(- μ_{T_{s}}, μ_{T_{s}})

within prescribed finite time

T_{s}

, where both

μ_{T_{s}}

and

T_{s}

are fully customizable. □

In order to clarify the design methodology of the adaptive prescribed performance control scheme in this paper, the algorithm design steps are given in Algorithm 1.

Algorithm 1 Adaptive Prescribed Performance Control Algorithm Design Procedure

Preliminary: Selecting and modeling system models and deception attacks. Design of funnel functions and coordinate transformations for backstepping.
Controller design stage: Using the backstepping method, the virtual controllers and adaptive laws are designed for the first, second and ith steps, respectively. Then, the final controller is derived from the previous i-step.
Design the event triggering mechanism and integrate it with the controller.
The design parameters are selected and the control scheme is thoroughly tested and validated using the simulation module of MATLAB.

4. Simulation Results

In the above formulation of the paper, the research work presented has been completed. In this section, the simulation examples will be used to verify the effectiveness of the proposed control strategy. The following electromechanical systems are considered to demonstrate the effectiveness and superiority of the developed control scheme in the physical system, shown in Figure 4:

\{\begin{matrix} M \ddot{q} + B \dot{q} + N sin (q) = I \\ L \dot{I} + R I = V_{ε} - K_{B} \dot{q} \end{matrix}

(65)

where

M = \frac{J}{K_{τ}} + \frac{m L_{0}^{2}}{3 K_{τ}} + \frac{M_{0} L_{0}^{2}}{K_{τ}} + \frac{2 M_{0} R_{0}^{2}}{5 K_{τ}}

,

N = \frac{m L_{0} G}{2 K_{τ}} + \frac{M_{0} L_{0} G}{K_{τ}}

,

B = \frac{B_{0}}{K_{τ}}

. G represents the gravity coefficient,

I (t)

is the motor armature current, and

q (t)

represents the angular position of the motor (i.e., the position of the load).

V_{ε}

represents the input control voltage. By designing the input voltage, the desired motion of the motor driving the load can be realized. The detailed explanation and values of the parameters of the electromechanical system are shown in Table 2.

Remark 4.

Figure 4 illustrates a conceptualized model of a small CPS that incorporates electromechanical systems, network communications and a computer operating system. The sensor part of the system is responsible for detecting and transmitting information about the state of the electromechanical system, such as armature current, motor angular position and angular velocity. The computer system utilizes the system status information to calculate the quantized value of the control input and transmits it to the programmable DC power supply module, which delivers the corresponding voltage value to the electromechanical system. Both sensor information and control input information are transmitted via network communication, thus completing the construction of the CPSs. It is worth noting that transmitting data in a shared network communication channel is vulnerable to adversarial cyber-attacks.

Via a coordinate transformation

x_{1} = q, x_{2} = \dot{q}, x_{3} = I

, the above kinetic model can be rewritten in the following form:

\{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = - \frac{N}{M} sin (x_{1}) - \frac{B}{M} x_{2} + \frac{1}{M} x_{3} + d_{2} (t) \\ {\dot{x}}_{3} = - \frac{K_{B}}{L} x_{2} - \frac{R}{L} x_{3} + \frac{1}{L} u + d_{3} (t) \end{matrix}

(66)

The deception attack signals suffered by the sensor network are chosen as

λ_{1} = 1 + 0.2 sin (t), λ_{2} = 1 + 0.1 cos (t), λ_{3} = 1 + 0.05 sin (t) cos (t)

.

In order to further validate the effectiveness of the control scheme designed in this paper when the actuator and sensor networks are subject to spoofing attacks and to comprehensively evaluate the performance of the proposed control method, we consider the following three different scenarios for illustration.

4.1. Control Performance Analysis and Validation

In this section, we verify and analyze the performance of the control scheme designed in this paper, and the controller parameters are designed as

g_{m 1} = 1, g_{m 2} = 1, g_{m 3} = 2

,

k_{1} = 12, k_{2} = 15, k_{3} = 25

,

γ_{1} = 4, γ_{2} = 5, γ_{3} = 2

,

γ_{11} = 5, γ_{21} = 8, γ_{31} = 15

,

τ_{1} = τ_{2} = τ_{3} = 0.1

,

σ_{1} = σ_{2} = σ_{3} = 0.001

. According to Definition 1, we design a prescribed-time funnel performance constraint function

μ (t)

as the form of (3), with its parameters set as

φ_{0} = 4, a = 0.5, κ = 4, μ_{T_{s}} = 0.1

. External interference to the system is taken as

d_{2} (t) = 0.2 sin (2 t), d_{3} (t) = 0.35 cos (1.2 t)

. The initial values of the adaptive parameters are chosen as

{[{\hat{θ}}_{1} (0), {\hat{θ}}_{2} (0), {\hat{θ}}_{3} (0)]}^{T} = {[0.5, 0.1, 0.5]}^{T}

. The actuator model subject to the deception attack is shown as

\tilde{u} = \{\begin{matrix} u, 0 \leq t < 4 \\ (0.5 + 0.2 sin (t)) u + 2 x_{1} x_{2} + 15, t \geq 4 \end{matrix}

(67)

In order to fully evaluate the performance of the proposed control scheme, simulations were performed using two representative initial points

x (0) = {(0.2, 0.1, 0.1)}^{T}

and

x (0) = {(- 0.3, 0.2, - 0.1)}^{T}

, which are labeled as Initial condition 1 and Initial condition 2. The control objective in this subsection is to make the system output y asymptotically track the desired target trajectory

y_{d} = 0.5 sin (t) + 0.5 sin (0.5 t)

and to ensure that the tracking error

e (t)

enters the preset accuracy range

(- 0.1, 0.1)

within a specified time

T_{s} = 2

.

The simulation results in this subsection are presented in Figure 5, Figure 6 and Figure 7. Figure 5 illustrates the schematic evolution of the system output y and tracking trajectory

y_{d}

as well as the tracking error

e (t)

under different initial conditions. From Figure 5, it can be seen that the controlled system can still ensure good tracking performance under deception attacks, and the tracking error can enter the predefined accuracy range within the prescribed time. Figure 6 depicts the trajectories of the system states

x_{2}

and

x_{3}

and the adaptive laws

{\hat{θ}}_{1}

,

{\hat{θ}}_{2}

and

{\hat{θ}}_{3}

for different initial values. Figure 7 depicts the trajectory of the control inputs

u (t), u_{r} (t)

in both scenarios.

4.2. Prescribed Performance Analysis under Different Predefined-Time Values

The objective of this part is to validate the effectiveness of the proposed control scheme in terms of the prescribed performance. The validity of the prescribed performance is verified by setting different values of the prescribed time and performing simulation experiments on system (66). In order to accurately demonstrate the performance of the proposed control scheme at different prescribed times, we set up six scenarios, which are

T_{s} = 0.5 s

;

T_{s} = 1 s

;

T_{s} = 1.5 s

;

T_{s} = 2 s

;

T_{s} = 2.5 s

; and

T_{s} = 3 s

. In this subsection, the actuator model subjected to deception attacks is updated to further reflect the effectiveness of the control scheme against deception attacks, in the following new form:

\tilde{u} = \{\begin{matrix} u, 0 \leq t < 2.8 \\ (0.4 + 0.2 sin (t)) u + 5 x_{1}^{2} x_{2} + 0.8 sin (t), t \geq 2.8 \end{matrix}

(68)

In this subsection, the six cases consider to be common a single system initial value

x (0) = {(0.4, - 0.2, - 0.3)}^{T}

, and the other design parameters are kept as the same values as in the previous section. The simulation results are shown in Figure 8, where we can clearly see that each of the system outputs tracks the desired trajectory signals within the corresponding prescribed time, and the tracking error also enters the predefined region within the corresponding time.

4.3. Comparison Analysis

To further conserve communication resources, reduce the frequency of data transmission, and prevent sudden pulse jumps in the control input, this paper uses a switching threshold event triggering mechanism. In this subsection, we choose the traditional fixed-threshold ETM to compare with the switching threshold ETM strategy in this paper, and choose the traditional ETM strategy as follows:

\begin{matrix} u (t) = v (t_{k}), t_{k} \leq t < t_{k + 1}, \\ t_{k + 1} = inf \{t \in R |g_{k} - |u - v| \leq 0\} \end{matrix}

(69)

In order to demonstrate the adaptability of the proposed control scheme for different desired target trajectories, the desired trajectories and the initial values of the system are set:

y_{d} = 2 sin (0.5 t) + cos (2 t)

,

x (0) = {(0.6, 0, - 0.1)}^{T}

. The associated controller and adaptive parameter values are the same as in the previous subsection, and the simulation sampling period is

0.002 s

.

The simulation results are displayed in Figure 9, Figure 10 and Figure 11. The comparisons of system outputs and tracking errors using two different event triggering mechanisms under the same control parameters are shown in Figure 9. From Figure 9, it can be seen that under the two different triggering mechanisms, there is only a very slight change in the system output and output tracking error, i.e., the effect of ETM on the system performance is limited. Therefore, it is reasonable to pursue a smaller number of triggers to save more communication resources while ensuring system performance. The time intervals for different event triggering strategies are given in Figure 10. As can be seen from Figure 11, the switching threshold ETM can further save communication resources compared to the conventional ETM. From the above simulation results, it can be concluded that the switching threshold ETM can further reduce the data transmission frequency while ensuring the control performance. Finally, the adaptive prescribed performance anti-deception attack control strategy designed in this paper is compared with the already existing related techniques as shown in Table 3.

5. Conclusions and Future Work

In this paper, we design a prescribed performance adaptive neural tracking control scheme for CPSs with unknown deception attacks on sensor and actuator networks. Combining backstepping and adaptive techniques overcomes controller design difficulties due to unknown nonlinear functions and deception attacks. The prescribed performance control scheme designed in this paper allows offline predefined system metrics such as settling time and tracking error in an environment subjected to deception attacks, and this characterization is verified by selecting a variety of different settling time values (0.5 s, 1 s, 1.5 s, 2 s, 2.5 s, and 3 s). Through verification, the switching threshold ETM used in this paper can ensure good system performance while further saving communication resources compared to the traditional fixed threshold ETM.

In real-world scenarios, CPSs typically contain multiple control loops. Moreover, it is an interesting challenge to validate the control scheme in this paper on a hardware platform due to its specificity. Therefore, in future work, we will explore the validation of security control strategies for large CPSs against multiple cyber-attacks on hardware platforms.

Author Contributions

Conceptualization, Y.L. (Yinguang Li) and C.L.; methodology, Y.L. (Yinguang Li); software, Y.L. (Yinguang Li); validation, C.L., Y.L. (Yinguang Li) and Y.L. (Yang Li); formal analysis, C.L.; investigation, Y.L. (Yinguang Li) and J.Z.; writing—original draft preparation, Y.L. (Yinguang Li); writing—review and editing, J.Z. and Y.L. (Yang Li); visualization, C.L.; supervision, J.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported in part by the National Natural Science Foundation (62203247) of China.

Data Availability Statement

No data were used for the research described in the article.

Conflicts of Interest

The authors declare no conflicts of interest.

References

Li, Z.; Zhao, J. Resilient adaptive control of switched nonlinear cyber-physical systems under uncertain deception attacks. Inf. Sci. 2021, 543, 398–409. [Google Scholar] [CrossRef]
Bi, Y.; Wang, T.; Qiu, J.; Li, M.; Wei, C.; Yuan, L. Adaptive Decentralized Finite-Time Fuzzy Secure Control for Uncertain Nonlinear CPSs Under Deception Attacks. IEEE Trans. Fuzzy Syst. 2023, 31, 2568–2580. [Google Scholar] [CrossRef]
Wu, J.; He, F.; He, X.; Li, J. Dynamic Event-Triggered Fuzzy Adaptive Control for Non-strict-Feedback Stochastic Nonlinear Systems with Injection and Deception Attacks. Int. J. Fuzzy Syst. 2023, 25, 1144–1155. [Google Scholar] [CrossRef]
Li, Y.; Zhu, Q.; Zhang, J. Distributed adaptive fixed-time neural networks control for nonaffine nonlinear multiagent systems. Sci. Rep. 2022, 12, 8459. [Google Scholar] [CrossRef] [PubMed]
Li, W.; Shi, F.; Li, W.; Yang, S.; Wang, Z.; Li, J. Review on Cooperative Control of Multi-Agent Systems. Int. J. Appl. Math. Control Eng. 2024, 7, 10–17. [Google Scholar] [CrossRef]
Han, H.; Wang, P.; Deng, Z. Research of Global Stabilization for Nonlinear Systems Based on Bounded Sampled-Data Control. Int. J. Appl. Math. Control. 2024, 7, 48–55. [Google Scholar] [CrossRef]
Gao, Z.; Zhao, N.; Zhao, X.; Niu, B.; Xu, N. Event-triggered prescribed performance adaptive secure control for nonlinear cyber physical systems under denial-of-service attacks. Commun. Nonlinear Sci. Numer. Simul. 2024, 131, 107793. [Google Scholar] [CrossRef]
Ma, Y.S.; Che, W.W.; Deng, C. Dynamic event-triggered model-free adaptive control for nonlinear CPSs under aperiodic DoS attacks. Inf. Sci. 2022, 589, 790–801. [Google Scholar] [CrossRef]
Ma, Y.S.; Che, W.W.; Deng, C.; Wu, Z.G. Model-Free Adaptive Resilient Control for Nonlinear CPSs with Aperiodic Jamming Attacks. IEEE Trans. Cybern. 2023, 53, 5949–5956. [Google Scholar] [CrossRef] [PubMed]
Chen, W.D.; Niu, B.; Wang, H.Q.; Li, H.T.; Wang, D. Adaptive Event-Triggered Control for Non-Strict Feedback Nonlinear CPSs with Time Delays against Deception Attacks and Actuator Faults. IEEE Trans. Autom. Sci. Eng. 2023, 1–11. [Google Scholar] [CrossRef]
Liu, S.; Wang, X.; Niu, B.; Song, X.; Wang, H.; Zhao, X. Adaptive Resilient Output Feedback Control Against Unknown Deception Attacks for Nonlinear Cyber-Physical Systems. IEEE Trans. Circuits Syst. II Express Briefs 2024. [Google Scholar] [CrossRef]
Li, T.; Chen, B.; Yu, L.; Zhang, W.A. Active Security Control Approach against DoS Attacks in Cyber-Physical Systems. IEEE Trans. Autom. Control 2021, 66, 4303–4310. [Google Scholar] [CrossRef]
Huang, J.; Zhao, L.; Wang, Q.G. Adaptive control of a class of strict feedback nonlinear systems under replay attacks. ISA Trans. 2020, 107, 134–142. [Google Scholar] [CrossRef] [PubMed]
Yang, Y.; Huang, J.; Su, X.; Deng, B. Adaptive control of cyber-physical systems under deception and injection attacks. J. Frankl. Inst. 2021, 358, 6174–6194. [Google Scholar] [CrossRef]
Tansel Yucelen, W.M.H.; Feron, E.M. Adaptive control architectures for mitigating sensor attacks in cyber-physical systems. Cyber-Phys. Syst. 2016, 2, 24–52. [Google Scholar] [CrossRef]
Li, M.; Chen, Y.; Zhang, Y.; Liu, Y. Adaptive sliding-mode tracking control of networked control systems with false data injection attacks. Inf. Sci. 2022, 585, 194–208. [Google Scholar] [CrossRef]
Ren, X.X.; Yang, G.H. Adaptive control for nonlinear cyber-physical systems under false data injection attacks through sensor networks. Int. J. Robust Nonlinear Control 2020, 30, 65–79. [Google Scholar] [CrossRef]
Zhao, N.; Tian, Y.; Zhang, H.; Herrera-Viedma, E. Fuzzy-based adaptive event-triggered control for nonlinear cyber-physical systems against deception attacks via a single parameter learning method. Inf. Sci. 2024, 657, 119948. [Google Scholar] [CrossRef]
Girard, A. Dynamic Triggering Mechanisms for Event-Triggered Control. IEEE Trans. Autom. Control 2015, 60, 1992–1997. [Google Scholar] [CrossRef]
Zhang, X.; Tan, J.; Wu, J.; Chen, W. Event-triggered-based fixed-time adaptive neural fault-tolerant control for stochastic nonlinear systems under actuator and sensor faults. Nonlinear Dyn. 2022, 108, 2279–2296. [Google Scholar] [CrossRef]
Zhang, H.; Chen, H.; Xiong, L.; Zhang, Y. Dynamic Event-Triggered Control for Delayed Nonlinear Markov Jump Systems under Randomly Occurring DoS Attack and Packet Loss. Mathematics 2024, 12, 1064. [Google Scholar] [CrossRef]
Chen, K.; Gu, Y.; Huang, W.; Zhang, Z.; Wang, Z.; Wang, X. Fixed-Time Adaptive Event-Triggered Guaranteed Performance Tracking Control of Nonholonomic Mobile Robots under Asymmetric State Constraints. Mathematics 2024, 12, 1471. [Google Scholar] [CrossRef]
Chen, J.; Hua, C.; Guan, X. Fast data-driven iterative event-triggered control for nonlinear networked discrete systems with data dropouts and sensor saturation. J. Frankl. Inst. 2020, 357, 8364–8382. [Google Scholar] [CrossRef]
Ma, H.; Li, H.; Lu, R.; Huang, T. Adaptive event-triggered control for a class of nonlinear systems with periodic disturbances. Sci. China Inf. Sci. 2020, 63, 150212. [Google Scholar] [CrossRef]
Li, Z.; Yan, H.; Zhang, H.; Lam, H.K.; Wang, M. Aperiodic Sampled-Data-Based Control for Interval Type-2 Fuzzy Systems via Refined Adaptive Event-Triggered Communication Scheme. IEEE Trans. Fuzzy Syst. 2021, 29, 310–321. [Google Scholar] [CrossRef]
Li, B.; Xia, J.; Zhang, H.; Shen, H.; Wang, Z. Event-triggered adaptive fuzzy tracking control for stochastic nonlinear systems. J. Frankl. Inst. 2020, 357, 9505–9522. [Google Scholar] [CrossRef]
Bu, X. Prescribed performance control approaches, applications and challenges: A comprehensive survey. Asian J. Control 2023, 25, 241–261. [Google Scholar] [CrossRef]
Chen, L.; Liang, H.; Pan, Y.; Li, T. Human-in-the-Loop Consensus Tracking Control for UAV Systems via an Improved Prescribed Performance Approach. IEEE Trans. Aerosp. Electron. Syst. 2023, 59, 8380–8391. [Google Scholar] [CrossRef]
Cheng, F.; Niu, B.; Zhang, L.; Chen, Z. Prescribed Performance-Based Low-Computation Adaptive Tracking Control for Uncertain Nonlinear Systems with Periodic Disturbances. IEEE Trans. Circuits Syst. II Express Briefs 2022, 69, 4414–4418. [Google Scholar] [CrossRef]
Zhang, L.; Che, W.W.; Deng, C.; Wu, Z.G. Prescribed Performance Control for Multiagent Systems via Fuzzy Adaptive Event-Triggered Strategy. IEEE Trans. Fuzzy Syst. 2022, 30, 5078–5090. [Google Scholar] [CrossRef]
Liang, H.; Zhang, Y.; Huang, T.; Ma, H. Prescribed Performance Cooperative Control for Multiagent Systems with Input Quantization. IEEE Trans. Cybern. 2020, 50, 1810–1819. [Google Scholar] [CrossRef] [PubMed]
Li, Y.; Zhang, J.; Li, Y. Command filter-based adaptive neural tracking control of nonlinear systems with multiple actuator constraints and disturbances. Complex Eng. Syst. 2024, 4. [Google Scholar] [CrossRef]
Gao, Y.; Niu, B.; Chen, W.; Wang, H.; Mu, C.; Wen, G. Adaptive Control of Constrained Nonlinear CPSs Under Deception Attacks Through Sensor and Actuator Networks. IEEE Trans. Circuits Syst. II Express Briefs 2024, 71, 1241–1245. [Google Scholar] [CrossRef]
Sui, S.; Chen, C.L.P.; Tong, S. A Novel Adaptive NN Prescribed Performance Control for Stochastic Nonlinear Systems. IEEE Trans. Neural Netw. Learn. Syst. 2021, 32, 3196–3205. [Google Scholar] [CrossRef] [PubMed]
Zhang, Z.; Dong, Y.; Duan, G. Global asymptotic fault-tolerant tracking for time-varying nonlinear complex systems with prescribed performance. Automatica 2024, 159, 111345. [Google Scholar] [CrossRef]
Zhang, J.; Li, Y. Adaptive RBF Neural Network Tracking Control of Stochastic Nonlinear Systems with Actuators and State Constraints. Mathematics 2024, 12, 1378. [Google Scholar] [CrossRef]
Chen, M.; Wang, H.; Liu, X. Adaptive Practical Fixed-Time Tracking Control with Prescribed Boundary Constraints. IEEE Trans. Circuits Syst. I Regul. Pap. 2021, 68, 1716–1726. [Google Scholar] [CrossRef]
Yuan, X.; Chen, B.; Lin, C. Neural Adaptive Fixed-Time Control for Nonlinear Systems with Full-State Constraints. IEEE Trans. Cybern. 2023, 53, 3048–3059. [Google Scholar] [CrossRef] [PubMed]
Zhang, Y.; Niu, B.; Zhao, X.; Duan, P.; Wang, H.; Gao, B. Global Predefined-Time Adaptive Neural Network Control for Disturbed Pure-Feedback Nonlinear Systems with Zero Tracking Error. IEEE Trans. Neural Netw. Learn. Syst. 2023, 34, 6328–6338. [Google Scholar] [CrossRef]

Figure 1. System architecture under sensor and actuator attacks.

Figure 2. Schematic illustration of the tracking error evolution.

Figure 3. Neural network structure.

Figure 4. Schematic of electromechanical system.

Figure 5. Trajectories of

y (t)

and

e (t)

under different initial conditions.

Figure 5. Trajectories of

y (t)

and

e (t)

under different initial conditions.

Figure 6. Trajectories of

x_{2}

,

x_{3}

and adaptive law under different initial conditions.

Figure 6. Trajectories of

x_{2}

,

x_{3}

and adaptive law under different initial conditions.

Figure 7. Curves of

u (t)

and

u_{r} (t)

under different initial conditions.

Figure 7. Curves of

u (t)

and

u_{r} (t)

under different initial conditions.

Figure 8. Trajectories of

y (t)

and

e (t)

under different settling times.

Figure 8. Trajectories of

y (t)

and

e (t)

under different settling times.

Figure 9. Trajectories of

y (t)

and

e (t)

under different ETM.

Figure 9. Trajectories of

y (t)

and

e (t)

under different ETM.

Figure 10. Event-triggered time intervals.

Figure 11. Comparison of the number of events triggered.

Table 1. Notations.

Symbol	Definition	Symbol	Definition
$\|*\|$	Absolute value	$R^{n}$	Real n-dimension space
$R^{+}$	Set of non-negative real numbers	$x^{T}$	The transpose of matrix x
$f_{i}$	Abbreviation for $f_{i} ({\bar{x}}_{i})$	$g_{i}$	Abbreviation for $g_{i} ({\bar{x}}_{i})$
$Ω_{Z}$	Compact set

Table 2. Parameter table for electromechanical systems.

Description	Value	Description	Value
J (rotor inertia)	$1.625 \times 10^{- 3} Kg \cdot m^{2}$	m (link mass)	$0.506 Kg$
$M_{0}$ (load mass)	$0.434 Kg$	$L_{0}$ (link length)	$0.305 m$
$R_{0}$ (radius of the load)	$0.023 m$	R (armature resistance)	$5.0 Ω$
$K_{B}$ (back-emf coefficient)	$0.9 N \cdot m / A$	L (armature inductance)	$25 \times 10^{- 3} H$
$B_{0}$ (coefficient of viscous friction)	$16.25 \times 10^{- 3} N \cdot m \cdot s / rad$	$K_{τ}$ (electromechanical conversion coefficient)	$0.9 N \cdot m / A$

Table 3. Comparison with existing technology.

Research Literature	Comparison
Refs. [18,33]	The settling time of the controlled system is not available for offline predefinition.
Refs. [11,18]	The case of simultaneous deception attacks on actuator and sensor networks was not investigated.
Refs. [3,10]	Failure to address the “complexity explosion” problem.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Li, C.; Li, Y.; Zhang, J.; Li, Y. Event-Triggered Adaptive Neural Prescribed Performance Tracking Control for Nonlinear Cyber–Physical Systems against Deception Attacks. Mathematics 2024, 12, 1838. https://doi.org/10.3390/math12121838

AMA Style

Li C, Li Y, Zhang J, Li Y. Event-Triggered Adaptive Neural Prescribed Performance Tracking Control for Nonlinear Cyber–Physical Systems against Deception Attacks. Mathematics. 2024; 12(12):1838. https://doi.org/10.3390/math12121838

Chicago/Turabian Style

Li, Chunyan, Yinguang Li, Jianhua Zhang, and Yang Li. 2024. "Event-Triggered Adaptive Neural Prescribed Performance Tracking Control for Nonlinear Cyber–Physical Systems against Deception Attacks" Mathematics 12, no. 12: 1838. https://doi.org/10.3390/math12121838

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Event-Triggered Adaptive Neural Prescribed Performance Tracking Control for Nonlinear Cyber–Physical Systems against Deception Attacks

Abstract

1. Introduction

2. Preliminaries

3. Controller Design and Stability Analysis

3.1. Controller Design

3.2. Stability Analysis

4. Simulation Results

4.1. Control Performance Analysis and Validation

4.2. Prescribed Performance Analysis under Different Predefined-Time Values

4.3. Comparison Analysis

5. Conclusions and Future Work

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI