Securing nonlinear networked Systems with model-based event-triggered control

Abstract

This paper proposes a novel model-based event-triggered control (MB-ETC) strategy for nonlinear networked delta operator systems subject to deception-type cyber-attacks. The approach uniquely integrates MB-ETC with the delta operator framework—marking the first such integration for nonlinear systems operating under cyber threats. The method leverages a two-layer neural network (NN)-based nonlinear controller, which outperforms traditional single-layer radial basis function (RBF) networks by eliminating dependence on predefined basis functions and enhancing approximation capability. Control signals and NN weights are updated only when a carefully designed event-triggering condition is violated, significantly reducing communication load and conserving network resources. The controller also adapts dynamically to variations in system model dynamics across triggering intervals. To avoid Zeno behavior, a strictly positive lower bound on inter-event time is analytically derived. Simulation results validate the proposed method’s effectiveness and superiority in maintaining system stability under attack conditions while optimizing resource usage.

Keywords

Cyber-attack delta operator event-triggered control networked control system neural network nonlinear system

Introduction

Networked control systems (NCSs) are systems where a control loop operates over a communication network, often shared with other systems (Mahmoud, 2016; Zhang et al., 2023). These systems provide several advantages, such as enhanced reliability, reduced installation, and maintenance costs, and flexible scalability. A common approach for implementing control in NCSs is sampled-data control (SDC), where signals are periodically sensed, actuated, and control actions executed at fixed time intervals. However, SDC often leads to inefficient use of limited network resources, including computational power, energy, and bandwidth (Aarzen, 1999). To address this inefficiency, two alternative methods have been proposed in the literature: event-triggered control (ETC) (Aarzen, 1999) and model-based control (Montestruque and Antsaklis, 2002).

In event-triggered control (ETC), data are transmitted aperiodically, only when the defined event-triggering condition is violated. This approach reduces the frequency of transmissions, leading to more efficient use of network resources. Extensive research has been conducted on regulating NCSs through ETC methods (see Mahmoud and Sabih (2014), Peng and Li (2018), and references therein). Conventional ETC continuously monitors the event-triggering condition, which may result in excessively fast sampling, potentially causing Zeno behavior (Tabuada, 2007). To address this issue, Heemels et al. (2013) proposed a periodic ETC approach, where the event-triggering condition is checked at regular time intervals. Yue et al. (2013) extended ETC to NCSs with time delays, while Bansal et al. (2018) introduced a periodic ETC for systems experiencing actuator faults. In Kumari et al. (2019), a delta operator approach was utilized to unify continuous-time systems (CTS) and discrete-time systems (DTS), resulting in the development of an event-triggered sliding mode controller. Similarly, Yuan and Liu (2021) designed an event-triggered fault detection strategy for NCSs under time-varying delays using the delta operator approach. Despite its advantages, ETC often employs a zero-order hold (ZOH) mechanism, which maintains a constant control signal between two triggering instants (Wang and Lemmon, 2011). This ZOH-based ETC operates in an open-loop fashion during these intervals, potentially compromising performance. To overcome this limitation, the model-based networked control system (MB-NCS) technique has been proposed (Montestruque and Antsaklis, 2002, 2003). In MB-NCS, the controller leverages a plant model to compute the control law and predict the system’s state between sampling instants. Using this estimated state, a control signal that offers better performance than ZOH while consuming fewer network resources is generated.

To combine the advantages of both ETC and MB-NCS, a model-based ETC (MB-ETC) technique has been developed in the literature (Garcia and Antsaklis, 2013; Heemels and Donkers, 2013; Liu et al., 2020; Song et al., 2022; Zhang et al., 2016). In MB-ETC, a plant model is used to estimate system dynamics, enabling aperiodic transmissions and updates based on an event-triggering condition. To further optimize network resource usage, the control law is adapted based on the plant model’s state between successive event-triggering instants. Heemels and Donkers (2013) introduced an MB-ETC method for perturbed linear systems, while Garcia and Antsaklis (2013) proposed an MB-ETC approach for systems with quantization and time-varying delays. Zhang et al. (2016) developed an adaptive MB-ETC framework for systems subject to external disturbances, and Song et al. (2022) designed a sliding-mode control strategy using MB-ETC for uncertain systems with external disturbances. For uncertain linear systems, Liu et al. (2020) proposed an MB-ETC approach featuring a dynamic event-triggering condition. However, these studies primarily address systems with well-understood dynamics (Heemels and Donkers, 2013; Zhang et al., 2016) or systems with limited uncertainty (Garcia and Antsaklis, 2013; Liu et al., 2020; Song et al., 2022). Accurately modeling the dynamics of real-world systems is often challenging, making it essential to develop control strategies that effectively handle systems with unknown or highly uncertain dynamics.

Neural network (NN)-based ETC techniques have gained prominence for systems with entirely unknown dynamics (Hu et al., 2018; Sahoo et al., 2013, 2016). Sahoo et al. (2013, 2016) developed NN-based ETC for single-input single-output (SISO) and multi-input multi-output (MIMO) nonlinear systems with uncertain dynamics, respectively. However, both studies employed a zero-order hold (ZOH)-based ETC approach. To address some limitations of ZOH-based methods, Chen et al. (2021) proposed an NN model-based ETC (NN MB-ETC) for nonlinear CTS. This approach utilized a single-layer NN controller with a radial basis function (RBF). While effective, the RBF design requires preselecting activation functions that correspond to a basis set of smooth nonlinear functions. This dependency can be mitigated by adopting a multi-layer NN design, which eliminates the need for basis set preselection and offers greater flexibility in handling complex nonlinear systems. Furthermore, a three-layer NN controller optimized via genetic algorithms and linear matrix inequalities is designed by Chen et al. (2025). It addresses external disturbances and network-induced time delays and implements a dynamic event-triggered scheme to reduce communication load. However, the ETC proposed in this paper is a ZOH-based ETC.

The utilization of networks for information sharing introduces the significant challenge of cyber-attacks, which can compromise the security of NCSs, degrade performance, or even lead to a complete system shutdown (De Sá et al., 2017). Consequently, it is critical to investigate and analyze the effects of cyber-attacks on the stability of nonlinear NCSs. Recent research has increasingly focused on addressing various types of cyber-attacks, including replay attacks, deception attacks, and denial-of-service (DoS) attacks, in the context of NCSs (De Sá et al., 2017). For instance, Dolk et al. (2017) proposed an event-triggered strategy to mitigate the effects of DoS attacks, while Wang et al. (2022) developed an ETC technique specifically for nonlinear systems subjected to jamming attacks. Liu et al. (2019) introduced an event-triggering approach for systems with distributed sensors under stochastic deception attacks. Similarly, Liu et al. (2018) proposed a hybrid triggering method for nonlinear NCSs vulnerable to deception attacks by combining time-triggering and event-triggering strategies. In addition, Wu et al. (2021) presented a novel ETC framework for NCSs under deception attacks, incorporating two distinct triggering conditions. Further contributions include Gu et al. (2021), who developed a unique event-triggering approach for nonlinear systems under deception attacks using T-S fuzzy approach, and Bansal and Mukhija (2020), who proposed a hybrid triggering approach combining self-triggering and ETC strategies for systems facing deception attacks. To address DoS attacks, Zhao et al. (2022) introduced a model-based dynamic ETC approach for system control, while Wang et al. (2023) designed a dynamic ETC framework for nonlinear systems affected by both DoS and deception attacks, employing an interval type-2 T-S fuzzy model. In addition, Wu et al. (2024) proposed a hybrid system approach combining periodic ETC and dynamic periodic ETC for controlling nonlinear systems under DoS attacks. Although the literature offers extensive findings on ETC strategies for networked systems prone to cyber-attacks, the majority of these studies focus on linear systems (Bansal and Mukhija, 2020; Dolk et al., 2017; Liu et al., 2019; Wu et al., 2021; Zhao et al., 2022). For nonlinear systems, researchers frequently use T-S fuzzy approximations to examine the impact of attacks (Gu et al., 2021; Liu et al., 2018; Wang et al., 2023). An adaptive NN control strategy ensuring CTS stability in the presence of unknown deception attacks on both actuators and sensors is developed by Li et al. (2024). The approach combines backstepping, funnel functions, and a ZOH-based event-triggering mechanism. A model-free control strategy using Goal representation Heuristic Dynamic Programming to handle sparse actuator attacks in discrete-time nonlinear systems is developed in Song et al. (2025). An ZOH-based event-triggered mechanism is designed to optimize control updates. However, to the best of authors’ knowledge, the impact of cyber-attacks on the networked systems with unknown nonlinear dynamics using model-based ETC (MB-ETC) for conserving network resources has not yet been explored.

Motivated by the preceding discussion, this paper introduces an MB-ETC technique for developing a NN-based controller for networked systems with unknown nonlinear dynamics, susceptible to disturbances and cyber-attacks. Prior works such as Liu et al. (2020) and Gu et al. (2021) contributed greatly to event-triggered control upon uncertain systems. Liu et al. (2020), however, focused on linear systems with known limited uncertainty and does not address cyber-attacks or nonlinear dynamics. Attacks are now considered, and T–S fuzzy approximation is employed by Gu et al. (2021), but then adaptive learning capability is lacking because pre-designed fuzzy rule bases are relied on. In contrast, our approach integrates a two-layer NN regarding online approximation as it targets a more general class for nonlinear systems including unknown dynamics. A probabilistic cyber-attack model accompanying the MB-ETC is proposed. This inclusion enables strong, efficient control in adversarial conditions and fills a critical gap that existing literature does not address. The primary focus of this paper is on deception-type cyber-attacks, which can corrupt transmitted data and thereby manipulate the controller’s actions, degrading system performance over time. A deception-type cyber-attack either happens at the moment when an event-triggered update is scheduled, or it occurs in between triggering events. If the attack occurs at an event-triggered time, the transmitted signal may get compromise. To handle this, a probabilistic model is used to represent the likelihood of an attack encoded as a Bernoulli random variable in this paper. Furthermore, instead of relying completely on the corrupted information, the controller uses a model-based prediction of the system’s state to compute the control action. In this way, even if the received data is compromised, the system can fall back on its internal prediction to maintain stability. On the other hand, if the attack happens between two event-triggered updates, it has almost no impact on the controller. This is due to the fact that during these periods, no new data is exchanged, and the controller does not rely on external input. It continues to operate using its internal model of the plant to estimate the system’s behaviour and apply the necessary control. Since there’s no communication, there’s nothing for the attacker to corrupt in that moment. Thus, by combining model-based predictions with event-triggering updates and updating only when necessary, the attack gets limited automatically thereby ensuring that the system remains stable, even under repeated or unpredictable attacks. The probability of such attacks is modeled as a random variable. To estimate the unknown nonlinearities, a two-layer NN is employed. This use of a two-layer NN marks a significant innovation over prior work that relied on single-layer RBF networks, by eliminating the need for predefined basis functions and offering greater flexibility in approximating complex dynamics. Furthermore, RBF NNs often need many neurons in case of complex nonlinearity while the two-layer NNs (with sigmoid activation) can approximate a broader class of functions with fewer neurons. The proposed MB-ETC technique integrates a model-based plant with the aperiodic transmission property of ETC to estimate the system’s actual states. Both NN weights and system states are updated only when a triggering condition is violated, thereby conserving network bandwidth, computational resources, and energy. Since in ZOH-ETC, the controller relies on the recently received state and maintains it constant until the next transmission occurs, this may lead to performance degradation if the system dynamics change significantly between two consecutive updates. However, in the MB-ETC approach, the plant model’s dynamic changes are used to compute the control law between two consecutive triggering events, thereby, maintaining accurate operation without requiring frequent updates. As a result, using MB-ETC instead of ZOH-based ETC, the consumption of network resources is significantly reduced. Furthermore, this work achieves a notable first by integrating the MB-ETC framework with the delta operator for nonlinear systems under cyber-attacks, allowing a unified treatment of both continuous-time (CTS) and discrete-time (DTS) dynamics. To prevent Zeno behavior, a positive lower bound on the minimum inter-event time (MIET) is derived. Simulation results are presented to validate the effectiveness of the proposed approach in maintaining system performance under both attacks and disturbances.

The main contributions of this paper are summarized as follows:

To effectively use network resources and control nonlinear systems under deception attack and disturbances, an MB-ETC strategy is proposed.

The proposed MB-ETC approach is developed for both CTS as well as DTS that are unified using delta operator approach, a key advancement that enables consistent design and analysis across time domains.

A two-layer NN is employed to approximate the unknown nonlinear dynamics within the system eliminating the dependency on predefined basis functions typically required in single-layer RBF networks.

A lower bound on MIET is obtained, which helps prevent Zeno behavior in a system.

Sections “Stability analysis” and “Simulation results” further explore several special cases of the proposed MB-ETC strategy, providing detailed analysis for nonlinear NCS.

The structure of the paper is as follows: section “Problem formulation” discusses the preliminaries, problem formulation, and system description. The stability analysis of the proposed approach is presented in section “Stability analysis.” The derivation of the condition on the minimum inter-event time (MIET) is provided in section “Minimum inter-event time.” Simulation results and a comparison of the proposed approach with other control strategies are discussed in section “Simulation results.” Finally, section “Conclusion” provides the conclusion.

Notations: The Frobenius norm of a matrix Z is defined as $∥ Z ∥_{F} = \sqrt{tr {Z^{T} Z}}$ , where $tr (\cdot)$ denotes the trace of a matrix. The induced two-norm for the matrices and euclidean norm of the vectors is represented by the notation ||⋅||, while $∥ \cdot ∥_{F}$ specifically indicates the Frobenius norm. An expectation operator is denoted by $E$ and matrix transpose or transpose of a vector is indicated by the superscript ′T′. The notation $P > 0$ ( $P \geq 0$ ) denotes the positive definite matrix (positive semi-definite matrix) of the proper dimensions, while a diagonal matrix is represented by the symbol $diag {\dots}$ .

Problem formulation

The nonlinear NCS under consideration is susceptible to disturbances and cyber-attacks. In Figure 1, a block diagram illustrating the proposed control approach is shown. At the sensor end, a communication network is present for data transmission, as seen in Figure 1. Combining ETC and MB-NCS for a nonlinear NCS under cyber-attack results in the proposed MB-ETC strategy. In the proposed control strategy, ETC is utilized for aperiodic transmission of signal and thereby, updates at the sensor end. A model-based technique is utilized to compute the control input during two adjacent trigger instants and approximate the behavior of the system in order to significantly reduce the number of transmissions compared to the ZOH ETC. Furthermore, the unknown dynamics are approximated using NN. The NN weights are updated, and the system state is transmitted to the model only when an event is triggered based on the specified triggering condition. The stochastic deception attacks could happen to the NCS. Deception attacks may alter the data that is transmitted, changing the control action and progressively decreasing the system’s efficiency. Because of its architecture, the suggested approach can be used for both CTS and DTS. While the CTS is useful for theoretical analysis, the DTS is suitable for computer realization. Consequently, the outcomes of CTS and DTS are combined using a delta operator (Goodwin et al., 1986).

Figure 1.

Block diagram of the MB-ETC strategy.

System description

Considering a nonlinear system whose dynamics are described by

\begin{matrix} \begin{matrix} δ x_{1} (t) & = x_{2} (t) \\ δ x_{2} (t) & = x_{3} (t) \\ ⋮ \\ δ x_{n} (t) & = f (x (t)) + u (t) + d (t) \\ y (t) & = x_{1} (t) \end{matrix} \end{matrix}

(1)

where $x^{T} (t) = {[x_{1} (t), x_{2} (t), \dots, x_{n} (t)]}^{T}$ is the system state vector, $u (t) \in ℛ$ is the control input and $y (t) \in ℛ$ is the system output. The unknown disturbance is denoted as $d (t)$ , $f (x (t))$ is an unknown smooth nonlinear function and δ is the delta operator which provides a unified representation for both CTS and DTS and is defined as follows

δx (t) = {\begin{matrix} \frac{dx (t)}{dt} & T = 0 \\ \frac{x (t + T) - x (t)}{T} & T \neq 0 \end{matrix}

(2)

where T is the sampling period.

The following assumption relates to the unknown disturbance affecting the system.

Assumption 1. The unknown disturbance $d (t)$ in equation (1) is assumed to be bounded by a positive constant $d_{\max}$ , such that $∥ d (t) ∥ \leq d_{\max}$ holds $\forall$ t (Khalil, 2000).

The nonlinear system (1) can be expressed in the Brunovsky canonical form (BCF)

\begin{matrix} \begin{matrix} δx (t) = & A_{δ} x (t) + B_{δ} f (x (t)) + B_{δ} d (t) + B_{δ} u (t) \end{matrix} \end{matrix}

(3)

where

A_{δ} = {[\begin{matrix} 0 & 1 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & 1 \\ 0 & 0 & \dots & 0 \end{matrix}]}_{n \times n}; B_{δ} = {[\begin{matrix} 0 \\ 0 \\ ⋮ \\ 1 \end{matrix}]}_{n \times 1}

The system $δx (t) = A_{δ} x (t) + B_{δ} u (t)$ is assumed to have a controllable pair $(A_{δ}, B_{δ})$ . Consequently, for an appropriate control gain $K_{2}$ , ensuring that $(A_{δ} + B_{δ} K_{2})$ is Hurwitz, there exists $P > 0$ and $Q > 0$ such that following condition holds

{(A_{δ} + B_{δ} K_{2})}^{T} P + P (A_{δ} + B_{δ} K_{2}) \leq - Q

(4)

The ideal control law for a system experiencing a deception attack and disturbances can be designed using the feedback linearization approach as

u_{d} (t) = - f (x (t)) + β (t) K_{1} h (x (t)) + (1 - β (t)) K_{2} x (t)

(5)

where the attack on the system is denoted by $h (x (t))$ , the probability of occurrence of deception attack is $β (t)$ , and the feedback gains are $K_{1}$ and $K_{2}$ .

Remark 1. The deception attack under consideration is stochastic. Therefore, $E {β (t)} = \bar{β}$ , wherein $β (t) \in {0, 1}$ is a random variable representing the probability with which the malicious assault affects the system performance. In this case, $β (t) = 1$ denotes an attack on the system, while $β (t) = 0$ implies no attack on the system.

Assumption 2. The deception attack $h (x (t))$ satisfies the following criterion, given that the attacker is assumed to have finite energy

∥ h (x (t)) ∥_{2} \leq ∥ Hx (t) ∥_{2}

(6)

where H is a known constant matrix (Liu et al., 2019).

Remark 2. In deception attacks, instead of totally jamming the network, the attacker seeks to gradually deteriorate system performance. As stated in Assumption 2, it is thus assumed that the attacker has a finite amount of energy.

It is not possible to directly apply the ideal control law (5) as the nonlinearity $f (x (t))$ in equation (3) is unknown. Therefore, Section “Controller design” introduces the system’s actual control law.

Neural network approximation

The unknown smooth nonlinear function is approximated via a two-layer NN structure. Figure 2 illustrates the output of the two-layer NN, which is used to approximate the function $f (x (t)) : ℛ^{n} \to ℛ$ and, this output is given by

f (x (t)) = W^{T} κ (V^{T} x) + ϵ

(7)

where ϵ is the approximation error, $V = [v_{1}, v_{2}, \dots, v_{l}] \in ℛ^{n \times l}$ and $W = [w_{1}, w_{2}, \dots, w_{m}] \in ℛ^{l \times m}$ are the ideal weight matrices between hidden layer and input layer and between output layer and hidden layer, respectively. The activation function of the hidden layer is denoted by $κ (\cdot)$ with $κ (z) = {[κ (z_{1}), κ (z_{2}), \dots, κ (z_{l})]}^{T}$ . The activation function is selected as a sigmoid function and is defined as follows

κ (z) = \frac{1}{1 + e^{- z}}

(8)

Figure 2.

Neural network structure.

Considering that the ideal weight matrices V and W are unknown, they cannot be directly used for controller design. Therefore, their estimates, $\hat{V}$ and Ŵ, of these ideal weight matrices is utilized instead. The weight estimation errors are thereby defined as

\begin{matrix} \begin{matrix} Ṽ = V - \hat{V} \\ \tilde{W} = W - Ŵ \end{matrix} \end{matrix}

(9)

The following assumptions are drawn regarding the weight matrices and the activation function.

Assumption 3. The bound on the ideal weight matrices is as follows (Sahoo et al., 2016)

∥ V ∥_{F} \leq V_{M}; ∥ W ∥_{F} \leq W_{M}

(10)

Assumption 4. The approximation error and activation function are bounded by a positive constants $ϵ_{\max}$ and $κ_{\max}$ , respectively, such that $∥ ϵ ∥ \leq ϵ_{\max}$ and $∥ κ (z) ∥ \leq κ_{\max}$ hold (Sahoo et al., 2016).

Assumption 5. Considering the Lipschitz condition, the activation function is assumed to satisfy

∥ κ (z) - κ (ẑ) ∥ \leq L_{κ} ∥ z - ẑ ∥

(11)

where $L_{κ}$ is a known constant (Sahoo et al., 2016).

Model-based design

The model-based plant for the nonlinear system (1) is designed as follows

δ \hat{x} (t) = A_{δ} \hat{x} (t) + B_{δ} \hat{f} (\hat{x} (t)) + B_{δ} u (t)

(12)

where $\hat{x} (t)$ is the model state and $\hat{f} (\hat{x} (t))$ is the estimate of the nonlinear function $f (\hat{x} (t))$ .

In the proposed MB-ETC approach, the event-triggering instants ${σ_{k}}_{k = 0}^{\infty}$ satisfy the condition $σ_{k + 1} > σ_{k}$ and the event-triggering error is expressed as

e (t) = x (t) - \hat{x} (t)

(13)

For aperiodic state transmission, the following event-triggering condition is considered

e {(t)}^{T} e (t) \leq αx {(t)}^{T} x (t)

(14)

where $α \in [0, 1)$ is the event-triggering parameter. The event is triggered only when the condition (14) is violated, prompting the update of the NN weights and model state. In addition, the error $e (t)$ becomes zero once $t = σ_{k}$ because $\hat{x} (t) = x (t)$ .

In the following subsection, an actual controller is developed for nonlinear NCS under cyber-attacks of the deception kind.

Controller design

Since, $f (x (t))$ is not exactly known and only the estimate $\hat{f} (\hat{x} (t))$ of the nonlinear function using NNs and model-based plant (12) is known. The intended control law $u_{d} (t)$ provided by equation (5) cannot be utilized. As a result, the actual control law becomes

u (t) = - \hat{f} (\hat{x} (t)) + β (t) K_{1} h (x (t)) + (1 - β (t)) K_{2} \hat{x} (t)

(15)

The approximation of the nonlinear function at estimated state is given by

\hat{f} (\hat{x} (t)) = Ŵ^{T} κ ({\hat{V}}^{T} \hat{x})

(16)

The suggested approach updates the NN weights only when an event is triggered on violation of the condition (14). As a result, the weight update laws for NN during the flow interval as well as jump instants are as follows.

For the flow interval $t \in (σ_{k}, σ_{k + 1}]$ ,

\overset{\cdot}{Ŵ} = 0; \overset{\cdot}{\hat{V}} = 0

(17)

And at the aperiodic jump instants (triggering instants) $t = σ_{k}$

\begin{matrix} \begin{matrix} Ŵ^{+} = Ŵ - M (κ ({\hat{V}}^{T} \hat{x}) - {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) {\hat{V}}^{T} \hat{x}) Λ^{T} e \\ {\hat{V}}^{+} = \hat{V} - N Λ^{T} e \hat{x} Ŵ^{T} {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) \end{matrix} \end{matrix}

(18)

given matrices M and N of appropriate dimensions, the updated NN weights upon an event trigger are denoted as $Ŵ^{+}$ and ${\hat{V}}^{+}$ and $Λ$ is a suitably chosen vector ensuring that $e (t)$ converges to zero.

Using equations (9), (17) and (18), the dynamics of the neural network weight estimation error are derived as follows.

For flow interval $t \in (σ_{k}, σ_{k + 1}]$

\overset{\cdot}{\tilde{W}} = 0; \overset{\cdot}{Ṽ} = 0

(19)

And at jump instants $t = σ_{k}$

\begin{matrix} \begin{matrix} {\tilde{W}}^{+} = \tilde{W} + M (κ ({\hat{V}}^{T} \hat{x}) - {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) {\hat{V}}^{T} \hat{x}) Λ^{T} e \\ Ṽ^{+} = Ṽ + N Λ^{T} e \hat{x} Ŵ^{T} {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) \end{matrix} \end{matrix}

(20)

The next section presents the development of stability conditions for the closed-loop system using the proposed MB-ETC strategy.

Stability analysis

The following theorem establishes the stability conditions for equation (1) under a deception attack using the proposed MB-ETC strategy.

Theorem 1. The nonlinear networked system (1), subjected to the attack (6), with actual control input (15), plant model (12), the NN weight update laws (17) and (18), under the triggering condition (14) and the Assumptions 1-5, is mean square stable and the estimation errors of NN weights remain bounded if there exists matrices $P > 0$ and $Q > 0$ of apt dimensions satisfying the following conditions.

For flow interval

\begin{matrix} \begin{matrix} ∥ x ∥ > {[\frac{Γ_{1}}{\frac{λ_{\min} (Q)}{2} - Γ_{2}}]}^{1 ∕ 2} \\ ∥ \hat{x} ∥ > {[\frac{Γ_{1}}{λ_{\min} (Q) - Γ_{3}}]}^{1 ∕ 2} \end{matrix} \end{matrix}

(21)

At triggering instants

∥ \hat{x} ∥ > \frac{Γ_{5} + \sqrt{4 Γ_{6} (λ_{\min} (P) - Γ_{4}) + Γ_{5}^{2}}}{2 (λ_{\min} (P) - Γ_{4})}

(22)

where

\begin{matrix} Γ_{1} & = \frac{2}{λ_{\min} (Q)} ∥ P_{1} ∥^{2} ∥ B_{δ} ∥^{2} (∥ \tilde{W} ∥_{F} κ_{\max} + ε_{\max} + d_{\max})^{2} \\ Γ_{2} & = 2 (1 - \bar{β}) \sqrt{α} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{2} ∥ + 2 \bar{β} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{2} ∥ \\ + 2 \bar{β} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{1} ∥ ∥ H ∥ + 2 L_{κ} ∥ P ∥ ∥ B_{δ} ∥ ∥ Ŵ ∥_{F} ∥ Ṽ ∥_{F} \\ + 2 \sqrt{α} L_{κ} ∥ P ∥ ∥ B_{δ} ∥ ∥ Ŵ ∥_{F} ∥ \hat{V} ∥_{F} + \frac{{\bar{β}}^{2}}{2} \\ Γ_{3} & = 2 \bar{β} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{2} ∥ + 2 ∥ P ∥^{2} ∥ B_{δ} ∥^{2} ∥ K_{1} ∥^{2} ∥ H ∥^{2} \\ Γ_{4} & = a_{2}^{2} ∥ e ∥^{2} ∥ Λ ∥^{2} (∥ M ∥ ∥ \hat{V} ∥_{F}^{2} + ∥ N ∥ ∥ Ŵ ∥_{F}^{2}) + ∥ P ∥ \\ Γ_{5} & = ∥ e ∥ ∥ Λ ∥ (2 c_{2} ∥ \hat{V} ∥_{F} + 4 a_{2} ∥ \hat{V} ∥_{F} ∥ Ŵ ∥_{F} + 2 c_{3} ∥ Ŵ ∥_{F}) \\ Γ_{6} & = 2 c_{1} ∥ e ∥ ∥ Λ ∥ + 2 a_{1} ∥ Ŵ ∥_{F} ∥ e ∥ ∥ Λ ∥ + a_{1}^{2} ∥ e ∥^{2} ∥ Λ ∥^{2} ∥ M ∥ \\ + ∥ P ∥ ∥ e ∥^{2} \\ a_{1} & = ∥ κ ({\hat{V}}^{T} \hat{x}) ∥_{F}; a_{2} = ∥ {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) ∥_{F}; c_{1} = W_{M} a_{1}; \\ c_{2} = W_{M} a_{2}; c_{3} = V_{M} a_{2} \end{matrix}

Proof. Consider the following Lyapunov candidate function

V (t) = V_{x} (x (t)) + V_{\hat{x}} (\hat{x} (t)) + V_{w} (\tilde{W}) + V_{v} (Ṽ)

(23)

where $V_{x} (x (t)) = x^{T} (t) Px (t)$ , $V_{\hat{x}} (\hat{x} (t)) = {\hat{x}}^{T} (t) P \hat{x} (t)$ , $V_{w} (\tilde{W}) = tr {{\tilde{W}}^{T} M^{- 1} \tilde{W}}$ and $V_{v} (Ṽ) = tr {Ṽ^{T} N^{- 1} Ṽ}$ .

For Flow interval:

From equation (19), using the infinitesimal operator (Bansal and Mukhija, 2020) on $V_{w} (\tilde{W})$ and $V_{v} (Ṽ)$ and taking the expectation on it yields

E {ℒ V_{w} (\tilde{W})} = 0; E {ℒ V_{v} (Ṽ)} = 0

(24)

Furthermore, using infinitesimal operator (Bansal and Mukhija, 2020) on $V_{x} (x (t))$ and $V_{\hat{x}} (\hat{x} (t))$ and taking the expectation on it gives

\begin{matrix} \begin{matrix} E {ℒ V_{x} (x (t))} = E {x^{T} (t) Pδx (t) + δ x^{T} (t) Px (t)} \\ E {ℒ V_{\hat{x}} (\hat{x} (t))} = E {{\hat{x}}^{T} (t) Pδ \hat{x} (t) + δ {\hat{x}}^{T} (t) P \hat{x} (t)} \end{matrix} \end{matrix}

(25)

From equations (1), (5), and (15), the flow dynamics of the nonlinear system are given by

\begin{matrix} \begin{matrix} δx (t) & = A_{δ} x (t) + B_{δ} f (x (t)) + B_{δ} [u (t) + u_{d} (t) - u_{d} (t)] + B_{δ} d (t) \\ = (A_{δ} + (1 - β (t)) B_{δ} K_{2}) x (t) \\ + B_{δ} [- \hat{f} (\hat{x} (t)) + β (t) K_{1} h (x (t)) \\ + (1 - β (t)) K_{2} \hat{x} (t) + f (x (t)) \\ - (1 - β (t)) K_{2} x (t)] + B_{δ} d (t) \end{matrix} \end{matrix}

(26)

From equations (7), (9), (13), and (16), equation (26) can be rewritten as

\begin{matrix} \begin{matrix} δx (t) & = (A_{δ} + B_{δ} K_{2}) x (t) - β (t) B_{δ} K_{2} x (t) - (1 - β (t)) B_{δ} K_{2} e (t) \\ + β (t) B_{δ} K_{1} h (x (t)) + B_{δ} ({\tilde{W}}^{T} κ (V^{T} x) + ϵ) + B_{δ} Ŵ^{T} (κ (V^{T} x) \\ - κ ({\hat{V}}^{T} \hat{x})) + B_{δ} d (t) \end{matrix} \end{matrix}

(27)

Similarly, from equations (12) and (15), dynamics of the plant model is expressed as

\begin{matrix} \begin{matrix} δ \hat{x} (t) = (A_{δ} + B_{δ} K_{2}) \hat{x} (t) - β (t) B_{δ} K_{2} \hat{x} (t) + β (t) B_{δ} K_{1} h (x (t)) \end{matrix} \end{matrix}

(28)

On utilizing equations (24), (25), (27), and (28), the expectation on the Lyapunov function (23) yields

\begin{matrix} \begin{matrix} E {ℒ V (t)} \\ = E {x^{T} (t) P (A_{δ} + B_{δ} K_{2}) x (t) + x^{T} (t) {(A_{δ} + B_{δ} K_{2})}^{T} Px (t) \\ - 2 x^{T} (t) Pβ (t) B_{δ} K_{2} x (t) - 2 x^{T} (t) P (1 - β (t)) B_{δ} K_{2} e (t) \\ + 2 x^{T} (t) Pβ (t) B_{δ} K_{1} h (x (t)) + 2 x^{T} (t) P B_{δ} (\tilde{W} κ (V^{T} x) + ϵ) \\ + 2 x^{T} (t) P B_{δ} Ŵ (κ (V^{T} x) - κ ({\hat{V}}^{T} \hat{x})) + 2 x^{T} (t) P B_{δ} d (t) \\ + {\hat{x}}^{T} (t) P (A_{δ} + B_{δ} K_{2}) \hat{x} (t) + {\hat{x}}^{T} (t) {(A_{δ} + B_{δ} K_{2})}^{T} P \hat{x} (t) \\ - 2 {\hat{x}}^{T} (t) Pβ (t) B_{δ} K_{2} \hat{x} (t) + 2 {\hat{x}}^{T} (t) Pβ (t) B_{δ} K_{1} h (x (t))} \end{matrix} \end{matrix}

(29)

On using equation (4) along with Assumptions 1, 2, 4, and 5 and taking Frobenius norm, equation (29) becomes

\begin{matrix} \begin{matrix} E {ℒ V (t)} \\ \leq - λ_{\min} (Q) ∥ x ∥^{2} + 2 \bar{β} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{2} ∥ ∥ x ∥^{2} \\ + 2 (1 - \bar{β}) ∥ x ∥ ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{2} ∥ ∥ e ∥ + 2 \bar{β} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{1} ∥ ∥ H ∥ ∥ x ∥^{2} \\ + 2 ∥ P ∥ ∥ B_{δ} ∥ (∥ \tilde{W} ∥_{F} κ_{\max} + ϵ_{\max} + d_{\max}) ∥ x ∥ \\ + 2 L_{κ} ∥ P ∥ ∥ B_{δ} ∥ ∥ Ŵ ∥_{F} ∥ Ṽ ∥_{F} ∥ x ∥^{2} - λ_{\min} (Q) ∥ \hat{x} ∥^{2} \\ + 2 L_{κ} ∥ x ∥ ∥ P ∥ ∥ B_{δ} ∥ ∥ Ŵ ∥_{F} ∥ \hat{V} ∥_{F} ∥ e ∥ + 2 \bar{β} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{2} ∥ ∥ \hat{x} ∥^{2} \\ + 2 \bar{β} ∥ \hat{x} ∥ ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{1} ∥ ∥ H ∥ ∥ x ∥ \end{matrix} \end{matrix}

(30)

Using triggering condition (14) and Young’s inequality leads to the following expression

\begin{matrix} \begin{matrix} E {ℒ V (t)} & \leq - [\frac{λ_{\min} (Q)}{2} - Γ_{2}] ∥ x ∥^{2} \\ - [λ_{\min} (Q) - Γ_{3}] ∥ \hat{x} ∥^{2} + Γ_{1} \end{matrix} \end{matrix}

(31)

Thus, $E {ℒ V (t)} \leq 0$ if $- [\frac{λ_{\min} (Q)}{2} - Γ_{2}] ∥ x ∥^{2} + Γ_{1} < 0$ and $- [λ_{\min} (Q) - Γ_{3}] ∥ \hat{x} ∥^{2} + Γ_{1} < 0$ . This yields condition (21).

At triggering instants:

Taking into account the aperiodic triggering instances, the Lyapunov function’s (23) first difference is given by

Δ V = Δ V_{x} + Δ V_{\hat{x}} + Δ V_{w} + Δ V_{v}

(32)

At jump instants $t = κ_{k}$ , the system dynamics as well as model-based plant dynamics are expressed as

x^{+} = x; {\hat{x}}^{+} = x

(33)

Hence, $Δ V_{x} + Δ V_{\hat{x}}$ can be expressed as

Δ V_{x} + Δ V_{\hat{x}} = x^{T +} P x^{+} - x^{T} Px + {\hat{x}}^{T +} P {\hat{x}}^{+} - {\hat{x}}^{T} P \hat{x}

(34)

Utilizing equation (33) and taking norm, equation (34) becomes

Δ V_{x} + Δ V_{\hat{x}} \leq - λ_{\min} (P) ∥ \hat{x} ∥^{2} + ∥ P ∥ ∥ x ∥^{2}

(35)

From equation (20), $Δ V_{w} = tr {{\tilde{W}}^{T +} M^{- 1} {\tilde{W}}^{+}} - tr {{\tilde{W}}^{T} M^{- 1}$ $\tilde{W}}$ can be rewritten as

\begin{matrix} \begin{matrix} Δ V_{w} & = 2 tr {{\tilde{W}}^{T} M^{- 1} M (κ ({\hat{V}}^{T} \hat{x}) - {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) {\hat{V}}^{T} \hat{x}) Λ^{T} e} \\ + tr {((κ ({\hat{V}}^{T} \hat{x}) - {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) {\hat{V}}^{T} \hat{x}) Λ^{T} e)^{T} M^{T} M^{- 1} M \\ ((κ ({\hat{V}}^{T} \hat{x}) - {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) {\hat{V}}^{T} \hat{x}) Λ^{T} e)} \end{matrix} \end{matrix}

(36)

Let $Θ_{1} = tr {{\tilde{W}}^{T} (κ ({\hat{V}}^{T} \hat{x}) - {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) {\hat{V}}^{T} \hat{x})}$ . Then, taking appropriate norm yields and using assumption 3

\begin{matrix} ∥ Θ_{1} ∥ \leq c_{1} + a_{1} ∥ Ŵ ∥_{F} + c_{2} ∥ \hat{V} ∥_{F} ∥ \hat{x} ∥ + a_{2} ∥ Ŵ ∥_{F} ∥ \hat{V} ∥_{F} ∥ \hat{x} ∥ \end{matrix}

(37)

In addition, taking the Frobenius norm of equation (36) and using (37) gives

\begin{matrix} \begin{matrix} Δ V_{w} & \leq (2 c_{1} + 2 a_{1} ∥ Ŵ ∥_{F} + 2 a_{2} ∥ Ŵ ∥_{F} ∥ \hat{V} ∥_{F} ∥ \hat{x} ∥ + 2 c_{2} ∥ \hat{V} ∥_{F} ∥ \hat{x} ∥) \\ ∥ Λ ∥ ∥ e ∥ + ∥ e ∥^{2} ∥ Λ ∥^{2} ∥ M ∥ (a_{1}^{2} + a_{2}^{2} ∥ \hat{V} ∥_{F}^{2} ∥ \hat{x} ∥^{2}) \end{matrix} \end{matrix}

(38)

Similarly, from equation (20), $Δ V_{v} = tr {Ṽ^{T +} N^{- 1} Ṽ^{+}} - tr {Ṽ^{T} N^{- 1} Ṽ}$ can be rewritten as

\begin{matrix} \begin{matrix} Δ V_{v} & = 2 tr {Ṽ^{T} N^{- 1} N Λ^{T} e \hat{x} Ŵ^{T} {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x})} + tr {(Λ^{T} e \hat{x} Ŵ^{T} {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}))^{T} \\ \times N^{T} N^{- 1} N (Λ^{T} e \hat{x} Ŵ^{T} {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}))} \end{matrix} \end{matrix}

(39)

Let $Θ_{2} = tr {Ṽ^{T} Λ^{T} e \hat{x} Ŵ^{T} {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x})}$ . Then, by applying the Frobenius norm and using Assumption 3 the following is obtained

∥ Θ_{2} ∥ \leq c_{3} ∥ \hat{x} ∥ ∥ Ŵ ∥_{F} ∥ Λ ∥ ∥ e ∥ + a_{2} ∥ \hat{x} ∥ ∥ Ŵ ∥_{F} ∥ \hat{V} ∥_{F} ∥ Λ ∥ ∥ e ∥

(40)

On taking Frobenius norm of equation (39) and utilizing (40), the following is obtained

\begin{matrix} \begin{matrix} Δ V_{v} & \leq 2 c_{3} ∥ \hat{x} ∥ ∥ Ŵ ∥_{F} ∥ Λ ∥ ∥ e ∥ + 2 a_{2} ∥ \hat{x} ∥ ∥ Ŵ ∥_{F} ∥ \hat{V} ∥_{F} ∥ Λ ∥ ∥ e ∥ \\ + a_{2}^{2} ∥ e ∥^{2} ∥ Λ ∥^{2} ∥ N ∥ ∥ Ŵ ∥_{F}^{2} ∥ \hat{x} ∥^{2} \end{matrix} \end{matrix}

(41)

Combining equations (35), (38) and (41), (32) is rewritten as

Δ V \leq - {(λ_{\min} (P) - Γ_{4}) ∥ \hat{x} ∥^{2} - Γ_{5} ∥ \hat{x} ∥ - Γ_{6}}

(42)

From equation (42), $Δ V < 0$ when

- {(λ_{\min} (P) - Γ_{4}) ∥ \hat{x} ∥^{2} - Γ_{5} ∥ \hat{x} ∥ - Γ_{6}} < 0

The condition (22) is derived by completing the squares.

Furthermore, for system dynamics (1) without deception attack, the desired and actual control law become

\begin{matrix} \begin{matrix} u_{d} (t) & = - f (x (t)) + Kx (t) \\ u (t) & = - \hat{f} (\hat{x} (t)) + K \hat{x} (t) \end{matrix} \end{matrix}

(43)

Remark 3. In Theorem 1, $Γ_{1}$ captures the total uncertainty energy introduced by the NN approximation error $ϵ_{\max}$ , unknown disturbances $d (t)$ , and weight estimation error ${\tilde{W}}_{F}$ . A large value of $Γ_{1}$ indicates strong disturbances or large approximation error, requiring more system energy to ensure stability. $Γ_{2}$ represents the net impact of model uncertainty, triggering sensitivity, and control actions on the flow dynamics of the system. A higher $Γ_{2}$ implies that approximation or attack-induced errors contribute more to destabilization, tightening the allowable bounds on state norm in the stability conditions. $Γ_{3}$ captures the cumulative impact of cyber-attacks $β (t)$ and controller gains on stability through their interaction with the system matrix $B_{δ}$ . Larger value of $Γ_{3}$ means stronger feedback or attack effect, which may reduce robustness to uncertainty. $Γ_{4}$ captures the cost of NN weight updates on the Lyapunov function jump, especially at event-triggering instants. A larger $Γ_{4}$ implies that more energy is going into the system due to parameter updates, potentially slowing convergence or increasing inter-event conservatism. $Γ_{5}$ represents the influence of weight norms and triggering error on the Lyapunov function change at triggering instants and a higher value of $Γ_{5}$ could make it more difficult to ensure an overall decrease in the Lyapunov function upon jumps, which may require more conservative triggering to compensate. $Γ_{6}$ represents the nonlinear contribution (second-order terms) of NN weight updates and triggering error to Lyapunov function evolution. A higher value of $Γ_{6}$ increases the nonlinearity and potential destabilization risk from large updates, thus requiring careful tuning of weight update matrices.

The plant model, NN weight update laws, and event-triggering conditions remain the same and are represented by equations (12), (17), (18), and (14), respectively.

Corollary 1. The nonlinear system (1) with desired and actual control input (43), plant model (12), the weight update laws (17) and (18) subject to Assumption 1, Assumptions 3–5 and the triggering condition (14) is asymptotically stable and the estimation errors of NN weights are bounded if there exists matrices $P > 0$ and $Q > 0$ of appropriate dimensions and satisfying the following conditions.

For flow interval

\begin{matrix} \begin{matrix} ∥ x ∥ & > {[\frac{Γ_{1}}{\frac{λ_{\min} (Q)}{2} - Γ_{2}}]}^{1 ∕ 2} and ∥ \hat{x} ∥ > {[\frac{Γ_{1}}{λ_{\min} (Q)}]}^{1 ∕ 2} \end{matrix} \end{matrix}

(44)

At triggering instants

∥ \hat{x} ∥ > \frac{Γ_{5} + \sqrt{4 Γ_{6} (λ_{\min} (P) - Γ_{4}) + Γ_{5}^{2}}}{2 (λ_{\min} (P) - Γ_{4})}

(45)

where

\begin{matrix} Γ_{1} & = \frac{2}{λ_{\min} (Q)} ∥ P ∥^{2} ∥ B_{δ} ∥^{2} (∥ \tilde{W} ∥_{F} κ_{\max} + ε_{\max} + d_{\max})^{2} \\ Γ_{2} & = 2 \sqrt{α} ∥ P ∥ ∥ B_{δ} ∥ ∥ K_{2} ∥ + 2 \sqrt{α} L_{κ} ∥ P ∥ ∥ B_{δ} ∥ ∥ Ŵ ∥_{F} ∥ \hat{V} ∥_{F} \\ + 2 L_{κ} ∥ P ∥ ∥ B_{δ} ∥ ∥ Ŵ ∥_{F} ∥ Ṽ ∥_{F} \\ Γ_{4} & = a_{2}^{2} ∥ e ∥^{2} ∥ Λ ∥^{2} (∥ M ∥ ∥ \hat{V} ∥_{F}^{2} + ∥ N ∥ ∥ Ŵ ∥_{F}^{2}) + ∥ P ∥ \\ Γ_{5} & = ∥ e ∥ ∥ Λ ∥ (2 c_{2} ∥ \hat{V} ∥_{F} + 4 a_{2} ∥ \hat{V} ∥_{F} ∥ Ŵ ∥_{F} + 2 c_{3} ∥ Ŵ ∥_{F}) \\ Γ_{6} & = 2 c_{1} ∥ e ∥ ∥ Λ ∥ + 2 a_{1} ∥ Ŵ ∥_{F} ∥ e ∥ ∥ Λ ∥ + a_{1}^{2} ∥ e ∥^{2} ∥ Λ ∥^{2} ∥ M ∥ \\ + ∥ P ∥ ∥ e ∥^{2} \\ a_{1} & = ∥ κ ({\hat{V}}^{T} \hat{x}) ∥_{F}; a_{2} = ∥ {\hat{κ}}^{'} ({\hat{V}}^{T} \hat{x}) ∥_{F} c_{1} = W_{M} a_{1}; \\ c_{2} = W_{M} a_{2}; c_{3} = V_{M} a_{2} \end{matrix}

Proof. By setting the attack probability $β (t) = 0$ and attack energy $h (x (t)) = 0$ in equations (21) and (22), respectively, one may derive the conditions (44) and (45).

Minimum inter-event time

In this section, a condition establishing a lower bound on the inter-event time is derived and presented in Theorem 2 to prevent Zeno phenomena in CTS.

Theorem 2. Under deception attack, the system (1) can be modeled as (27) and (33) with NN weight update law (17) and (18), control law (15), and event-triggering condition (14). The condition on MIET can be given by

τ_{k} = κ_{k + 1} - κ_{k} \geq \frac{1}{Υ_{1}} \ln (1 + \frac{\sqrt{α} Υ_{1}}{Υ_{2} + Υ_{3}})

(46)

where

\begin{matrix} Υ_{1} & = ∥ A_{δ} ∥ + L_{κ} ∥ B_{δ} Ŵ \hat{V} ∥_{F}; Υ_{2} = L_{κ} ∥ B_{δ} ŴṼx (t) ∥_{F} \\ Υ_{3} & = ∥ B_{δ} ∥ (∥ \tilde{W} ∥_{F} κ_{\max} + ϵ_{\max} + d_{\max}) \end{matrix}

Proof. Considering the error derivative

\frac{d}{dt} e (t) \leq ∥ ė (t) ∥ = ∥ ẋ (t) - \overset{\cdot}{\hat{x}} (t) ∥

(47)

On utilizing equations (27) and (28), equation (47) becomes

\begin{matrix} \begin{matrix} \frac{d}{dt} e (t) & \leq ∥ A_{δ} ∥ ∥ e (t) ∥ + L_{κ} ∥ B_{δ} Ŵ \hat{V} ∥_{F} ∥ e (t) ∥ + L_{κ} ∥ B_{δ} ŴṼx (t) ∥_{F} \\ + ∥ B_{δ} ∥ (∥ \tilde{W} ∥_{F} κ_{\max} + ϵ_{\max} + d_{\max}) \end{matrix} \end{matrix}

(48)

Reorganizing the terms in equation (48) results in

\frac{d}{dt} e (t) \leq Υ_{1} ∥ e (t) ∥ + Υ_{2} + Υ_{3}

(49)

On employing Comparison lemma (Khalil, 2000) considering the initial condition $e^{+} = 0$ and $t = σ_{k + 1}$ and utilizing (14), we get

\sqrt{α} \leq (e^{Υ_{1} τ_{k}} - 1) (\frac{Υ_{2} + Υ_{3}}{Υ_{1}})

(50)

On solving equation (50) for $τ_{k}$ , MIET condition (46) is obtained.

Simulation results

This section presents simulation examples that validate the effectiveness of the suggested MB-ETC control strategy.

Example 1: As a representation of a CTS, consider an inverted pendulum, whose system dynamics are as follows

\begin{matrix} \begin{matrix} δ x_{1} (t) & = x_{2} (t) \\ δ x_{2} (t) & = \frac{g}{l} \sin (x_{1} (t)) + \frac{1}{m l^{2}} u (t) + d (t) \end{matrix} \end{matrix}

(51)

From equation (2), since $T = 0$ for CTS, equation (51) can be rewritten as

\begin{matrix} \begin{matrix} ẋ_{1} (t) & = x_{2} (t) \\ ẋ_{2} (t) & = \frac{g}{l} \sin (x_{1} (t)) + \frac{1}{m l^{2}} u (t) + d (t) \end{matrix} \end{matrix}

(52)

assuming $l = 2$ m, $m = 1$ kg, $g = 10$ m/s² and number of NN nodes $= 10$ . The bound on attack energy is $H = diag {0.4, 0.5}$ with $\bar{β} = 0.45$ being the probability of attack occurrence. The event-triggering parameter $α = 0.1$ . Figure 3 illustrates the simulation results for the system under deception attack along with disturbance. It is evident from Figure 3(a) that system stability results as the estimated state and system state both converging to zero. The NN weights are adjusted in an aperiodic fashion, as can be observed from Figure 3(b) and (c). The number of state transmissions for the suggested MB-ETC approach (Theorem 1), T-S Fuzzy ETC, ZOH-ETC strategy (Bansal and Mukhija, 2020) (without model-based technique) and the feedback-linearized system (Khalil, 2000) with an ETC strategy is illustrated in Figure 4. Table 1 presents a comprehensive comparison of T-S Fuzzy ETC, ZOH-ETC, feedback-linearized ETC, MB-ETC (Theorem 1), and time-triggered control (with $α = 0$ ). It is evident that even in the face of attacks and disturbances, the MB-ETC method stabilizes the system with fewer state transmissions and better settling times ( $t_{s 1}$ and $t_{s 2}$ ) and peak value.

Figure 3.

System under deception attack (Example 1): (a) system state and estimated state, (b) NN weights $\hat{V}$ norm history, and (c) NN weights Ŵ norm history.

Figure 4.

Transmission instants for Example 1: (a) transmission instants for the proposed MB-ETC strategy (Theorem 1), (b) transmission instants using ZOH-ETC strategy, (c) transmission instants for the feedback-linearized system using the ETC strategy, and (d) transmission instants using T-S Fuzzy based ETC strategy.

Table 1.

Comparative analysis of different control strategies for $\bar{β} = 0.45$ and $α = 0.1$ (Example 1).

Strategy/Performance measure	Theorem 1	T-S Fuzzy ETC	ZOH-based ETC	Feedback linearized ETC	Time-triggered control
$t_{s 1}$	4.02 s	4.26 s	4.32 s	4.5 s	3.5 s
$t_{s 2}$	4.3 s	4.6 s	4.75 s	5.66 s	4.01 s
Peak	4.1	4.9	5.3	5.6	4.33
Trigger instants	77	95	79	113	continuous

Furthermore, Table 2 presents a comparison of the suggested control strategy with various control strategies for a system without attack in the presence of disturbances. $α = 0.25$ is the parameter that initiates the event. Compared to other control strategies, it is noted that the system is stabilized with less transmissions with the proposed MB-ETC technique. Also, a trade-off between the triggering threshold α, number of triggering instants, and control performance for sensitivity analysis of the system is provided in Table 3, and it can be observed that as the value of α increases, the number of transmissions and thereby, the number of times control action takes place decreases and system performance degrades, MIET increases and attack resilience also decreases.

Table 2.

Comparative analysis of various control strategies for $α = 0.25$ and $\bar{β} = 0$ (Example 1).

Strategy/Performance measure	Corollary 1	T-S Fuzzy ETC	ZOH-based ETC	Feedback linearized ETC	Time-triggered control
$t_{s 1}$	3.59 s	1.38 s	2.5 s	2.5 s	2.9 s
$t_{s 2}$	3.46 s	1.45 s	2.7 s	2.6 s	3.05 s
Peak	4.24	6.12	6.5	6.6	4.3
Trigger instants	74	84	76	77	continuous

Table 3.

Sensitivity Analysis for the proposed NN-based MB-ETC strategy.

Performance	$t_{s 1}$ (s)	$t_{s 2}$ (s)	Peak	Trigger Instants	$t_{s 1}$ (s)	$t_{s 2}$ (s)	Peak	Trigger Instants
α	For system under attack ( $\bar{β} = 0.25$ )				For system without attack ( $\bar{β} = 0$ )
0	2.45	2.3	4.1	700	3.31	3.23	2.05	700
0.1	2.6	2.3	4.1	73	3.59	3.46	4.24	74
0.3	3.9	3.7	4.18	70	4.02	3.52	4.11	63
0.5	4.2	3.9	4.18	53	4.30	3.68	4.12	46
0.7	4.59	4.13	4.18	45	4.41	3.96	4.12	38
0.9	4.9	4.29	4.18	39	4.62	4.25	4.12	34

Example 2: Considering the nonlinear mass spring DTS wherein the dynamics of the system are explained as follows

\begin{matrix} \begin{matrix} δ x_{1} (t) & = x_{2} (t) \\ δ x_{2} (t) & = - 0.01 x_{1} (t) - 0.67 x_{1}^{3} (t) + u (t) + d (t) \end{matrix} \end{matrix}

(53)

From equation (2) the DTS (53) can be rewritten as

\begin{matrix} \begin{matrix} x_{1} ((k + 1) T) & = x_{1} (kT) + T x_{2} (kT) \\ x_{2} ((k + 1) T) & = x_{2} (kT) - 0.01 T x_{1} (kT) - 0.67 T x_{1}^{3} (kT) \\ + Tu (kT) + Td (kT) \end{matrix} \end{matrix}

(54)

In the example above, the number of NN nodes is chosen to be 12. The attack’s probability of occurrence is estimated to be $\bar{β} = 0.2$ , and its energy bound is $H = diag {0.1, 0.3}$ , with a sampling time of $T = 0.1$ s and an event-triggering parameter of $α = 0.001$ . Figure 5 illustrates the estimated model and the real system state, the norm history of the NN weights $\hat{V}$ and Ŵ and the number of triggering instants when employing the suggested approach (Theorem 1), the ZOH-ETC method, and the feedback-linearized ETC approach. It is observed that as the estimated state and system state approach zero, the system becomes stable and the NN weights (17) and (18) are being updated aperiodically. Furthermore, the number of transmission instants in Figure 5(d) illustrates the aperiodic nature of the suggested control approach. Table 4 provides a comprehensive comparison between T-S Fuzzy ETC, ZOH-ETC (Bansal and Mukhija, 2020), feedback-linearized system, MB-ETC (Theorem 1), and time-triggered control. Table 4 suggests that the MB-ETC strategy stabilizes the system with fewer transmitting instants, improved settling times ( $t_{s 1}$ and $t_{s 2}$ ), and peak values even in the presence of deception attacks and disturbances.

Figure 5.

Example 2 under deception attack: (a) system state and estimated state, (b) NN weight $\hat{V}$ norm history, (c) NN weight Ŵ norm history, (d) instants for the proposed MB-ETC strategy, (e) event-triggering instants using ZOH-ETC strategy, and (f) event-triggering instants for the feedback-linearized system using the ETC strategy.

Table 4.

Comparative analysis of various control strategies for Example 2 with $α = 0.001$ and $\bar{β} = 0.2$ .

Strategy/Performance measure	Theorem 1	T-S Fuzzy ETC	ZOH-based ETC	Feedback linearized ETC	Time-triggered control
$t_{s 1}$	148.79 s	165.8 s	188.90 s	90.05 s	172.02 s
$t_{s 2}$	84.68 s	102.5 s	138.19 s	16.16 s	4.49 s
Peak	1.3	1.33	1.34	1.12	1.3
Trigger instants	151	197	267	142	2000

In addition, Example 2 is also examined without an attack ( $β (t) = 0$ ), with $α = 0.01$ as the event-triggering parameter and $T = 0.1$ s as the sampling duration. Figure 6 illustrates the state of the real system and the estimated model, the norm history of the NN weights and the state transmission over the network for Example 2 (Corollary 1). Furthermore, Table 5 presents a comparative analysis between the proposed technique (Corollary 1), T-S Fuzzy ETC, ZOH-based ETC (with $T = 0.001$ s), feedback-linearized system with ETC, and time-triggered control for system (Example 2) without deception attack in the presence of disturbances. Even in the presence of disturbances, it is observed that the system becomes stable with fewer transmissions than ZOH-ETC and time-triggered control and the NN weights are only updated in the case that the event-triggering condition is violated. Tables 4 and 5 show that, in contrast to the proposed technique, there are less state transmissions utilizing the feedback-linearized strategy. This is due to the fact that, in the case of a feedback-linearized system, the nonlinearity is regarded as known, which represents a significant drawback of this method. Nevertheless, the method suggested in this work considers unknown nonlinear function and uses NN to cope with this unknown nonlinearity.

Figure 6.

Example 2 without attack: (a) System state and estimated state, (b) NN weight norm history $\hat{V}$ , (c) NN weight Ŵ norm history for system without attack and (d) Instants for the MB-ETC strategy (Corollary 1).

Table 5.

Comparative analysis of different control strategies for Example 2 with $α = 0.01$ and $\bar{β} = 0$ .

Strategy/Performance measure	Corollary 1	T-S Fuzzy ETC	ZOH-based ETC	Feedback linearized ETC	Time-triggered control
$t_{s 1}$	56.06 s	57.3 s	59.19 s	55.86 s	64.86 s
$t_{s 2}$	49.27 s	45.6 s	38.22 s	39.8 s	8.06 s
Peak	1.15	1.23	1.19	1.6	1.19
Trigger instants	121	178	221	89	2000

Conclusion

An MB-ETC technique is presented in this research to control nonlinear delta operator networked systems under stochastic deception attacks and in the presence of disturbances. This technique uses aperiodic state transmission by employing ETC at the sensor end. Model-based methods are utilized for system approximation in between trigger instants in order to further minimize the number of transmissions. NNs are used to approximate the system’s unknown nonlinearity. NN weight update rules are computed and a stability condition is derived. Furthermore, a condition on MIET is devised to prevent the occurrence of Zeno phenomenon. The proposed control approach renders the system stable while lowering the transmission rate and maximizing the use of energy, computing power, and communication bandwidth. The proposed MB-ETC framework unifying CTS and DTS using delta δ operator can be extended to multi-agent systems, where each agent may share information aperiodically over a network while being resilient to cyber-attacks leading to an important extention of the proposed algorithm as a future direction.

Footnotes

Declaration of conflicting interests

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

Funding

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

ORCID iD

Kritika Bansal

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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