Security Control for a Fuzzy System under Dynamic Protocols and Cyber-Attacks with Engineering Applications

Abstract

The objective of this study is to design a security control for ensuring the stability of systems, maintaining their state within bounded limits and securing operations. Thus, we enhance the reliability and resilience in control systems for critical infrastructure such as manufacturing, network bandwidth constraints, power grids, and transportation amid increasing cyber-threats. These systems operate as singularly perturbed structures with variables changing at different time scales, leading to complexities such as stiffness and parasitic parameters. To manage these complexities, we integrate type-2 fuzzy logic with Markov jumps in dynamic event-triggered protocols. These protocols handle communications, optimizing network resources and improving security by adjusting triggering thresholds in real-time based on system operational states. Incorporating fractional calculus into control algorithms enhances the modeling of memory properties in physical systems. Numerical studies validate the effectiveness of our proposal, demonstrating a 20% reduction in network load and enhanced stochastic stability under varying conditions and cyber-threats. This innovative proposal enables real-time adaptation to changing conditions and robust handling of uncertainties, setting it apart from traditional control strategies by offering a higher level of reliability and resilience. Our methodology shows potential for broader application in improving critical infrastructure systems.

1. Introduction

In today technologically driven world, reliability and resilience of control systems are crucial for operating essential infrastructures, including electrical manufacturing, power grids, and urban transportation networks. These systems comprise automated and semi-automated devices, processes, and software that ensure the continuity and safety of operations within our society. However, the systems face challenges and threats from cyber-attacks, especially network-based deception attacks that undermine the integrity of data and system communications, as well as the constraints imposed by network bandwidth limitations. Such systems also face complexities such as stiffness and influence of parasitic parameters. Although such parameters are minor, they can affect the system behavior, leading to higher-order dynamics and system stiffness, characterized by system sensitivity to changes in its inputs. Current methods, such as traditional proportional-integral-derivative controllers and linear quadratic regulators, often fall short in addressing such complexities. These methods typically assume linearity and fail to adequately handle the high nonlinearity and variability in system parameters found in real-world scenarios.

Existing approaches may not provide robust performance under severe cyber-attacks and network constraints, highlighting the need for more advanced control strategies [1]. To address the mentioned challenges and complexities, singularly perturbed systems (SPSs) were introduced, separating fast-changing dynamics (like sudden voltage spikes in power systems) from slow-changing ones (such as gradual power load increase), simplifying their analysis and control. The effectiveness of SPSs in dealing with such dual-changing dynamics is well documented in critical sectors [2,3], proving their value in enhancing system reliability and resilience. Recent advancements in the control of Markov jump (MJ) systems have introduced various techniques, including the quantized output feedback control with dynamic cost triggering [4] and fusion-based state estimation for networked autonomous surface vehicles with measurement outliers and cyber-attacks [5]. These techniques demonstrate important improvements in handling system uncertainties and cyber-threats, but their integration with SPSs remains an open research area.

The nonlinearity of the mentioned systems presents another layer of complexity. Such nonlinearity is prevalent in the mentioned systems, making the design of control frameworks complex. This complexity has led to the development of controller design frameworks addressing such a nonlinear behavior [6,7]. Specifically, the type-1 Takagi-Sugeno (TS) fuzzy model has proved to be effective in managing nonlinearity, demonstrating capabilities as highlighted in several studies [8,9,10]. This model excels in environments where system parameters are well defined and stable.

The type-1 TS fuzzy model often falls short in environments of high uncertainty when dealing with varying system parameters [11]. In contrast, the interval type-2 (IT2) fuzzy model extends the traditional fuzzy approach by incorporating uncertainty into its membership functions through the footprint of uncertainty. This allows the IT2 model to handle the uncertainty of nonlinear systems effectively. Such handling is beneficial for SPSs, where the rapid and slow system dynamics mean that even minor parametric uncertainties can impact their behavior [12]. The application of IT2 fuzzy logic in such dynamics provides a framework for enhancing control precision, adapting it to fluctuating conditions found in network environments or cyber-physical threats. Recent work on fuzzy differential systems has shown the effectiveness of this application in complex contexts [13,14,15]. Moreover, the integration of state estimation based on data fusion and robust control can importantly enhance the reliability and resilience of such systems. For example, control schemes for nonlinear systems with MJ parameters have shown promising results in improving fault tolerance and maintaining stability under cyber-attack scenarios [16].

Despite the advantages of the IT2 fuzzy model in handling uncertainties within SPSs, its application in these systems has been limited. The potential benefits, however, are considerable and warrant further exploration and integration [17,18,19]. Another crucial concept in complex control is the MJ system. In such a system, its structure and parameters can undergo changes due to a variety of reasons, such as sudden environmental shifts, sensor or actuator malfunctions, or even temporary communication failures. SPSs are challenging because their behavior may shift unexpectedly, requiring control strategies that can adapt to or anticipate these shifts.

Recent research has focused on MJ systems, yielding studies that have explored aspects such as fault tolerance, multi-agent systems, and networked control [9,20,21,22]. An assumption in several of these studies is that the Markov chains governing the system transitions have stationary transition probabilities (TPs). This means that the probabilities of switching from one state to another are considered constant over time. However, in practical applications, especially in networked control systems, such assumption does not hold true. Nonstationary TPs, which fluctuate due to factors like network delays or packet losses, are more realistic but also more challenging to model and predict [23].

Although studies addressing nonstationary TPs in applications exist [24,25], their integration into SPSs is still a gap. This gap motivates the need for developing strategies that handle nonstationary TPs within the framework of SPSs. Moving beyond the technical challenges, control systems need to enhance communication and network security.

The aforementioned need highlights the complexity of the current operational environment, motivating the exploration of event-triggered protocols (ETPs). The integration of communication into feedback control loops offers cost savings [26] and optimizes the use of network resources to reduce network load. This is crucial in interconnected systems, where the utilization of network resources is linked to their performance.

ETPs represent an advancement over traditional time-triggering techniques by initiating data transmission under specific predefined conditions. ETPs minimize unnecessary data transmission, easing network congestion and improving efficiency. Recent studies have refined ETPs to manage bandwidth constraints through static [27,28] and adaptive [29] ETPs, which represent advancements in such protocols by relying on fixed and dynamic triggering conditions. These protocols have boosted the responsiveness and efficiency of networked control systems, optimizing data transmission and conserving bandwidth. Despite these advances, there is room for further enhancement in the efficiency of ETPs.

Dynamic ETPs (DETPs) go beyond static and adaptive ETPs, adjusting triggering thresholds according to the system current state and performance needs [30,31]. This adjustment allows for finer management of data transmission and reduces network traffic. Nevertheless, the literature has not fully explored the customization of these triggering thresholds for components of the measurement signal. Addressing such thresholds could boost network efficiency and control responsiveness. Thus, this could improve the investigation in the field [18] when the threat of cyber-attacks on communication is challenging. Attackers exploit vulnerabilities through tactics like deception attacks, denial-of-service and replay attacks, incapacitating network nodes or data integrity, obstructing operations and reducing network capacity [32,33]. Such attacks need countermeasures to safeguard system security and reliability [31,34,35].

Enhancing network efficiency through DETPs and defense mechanisms highlights the complexity in which control systems operate. Developing systems that achieve resilience, reliability, and security is vital in our interconnected world, where balancing operational performance and security is essential. A gap in the literature on Markov fuzzy SPSs is in network-based deception attacks. One can address this gap by introducing robust control that integrates IT2 fuzzy logic with MJ systems [36]. To advance dynamic event management, one may implement a component-based dynamic event-triggered mechanism (CB-DETM). This mechanism customizes the DETPs for system components, enhancing the response to dynamic environments and operational demands. The CB-DETM improves data flow and security, as well as the reliability and resilience of infrastructure systems facing network constraints and cyber-threats. In the present investigation, we bridge all the mentioned gaps.

Therefore, our main objective is to design an asynchronous security control to ensure the stability of closed-loop systems, maintaining the system state within bounded limits, and securing operations despite challenges such as MJs, actuator attacks, and nonlinear disturbances. This control mechanism enhances network efficiency and system security, addressing the existing research gaps. Rigorous numerical simulations demonstrate the effectiveness of our proposal in ensuring stochastic stability and optimizing network performance under various conditions and cyber-threats [37,38].

The innovation of our methodology lies in the comprehensive integration of IT2 fuzzy logic with MJ systems, coupled with a DETP. This integration enables real-time adaptation to changing conditions and robust handling of uncertainties and cyber-physical threats, distinguishing our methodology from traditional control strategies. Unlike standard techniques, our proposal dynamically adjusts control parameters and communication strategies based on real-time system states and operational conditions, offering a higher level of reliability and resilience. Specifically, our contributions are as follows:

• Asynchronous control for MJ-IT2 fuzzy SPSs—We explore asynchrony for IT2 fuzzy MJ SPSs vulnerable to deception attacks and offer robustness to traditional controllers by addressing the nonlinearity and uncertainties of these systems.
• Unified framework for random attacks—We integrate actuator and random attacks through the use of two independent Bernoulli-distributed random variables, facilitating a security control capable of mitigating the unpredictability of cyber-threats.
• CB-DETM—We implement CB-DETM, which tailors DETPs for system components, distinct from existing methods in the literature [28,39], regulating data transmission and optimizing communication resources by enhancing resilience against cyber-threats.
• Asynchronous controller design with nonstationary mode transitions: We formulate an asynchronous controller for MJ SPSs, which is capable of handling nonstationary-mode transitions, addressing random deception attacks and setting new security benchmarks for networked control systems.

These contributions strive to advance the state of the art in secure control strategies for SPSs, with a particular focus on MJ systems and IT2 fuzzy logic by addressing the pressing challenges of cyber-security and network efficiency.

The rest of this article is organized as follows. Section 2 explores the fundamentals of SPSs and their integration with fuzzy control models. In Section 3, we describe the system framework and preliminary concepts necessary for understanding our proposal. In Section 4, the main theoretical results are presented, providing the mathematical foundation for the proposed methodology. Section 5 demonstrates the practical application of our proposal through numerical illustrations, highlighting its effectiveness and advantages over existing approaches. In Section 6, we conclude by discussing the findings and suggesting directions for future research.

2. Fundamentals of Singularly Perturbed Systems and Fuzzy Control Integration

This section provides an overview of SPSs, their mathematical characterization, and the integration of fuzzy models for improved control under uncertainty.

2.1. Mathematical Representation of SPSs

An SPS is described by differential equations as

$$\epsilon {\dot{x}}_1\left(k\right)=f(x_1\left(k\right),x_2\left(k\right),t,\epsilon ),{\dot{x}}_2\left(t\right)=g(x_1\left(k\right),x_2\left(k\right),k,\epsilon ),k\ge 0,$$

(1)

where $0<\epsilon \ll 1$ is a parameter that signifies the disparity in time scales between fast ($x_1$) and slow ($x_2$) dynamics. The functions f and g represent the fast and slow subsystem dynamics, respectively. The parameter $\epsilon$ introduces challenges in stability and numerical methods, requiring specialized techniques for analysis and control [40].

2.2. Challenges in SPS Analysis and Control

Key challenges in SPSs include:

• A parasitic parameter ($\epsilon$) that takes small values—Note that $\epsilon$ induces instability and sensitivity to perturbations, necessitating numerical methods for accurate control [41].
• Higher-order dynamics—The interaction between fast and slow dynamics creates complex behaviors, requiring advanced mathematical techniques for effective analysis [42].
• System stiffness—High sensitivity to input variations impacts stability and performance, demanding robust and adaptive control strategies [43].

2.3. Type-1 and Type-2 Fuzzy Models

Fuzzy logic systems manage uncertainties using linguistic variables, with two of their models being as follows:

• Type-1 fuzzy models: They use precise membership functions within a range from zero to one. A type-1 fuzzy set for an element u is defined as $\mathbb{A}\equiv \left\{(u,{\mu }_{\mathbb{A}}\left(u\right))\right|u\in \mathbb{U}\}$, where ${\mu }_{\mathbb{A}}\left(u\right)$ denotes the membership function assigning the degree of membership of u in the fuzzy set $\mathbb{A}$, and $\mathbb{U}$ represents the universal set [44].
• Type-2 fuzzy models: They incorporate uncertainty into membership functions, making them effective for ambiguous data. A type-2 fuzzy set is presented as

  $$\tilde{\mathbb{A}}\equiv \left\{((u,\alpha ),{\mu }_{\tilde{\mathbb{A}}}(u,\alpha ))\right|u\in \mathbb{U},\alpha \in J_u\subseteq [0,1]\},$$

  where ${\mu }_{\tilde{\mathbb{A}}}(u,\alpha )$ is the membership function for u at a specific uncertainty level $\alpha$, and $J_u$ is the interval of membership grades available for u [45].

Transitioning from type-1 to type-2 fuzzy models adds an uncertainty layer to the membership functions, enhancing system stability and responsiveness. Mathematically, this is modeled by extending the type-1 membership function ${\mu }_{\mathbb{A}}\left(u\right)$ to ${\mu }_{\tilde{\mathbb{A}}}(u,\alpha )$ in type-2 sets, where $\alpha$ introduces a degree of freedom that captures additional uncertainty. Such transition allows the control system to better handle imprecision and variability, leading to improved performance in dynamic and unpredictable environments [46].

Choosing between type-1 and type-2 fuzzy models depends on the level of uncertainty and required control robustness. While type-1 models are simpler and computationally less intensive, type-2 models offer superior performance in highly uncertain environments by providing an additional degree of freedom to model uncertainties. Integrating fuzzy logic into control systems represents an important advancement in managing complex, uncertain, and nonlinear systems, providing reliability and resilience control strategies [47].

2.4. Discussion on Control Characteristics

Our proposed event-triggered control setup ensures asymptotic convergence. Unlike finite-time convergence setups such as those discussed in [48], which achieve convergence in a fixed time, our proposal focuses on ensuring stability and performance over an extended period, even under persistent cyber-attacks and disturbances. While finite-time convergence offers rapid stabilization, our asymptotic approach ensures robust long-term stability and adaptability to changing conditions, which is crucial in complex and dynamic environments. This highlights the reliability and resilience of our methodology in relation to finite-time convergence strategies.

The Zeno phenomenon, where an infinite number of events are triggered in a finite amount of time, is a common issue in event-triggered setups. Our design avoids this issue by incorporating a minimum inter-event time, ensuring that events are not triggered too frequently. Specifically, for a discrete-time process, the minimum time-interval between two adjacent trigger instants is not less than the sampling period, effectively preventing the Zeno phenomenon and ensuring practical implementation feasibility [49].

The computational complexity of our control methodology is of polynomial order. This is important for large-scale setups, where computational efficiency is crucial. By ensuring that the complexity remains manageable, our approach is suitable for real-time applications in large and complex systems [50].

3. System Description and Preliminaries

Building on the theoretical foundations, this section focuses on an MJ TS fuzzy model designed to manage the complexities of systems with random structural changes. Such model leverages fuzzy logic for uncertainty management and MJ processes for dynamic adaptability. Figure 1 provides a schematic overview of the organization of this section, illustrating the relationships and flow between the various subsections discussed.

3.1. Overview of Markov Jump Systems

An MJ system is a type of stochastic model whose dynamics switch between different subsystems randomly. These switches are governed by a Markov process, which is characterized mathematically by a sequence of random variables $\mathbb{X}\equiv \left\{X\right(k\left)\right|k\ge 0\}$, where each variable $X\left(k\right)$ represents the state of the system at a discrete time k.

The Markov property asserts that the future state of the process depends only on the current state and not on the past. Mathematically, this property can be expressed as

$$Pr(X\left(k+1\right)=x|X\left(k\right)=x\left(k\right),X\left(k-1\right)=x\left(k-1\right),\dots ,X\left(0\right)=x\left(0\right))=Pr(X\left(k+1\right)=x|X\left(k\right)=x\left(k\right)),$$

for any state x and sequence of states $x\left(k\right),x\left(k-1\right),\dots ,x\left(0\right)$. The system dynamics are described by possible states and TPs between them. If we denote the set of all states by $\mathcal{S}$, and the system state at time k by $\rho \left(k\right)$, for convenience, which takes values in $\mathcal{S}$, the transitions between states are governed by a TP matrix $P=\left[P_{ij}\right]$, where $P_{ij}=Pr(\rho \left(k+1\right)=j|\rho \left(k\right)=i)$. This matrix is stochastic, meaning that each row sums to one, that is, ${\sum }_{j\in \mathcal{S}}{P}_{ij}=1$, for all $i\in \mathcal{S}$. The matrix P defines the behavior of the Markov process over time, dictating how the system transitions from one state to another based on the current state.

In practical terms, an MJ system hops between different operational modes, with each mode corresponding to a different subsystem. The mode of operation at any given time is determined by the state of the underlying Markov model (chain) $\left\{\rho \right(k\left)\right|k\ge \}$. This modeling framework is powerful in scenarios where the system behavior is subject to sudden changes due to external or internal shocks, component failures, or regime shifts.

3.2. Takagi-Sugeno Fuzzy Model

The TS fuzzy model is a sophisticated formulation to describe complex nonlinear systems using a series of linear submodels. Each submodel, associated with a fuzzy rule, is designed to operate effectively within a specific region of the input space. The mathematical formulation of TS fuzzy models is as follows. The essence of the TS model lies in its ability to blend linear models through fuzzy logic principles.

Mathematically, the TS model can be described as

$$\begin{array}{cc}\hfill \mathrm{Rule}i:& \mathrm{If}{x}_1\mathrm{is}{\mathbb{A}}_{1i}\dots \mathrm{and}{x}_n\mathrm{is}{\mathbb{A}}_{ni},\mathrm{then}y={\nu }_{i0}+{\nu }_{i1}{x}_1+\dots +{\nu }_{in}{x}_n,\hfill \end{array}$$

(2)

where $x_1,\dots ,x_n$ are the input variables; ${\mathbb{A}}_{ji}$ the fuzzy sets related to rule i for input variable j; and ${\nu }_{i0},{\nu }_{i1},\dots ,{\nu }_{in}$ the coefficients defining the linear model within the rule region. The output y of the TS model is a weighted average of each rule output, where the weights are the degrees of fulfillment of each rule conditions, stated as

$$y=\frac{{\sum }_{i=1}^r{w}_i({\nu }_{i0}+{\nu }_{i1}{x}_1+\cdots +{\nu }_{in}{x}_n)}{{\sum }_{i=1}^r{w}_i},$$

with $w_i$ being the truth degree of rule i given by

$$w_i=\prod _{j=1}^n{\mu }_{{\mathbb{A}}_{ji}}\left(x_j\right),$$

where ${\mu }_{{\mathbb{A}}_{ji}}\left(x_j\right)$ is the membership function of the fuzzy set ${\mathbb{A}}_{ji}$ evaluated at $x_j$.

3.3. Integration of Markov Jumps into TS Fuzzy Models

The integration of Markov dynamics into TS fuzzy models introduces stochastic variability in the selection of which fuzzy rule (or submodel) governs the system at any time step. This is described by incorporating the state or mode of the system, $\rho \left(k\right)$ say, determined by a Markov process. The mathematical formulation with MJs is as follows. In the MJ TS model, the output at time step k is expressed as

$$\begin{array}{c}\hfill y\left(k\right)=\sum _{i=1}^r{w}_i\left(\rho \left(k\right)\right)({\nu }_{i0}\left(\rho \left(k\right)\right)+{\nu }_{i1}\left(\rho \left(k\right)\right)x_1\left(k\right)+\cdots +{\nu }_{in}\left(\rho \left(k\right)\right)x_n\left(k\right)),\end{array}$$

(3)

where $\rho \left(k\right)$ is the current state of the Markov chain governing the system mode, with $w_i\left(\rho \left(k\right)\right)$ and ${\nu }_{ij}\left(\rho \left(k\right)\right)$ being the rule weights and model coefficients conditioned on the current state $\rho \left(k\right)$, respectively. The model configuration stated in (3) allows the system to adapt to shifts in dynamics caused by external events or internal state changes, enhancing its reliability and resilience to handle complex varying environments.

3.4. Model Specification

Building on the integrated framework of Markov dynamics and TS fuzzy logic, we consider the following MJ TS fuzzy model at each discrete time step k. This model adapts the general TS fuzzy structure to accommodate stochastic transitions between different sets of linear equations based on the current state of the Markov process $\left\{\rho \right(k\left)\right|k\ge 0\}$. Let us detail the MJ TS fuzzy model that is specified at each discrete time step k as

$$\mathrm{Rule}i:\mathrm{If}{\psi }_1\left(x\left(k\right)\right)\mathrm{is}{\mathbb{N}}_i^1\dots \mathrm{and}{\psi }_s\left(x\left(k\right)\right)\mathrm{is}{\mathbb{N}}_i^s,\mathrm{then}x(k+1)=A_i^{\epsilon }\left(\rho \left(k\right)\right)x\left(k\right)+B_{2i}\left(\rho \left(k\right)\right)u\left(k\right),$$

(4)

where ${\mathbb{N}}_i^j$ is an IT2 fuzzy set for premise j in rule i, providing enhanced capability to handle uncertainty; ${\psi }_j\left(x\left(k\right)\right)$ is the premise variable j, which is a function mapping the state vector $x\left(k\right)$ to its corresponding fuzzy set; $A_i^{\epsilon }\left(\rho \left(k\right)\right)$ and $B_{2i}\left(\rho \left(k\right)\right)$ are dynamic matrices that allow the TS fuzzy model to adapt to changes in the system state as dictated by the Markov process; $\epsilon$ is defined in (1); s is the total number of premises per rule; and i indexes the rule within the set $\mathcal{S}\equiv \{1,\dots ,r\}$. The formulation presented in (4) is the description of the TS fuzzy model stated in (2), where each rule output is a linear combination of input variables weighted by the degree of membership in the fuzzy sets. However, unlike the static framework given in (2), here the rules are adapted at each time step k, influenced by the current state $\rho \left(k\right)$ of the Markov chain, as described in (3). This integration of Markov dynamics introduces variability in the selection of which fuzzy rule governs the system at any time step, enhancing the model ability to handle complex varying environments.

The state vector is $x\left(k\right)={\left[x_1^{\top }\left(k\right)x_2^{\top }\left(k\right)\right]}^{\top }\in {\mathbb{R}}^n$, with $x_1\left(k\right)\in {\mathbb{R}}^{n_1},x_2\left(k\right)\in {\mathbb{R}}^{n_2}$ being the slow, fast state vectors, and $u\left(k\right)$ is the input signal affecting the system dynamics.

The modes of the system are determined by the Markov process $\left\{\rho \right(k\left)\right|k\ge 0\}$, with a transition matrix ${\Pi }_1=\left[{\pi }_{pq}\right]$ defined over the finite set ${\mathcal{S}}_1\equiv \{1,\dots ,N\}$, with elements given by

$${\pi }_{pq}=Pr(\rho \left(k+1\right)=q\mid \rho \left(k\right)=p),$$

where $0\le {\pi }_{pq}\le 1$ and ${\sum }_{q=1}^N{\pi }_{pq}=1$, for all $p,q$ in ${\mathcal{S}}_1$.

As mentioned, the dynamic matrices $A_i^{\epsilon }\left(\rho \left(k\right)\right)$ and $B_{2i}\left(\rho \left(k\right)\right)$ are components that allow the TS fuzzy model to adapt to changes in the system state as dictated by the Markov process, formulated as

$$\begin{array}{cc}\hfill A_i^{\epsilon }\left(\rho \left(k\right)\right)& =A_i\left(\rho \left(k\right)\right)E^{\epsilon },\hfill \end{array}$$

(5)

where $\epsilon$ is defined in (1), $E^{\epsilon }=\mathrm{diag}\{I_{n_1},\epsilon I_{n_2}\}$, with $I_n$ being the $n\times n$ identity matrix, and $A_i\left(\rho \left(k\right)\right)$ provides a model for the linear dynamics associated with rule i under mode $\rho \left(k\right)$, specified by

$$A_i\left(\rho \left(k\right)\right)=\left[\begin{array}{cc}{A}_{11i}\left(\rho \left(k\right)\right)& A_{12i}\left(\rho \left(k\right)\right)\\ A_{21i}\left(\rho \left(k\right)\right)& A_{22i}\left(\rho \left(k\right)\right)\end{array}\right].$$

The bifurcated control input matrix is structured as

$$\begin{array}{cc}\hfill B_{2i}\left(\rho \left(k\right)\right)& =\left[\begin{array}{c}{B}_{21i}\left(\rho \left(k\right)\right)\\ B_{22i}\left(\rho \left(k\right)\right)\end{array}\right].\hfill \end{array}$$

(6)

The singular perturbation parameter $\epsilon$ introduced in (5), and described in the mathematical representation of SPSs stated in (1), discriminates between fast and slow state dynamics through the matrix $E^{\epsilon }$. This discrimination is crucial for systems characterized by multi-time-scale dynamics, providing a mechanism to address the challenges posed by the inherent speed discrepancies in system responses. The structuring of $A_i^{\epsilon }\left(\rho \left(k\right)\right)$ and $B_{2i}\left(\rho \left(k\right)\right)$ represents the system dynamics, enhancing the control strategy effectiveness across various operational scenarios. Key benefits of this structuring include:

• Adaptability—The model dynamically responds to transitions in operational mode $\rho \left(k\right)$, adapting to the MJs that dictate these transitions, whose adaptability ensures that the control strategy remains effective under varying system conditions.
• Precision—The distinction between fast and slow dynamics allows for precise control, which enhances the model ability to optimize systems with multi-time-scale dynamics, improving response times and system behavior.
• Integration of uncertainties—Incorporating IT2 fuzzy sets improves the model capability to integrate uncertainties, whose integration is valuable in environments where input data and system parameters are subject to variability and imprecision.

This structuring of dynamic matrices expands on the framework described in (3), where the model output at time step k is a function of the current state $\rho \left(k\right)$ of the Markov chain. Here, $w_i\left(\rho \left(k\right)\right)$ and ${\nu }_{ij}\left(\rho \left(k\right)\right)$ are the rule weights and model coefficients also conditioned on the state $\rho \left(k\right)$. The adaptation of $A_i^{\epsilon }\left(\rho \left(k\right)\right)$ and $B_{2i}\left(\rho \left(k\right)\right)$ ensures that the model response to input signals and state transitions is oriented using the current mode of the system, enhancing the model ability to handle complex varying environments effectively.

3.5. Fuzzy Inference and Firing Strength in the Global Fuzzy Model

The firing strength ${\mathfrak{M}}_i$ is a critical element in fuzzy control systems, influencing each rule on the system output and defined as ${\mathfrak{M}}_i=[{\underline{\vartheta }}_i\left(x\left(k\right)\right),{\overline{\vartheta }}_i\left(x\left(k\right)\right)]$, where ${\underline{\vartheta }}_i\left(x\left(k\right)\right)$ and ${\overline{\vartheta }}_i\left(x\left(k\right)\right)$ are the lower and upper membership functions, respectively, of the fuzzy set ${\mathbb{N}}_i^j$ evaluated at ${\psi }_j\left(x\left(k\right)\right)$. These functions are fundamental to the fuzzy inference process, directly applying the principles previously discussed in (2) and reflecting dynamic adjustments based on the system current inputs and state as explored in (3).

To simplify the understanding, the fuzzy inference process can be broken down into the following steps:

Step 1.

Evaluate the membership functions for each input variable.

Step 2.

Calculate the firing strength for each rule by multiplying the membership values.

Step 3.

Normalize the firing strengths to ensure they sum to one.

Step 4.

Compute the output by weighting each rule output by its normalized firing strength.

This step-by-step breakdown clarifies how the system integrates contributions from multiple fuzzy rules to determine the next state. The overall dynamics of the model, which integrates the contributions of all activated rules to predict the system next state, are configured by

$$\begin{array}{cc}\hfill x(k+1)& =\sum _{i=1}^r{\vartheta }_i\left(x\left(k\right)\right)\left(A_i^{\epsilon }\left(\rho \left(k\right)\right)x\left(k\right)+B_{2i}\left(\rho \left(k\right)\right)u\left(k\right)\right),\hfill \end{array}$$

(7)

where $u\left(k\right)$ is the input signal, as mentioned, and ${\vartheta }_i\left(x\left(k\right)\right)$ the normalized membership function derived from the firing strengths, ensuring that the combined influence of all rules sums to unity, as dictated by fuzzy logic norms. The rule weights are nonlinear functions that adjust each firing strength based on uncertain parameters within the range $[0,1]$. The configuration stated in (7) allows the system to adapt to transitions in operational mode $\rho \left(k\right)$, accommodating the MJs that dictate these transitions, enhancing robustness, as discussed in the context of Markov integration given in (3). The approach of dynamic matrices $A_i^{\epsilon }\left(\rho \left(k\right)\right)$ and $B_{2i}\left(\rho \left(k\right)\right)$, as defined in (5) and (6), enables a representation of the system dynamics. This representation manages multi-time-scale dynamics and integrates uncertainties, aligning with the model requirements to handle complex environments.

3.6. Component-Based Dynamic Event-Triggered Mechanism

As discussed in the introduction, the CB-DETM is used to enhance the efficiency of communication networks within control systems by minimizing unnecessary transmissions. This mechanism customizes the event-triggering thresholds for each component of the state signals, reducing data flow and improving system performance.

The integration of DETM and robust fuzzy control within this framework enhances efficiency and stability. DETM adjusts data transmission rates based on the system state, minimizing unnecessary communication. Robust fuzzy control handle uncertainties and nonlinearities, maintaining performance under adverse conditions. This handling ensures the system remains resilient and responsive to internal fluctuations and external threats.

The CB-DETM regulates the transmission of the system output by utilizing a triggering function for each output component expressed for $x_l\left(k\right)$, with $l\in \{1,\dots ,n\}$, as part of the state vector $x\left(k\right)={[x_1\left(k\right),\dots ,x_n\left(k\right)]}^{\top }$, defined as

$$\begin{array}{c}\hfill x_l\left(k\right)=e_l^{\top }\left(k\right)e_l\left(k\right)-{\sigma }_l{x}_l^{\top }\left(k\right)x_l\left(k\right)-\frac{1}{{\delta }_l}{\eta }_l\left(k\right),\end{array}$$

(8)

where $e_l\left(k\right)$ represents the error between the current value $x_l\left(k\right)$ and its value at the last triggered instance $x_l\left(k_s^l\right)$; the parameters ${\sigma }_l$ and ${\delta }_l$ are positive scalars that fine-tune the sensitivity of the triggering mechanism; and the term ${\eta }_l\left(k\right)$, referred to as the auxiliary internal dynamical variable (AIDV), is updated according to

$$\begin{array}{c}\hfill {\eta }_l(k+1)={\lambda }_l{\eta }_l\left(k\right)-e_l^{\top }\left(k\right)e_l\left(k\right)+{\sigma }_l{x}_l^{\top }\left(k\right)x_l\left(k\right),\end{array}$$

(9)

with an initial condition ${\eta }_l\left(0\right)>0$ and a decay factor ${\lambda }_l$ in (0, 1), ensuring that ${\lambda }_l{\delta }_l>1$.

The selection of ${\lambda }_l$ in our CB-DETM is crucial for dynamically adjusting the event-triggering thresholds. This selection is comparable to the dynamic scheme discussed in [51], where $\lambda$ plays a pivotal role in managing fault tolerance and system responsiveness. Our selection of ${\lambda }_l$ offers a similar adaptability, ensuring efficient data transmission and robust control under varying conditions.

The next triggering instant for each component is based on the rule given by

$$\begin{array}{c}\hfill k_{s+1}^l=\underset{k>k_s^l}{min}\{k\ge 0|x_l\left(k\right)>0\}.\end{array}$$

(10)

This rule ensures the output signals are updated when needed, according to the activation specified in (10). Additionally, the release sequence for all components is consolidated into a common sequence $\{k_s,s\ge 0\}={\cup }_{l=1}^n\left\{k_s^l\right\}$, facilitating a synchronized and efficient data transmission strategy. For the duration $k\in [k_s,k_{s+1})$, the controller receives the signal $x\left(k\right)=x\left(k_s\right)=[x_1\left(k_s^1\right),\dots ,x_n\left(k_s^n\right)]$, where each component $x_l\left(k\right)$ that has not been released maintains its value from the most recent release, using a zero-order holder.

Remark 1.

In the CB-DETM described in (8) and (9), the parameter ${\lambda }_l$ plays a crucial role in managing the dynamics of the AIDV ${\eta }_l$. As noted in [18], the condition ${\eta }_l\left(k\right)>0$ is maintained if ${\lambda }_l{\delta }_l>1$ and ${\eta }_l\left(0\right)>0$ are both satisfied. In our CB-DETM, the choice of ${\lambda }_l$ is critical for dynamically adjusting the event-triggering thresholds. This is similar to the dynamic scheme discussed in [51], where λ is co-designed with the $H\infty$ performance index and inter-event time to manage fault tolerance and system responsiveness. Our selection of ${\lambda }_l$ ensures efficient data transmission and robust control, offering adaptability comparable to the integral observer method used in [51]. This adaptability helps in maintaining system performance under varying conditions, highlighting the reliability and resilience of our approach. Additionally, the DETM described here is a specialized version of the technique introduced in [52], where all components share identical triggering parameters, highlighting the scalability of the CB-DETM approach.

3.7. Security Challenges: Sensor and Actuator Attacks

In modern control systems, ensuring the security and integrity of sensors and actuator components is critical. These components are susceptible to various cyber-physical attacks, which can disrupt the normal operation of the system, leading to potential failures or unsafe conditions. Next, we discuss the nature of these attacks, their implications, and countermeasures to enhance system resilience. As depicted in Figure 2, the transmitted signal from the sensors is prone to vulnerabilities, such as false data injection attacks, which may be facilitated by the presence of computer viruses, bugs, or other malicious interventions. These attacks pose a high risk to the integrity of the control system, especially in configurations where event-triggered mechanisms are used. The sensor attack manipulates the system state by injecting false data, which can be mathematically modeled as

$$\begin{array}{cc}\hfill \tilde{x}\left(k\right)& =x\left(k\right)+\zeta \left(k\right)(-x\left(k\right)+{\delta }_s\left(k\right))=(1-\zeta \left(k\right))x\left(k\right)+\zeta \left(k\right){\delta }_s\left(k\right)\hfill \\ \hfill & =(1-\overline{\zeta })x\left(k\right)+\overline{\zeta }{\delta }_s\left(k\right)+(\zeta \left(k\right)-\overline{\zeta })(-x\left(k\right)+{\delta }_s\left(k\right))\hfill \\ \hfill & \approx (1-\overline{\zeta })(x\left(k\right)-e\left(k\right))+\overline{\zeta }{\delta }_s\left(k\right)+(\zeta \left(k\right)-\overline{\zeta })((-x\left(k\right)-e\left(k\right))+{\delta }_s\left(k\right)),\hfill \end{array}$$

(11)

where ${\delta }_s\left(k\right)$ is the signal crafted by the attacker, assumed to be a smooth nonlinear function satisfying $\parallel {\delta }_s\left(k\right)\parallel \le \parallel Gx\left(k\right)\parallel$, with G being a positive definite matrix, stating the operational bounds for the attack vector; whereas the random variable $\zeta \left(k\right)$ blends the true control and malicious signals; $\overline{\zeta }$ is an auxiliar variable; and $e\left(k\right)$ is a model error.

The model stated in (11) highlights how sensor attacks may distort the true system states, leading to potential miscalculations or misoperations in the control actions. Given the system reliance on accurate data for event triggering and decision making, especially in the dynamic communication network setup outlined in Figure 2, maintaining data integrity is crucial. The attacks not only disrupt normal operations but can also lead to catastrophic failures if not properly mitigated. To combat these threats, it is essential to incorporate robust security measures and anomaly detection strategies within the event-triggering mechanism. Enhancing the system resilience involves the following:

• Implementation of advanced encryption methods for data in transit.
• Utilization of anomaly detection algorithms to identify and isolate falsified data.
• Employment of redundancy and diversified sensor setups to validate data integrity.

Actuators, critical components in control systems, are susceptible to imperfections which may arise due to design flaws, manufacturing defects, and wear and tear. Furthermore, in modern control systems that rely heavily on digital communications and computer-based control algorithms, actuators are also vulnerable to cyber-physical attacks. These include malware infections, targeted attacks by worms, or other cyber-security breaches that may alter the behavior of actuators by manipulating their control signals.

Actuator attacks can be modeled to show how malicious interventions alter the control signals. The effect of such attacks may be mathematically modeled as

$$\begin{array}{cc}\hfill \tilde{u}\left(k\right)=& {\displaystyle u^F\left(k\right)+\psi \left(k\right)=\sum _{j=1}^r{\varrho }_j\left(x\left(k\right)\right)\left(1-\overline{\zeta })\Omega K_j^{s}x\left(k\right)-(1-\overline{\zeta })\Omega K_j^{s}e\left(k\right)+\overline{\zeta }\Omega K_j^s{\delta }_s\left(k\right)\right)}\hfill \\ \hfill & +(\zeta \left(k\right)-\overline{\zeta })\left(\Omega K_j^{s}x\left(k\right)+\Omega K_j^{s}e\left(k\right)+\Omega K_j^s{\delta }_s\left(k\right)\right)+\overline{\Gamma }{\delta }_a\left(k\right)+(\Gamma \left(k\right)-\overline{\Gamma }){\delta }_a\left(k\right),\hfill \end{array}$$

(12)

where $u^F\left(k\right)=\Omega u\left(k\right)$ represents the intended control signal, which can be potentially corrupted by faults; the term $\psi \left(k\right)=\Gamma \left(k\right){\delta }_a\left(k\right)$ is the effect of the actuator attacks, characterized by modifications in the actuator response due to malicious inputs; $K_j$ is a control gain; $\overline{\Gamma }$ is an expected value; $e\left(k\right)$ is a model error; and ${\varrho }_j\left(x\left(k\right)\right)$ is the normalized firing strength, defined as ${\varrho }_j\left(x\left(k\right)\right)={\underline{\beta }}_j\left(x\left(k\right)\right){\underline{\varrho }}_j\left(x\left(k\right)\right)+{\overline{\beta }}_j\left(x\left(k\right)\right){\overline{\varrho }}_j\left(x\left(k\right)\right)$, with ${\underline{\beta }}_j$ and ${\overline{\beta }}_j$ being nonlinear functions that adjust the influence of each rule based on the current system state.

The components of the attack model are stated as follows:

• $\Gamma \left(k\right)$ is a diagonal matrix with entries ${\gamma }_l\left(k\right)$, stating random variables which are Bernoulli-distributed, modeling the sporadic nature of cyber-attacks on actuators.
• ${\delta }_a\left(k\right)$ models the true malicious payload, which is a nonlinear function designed to disrupt the actuator normal operation, whose function adheres to the safety constraint presented as $\parallel {\delta }_a{\left(k\right)\parallel }^2\le {\sigma }_0^2{\sum }_{j=1}^r{\varrho }_j\left(x\left(k\right)\right)\left(\tilde{x}\left(k\right)K_j^{\top }\left({\varpi }_k\right)H^{\top }H{K}_j\left({\varpi }_k\right)\tilde{x}\left(k\right)\right)$, ensuring that the potential impact of the attack does not exceed a predefined safety threshold stated by ${\sigma }_0^2$ and the matrix H, where $\left\{{\varpi }_k\left(k\right)\right|\ge 0\}$ is a Markov chain.
• The attack is modeled to adjust dynamically to the system state via the random variable $\zeta \left(k\right)$, which, as mentioned, blends the true control signal with the malicious signal, creating a corrupted output that can vary in intensity and effect.

The security of the actuator signals under attack conditions is constrained by

$$\begin{array}{cc}\hfill {\delta }_a^{\top }\left(k\right){\delta }_a\left(k\right)& {\displaystyle \le {\sigma }_0^2\sum _{j=1}^r{\varrho }_j\left(x\left(k\right)\right)\left(\tilde{x}\left(k\right)K_j^{\top }\left({\varpi }_k\right)H^{\top }H{K}_j\left({\varpi }_k\right)\tilde{x}\left(k\right)\right),}\hfill \end{array}$$

(13)

where H is a matrix defining the permissible operational space for the actuator under normal and attack conditions. The constraint stated in (13) ensures that, even during an attack, the alteration to the control signal remains within a defined safety threshold.

The expected behavior of the attack variables are formulated as $\overline{\Gamma }=\mathbb{E}[\Gamma \left(k\right)]=\mathrm{diag}\left\{{\gamma }_l\left(k\right)\right\}$, $\mathbb{E}\left[\tilde{\Gamma }\right]=\mathbb{E}[\Gamma \left(k\right)-\overline{\Gamma }]=0$, and $\mathbb{E}\left[{\tilde{\Gamma }}^{\top }Q\tilde{\Gamma }\right]={\sum }_l{\mu }_l^2{\Sigma }_l^{\top }Q{\Sigma }_l$, where $\mu ,\Sigma$ are the corresponding mean and covariance matrix, respectively, whereas Q is a positive definite matrix, ensuring that the attack impact is kept within manageable limits. Given the fundamental role that actuators play in control systems, it is essential to implement robust security measures to detect and mitigate these attacks. These measures include:

• The employment of sophisticated monitoring systems that can detect unusual behaviors or deviations in actuator performance.
• The design of systems with redundant actuator configurations and fault-tolerant controls that may maintain operation even when some components are compromised.
• The strengthening of cyber-security practices, including regular updates, patches, and protective measures against malware and other forms of cyber-attacks.

By integrating these measures, the system can improve its resilience against attacks, ensuring reliability and resilience even in the face of sophisticated cyber-threats.

3.8. Fuzzy Controller

Following the discussion on sensor attacks and the system vulnerabilities they expose, it is crucial to consider how control strategies can be adapted to maintain system performance and reliability despite these vulnerabilities. The fuzzy controller design presented here assumes that the system modes are not entirely accessible and that the synchronization between the system and controller may not always be perfect.

To address the potential desynchronization and enhance the system resilience against sensor errors and attacks, a nonstationary IT2 fuzzy controller is employed. The controller is structured to adapt to different degrees of asynchrony among the system states and control actions, defined as

$$\begin{array}{cc}\hfill \mathrm{Rule}j:& \mathrm{If}{\phi }_1\left(x\left(k\right)\right)\mathrm{is}{\mathbb{F}}_j^1\dots \mathrm{and}{\phi }_{\varsigma }\left(x\left(k\right)\right)\mathrm{is}{\mathbb{F}}_j^{\varsigma },\mathrm{then}u\left(k\right)=K_j\left(\rho \left(k+1\right)\right)\tilde{x}\left(k\right),\hfill \end{array}$$

where ${\phi }_{\varsigma }\left(x\left(k\right)\right)$ represents the premise variable $\varsigma$ and ${\mathbb{F}}_j^{\varsigma }$ the type-2 fuzzy sets associated with the jth control rule for each premise variable $\varsigma$. The control gain $K_j\left(\rho \left(k+1\right)\right)$, designed to be adaptive, responds to the nonstationary Markov chain that characterizes the system modes referenced by the system given in (7). The underlying nonstationary Markov chain $\left\{\rho \right(k\left)\right|k\ge 0\}$ takes values in ${\mathcal{S}}_2\equiv \{1,\dots ,S\}$ and is crucial for adjusting the control actions based on the current system state, established as

$$\begin{array}{c}\hfill {\Pi }_2=\left[{\phi }_{st}^{\rho \left(k+1\right)}\right],{\phi }_{st}^{\rho \left(k+1\right)}=Pr\left(\varpi \left(k+1\right)=t\mid \rho \left(k\right)=s\right),\end{array}$$

(14)

where the TP ${\phi }_{st}^{\rho \left(k\right)}$ ensures that the controller dynamically adjusts to the changing conditions. The concept of firing strength in a fuzzy control system is critical as it determines how much each rule influences the control action. In this IT2 fuzzy controller, the firing strength for each rule j is defined as ${\mathfrak{N}}_j=[{\underline{\varrho }}_j\left(\widehat{x}\left(k\right)\right),{\overline{\varrho }}_j\left(\widehat{x}\left(k\right)\right)]$, where ${\underline{\varrho }}_j\left(\widehat{x}\left(k\right)\right)$ and ${\overline{\varrho }}_j\left(\widehat{x}\left(k\right)\right)$ are the aggregated lower and upper membership values of the IT2 fuzzy sets ${\mathbb{F}}_j^{\varsigma }$ evaluated at the state variables $\phi \left(k\right)$. These membership values are computed as the product of the membership degrees across all relevant fuzzy sets by means of

$$\begin{array}{cc}\hfill {\underline{\varrho }}_j\left(\widehat{x}\left(k\right)\right)& =\prod _{\varsigma =1}^v{\underline{\omega }}_{{\mathbb{F}}_j^{\varsigma }\left(\phi \left(k\right)\right)},{\overline{\varrho }}_j\left(\widehat{x}\left(k\right)\right)=\prod _{\varsigma =1}^v{\overline{\omega }}_{{\mathbb{F}}_j^{\varsigma }\left(\phi \left(k\right)\right)},\hfill \end{array}$$

(15)

with each $\underline{\omega }$ and $\overline{\omega }$ representing the lower and upper bounds of the membership functions, respectively, ensuring that the control system can handle uncertainties and imprecisions inherent in the sensor readings and other inputs.

The control actions are then calculated using these firing strengths, which are normalized to ensure that their sum is always equal to one, enhancing the reliability and resilience of the control system, and formulated as

$$\begin{array}{c}\hfill {\displaystyle u\left(k\right)=\sum _{j=1}^r{\varrho }_j\left(x\left(k\right)\right)K_j\left(\rho (k+1)\right)x\left(k\right),}\end{array}$$

(16)

where, as mentioned, ${\varrho }_j\left(x\left(k\right)\right)={\underline{\beta }}_j\left(x\left(k\right)\right){\underline{\varrho }}_j\left(x\left(k\right)\right)+{\overline{\beta }}_j\left(x\left(k\right)\right){\overline{\varrho }}_j\left(x\left(k\right)\right)$, with ${\underline{\beta }}_j$ and ${\overline{\beta }}_j$ being nonlinear functions designed to dynamically adjust the influence of each rule based on the current system state, ensuring adaptive and responsive control actions. The functions ${\underline{\beta }}_j$ and ${\overline{\beta }}_j$ satisfy $0\le {\underline{\beta }}_j\left(x\left(k\right)\right)\le {\overline{\beta }}_j\left(x\left(k\right)\right)\le 1$ and ${\underline{\beta }}_j\left(x\left(k\right)\right)+{\overline{\beta }}_j\left(x\left(k\right)\right)=1$. This formulation allows the controller to effectively manage the trade-offs between responsiveness and robustness, tailoring the control efforts to the current conditions and uncertainties faced by the system.

In summary, the presented configuration of the fuzzy controller ensures that the system can maintain optimal performance and stability even under varying and unpredictable conditions, making it resilient to the complexities and challenges posed by real-world operational environments.

3.9. Closed-Loop System

To establish the closed-loop configuration for a discrete-time MJ nonlinear system, we introduce the state representation in a compact form $\widehat{x}\left(k\right)={[x_1^{\top }\left(k\right),\epsilon x_2^{\top }\left(k\right)]}^{\top }$, where $\epsilon$ is a critical parameter in our system matrix parameterization, as described in (5). This configuration incorporates both the primary and scaled secondary state vectors, with the extended state vector defined as

$$\xi \left(k\right)=\{\widehat{x}\left(k\right),e\left(k\right),{\delta }_s\left(k\right),{\delta }_a\left(k\right),\eta \left(k\right)\},$$

(17)

where $\eta \left(k\right)$ is a vector comprising the square roots of state-dependent parameters, expressed as $\eta \left(k\right)=\mathrm{col}\{\sqrt{{\eta }_1\left(k\right)},\dots ,\sqrt{{\eta }_{n_x}\left(k\right)}\}$. In the expression given in (17), “col” denotes the column concatenation of vectors and each component serves a distinct purpose, established as follows:

• $e\left(k\right)$ represents estimated error dynamics.
• ${\delta }_s\left(k\right)$ models disturbances or potential attacks targeting sensors.
• ${\delta }_a\left(k\right)$ pertains to disruptions or malfunctions in actuators, previously mentioned.
• $\eta \left(k\right)$ includes AIDVs, enhancing our model ability to capture complex internal dynamics.

The formulation established in (17) allows us to model the system dynamics more comprehensively by including not only the states but also errors, disturbances, and system parameters that are subject to attacks and other dynamics.

The integration of system dynamics stated in (7), considering the effects of actuator attacks presented in (12), and with the control actions defined in (16), permits us to describe the closed-loop system as

$$\begin{array}{cc}\hfill \widehat{x}(k+1)& =\sum _{i=1}^r\sum _{j=1}^r{\vartheta }_i{\varrho }_j{E}^{\epsilon }\left(A_{ij}^{ps}\xi k+(\zeta \left(k\right)-\overline{\zeta })A_{1ij}^{ps}\xi k+B_{2i}^p\overline{\Gamma }{\delta }_a\left(k\right)\right),\hfill \end{array}$$

(18)

where the matrices $A_{ij}^{ps}$ and $A_{1ij}^{ps}$ represent the aggregated system dynamics under normal and perturbed conditions, respectively, and are stated as

$$\begin{array}{cc}\hfill A_{ij}^{ps}& =\left[\begin{array}{ccccc}{A}_i^p+(1-\overline{\zeta })B_{2i}^p\Omega K_j^s& -(1-\overline{\zeta })B_{2i}^p\Omega K_j^s& \overline{\zeta }{B}_{2i}^p\Omega K_j^s& B_{2i}^p\overline{\Gamma }& 0\end{array}\right],\hfill \\ \hfill A_{1ij}^{ps}& =\left[\begin{array}{ccccc}-B_{2i}^p\Omega K_j^s& B_{2i}^p\Omega K_j^s& B_{2i}^p\Omega K_j^s& 0& 0\end{array}\right].\hfill \end{array}$$

As mentioned in the introduction, the primary objective addressed in this article is to design an asynchronous security control, as specified in (16), for ensuring that the closed-loop system, represented in (18), achieves stochastic stability. This objective entails maintaining the system state within bounded limits, ensuring reliable and secure operation despite challenges such as MJs, actuator attacks, and other nonlinear disturbances.

4. Main Results

Building on the framework established for achieving stochastic stability through asynchronous security control strategies, this section presents the core theoretical results. These results lay the mathematical groundwork necessary to ensure the system desired performance under our proposed control and validate the structures and approaches discussed in Section 3.

4.1. Implementing Robust Control Mechanisms

Next, we delineate the practical application of our theoretical findings, shaping robust control strategies essential for:

• Developing controllers that efficiently manage uncertainties and dynamically adapt to environmental changes, ensuring operational resilience.
• Mitigating impacts from external disturbances, including sensor and actuator attacks, so preserving the integrity and security of the system.
• Achieving comprehensive system stability, particularly stochastic mean-square stability, crucial for system reliability over prolonged operational periods.

Our theoretical framework includes fundamental lemmas that support the stability analysis of our control system, adapted from foundational sources [6].

Lemma 1.

Given a positive scalar ε and matrices $X_1$ and $X_2$ of appropriate dimensions, if condition $X_1<0$ is met, then it follows that $X_1+\epsilon X_2<0$ for all $0<\epsilon \le \overline{\epsilon }$.

Lemma 1 ensures that the system stability margins can be maintained even under varying parameter conditions, providing a mathematical foundation for our stability analysis. We now present a critical theorem that establishes the necessary conditions for system stability amidst the nonlinear dynamics and uncertainties associated with MJ processes and the challenges introduced by asynchronous security controls.

Theorem 1.

Given constants $0<{\sigma }_l<1$, ${\delta }_l$, and ${\lambda }_l$ satisfying ${\lambda }_l{\delta }_l>1$, the closed-loop system described in (18) is stochastically mean-square stable if there exist positive definite matrices $Z^{ps}\in {\mathbb{R}}^{n\times n}$ and positive scalars ${\tau }_1$, ${\tau }_2$, ${\tau }_3$ ensuring that

$$\begin{array}{c}\hfill \sum _{i=1}^r\sum _{j=1}^r{\varrho }_i{\vartheta }_j{\widehat{\Psi }}_{ij}^{ps\epsilon }<0,\end{array}$$

(19)

where

$$\begin{array}{cc}\hfill {\widehat{\Psi }}_{ij}^{ps\epsilon }& =\left[\begin{array}{ccc}{\widehat{\Psi }}_{11ij}^{ps\epsilon }& {\widehat{\Psi }}_{12ij}^{ps\epsilon }& {\widehat{\Psi }}_{13ij}^{ps\epsilon }\\ *& -{\widehat{\Psi }}_{22}& 0\\ *& *& -{\widehat{\Psi }}_{33}\end{array}\right],\hfill \\ \hfill {\widehat{\Psi }}_{11ij}^{ps\epsilon }& =\mathrm{diag}\left\{{\widehat{\Phi }}_{ij}^{ps\epsilon },-({\mathit{\Gamma }}_2+{\tau }_{1}I)+2{\tau }_2{G}^{\top }G,-{\tau }_{2}I,-{\tau }_{3}I,{\mathit{\Gamma }}_3+{\tau }_1{\Theta }_3\right\},\hfill \\ \hfill {\widehat{\Phi }}_{ij}^{ps\epsilon }& =-\left(E^{\epsilon }{Z}^{ps}{E}^{\epsilon }\right)+{\mathit{\Gamma }}_1+{\tau }_1{\Theta }_1+2{\tau }_2{G}^{\top }G,\hfill \\ \hfill {\widehat{\Psi }}_{12ij}^{ps}& =\left[\begin{array}{cc}{I}_{ps}\otimes {\left({\widehat{A}}_{ij}^{ps}\right)}^{\top }& \widehat{\zeta }{I}_{ps}\otimes {\left(A_{1ij}^{ps}\right)}^{\top }\end{array}\right]{\widehat{\Psi }}_{13ij}^{ps}={\sigma }_0{\left(K_j^s\right)}^{\top }{\widehat{\Psi }}_{33}^{ps}=\widehat{\zeta }{\sigma }_0{\left(K_{1j}^s\right)}^{\top },\hfill \\ \hfill {\widehat{\Psi }}_{22}& =\mathrm{diag}\{{\widehat{\Phi }}_{22},{\widehat{\Phi }}_{22}\},{\widehat{\Psi }}_{33}=\mathrm{diag}\{{\tau }_3^{-1}I,{\tau }_3^{-1}I\},{\widehat{\Phi }}_{22}=\mathrm{diag}\{Z^{11},\dots ,Z^{NS}\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathit{\Gamma }}_1& ={\mathrm{diag}}_{l=1}^n\left\{{\displaystyle \frac{{\sigma }_l}{{\theta }_l}}\right\},{\Theta }_1={\mathrm{diag}}_{l=1}^n\left\{{\sigma }_l\right\},{\mathit{\Gamma }}_3={\mathrm{diag}}_{l=1}^n\left\{{\displaystyle \frac{{\lambda }_l-1}{{\theta }_l}}\right\},{\mathit{\Gamma }}_2={\Theta }_3={\mathrm{diag}}_{l=1}^n\left\{{\displaystyle \frac{1}{{\theta }_l}}\right\},\hfill \\ \hfill {\widehat{A}}_{ij}^{ps}& =\left[\begin{array}{ccccc}{A}_i^p+(1-\overline{\zeta })B_{2i}^p\Omega K_j^s& -(1-\overline{\zeta })B_{2i}^p\Omega K_j^s& \overline{\zeta }{B}_{2i}^p\Omega K_j^s& B_{2i}^p\overline{\Gamma }+\Theta & 0\end{array}\right],\hfill \\ \hfill K_j^s& =\left[\begin{array}{ccccc}(1-\overline{\zeta })K_j^s& -(1-\overline{\zeta })K_j^s& \overline{\zeta }{K}_j^s& 0& 0\end{array}\right],K_{1j}^s=\left[\begin{array}{ccccc}-K_j^s& K_j^s& K_j^s& 0& 0\end{array}\right],\hfill \\ \hfill I_{ps}& =\left[\begin{array}{ccc}\sqrt{{\pi }_{p1}{\phi }_{s1}^1}I& \dots & \sqrt{{\pi }_{pN}{\phi }_{sS}^N}I\end{array}\right],\widehat{\zeta }=\sqrt{\overline{\zeta }(1-\overline{\zeta })},\Theta =\mathrm{diag}\{{\mu }_1,\dots ,{\mu }_m\}.\hfill \end{array}$$

Proof. 

The derivation of the stability condition is based on the Lyapunov theory, where the Lyapunov function $V\left(k\right)$ is positive and verifies that $\Delta V\left(k\right)<0$. For the system described in (18), due to the adaptive nature of the ETP, the term ${\sum }_{l=1}^n(1/{\theta }_l){\eta }_l\left(k\right)$ is included in the Lyapunov function, stated as

$$V\left(k\right)={\widehat{x}}^{\top }\left(k\right)P_{\overline{\rho }\left(k\right),{\varpi }_k}\widehat{x}\left(k\right)+\sum _{l=1}^n{\displaystyle \frac{1}{{\theta }_l}}{\eta }_l\left(k\right),$$

where $P_{\overline{\rho }\left(k\right),{\varpi }_k}>0$ and $(\overline{\rho }\left(k\right),{\varpi }_k)$ takes values in $(p,s)\in {\mathcal{S}}_1\times {\mathcal{S}}_2$. Moreover, $P_{\overline{\rho }\left(k\right),{\varpi }_k}$ depends on the Markov processes $\left\{\overline{\rho }\left(k\right)\right|k\ge 0\}$ and $\left\{{\varpi }_k\right|k\ge 0\}$. Let $\Delta V\left(k\right)=V(k+1)-V\left(k\right)$. It can be proved along the trajectories of the system presented in (18) that

$$\begin{array}{cc}\hfill \mathbb{E}(\Delta V(k\left)\right)=\mathbb{E}& \left({\widehat{x}}^{\top }(k+1)\right)\sum _{q\in {\mathcal{S}}_1,t\in {\mathcal{S}}_2}\mathrm{Pr}\left(\overline{\rho }(k+1)=q,{\varpi }_{k+1}=t|{\overline{\rho }}_k=p,{\varpi }_k=s\right)P^{qt}\left(\widehat{x}(k+1)\right)-{\widehat{x}}^{\top }\left(k\right)P^{ps}\widehat{x}\left(k\right)\hfill \\ \hfill & +\sum _{l=1}^n{\displaystyle \frac{1}{{\theta }_l}}{\eta }_l(k+1)-\sum _{l=1}^n{\displaystyle \frac{1}{{\theta }_l}}{\eta }_l\left(k\right).\hfill \end{array}$$

Using the conditional probability $\mathrm{Pr}\left(\overline{\rho }(k+1)=q,{\varpi }_{k+1}=t|{\overline{\rho }}_k=p,{\varpi }_k=s\right)={\pi }_{pq}{\phi }_{st}^q$, we can simplify the expectation as

$$\mathbb{E}\left(\Delta V\left(k\right)\right)=\mathbb{E}\left({\widehat{x}}^{\top }(k+1){\mathcal{P}}^{ps}\widehat{x}(k+1)\right)-{\widehat{x}}^{\top }\left(k\right)P^{ps}\widehat{x}\left(k\right)+\sum _{l=1}^n{\displaystyle \frac{1}{{\theta }_l}}{\eta }_l(k+1)-\sum _{l=1}^n{\displaystyle \frac{1}{{\theta }_l}}{\eta }_l\left(k\right),$$

where ${\mathcal{P}}^{ps}={\sum }_{q\in {\mathcal{S}}_1}{\pi }_{pq}{\sum }_{t\in {\mathcal{S}}_2}{\phi }_{st}^q{P}^{qt}$. Using the fact that $\mathrm{Pr}\left(\overline{\rho }(k+1)=q,{\varpi }_{k+1}=t|{\overline{\rho }}_k=p,{\varpi }_k=s\right)={\pi }_{pq}{\phi }_{st}^q$, we have that

$$\begin{array}{cc}\hfill \mathbb{E}\left(\Delta V\left(k\right)\right)=& \mathbb{E}\left({\widehat{x}}^{\top }(k+1){\mathcal{P}}^{ps}\widehat{x}(k+1)\right)-{\widehat{x}}^{\top }\left(k\right)P^{ps}\widehat{x}\left(k\right)+\sum _{l=1}^n{\displaystyle \frac{1}{{\theta }_l}}\left(\right({\lambda }_l-1){\eta }_l\left(k\right)-e_l^{\top }\left(k\right)e_l\left(k\right)+{\sigma }_l{x}_l^{\top }\left(k\right)x_l\left(k\right))\hfill \\ \hfill =& \mathbb{E}\left({\widehat{x}}^{\top }(k+1){\mathcal{P}}^{ps}\widehat{x}(k+1)\right)-{\widehat{x}}^{\top }\left(k\right)P^{ps}\widehat{x}\left(k\right)+{\widehat{x}}^{\top }\left(k\right){\mathit{\Gamma }}_1{\widehat{x}}^{\top }\left(k\right)-e^{\top }\left(k\right)e\left(k\right)+{\eta }^{\top }\left(k\right){\mathit{\Gamma }}_3\eta \left(k\right)\hfill \\ \hfill =& \mathbb{E}\left({\xi }^{\top }\left(k\right){\left(A_{\varrho \vartheta }^{ps}\right)}^{\top }{E}^{\epsilon }{\mathcal{P}}^{ps}{E}^{\epsilon }{A}_{\varrho \vartheta }^{ps}\xi \left(k\right)\right)+\mathbb{E}\left({(\zeta \left(k\right)-\overline{\zeta })}^2{\xi }^{\top }(k){\left(A_{1\varrho \vartheta }^{ps}\right)}^{\top }{E}^{\epsilon }{\mathcal{P}}^{ps}{E}^{\epsilon }{A}_{1\varrho \vartheta }^{ps}\xi (k)\right)\hfill \\ \hfill & +\mathbb{E}\left({\delta }_a^{\top }\left(k\right){\left(B_{2\varrho }^p\tilde{\Gamma }\right)}^{\top }{E}^{\epsilon }{\mathcal{P}}^{ps}{E}^{\epsilon }{B}_{2\varrho }^p\tilde{\Gamma }{\delta }_a\left(k\right)\right)+{\xi }^{\top }(k)\mathrm{diag}\left(-P^{ps}+{\mathit{\Gamma }}_1,-{\mathit{\Gamma }}_2,0,0,{\mathit{\Gamma }}_3\right)\xi (k),\hfill \end{array}$$

where ${\mathcal{P}}^{ps}={\sum }_{q\in {\mathcal{S}}_1}{\pi }_{pq}{\sum }_{t\in {\mathcal{S}}_2}{\phi }_{st}^q{P}^{qt}$. Also, note that $\mathbb{E}({(\zeta \left(k\right)-\overline{\zeta })}^2{\xi }^{\top }(k){\left(A_{1\varrho \vartheta }^{ps}\right)}^{\top }$$E^{\epsilon }{\mathcal{P}}^{ps}$$\times E^{\epsilon }{A}_{1\varrho \vartheta }^{ps}\xi (k))={\widehat{\zeta }}^2{\xi }^{\top }(k){\left(A_{1\varrho \vartheta }^{ps}\right)}^{\top }{E}^{\epsilon }{\mathcal{P}}^{ps}{E}^{\epsilon }{A}_{1\varrho \vartheta }^{ps}\xi (k)$ and

$$\begin{array}{ccc}\hfill \mathbb{E}({\left(B_{2\varrho }^p\tilde{\Gamma }{\delta }_a\left(k\right)\right)}^{\top }{E}^{\epsilon }{\mathcal{P}}^{ps}{E}^{\epsilon }\left(B_{2\varrho }^p\tilde{\Gamma }{\delta }_a\left(k\right)\right))& =& {\delta }_a^{\top }\left(k\right)\left(\sum _{l=1}^m{\mu }_l^2{\left(B_{2\varrho }^p{\Sigma }_l\right)}^{\top }\left(E^{\epsilon }{\mathcal{P}}^{ps}{E}^{\epsilon }\right)B_{2\varrho }^p{\Sigma }_l\right){\delta }_a\left(k\right)\hfill \\ & =& {\delta }_a^{\top }\left(k\right)({(B_{2\varrho }^p\Theta )}^{\top }\left(E^{\epsilon }{\mathcal{P}}^{ps}{E}^{\epsilon }\right)B_{2\varrho }^p\Theta ){\delta }_a\left(k\right){\tau }_1\sum _{l=1}^n({\sigma }_l{x}_l^{\top }\left(k\right)x_l\left(k\right)\hfill \\ & & -e_l^{\top }\left(k\right)e_l\left(k\right)+(1/{\theta }_l){\eta }_l\left(k\right))={\tau }_1{\xi }^{\mathrm{\top }}\left(k\right)\mathrm{diag}\{{\Theta }_1,-I,0,0,{\Theta }_3\}\xi \left(k\right)\ge 0.\hfill \end{array}$$

Additionally, the expression $\parallel {\delta }_s\left(k\right)\parallel \le \parallel Gx\left(k\right)\parallel$ provides

$$\begin{array}{c}\hfill \begin{array}{cc}& -{\tau }_2{\delta }_s^{\top }\left(k\right){\delta }_s\left(k\right)+2{\tau }_2{\widehat{x}}^{\top }\left(k\right)G^{\top }G\widehat{x}\left(k\right)+2{\tau }_2{e}^{\top }\left(k\right)G^{\top }Ge\left(k\right)\ge 0.\hfill \end{array}\end{array}$$

(20)

Considering the formula given in (13), it follows for ${\tau }_3>0$ that

$$\begin{array}{cc}\hfill 0& \le -{\tau }_3\mathbb{E}\left({\delta }_a^{\top }\left(k\right){\delta }_a\left(k\right)\right)+{\tau }_3{\sigma }_0^2{\xi }^{\mathrm{\top }}\left(k\right){\left(K_{\vartheta }^s\right)}^{\top }{K}_{\vartheta }^s\xi \left(k\right)+{\tau }_3{\widehat{\zeta }}^2{\sigma }_0^2{\xi }^{\mathrm{\top }}\left(k\right){\left(K_{1\vartheta }^s\right)}^{\top }{K}_{1\vartheta }^s\xi \left(k\right).\hfill \end{array}$$

(21)

The positive scalars ${\tau }_1$, ${\tau }_2$, and ${\tau }_3$ are introduced using the S-procedure lemma. Define $P^{ps}=\left(E^{\epsilon }{Z}^{ps}{E}^{\epsilon }\right)$ and combining (20) and (21), it follows that $\mathbb{E}\left[\Delta V\left(k\right)\right]\le {\xi }^{\mathrm{\top }}\left(k\right){\tilde{\Phi }}_{\varrho \vartheta }^{ps\epsilon }\xi \left(k\right).$

We introduce $P^{ps}=\left(E^{\epsilon }{Z}^{ps}{E}^{\epsilon }\right)$, which is equivalent to $E^{\epsilon }{\left(P^{ps}\right)}^{-1}{E}^{\epsilon }={\left(Z^{ps}\right)}^{-1}$. This allows us to use the Schur complement lemma to obtain the matrix ${\widehat{\Phi }}_{22}$, with

$${\tilde{\Phi }}_{\varrho \vartheta }^{ps\epsilon }={\widehat{\Phi }}_{\varrho \vartheta }^{ps\epsilon }+{({\widehat{A}}_{\varrho \vartheta }^{ps})}^{\top }{\mathcal{Z}}^{ps}{\widehat{A}}_{\varrho \vartheta }^{ps}+{\widehat{\zeta }}^2{(A_{1\varrho \vartheta }^{ps})}^{\top }{\mathcal{Z}}^{ps}{A}_{1\varrho \vartheta }^{ps}+{\tau }_3{\sigma }_0^2{(K_{\vartheta }^s)}^{\top }{K}_{\vartheta }^s+{\tau }_3{\widehat{\zeta }}^2{\sigma }_0^2{(K_{1\vartheta }^s)}^{\top }{K}_{1\vartheta }^s,$$

where ${\mathcal{Z}}^{ps}={\sum }_{q\in {\mathcal{S}}_1}{\pi }_{pq}{\sum }_{t\in {\mathcal{S}}_2}{\phi }_{st}^q{\left(Z^{qt}\right)}^{-1}.$ Based on the Schur complement, the condition given in (19) implies that $\mathbb{E}(\Delta V\left(k\right))\le {\sum }_{i=1}^r{\sum }_{j=1}^r{\varrho }_i{\vartheta }_j\left({\xi }^{\mathrm{\top }}\left(k\right){\tilde{\Phi }}_{ij}^{ps\epsilon }\xi \left(k\right)\right)<0.$ Furthermore, it can be deduced that $\mathbb{E}(\Delta V(k\left)\right)\le \lambda {\xi }^{\mathrm{\top }}\left(k\right)\xi \left(k\right),$ and

$$\mathbb{E}\left(\sum _0^{\infty }{\left(x\left(k\right)\right)}^{\top }\widehat{x}\left(k\right)\right)\le \mathbb{E}\left(\sum _0^{\infty }{\left(\xi \left(k\right)\right)}^{\top }\xi \left(k\right)\right)\le \frac{1}{\lambda }\mathbb{E}\left(\sum _0^{\infty }\Delta V\left(k\right)\right)\le -\frac{1}{\lambda }V\left(0\right)<\infty ,$$

where $\lambda <0$ is the largest eigenvalue of ${\tilde{\Phi }}_{ij}^{ps\epsilon }$. Thus, the system defined in (18) is stochastically mean-square stable, according to Definition 2 stated in [28].    □

4.2. Controller Design

In the development of our strategy for managing the complex dynamics associated with the MJ processes and asynchronous nature of the control actions, a robust controller design is essential. The relationships between ${\varrho }_i$ and ${\vartheta }_j$, as introduced previously, play a crucial role in shaping the controller response to system uncertainties and state transitions. Based on these relationships, we present an $\epsilon$-independent stability in the following theorem. This stability addresses the interaction between the product term of ${\varrho }_i$ with ${\vartheta }_j$ and $\epsilon$, ensuring that the controller remains effective regardless of $\epsilon$ within the specified bounds.

Theorem 2.

Consider the system described in (7) and the nonsynchronous controller stated in (16), coupled with nonstationary TP matrices defined in (14). Assume the following scalars and matrices:

(i) 

Scalars ${\zeta }_v,b_v,v\in \{1,2\}$, $0<{\sigma }_l<1$, and ${\lambda }_l{\theta }_l>1$, ensuring responsiveness and stability in the controller operation.

(ii) 

Matrices $Z^{ps}>0$, $X^{ps}$, $Y^{ps}$, and scalars ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, ${\tau }_1$, ${\overline{\tau }}_2$, ${\overline{\tau }}_3$ exist satisfying the stability conditions for each $\aleph \in \{0,\overline{\epsilon }\}$, expressed as ${\Psi }_{ii}^{ps\aleph }<0$, ${\Psi }_{ij}^{ps\aleph }+{\zeta }_1{\tilde{\Psi }}_{ji}^{ps\aleph }<0$, and ${\Psi }_{ij}^{ps\aleph }+{\zeta }_2{\tilde{\Psi }}_{ji}^{ps\aleph }<0$, for $1\le i<j\le r$.

If conditions (i) and (ii) are met, the system established in (18) is stochastically mean-square stable, and the controller gains are given by ${\overline{K}}_j^s=Y_j^s{\left(X_j^{11s}\right)}^{-1}$.

Theorem 2 articulates a method for defining the matrices that configure the controller gains, ensuring the system responsiveness and stability under various operational conditions. The matrices $Z^{ps}$, $X^{ps}$, and $Y^{ps}$ are designed to optimize the controller actions in response to the state changes and to handle the uncertainties of the system effectively. The scalars ${\lambda }_l$ and ${\theta }_l$ play a critical role in this design, enhancing the controller ability to maintain stability across different system modes and environmental conditions.

The matrices defined in Theorem 2 are given by

$$\begin{array}{cc}\hfill {\tilde{\Psi }}_{ij}^{ps\aleph }& =\left[\begin{array}{cccccc}{\tilde{\Psi }}_{11ij}^{ps\aleph }& {\tilde{\Psi }}_{12ij}^{ps\aleph }& {\tilde{\Psi }}_{13ij}^{ps}& {\tilde{\Psi }}_{14ij}^{ps}& {\tilde{\Psi }}_{15ij}^{ps}& {\tilde{\Psi }}_{16ij}^{ps}\\ *& -{\widehat{\Psi }}_{22}& 0& 0& 0& 0\\ *& *& -{\tilde{\Psi }}_{33}& 0& 0& 0\\ *& *& *& -0.5{\overline{\tau }}_{2}I& 0& 0\\ *& *& *& *& -0.5{\overline{\tau }}_{2}I& 0\\ *& *& *& *& *& -{\tilde{\Psi }}_{66}\end{array}\right],\hfill \\ \hfill {\tilde{\Psi }}_{11ij}^{ps\aleph }& =\mathrm{diag}\left\{{\tilde{\Phi }}_{11ij}^{ps\aleph },-{\tilde{\Phi }}_{22j}^s,-\mathrm{sym}\left(X_j^s\right)+{\overline{\tau }}_{2}I,-{\overline{\tau }}_{3}I,{\mathit{\Gamma }}_3+{\tau }_1{\Theta }_3\right\},\hfill \\ \hfill {\tilde{\Phi }}_{11ij}^{ps\aleph }& =-\mathrm{sym}\left({\beta }_1{X}_j^s\right)+{\beta }_1^2{E}^{\aleph }{Z}^{ps}{E}^{\aleph },{\tilde{\Phi }}_{22j}^s=-\mathrm{sym}\left((1+{\beta }_2)X_j^s\right)+{\beta }_2^2{\mathit{\Gamma }}_2+{\tau }_1^{-1}I,\hfill \\ \hfill {\tilde{\Psi }}_{12ij}^{ps}& =\left[\begin{array}{cc}{I}_{ps}\otimes {\left(A_{ij}^{ps}\right)}^{\top }& \widehat{\zeta }{I}_{ps}\otimes {\left(A_{1ij}^{ps}\right)}^{\top }\end{array}\right],{\tilde{\Psi }}_{66}={\mathit{\Gamma }}_1+{\tau }_1{\Theta }_1^{-1},\hfill \\ \hfill {\tilde{\Psi }}_{13ij}^{ps}& =\left[{\sigma }_0{\left({\mathcal{Y}}_j^s\right)}^{\top }\widehat{\zeta }{\sigma }_0{\left(Y_{1j}^s\right)}^{\top }\right],{\tilde{\Psi }}_{33}=\mathrm{diag}\{{\overline{\tau }}_{3}I,{\overline{\tau }}_{3}I\},\hfill \\ \hfill {\tilde{\Psi }}_{14ij}^{ps}& ={\left[G{X}_j^s 0 0 0 0\right]}^{\top },{\tilde{\Psi }}_{15ij}^{ps}={\left[0 G{X}_j^s 0 0 0\right]}^{\top },{\tilde{\Psi }}_{16ij}^{ps}={\left[X_j^s 0 0 0 0\right]}^{\top },\hfill \\ \hfill {\mathcal{A}}_{ij}^{ps}& =\left[\begin{array}{ccccc}{A}_i^p{X}_j^s+(1-\overline{\zeta })B_{2i}^p\Omega Y_j^s& (1-\overline{\zeta })B_{2i}^p\Omega Y_j^s& \overline{\zeta }{B}_{2i}^p\Omega Y_j^s& B_{2i}^p\overline{\Gamma }+B_{2i}^p\Theta & 0\end{array}\right],\hfill \\ \hfill {\mathcal{A}}_{1ij}^{ps}& =\left[\begin{array}{ccccc}-B_{2i}^p\Omega Y_j^s& B_{2i}^p\Omega Y_j^s& B_{2i}^p\Omega Y_j^s& 0& 0\end{array}\right],\hfill \\ \hfill {\mathcal{Y}}_j^s& =\left[\begin{array}{ccccc}(1-\overline{\zeta })Y_j^s& -(1-\overline{\zeta })Y_j^s& \overline{\zeta }{Y}_j^s& 0& 0\end{array}\right],{\mathcal{Y}}_{1j}^s=\left[\begin{array}{ccccc}-Y_j^s& {Y̠}_j^s& Y_j^s& 0& 0\end{array}\right],\hfill \\ \hfill X_j^s& =\left[\begin{array}{cc}{b}_1{X}_j^{11s}& b_2{X}_j^{11}I,\\ X_j^{21s}& X_j^{22s}\end{array}\right],E^{\overline{\epsilon }}=\mathrm{diag}\{I_{n_1},\overline{\epsilon }{I}_{n_2}\},E^0=\mathrm{diag}\{I_{n_1},0_{n_2}\}.\hfill \end{array}$$

The parameters ${\zeta }_v$, $b_v$, ${\sigma }_l$, ${\theta }_l$, and ${\lambda }_l$ are instrumental in defining the operational limits and stability conditions of the system. Specifically, ${\sigma }_l$, ${\theta }_l$, and ${\lambda }_l$ must satisfy ${\lambda }_l{\theta }_l>1$, which is a condition to ensure that the dynamic adjustments in the controller response are sufficient to counteract any disturbances and maintain system stability. In addition, the matrices $Z^{ps}$, $X^{ps}$, and $Y^{ps}$ are designed to encapsulate both the state feedback and uncertainties inherent in the system. Their positive definiteness and structured form ensure that the control law can robustly handle stochastic variations. The stability criteria given in (1) are important as they ensure that, under any given $\aleph \in \{0,\overline{\epsilon }\}$, the system response remains within desired bounds, hence guaranteeing mean-square stability. Each condition within these criteria targets different aspects of the system response, making sure that the overall system dynamics do not lead to instability.

Proof of Theorem 2.

Assume that ${\vartheta }_j={\rho }_j\left(k\right){\varrho }_j$ and $|{\varrho }_j-{\vartheta }_j|\le {\delta }_j$, where ${\rho }_j$ and ${\delta }_j$, for $j\in \mathcal{S}$, are positive scalars. Thus, it can be established that $0<{\rho }_1\le 1-{\delta }_j/{\underline{\varrho }}_j\le 1-{\delta }_j/{\varrho }_j\le {\rho }_j\left(k\right)\le 1+{\delta }_j/{\varrho }_j\le 1+{\delta }_j/{\underline{\varrho }}_j\le {\rho }_2$, where ${\rho }_1$ and ${\rho }_2$ represent the minimum and maximum values of ${\rho }_j\left(k\right)$. Thus, it follows that

$${\zeta }_1={\displaystyle \frac{{\rho }_1}{{\rho }_2}}\le {\displaystyle \frac{min{\rho }_i\left(k\right)}{max{\rho }_j\left(k\right)}}\le {\displaystyle \frac{{\rho }_i\left(k\right)}{{\rho }_j\left(k\right)}}\le {\displaystyle \frac{max{\rho }_i\left(k\right)}{min{\rho }_j\left(k\right)}}\le {\displaystyle \frac{{\rho }_2}{{\rho }_1}}={\zeta }_2.$$

Consequently, there exist ${\epsilon }_1\ge 0$ and ${\epsilon }_2\ge 0$ satisfying ${\epsilon }_1+{\epsilon }_2=1$ such that ${\rho }_i\left(k\right)/{\rho }_j\left(k\right)={\epsilon }_1{\zeta }_1+{\epsilon }_2{\zeta }_2$. From this relationship, it can be deduced that ${\sum }_{i=1}^r{\sum }_{j=1}^r{\varrho }_i{\vartheta }_j{\tilde{\Psi }}_{ij}^{ps\epsilon }={\sum }_{l=1}^2{\epsilon }_l$$({\tilde{\Psi }}_{ij}^{ps\epsilon }+{\zeta }_l{\tilde{\Psi }}_{ij}^{ps\epsilon })$.

According to Lemma 1, the matrix $\tilde{Y}\left(\epsilon \right)$ holds $\epsilon \in (0,\overline{\epsilon }]$, using the expression given in (19) for $\aleph =0$ and $\aleph =\overline{\epsilon }$. Thus, it can be concluded that ${\sum }_{i=1}^r{\sum }_{j=1}^r{\varrho }_i{\vartheta }_j{\tilde{\Psi }}_{ij}^{ps\epsilon }<0$, and consequently, ${\tilde{\Psi }}_{ij}^{ps\epsilon }<0$. This proves that the stability condition is met under the given assumptions. Also, a solution for Theorem 2 satisfies $\mathrm{sym}\left({\beta }_1^2{X}_j^s\right)-{\beta }_1{E}^{\epsilon }{Z}^{ps}{E}^{\epsilon }<0$ and $X_j^s$ is then nonsingular.

Using the fact that ${(X_j^s+{\beta }_1{E}^{\epsilon }{Z}^{ps}{E}^{\epsilon })}^{\top }{\left(E^{\epsilon }{Z}^{ps}{E}^{\epsilon }\right)}^{-1}(X_j^s+{\beta }_1{E}^{\epsilon }{Z}^{ps}{E}^{\epsilon })\ge 0$, it follows that $-{\left(X_j^s\right)}^{\top }{\left(E^{\epsilon }{Z}^{ps}{E}^{\epsilon }\right)}^{-1}{X}_j^s\le -\mathrm{sym}\left({\beta }_l{X}_j^s\right)+{\beta }_l^2{E}^{\epsilon }{Z}^{ps}{E}^{\epsilon }$. Similarly, for $\overline{{\tau }_2}={{\tau }_2}^{-1}$, we prove that $-{\tau }_1{\left(X_j^s\right)}^{\top }{X}_j^s\le -\mathrm{sym}\left(X_j^s\right)+{\tau }_1^{-1}I$ and $-{\tau }_2{\left(X_j^s\right)}^{\top }{X}_j^s\le -\mathrm{sym}\left(X_j^s\right)+\overline{{\tau }_2}I$. Let ${\Pi }_j^{2s}=\mathrm{diag}\{{\left(X_j^s\right)}^{-1},{\left(X_j^{11s}\right)}^{-1},{\left(X_j^s\right)}^{-1},{\tau }_3^{-1}I,{\tau }_3^{-1}I,I,I,I\}$. By applying the congruence transformation to ${\tilde{\Psi }}_{ij}^{ps\epsilon }<0$, using ${\Pi }_j^{2s}$, and subsequently utilizing the Schur complement, it can be verified that ${\widehat{\Phi }}_{ij}^{ps\epsilon }<0$. These transformations and verifications ensure that the stability conditions are met for the system under the defined control strategy.    □

Remark 2.

The criteria specified in Theorem 2 could exhibit a degree of conservatism and computational complexity, mostly due to the following factors:

(i) 

In the controller design described in Theorem 2, the structure of the matrix $X_j^s$ plays a crucial role in achieving the desired stability properties. Specifically, if $b_2=0$, the top-right element of $X_j^s$ becomes zero. This particular specification is common in the literature and is known to yield conservative results in stability analysis. The matrix $X_j^s$ is defined as

$$X_j^s=\left[\begin{array}{cc}{b}_1{X}_j^{11s}& b_2{X}_j^{11s}I\\ X_j^{21s}& X_j^{22s}\end{array}\right],$$

where I represents an appropriately dimensioned identity matrix, ensuring compatibility across the matrix dimensions. This matrix allows for flexibility in adjusting the controller gains by modifying the constants $b_1$ and $b_2$. Adjusting these constants can enhance the system performance by optimizing the response of the controller to changes in system states and disturbances. Therefore, careful selection of $b_1$ and $b_2$ is recommended to tailor the controller performance to specific operational requirements, thereby potentially overcoming the conservatism associated with standard configurations.

(ii) 

Theorem 2 provides a direct correlation between the expansion of fuzzy rules and MJ modes as well as the subsequent expansion in computing complexity. In this article, the linear matrix inequalities (LMIs) matrices are used to calculate the controller gains, in which the corresponding algorithm dealing with the LMIs has polynomial time complexity. Fortunately, with the increasing power of computers and accelerating advances in LMI optimization techniques, the complex LMIs are well solved.

5. Empirical Applications

This section illustrates the feasibility and merits of the proposed methodology by employing two applications.

5.1. Application I: Control of a Tunnel Diode Circuit

The dynamics of SPSs pose important challenges in control theory, particularly under conditions involving cyber-physical threats and the necessity for efficient communication within networked control systems. Next, we illustrate the application of an advanced control strategy employing IT2 fuzzy models within an MJ framework to manage uncertainties and enhance both the reliability and resilience of networked systems. The circuit selection and characteristics are as follows. The tunnel diode is chosen for its unique electrical properties, ideal for applications in real-world industrial scenarios characterized by frequent operational fluctuations. Tunnel diodes exhibit a phenomenon known as negative differential resistance, where the current decreases as the voltage increases within a specific range. This phenomenon is due to the quantum tunneling effect that allows electrons to pass through a potential barrier at low voltages. The nonlinear current-voltage (I–V) relationship of tunnel diodes is crucial for their application in circuits where rapid response to voltage changes is needed and less common in other semiconductor devices. This nonlinear response is vital for evaluating control strategies under dynamic conditions, where system performance is susceptible to unpredictable parameter variations and cyber-threats.

The parameters used in our model are grounded in empirical data and circuit specifications. Specifically, the values for inductance, capacitance, and resistance were chosen based on typical characteristics of commercially available tunnel diodes and components used in high-speed circuit applications. The I–V characteristics of the tunnel diode, described by the parameter ${\alpha }_0$, were derived from experimental measurements and are documented in the literature. This ensures that our model accurately reflects the behavior of real-world tunnel diodes, providing a reliable basis for evaluating the proposed control strategies.

The mathematical representation of I–V characteristics is as follows. The current $i_D\left(t\right)$ through the diode as a function of the voltage across it, $v_D\left(t\right)$ say, is empirically derived to capture the tunnel diode dynamic behavior, particularly its ability to handle rapid changes in voltage. This relationship is described by $i_D\left(t\right)=0.002v_D\left(t\right)+{\alpha }_0{v}_D^3\left(t\right)$, where the parameter ${\alpha }_0$, ranging from $0.001$ to $0.004$, quantifies the uncertainty in the model and is based on empirical observations.

To accurately simulate the behavior of the tunnel diode, specific values for inductance and capacitance are selected: (i) inductance, $L=0.1\mathrm{mH}$, and (ii) capacitance, $C=20\mathrm{mF}$. These values help to model the temporal response of the circuit to external stimuli and enhance the fidelity of simulations in dynamic and potentially adverse environments. The time-varying resistance, denoted by R, complements the diode rapid response characteristics by facilitating an accurate representation of the system dynamic response to voltage and current variations.

To evaluate the control strategy in a digital setting, we discretized the continuous model formulated previously. The discrete-time model uses state variables $x_1\left(k\right)$ and $x_2\left(k\right)$ to represent the capacitor voltage and inductor current, respectively. The model is given by

$$x_1(k+1)=\gamma ({\alpha }_0,x_1\left(k\right))x_1\left(k\right)+x_2\left(k\right),x_2(k+1)=-x_1\left(k\right)-x_2\left(k\right)+\delta \left(\rho \left(k\right)\right)u\left(k\right),$$

where $\gamma ({\alpha }_0,x_1\left(k\right))=0.002+{\alpha }_0{x}_1^2\left(k\right)$ models the nonlinear I-V characteristics affected by ${\alpha }_0$. Additionally, $\delta \left(\rho \right(k\left)\right)$ adjusts the control input $u\left(k\right)$ based on the operational state of the system, which is governed by a Markov process. The operational states $\delta \left(1\right)=1$ and $\delta \left(2\right)=0.75$ represent full and reduced functionality, respectively, reflecting potential actuator failures. These states are part of the MJ system described in (7), illustrating how system dynamics are integrated with control inputs to manage state transitions effectively.

Given the operational constraints of the system, which limit the state variable $x_1\left(k\right)$ such that $|x_1\left(k\right)|\le 3$, the sector nonlinearity technique is employed to approximate the nonlinear behavior exhibited by the tunnel diode circuit. This technique transforms the nonlinear circuit model into an equivalent IT2 fuzzy singularly perturbed model, similar to the parameterization of system matrices described in (5) and (6).

The linear components of this transformed model are configured to adapt to the fuzzy rules and Markov transitions, mirroring the approach used in our TS fuzzy model, for $p\in \{1,2\}$, stated as

$$A_1^p=\left[\begin{array}{cc}1.0017& 0.3711\\ 0.5344& 0.6902\end{array}\right],A_2^p=\left[\begin{array}{cc}1.0108& 0.2177\\ 0.1121& 1.0355\end{array}\right],$$

$$B_{21}^1=\left[\begin{array}{c}0.5046\\ 0.0391\end{array}\right],B_{22}^1=\left[\begin{array}{c}0.37845\\ 0.029325\end{array}\right],B_{23}^2=\left[\begin{array}{c}0.2944\\ 0.0236\end{array}\right],B_{24}^2=\left[\begin{array}{c}0.2208\\ 0.0177\end{array}\right].$$

where the matrices are parameterized in accordance with the current Markov state, $\rho \left(k\right)$ namely, influencing how the system dynamics are linearly combined with control inputs under varying operational conditions, as discussed in Section 3.3.

While the parameters used in our model are grounded in empirical data and circuit specifications, due to failures and attacks which affect the system, we assume that the input matrix $B_{2ip}$ may randomly change according to a Markov process, with operational states $\delta \left(1\right)=1$ and $\delta \left(2\right)=0.75$ representing full and reduced functionality, respectively.

In our methodology, to control the tunnel diode circuit we adapt techniques from the TS fuzzy model framework.

Similar to the dynamic calculation of firing strengths based on system states and uncertainties in the TS model stated in Section 3.5, we define the lower and upper membership function bounds, ${\underline{\varrho }}_i$ and ${\overline{\varrho }}_i$ say, based on the nonlinear characteristics captured by $\gamma ({\alpha }_0,x_1\left(k\right))$. This ensures that our IT2 fuzzy controller dynamically adjusts its strategy to optimally handle the nonlinear and uncertain dynamics of the tunnel diode, as seen in the calculations presented in (15). The membership functions ${\underline{\vartheta }}_i$ and ${\overline{\vartheta }}_i$, specific to the state variables, enable precise modulation of the controller response, ensuring robust control even under high system variations. This method mirrors the adaptive response strategies employed in the TS fuzzy model, where rule contributions are adjusted according to their relevance and the current system conditions. To adeptly model these nonlinear dynamics and uncertainties within the fuzzy logic framework, we proceed to define the lower and upper bounds of the membership functions as

$${\underline{\varrho }}_1\left(x\left(k\right)\right)=\frac{{\gamma }_{\max }-\gamma (0.04,x_1\left(k\right))}{{\gamma }_{\max }-{\gamma }_{\min }},{\overline{\varrho }}_1\left(x\left(k\right)\right)=\frac{{\gamma }_{\max }-\gamma (0.01,x_1\left(k\right))}{{\gamma }_{\max }-{\gamma }_{\min }}$$

$${\underline{\varrho }}_2\left(x\left(k\right)\right)=\frac{\gamma (0.01,x_1\left(k\right))-{\gamma }_{\min }}{{\gamma }_{\max }-{\gamma }_{\min }},{\overline{\varrho }}_2\left(x\left(k\right)\right)=\frac{\gamma (0.04,x_1\left(k\right))-{\gamma }_{\min }}{{\gamma }_{\max }-{\gamma }_{\min }}.$$

By establishing the bounds in the expressions above, the controller can adjust its strategy based on the system current state, enhancing its adaptability and response to nonlinearities and uncertainties. The IT2 fuzzy logic controller is modeled through the membership functions specific to each state variable in the vector $x\left(k\right)$, whose elements are given by

$${\underline{\vartheta }}_{1i}\left(x\left(k\right)\right)=0.8exp(-x_i^2),{\overline{\vartheta }}_{1i}\left(x\left(k\right)\right)={\underline{\vartheta }}_{1i}\left(x\left(k\right)\right),$$

$${\underline{\vartheta }}_{2i}\left(x\left(k\right)\right)=1-{\underline{\vartheta }}_{1i}\left(x\left(k\right)\right),{\overline{\vartheta }}_{2i}\left(x\left(k\right)\right)={\underline{\vartheta }}_{2i}\left(x\left(k\right)\right),$$

where i indexes the components of $x\left(k\right)$. These functions provide the controller with a flexible mechanism to adeptly manage the system dynamics and the uncertainties it encounters, applying tailored control strategies to each dimension of the state vector.

Building upon the Markov framework outlined in Section 3.1 and Section 3.2, this control model leverages the TP matrices ${\Pi }_1$ and ${\Pi }_2^{\rho (k+1)}$ to deal with the operational states of actuators, aligning with the Markov dynamics previously discussed. These matrices, elaborated in Section 3.3 and Section 3.8, facilitate a proactive and dynamic response to the actuator current conditions, thereby enhancing the system resilience to uncertainties and external perturbations. The model incorporates TPs to manage Markov transitions between different states of the actuators. These transitions are defined within the stationary TP matrix ${\Pi }_1$ and the piecewise-stationary matrices ${\Pi }_2^{\rho (k+1)}$ given in (14), which are structured as

$${\Pi }_1=\left[\begin{array}{cc}0.7& 0.3\\ 0.55& 0.45\end{array}\right],{\Pi }_2^1=\left[\begin{array}{cc}0.4& 0.6\\ 0.8& 0.2\end{array}\right],{\Pi }_2^2=\left[\begin{array}{cc}0.85& 0.15\\ 0.5& 0.5\end{array}\right],$$

providing a representation of the actuator state dynamics under different operational conditions. After introducing IT2 fuzzy models and the MJ framework for tunnel diode circuit complexities, we move to test our control strategy, checking its reliability and resilience. We now proceed to the practical implementation phase, starting with Case I. This phase is designed to test the resilience and adaptability of our control strategy against cyber-physical threats, with a particular focus on the CB-DETM.

Case I.

Next, we explore the strategic development of the CB-DETM controller, designed to protect the system from sensor and actuator deception attacks. This case demonstrates the practical application of IT2 fuzzy models and the MJ framework, highlighting the challenges and implications of applying our methodology in real-world scenarios. Furthermore, this case bridges the gap between theoretical foundations and practical applications, underscoring the relevance of our research in addressing contemporary cyber-security challenges in control systems.

To optimize the performance of the CB-DETM controller, we select key design parameters (see Section 3.7). These parameters include $\Omega =0.8$, reflecting the system sensitivity to changes; $G=\mathrm{diag}\{0.3,0.4\}$, outlining the feedback loop gain matrix; $H_0=1$, which sets the threshold for event triggering; and ${\sigma }_0=0.5$, quantifying the cyber-attack impact strength. These selections enable the CB-DETM to dynamically adapt its response, ensuring the system stability and effective performance under various attack conditions.

Table 1. Parameters of the DETP for each sensor.

Parameter	$\mathit{\sigma }$	$\mathit{\theta }$	$\mathit{\lambda }$	${\mathit{\eta }}_0$
Sensor 1	0.5	3	0.8	5
Sensor 2	0.4	4	0.6	5

details specific parameters for each sensor within the system. The parameter $\sigma$ represents the sensitivity related to the sensor measurements, affecting how the sensor perceives changes in its environment. The parameter $\theta$ is a specific adjustment factor for each sensor, impacting their sensitivity to detected changes. The parameter $\lambda$ is another adjustment factor, influencing the rate at which the sensor updates or responds to changes, while ${\eta }_0$ is a baseline parameter that applies to both sensors representing an initial state used for calibration and consistency across different operational scenarios.

The parameters in Table 1 are chosen based on theoretical insights, empirical evidence, and simulations. While some choices are common in similar applications, experimentation and adjustment are crucial. Different scenarios may require parameter optimization to enhance resilience against cyber-threats. Thus, we recommend periodic re-evaluation and optimization of these parameters based on updated research and system performance.

To calculate the controller gain matrices, we employ Yalmip, a MATLAB toolbox designed for optimization, alongside Mosek, a solver renowned for its efficiency and precision in tackling large-scale optimization challenges. Our methodology facilitates the precise fine-tuning of the CB-DETM response, ensuring it aligns with the system operational demands in the face of simulated deception attacks. Using Yalmip and Mosek, we derive the controller gain matrices as $K_1^1=-0.9303$, $K_2^1=-1.1696$, $K_1^2=-0.93036$, and $K_2^2=-1.1694$. To simulate realistic cyber-physical threats, we define models for sensor and actuator attacks. The sensor attack model, ${\delta }_s\left(k\right)$, simulates disruptions in sensor readings and is defined as

$$\begin{array}{c}\hfill {\delta }_s\left(k\right)=\left\{\begin{array}{cc}0,\hfill & k<1;\hfill \\ 0.1kx,\hfill & 1\le k\le 3;\hfill \\ -bsin\left(10k\right)\sqrt{x_1^2+x_2^2+0.1},\hfill & 13\le k\le 16;\hfill \end{array}\right.\end{array}$$

where $b=0.5$ is the magnitude of the attack, incorporating both continuous and intermittent disruptions within a simulation period. The actuator attack signal, ${\delta }_a\left(k\right)$, represents adversarial actions manipulating actuator behavior. For every time step $k\ge 0$, the actuator attack model is expressed as ${\delta }_a\left(k\right)={\sigma }_{0}tanh(-3u\left(k\right))cos\left(1000k\right)$. This model helps to analyze how external manipulations could potentially alter the behavior of the system actuators. Furthermore, the probabilities of encountering sensor and actuator attacks are assigned as $\overline{\zeta }=0.25$ and $\overline{\Gamma }=0.45$, respectively. These values reflect our assumptions about the probability of attacks occurring within the system operational environment.

Setting the initial condition of the system as $x\left(0\right)={[-2.5,-1.5]}^{\top }$, 50 random independent simulations were conducted to robustly test the system resilience. This simulation allows us to observe the system behavior under a variety of attack scenarios.

Our numerical studies validated the effectiveness of our proposal, demonstrating important improvements in system performance. The results indicated a 20% reduction in network load and enhanced stochastic stability under varying conditions and cyber-threats. Figure 3 displays the results of these simulations, focusing on the closed-loop system performance with the applied control strategy, as defined in (12).

Figure 3 shows the system behavior with the CB-DETM control strategy implemented, illustrating the system stability and resilience across simulated attack scenarios, highlighting the effectiveness of our proposal in maintaining operational integrity and mitigating cyber-attack impacts. The figure shows that, despite the presence of deception attacks targeting both sensors and actuators, the system remains stable over time. This stability shows the effectiveness of our control strategy.

Figure 4 displays the open-loop response of the system without the CB-DETM, serving as a baseline for comparison. This figure provides a clear reference point to evaluate the essential improvements and benefits obtained by implementing the control strategy, illustrating the system vulnerability to cyber-physical attacks in the absence of advanced control mechanisms.

By comparing the outcomes in Figure 3 and Figure 4, the success and efficiency of the implemented control approach in mitigating the effects of cyber-physical attacks are evident. Through such an analysis, we can clearly see how our strategy ensures the system robustness and operational integrity, even in the face of sophisticated cyber-threats. To evaluate our strategy against cyber-physical threats, we show the dynamics of the controller using simulations.

Figure 5 sketches timelines of (a) a sequence of system and asynchronous controller modes, illustrating the state transitions and control actions, and (b) deception attacks encountered during the simulations, indicating when and how the system is targeted. This helps to understand the frequency and nature of these threats, as well as the temporal distribution and impact of these attacks, providing insights into the adaptability and responsiveness of the system to conditions induced by cyber-threats, illustrating the system and controller abilities to maintain operational integrity amidst such threats.

Figure 6 focuses on the instants and behavior of internal variables, ${\eta }_1\left(k\right),{\eta }_2\left(k\right)$ say, as governed by the adaptive equation stated in (9), showing the moments when the controller decides to transmit signals from sensors to actuators, which is essential for managing communication while ensuring stability. From Figure 6, we find that sensor 1 transmits 48 signals, achieving a transmission rate of 19.12%. Similarly, sensor 2 transmits 74 signals, resulting in a transmission rate of 29.48%. These findings show the efficacy of our strategy in optimizing signal transmission under varying operational conditions and cyber-threats.

The results garnered from the simulations confirm the reliability and resilience of our methodology. By managing communication between sensors and actuators, the system demonstrates resilience to cyber-physical attacks, ensuring stable and efficient operation even under adversity. After validating the effectiveness of the CB-DETM control strategy against cyber-physical threats through simulations, our attention now turns to Case II.

Case II.

Now, we explore the feasibility of constructing a dynamic event trigger (DET)-based control system for components with identical triggering parameters, as discussed in Remark 1 and outlined in [52]. We analyze two distinct scenarios characterized by the minimum and maximum values of the parameters. The outcomes of this investigation are succinctly summarized in

Table 2. Results of DET-based control.

	Minimum Values	Maximum Values
Gains	$\begin{array}{cccc}& {\overline{K}}_1^1=-0.82877\hfill & & {\overline{K}}_2^1=-1.0369,\hfill \\ & {\overline{K}}_1^2=-0.82852\hfill & & {\overline{K}}_2^2=-1.037\hfill \end{array}$	$\begin{array}{cccc}& {\overline{K}}_1^1=0.92125\hfill & & {\overline{K}}_2^1=-1.1582,\hfill \\ & {\overline{K}}_1^2=-0.92227\hfill & & {\overline{K}}_2^2=-1.1581\hfill \end{array}$
Number of triggering	73	76
Communication rate	29.08%	30.27%

. The DET-based control system effectively maintains communication efficiency and control gains by applying uniform triggering parameters. The data, particularly from the comparison with the protocol, show that sensor 1 could remain inactive for up to 28 sample periods. This inactivity indicates the ability of our methodology to reduce redundant communications without affecting system performance. The results from Case II highlight the benefits of uniform triggering parameters in a DET-based control system, pointing to opportunities for enhancing the management of communications within cyber-physical systems, aiming for greater efficiency and optimization.

Case III.

In this case, we evaluate the resilience of the proposed methodology by increasing the intensity of cyber-attacks. The actuator attack intensity is increased by a factor of 15, and the sensor by a factor of 20. Figure 7 illustrates the system state trajectories under these intensified attacks, demonstrating that stability is maintained despite the escalated threat level.

Furthermore, we compare our control strategy with the scheme developed in [17] to emphasize the superiority of our methodology. Implementing the control scheme presented in [17] under identical initial conditions and system parameters results in diminished system efficacy, as evident from Figure 8. Given the advancements demonstrated in the present study, our control methodology notably outperforms the strategy presented in [17], especially in managing increased attack intensities. This superior performance highlights the relevance of event-triggered mechanisms in bolstering system resilience against complex cyber-threats, contributing to the field of adversarial-condition control systems.

Cases I, II, and III collectively underscore the effectiveness and superiority of our proposed methodology. Case I establishes the foundational effectiveness of the CB-DETM control strategy. Case II further validates the methodology under uniform triggering parameters, showing improved communication efficiency. Case III confirms the system robustness against strongly intensified attacks. Armed with the conclusive evidence from these investigations, we advance to explore additional applications of our methodology.

5.2. Discussion and Limitations

While the numerical results validate the effectiveness of the proposed CB-DETM control strategy, several limitations must be acknowledged. First, the simulations assume ideal conditions that may not fully capture the complexities and uncertainties present in real-world applications.

Factors such as sensor noise, communication delays, and hardware limitations are not explicitly considered in our simulations. Furthermore, the chosen parameters for the simulations, although based on empirical data and realistic specifications, may not cover the entire range of possible operational scenarios.

Therefore, the robustness of the proposed methodology should be further tested under more diverse and challenging conditions to ensure its applicability in a wider array of real-world situations. The potential real-world applications of this research are relevant, particularly in areas requiring high reliability and resilience against cyber-physical threats, such as industrial control systems, smart grids, and autonomous vehicles.

The ability to maintain system stability and performance under adversarial conditions is crucial for the safety and efficiency of these systems. In comparison with existing methods, the proposed CB-DETM methodology offers enhanced reliability and resilience.

Unlike traditional control strategies, which may not effectively handle the dynamic nature of cyber-attacks and operational uncertainties, our methodology leverages the flexibility of IT2 fuzzy logic and the dynamic adaptation capabilities of MJ systems. This allows for a more reliable and resilient control mechanism, as demonstrated by the superior performance observed in our simulations compared to the control scheme presented in [17]. Future work should focus on addressing the identified limitations by incorporating more realistic system models and exploring the integration of additional factors such as network constraints and multi-agent coordination. This will help to validate the applicability of the proposed control strategy and enhance its effectiveness in real-world scenarios.

5.3. Controller Performance Results

The proposed state feedback controller was implemented, and its performance was evaluated using various metrics. The results demonstrated important improvements in system stability and responsiveness under different operational conditions. The figures presented next illustrate the effectiveness of the state feedback controller. Figure 9a shows the system response with damped oscillations in the state variables. Figure 9b displays the dynamic variation in the control input. Figure 9c highlights the periods where the system is subject to sensor and actuator attacks. These results validate the reliability and resilience of the controller in managing the dynamic behavior and maintaining stability under different operational conditions and attack scenarios.

Figure 9 complements Figure 3 by providing a detailed analysis of the simulated attack scenarios and the system response to these attacks. While Figure 3 focuses on the overall closed-loop performance of the system, Figure 9 breaks down the specific effects of the sensor and actuator attacks, as well as the dynamic variations in the control input and system state variables. This detailed analysis is crucial for demonstrating the controller effectiveness under attack conditions, thereby validating its reliability and resilience in maintaining system stability and performance in the presence of cyber-physical threats.

5.4. Application II: Another Operational Context

Next, we evaluate the effectiveness of our control system of an IT2 fuzzy SPS in a distinct operational context. This evaluation demonstrates the reliability and resilience of our system, facilitating a comparison with the methodologies given in [18].

Through such an evaluation, we aim to highlight the robustness and adaptability of our methodology under varying conditions. The system parameters and configuration for this illustration are specified as

$$\begin{array}{cc}\hfill & A_1=\left[\begin{array}{cc}0.8& 0.3\\ 0.1& 1.0\end{array}\right],A_2=\left[\begin{array}{cc}1.1& 0.4\\ 0.2& 1.0\end{array}\right],A_{21}=\left[\begin{array}{c}0.1\\ 0.2\end{array}\right],A_{22}=\left[\begin{array}{c}0.1\\ 0.5\end{array}\right].\hfill \end{array}$$

Considering the range of $x_1\left(k\right)\in [-2,2]$, under varying conditions, we define the lower and upper bounds of the membership functions to model the system stated as

$${\underline{\varrho }}_1\left(x\left(k\right)\right)=\frac{0.2(x_1^2\left(k\right)+2)}{3},{\overline{\varrho }}_1\left(x\left(k\right)\right)=\frac{0.2(3x_1^2\left(k\right)+2)}{3},{\underline{\varrho }}_2\left(x\left(k\right)\right)=1-{\overline{\varrho }}_1\left(x\left(k\right)\right),{\overline{\varrho }}_2\left(x\left(k\right)\right)=1-{\underline{\varrho }}_1\left(x\left(k\right)\right).$$

Functions are crucial for tailoring the IT2 fuzzy controller as

$${\underline{\vartheta }}_1\left(x_1\right)=0.6{\underline{\varrho }}_1\left(x\left(k\right)\right)+0.4{\overline{\varrho }}_1\left(x\left(k\right)\right),{\underline{\vartheta }}_2\left(x_1\right)=1-{\underline{\vartheta }}_1\left(x_1\right),{\overline{\vartheta }}_2\left(x_1\right)={\underline{\vartheta }}_2\left(x_1\right).$$

For a comprehensive understanding, the dynamic event-triggered parameters for each sensor, as defined in the comparative study given in [18], are listed in

Table 3. Parameters of the dynamic event trigger for each sensor.

Parameter	$\mathit{\sigma }$	$\mathit{\theta }$	$\mathit{\lambda }$	${\mathit{\eta }}_0$
Sensor 1	0.2	10	0.4	0.1
Sensor 2	0.1	8	0.5	0.2

. These parameters are similar to those listed in Table 1, which detail the parameters for each sensor in a different operational context. This helps to compare the performance and adaptability of the system under varying conditions.

The model for a sensor attack, ${\delta }_s\left(k\right)$ say, simulates cyber-threats affecting sensor readings, described as ${\delta }_s\left(k\right)=0.2\sqrt{\parallel 2x_1{x}_2\parallel }$. Given the parameters $\overline{\epsilon }=0.01$, $\Omega =0.6$, $G=\mathrm{diag}\{0.2,0.2\}$, $H_0=1$, and ${\sigma }_0=0.1$, we apply Theorem 2 to find a viable control solution. This results in the gain matrices: $K_1^1=-11.3217$, $K_2^1=-15.8528$, $K_1^2=-11.3198$, and $K_2^2=-15.8528.$ The probabilities of sensor and actuator attacks are set to $\overline{\zeta }=0.3$ and $\overline{\Gamma }=0.2$, respectively. The dynamics of $\zeta \left(k\right)$ and $\Gamma \left(k\right)$, illustrated in Figure 10, show the system behavior under cyber-attacks with the initial condition $x\left(0\right)={[-2,-1.5]}^{\top }$. The simulation results demonstrate the stability and performance of the system under the CB-DETM controller, even in the presence of cyber-attacks. The system states $x_1\left(k\right)$ and $x_2\left(k\right)$ maintain their trajectories effectively, while the input $u\left(k\right)$ adapts to the changes induced by the attacks. This illustrates how the controller manages the probabilistic nature of the attacks. The states $x_1\left(k\right)$ and $x_2\left(k\right)$ achieved triggering numbers of 27 and 29, respectively, indicating that the communication and control mechanisms are functioning efficiently to mitigate the effects of the cyber-attacks.

To highlight the superiority of our methodology, we employ the control method suggested in [18], which is rooted in the sliding mode control scheme. This method utilizes the gains $K_1=[1.1706,-40553]$ and $K_2=[-1.1703,-3.5647]$. We compare the performance of the method stated in [18] under identical initial conditions and system parameters in Figure 11. This comparison shows the enhanced reliability and resilience of our methodology, particularly when evaluating system performance under equivalent operational scenarios. However, note that the method given in [18] did not consider the complexities introduced by MJs and cyber-attacks, which are critical aspects of our study. Thus, we conclude that our strategy offers important advantages when handling the uncertainties and dynamics introduced by MJs and cyber-attacks over the method suggested in [18].

Throughout the present study, we have undertaken a thorough examination of an advanced control strategy designed to enhance the reliability and resilience of systems facing cyber-physical threats. Application I, with its cases I, II, and III, as well as Application II, have clearly demonstrated the effectiveness of our proposal in mitigating diverse and intensified cyber-attacks. The exploration began with a tunnel diode circuit application, showing the methodology capability to handle unexpected parameter variations and sophisticated cyber-threats.

6. Conclusions and Future Work

This article presented a control methodology for discrete-time Markov jump singularly perturbed systems under cyber-physical threats, integrating a dynamic event-triggered protocol with interval type-2 fuzzy models. Our methodology effectively handled system uncertainties and cyber-attacks, demonstrating high reliability and resilience improvements. Key contributions of our study included:

• Robust framework integration—It effectively handled uncertainties and dynamic changes, critical for managing stochastic cyber-attacks.
• Enhanced communication efficiency—It optimized data transmission, reducing unnecessary communications while maintaining stability.
• Resilience under cyber-attacks—It maintained system stability under intense attacks.
• Comparative superiority: It demonstrated enhanced stability and robustness compared to existing methods.
• Adaptive control mechanisms—It dynamically adjusted control parameters in response to real-time conditions, ensuring resilience and operational integrity.

The innovation of our methodology lies in the comprehensive integration of interval type-2 fuzzy logic and Markov jump systems, coupled with a dynamic event-triggered protocol. This integration enabled real-time adaptation to changing conditions and robust handling of uncertainties and cyber-physical threats, setting our methodology apart from traditional control strategies. Unlike standard techniques, our methodology dynamically adjusted control parameters and communication strategies based on real-time system states and operational conditions, offering a higher level of reliability and resilience.

Future research should explore machine learning techniques to optimize event-triggered parameters dynamically, enhancing network resource utilization and system response. Investigating semi-Markovian jump systems and integrating fractional calculus and fractal dynamics could provide more realistic modeling and control of complex systems under nonstationary conditions. Further investigation should address scalability and applicability across different systems and industries to ensure robustness against evolving cyber-threats. Additionally, while our study guarantees asymptotic stability in the mean-square sense using a mode-dependent Lyapunov function technique, exploring finite-time stability, as suggested in [48], could offer practical advantages in scenarios where rapid stabilization is critical.

In conclusion, this study addressed important gaps that are present in the literature on the topic controlling Markov jump singularly perturbed systems under cyber-physical threats, providing a foundation for future advancements. Ongoing innovation is essential to bridge the gap between theoretical developments and practical applications in safeguarding interconnected systems.