Resilient adaptive fuzzy SVM control design for uncertain nonlinear Markov jump cyber-physical systems against replay attacks

Abstract

This paper presents a novel control strategy framework using adaptive fuzzy support vector machine (FSVM) for enhancing the detection and resilience of Uncertain Nonlinear Markov Jump Cyber-Physical Systems (UNMJCPS) under replay attacks. At first, an introduction to FSVM method is given. Then, the dynamic of UMJCPS with replay attack is presented. In such condition, a resilient adaptive FSVM control mechanism for stabilization of system in condition of abnormal behavior due to replay attacks is proposed. The proposed new approach ensures system stability and performance during and after the attack. Simulation results demonstrate the effectiveness of the control design in mitigating the effects of replay attacks and maintaining system functionality.

Keywords

Adaptive fuzzy SVM control uncertain nonlinear Markov jump cyber-physical systems replay attacks

Introduction

Cyber-Physical Systems (CPS) have become an integral part of modern infrastructures, with applications in sectors such as transportation, energy, healthcare, and industrial automation. These systems seamlessly integrate computational elements with physical processes, enabling real-time data collection, analysis, and control (Duo et al., 2022; Tan et al., 2020). Because many real-world phenomena have stochastic, uncertain and nonlinear nature, Uncertain Nonlinear Markov Jump Cyber-Physical Systems (UNMJCPS) has attracted many attentions in dynamical modeling of CPS. UNMJCPSs are challenging to analyze and control due to the interplay of nonlinearity, Markovian switching, and uncertainties. Specifically, the nonlinearity introduces complexities in stability analysis and controller design, while the Markovian switching introduces randomness and requires handling the system’s evolution across different modes. These systems characterized by discrete modes that switch according to a Markov chain, involve both continuous dynamics within each mode and stochastic switching behavior. On the other hand, these systems are often subject to cyber-attacks, uncertainties, and external disturbances, making their analysis and control challenging.

In general, cyber-attacks imposed on CPS are divided into three general categories: (1) false data injection (FDI) or deception attack, (2) denial of service (DoS) attack, and (3) Replay attack. FDI attacks alter the integrity of data, replay attacks retransmit captured data to deceive systems, and DoS attacks overwhelm systems to make them unavailable. FDI attacks compromise the accuracy of data, while DoS attacks aim to disrupt availability. Replay attacks can be used to compromise both, depending on the context. In this study, replay attack will be considered in the dynamic of UNMJCPS.

As an example for replay attack, in this attack, previous recorded valid sensor data is re-injected into the system, causing the controller to operate based on outdated information, potentially leading to catastrophic consequences (Naha et al., 2022). Replay attacks can have devastating consequences on CPS performance, leading to significant operational disruptions, resource wastage, and even safety hazards. If such cyber threats are not effectively countered, they can lead to significant degradation in system performance and potentially cause severe consequences. A common other example of a replay attack is seen in the remote keyless entry systems used in vehicles (Merco et al., 2018). These systems are designed to enhance user convenience but are vulnerable to this type of attack. By positioning a device that can intercept and send radio signals near the vehicle, the attacker disrupts the legitimate unlocking signal, stores it, and replays an earlier signal. This creates a sequence that stays ahead of the vehicle’s system, allowing the attacker to unlock the vehicle using the stored code. Traditional detection mechanisms, such as anomaly detection and signature-based methods, often fall short in identifying such attacks, especially in nonlinear and stochastic environments that characterize many real-world CPS (Makkar and Jong, 2022; Chu et al., 2024).

Based on aforementioned explanations, because of importance of subjects of MJCPS and cyber-attacks, recently, some researchers have studied in this area. For instance, in Jiyang Wang et al. (2025b), observer-based asynchronous sliding mode control (SMC) strategy has been proposed for MJCPS against DoS attack. Deteng Wang et al. (2025a) proposed dynamic event-triggering asynchronous dissipative control for MJCPS with DoS and deception attack. In Ye et al. (2023), event-triggered-based control for MJCPS against DoS attacks has been investigated. Cao et al. (2024) proposed observer-based adaptive neural H∞ synchronous control for fuzzy MJCPS under FDI attacks. Xu et al. (2023) considered the security-based passivity problem for a class of discrete-time MJCPS in the presence of deception attacks. In Zeng et al. (2022), the event-triggered resilient L∞ control problem for MJCPS in the presence of DoS jamming attacks has been discussed. Neural network (NN)-based event-triggered control problem for discrete-time MJCPS with DoS and deception attacks has been considered in Gao et al. (2023). Wang et al. (2024) discussed the problem of composite H∞ control for hidden Markov jump systems subject to replay attacks. Dynamic-event-based reachable set synthesis for nonlinear delayed MJCPS has been developed in Zhang et al. (2025). Yao et al. (2024) investigated the security control of a class of discrete-time Markov jump systems (DMJS) against DoS and deception attack via switching-Like Adaptive Law. In Liu et al. (2023), security control for MJCPS against actuator failures, FDI attack, and inaccessible states by virtue of state estimator-based adaptive SMC strategy has been developed. Gu et al. (2025) developed dynamic event-triggered H∞ filtering for fuzzy Markov Jump systems subject to mismatched quantization that compromises the data integrity of system.

As per examples of the mentioned papers in the fields of MJCPS and cyber-attacks, one can find that although this area has been very applicable with high potential of development in modeling and control strategy, research studies in this area are limited and bounded in few methods such as robust, adaptive, or neural networks. To mitigate the impact of cyber-attacks like replay attacks, various control strategies have been developed. Among them, Support Vector Machine (SVM) can be integrated with Markov models to enhance their predictive capabilities. Also, SVMs are a powerful machine learning technique widely used for attack detection in various systems. In addition, Fuzzy SVM (FSVM) offer advantages over traditional SVM by being more robust to noisy data and outliers. FSVM assigns fuzzy membership values to data points, making it less sensitive to noise near the decision boundary. In addition, by adding adaptive term to proposed controller, the effects of unknown nonlinearity of system can be eliminated. On the other hands, adaptive FSVM has advantages against methods that until now has proposed for control of MJCPS (see Remark 3 of this paper). Such issues, motivated us to present this work.

Per aforementioned explanations, this paper proposes an adaptive FSVM controller for UNMJCPS. The innovations of this research are:

Designing an FSVM classifier to enhance the detection of replay attacks in UNMJCPS.

Developing a resilient adaptive FSVM control strategy that ensures UNMJCPS stability and performance during and after an attack. Until know, this subject has not been reported in any paper.

The remainder of this paper is structured as follows: The Preliminaries section introduces required Lemmas and the FSVM framework. The Problem formulation section provides a mathematical model of UNMJCPS. In the Resilient controller design section, the proposed mechanism is proved. The Simulation section presents simulation results to validate the effectiveness of the proposed approach. Finally, the Conclusion and Future work direction sections conclude the paper with a discussion on future research directions and potential improvements.

Preliminaries

Applied lemmas

In this work, we use some Lemmas as:

Lemma 1. Wei et al. (2017): for given positive integers $p, q, a$ scalar δ in intervals $(0, 1)$ , a given positive defined matrix, $R \in R^{p}$ two matrices $M_{1}$ and $M_{2}$ in $R \in R^{p \times q}$ define for all vectors v the function $p, q, a$ given by:

ℑ (δ, R) = \frac{1}{δ} v^{T} {M_{1}}^{T} R M_{1} v + \frac{1}{1 - δ} v^{T} {M_{2}}^{T} R M_{2} v

Then if there exist a matrix $U \in R^{p \times q}$ such that $[\begin{matrix} R & U^{T} \\ * & R \end{matrix}] > 0$ the following holds

\min_{δ \in (0, 1)} ℑ (δ, R) \geq {[\begin{matrix} M_{1} v \\ M_{2} v \end{matrix}]}^{T} [\begin{matrix} R & U^{T} \\ * & R \end{matrix}] [\begin{matrix} M_{1} v \\ M_{2} v \end{matrix}],

Lemma 2. Zhang et al. (2017): For given matrices $D, E (t)$ and F of appropriate dimension with $D, E (t)$ satisfying $E^{T} (t) E (t) \leq I$ for any $ε > 0$ the following inequality holds:

DE (t) F + FTET (t) DT \leq ε D D_{T} + ε^{- 1} F^{T} F

Lemma 3. Zhang et al. (2014): the following two inequalities are equivalent:

(a) There exist a symmetric and positive definite matrix P satisfying:

[\begin{matrix} - P & A^{T} \\ * & - P^{- 1} \end{matrix}] < 0,

(b) There exist a symmetric and positive definite matrix P and matrix Y satisfying:

[\begin{matrix} - P & {(YA)}^{T} \\ * & He (- Y) + P \end{matrix}] < 0,

Support vector machine

Consider a set S of labeled training points $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})$ , where each point $x_{i} \in R^{n}$ is assigned a label $y_{i} \in {- 1, 1}$ and belongs to one of two classes. Often, finding an appropriate hyper-plane directly in the input space is too restrictive for practical purposes. To overcome this limitation, the input space is mapped to a higher-dimensional feature space where we search for an optimal hyper-plane. Let $z = ϕ (x)$ denote the feature space vector, where ϕ is a mapping from $R^{n}$ to the feature space Z. The goal is to find a hyper-plane described by

w^{T} z + b = 0

(1)

where w and b define the hyper-plane, allowing us to classify points based on the function

f (x) = sign (w^{T} z + b)

(2)

with $w \in Z$ and $b \in R$ . The set S is linearly separable if there exist values of w and b such that (Liu, 2021)

{\begin{cases} y_{i} (w^{T} z_{i} + b) \geq 1, i f y_{i} = 1 \\ y_{i} (w^{T} z_{i} + b) \leq - 1, i f y_{i} = - 1 \end{cases}

(3)

These inequalities must hold for all elements in S. For linearly separable data, it is possible to find a unique optimal hyper-plane that maximizes the margin between the projections of points from the two classes. When the data set S is not linearly separable, classification errors are allowed, and the SVM approach must be modified. This generalization introduces non-negative variables $ξ_{i} \geq 0$ , modifying the inequalities to

y_{i} (w^{T} z_{i} + b) \geq 1 - M_{i}

(4)

Non-zero values of $M_{i}$ indicate points that do not meet the original separability conditions, and $\sum_{i = 1}^{n} ξ_{i}$ , can be viewed as a measure of the degree of misclassification.

The task of determining the optimal hyper-plane can be structured as follows (Liu, 2021)

\begin{matrix} objective : min (1 / 2 w^{T} w + C) \sum_{i = 1}^{n} ξ_{i}), \\ constrains : y_{i} (w^{T} z_{i} + b) \geq 1 - ξ_{i}, ξ_{i} \geq 0, i = 1, \dots, n \end{matrix}

(5)

Here, C is a regularization parameter that controls the trade-off between maximizing the margin and minimizing classification error. This problem can be solved via the Lagrangian method, which leads to the following dual optimization problem

\begin{matrix} objective : \max W (α) = \sum_{i = 1}^{n} α_{i} - 1 / 2 \sum_{i, j = 1}^{n} y_{i} y_{j} α_{i} α_{j} (x_{i}, x_{j})), \\ constrains : \sum_{i = 1}^{n} y_{i} α_{i} = 0, 0 \leq α_{i} \leq C, i = 1, \dots, n \end{matrix}

(6)

The Kuhn-Tucker conditions are essential in this context. According to these conditions, the solution $α_{i}$ must satisfy (Liu, 2021)

\begin{matrix} α_{i} [y_{i} (w^{T} x_{i} + b) - 1 + ξ_{i}] = 0, i = 1, \dots, n \\ (C - α_{i}) ξ_{i} = 0, i = 1, \dots, n \end{matrix}

(7)

These conditions imply that non-zero $α_{i}$ values correspond only to points that satisfy the constraints exactly. Such points are called support vectors. In the case of non-separable data, there are two types of support vectors: those for which $0 < α_{i} < C$ and $ξ_{i} = 0$ , and those where $α_{i} = C$ and $ξ_{i} > 0$ . The latter represent misclassification errors. Points with $α_{i} = 0$ are correctly classified and lie outside the margin boundary.

To find the optimal hyper-plane, the following expression is used

\bar{w} = \sum_{i = 1}^{n} {\bar{α}}_{i} y_{i} z_{i}

(8)

The bias term b is derived using the Kuhn-Tucker conditions. Thus, the decision function is written as

f (x) = sign ({\bar{w}}^{T} z + b) = sign (\sum_{i = 1}^{n} (α_{i} y_{i} z_{i})^{T} z + b)

(9)

On the other hand, one of the strengths of SVM is that a kernel function $K (0, 0)$ is employed to compute the dot product in the feature space Z, enabling us to work in the transformed space. The kernel function is defined as

{z_{i}}^{T} z_{j} = ϕ (x_{i})^{T} ϕ (x_{j}) = K (x_{i}, x_{j})

(10)

Any function that satisfies Mercer’s theorem can serve as a kernel. For example, the polynomial kernel of degree d is expressed as

K (x_{i}, x_{j}) = (1 + {x_{i}}^{T} x_{j})^{d}

(11)

To create an SVM classifier, the following optimization problem is solved (Oyetade et al., 2022)

\begin{matrix} objective : \max W (α) = \sum_{i = 1}^{n} α_{i} - 1 / 2 \sum_{i, j = 1}^{n} y_{i} y_{j} α_{i} α_{j} K (x_{i}, x_{j})), \\ constrains : \sum_{i = 1}^{n} y_{i} α_{i} = 0, 0 \leq α_{i} \leq C, i = 1, \dots, n \end{matrix}

(12)

The decision function for classifying new data points is given by

f (x) = sign (w^{T} z + b) = sign (\sum_{i = 1}^{n} (α_{i} y_{i} K (x_{i}, y_{i}))^{T} z + b)

(13)

Fuzzy property of input

SVMs are a powerful tool for solving classification problems, but they have some limitations. Traditionally, each training point is assigned to one of two classes, and all points in a class are treated equally. However, in real-world situations, the importance of training points can vary. Some points are more critical for classification, while others, such as noisy data, are less significant.

In practice, a training point may not entirely belong to a single class. For instance, a point might be 90% relevant to one class and 10% irrelevant, or 20% relevant to one class and 80% irrelevant. This introduces the concept of fuzzy membership, represented as $0 < s_{i} \leq 1$ , for each training point $x_{i}$ . The fuzzy membership $s_{i}$ indicates the degree to which a training point belongs to a class, while $(1 - s_{i})$ represents its irrelevance.

To address this, we extend the SVM framework to account for fuzzy membership, creating what is known as Fuzzy Support Vector Machines (FSVMs).

Combination of fuzzy and SVM

Consider a set S of labeled training points with associated fuzzy memberships

(y_{1}, x_{1}, x_{1}), \dots, (y_{n}, x_{n}, s_{n})

(14)

Each training point $x_{i} \in R^{n}$ is assigned a label $y_{i} \in (- 1, 1)$ and a fuzzy membership $0 < s_{i} \leq 1$ . The fuzzy membership $s_{i}$ reflects the importance of the training point $x_{i}$ in the classification problem.

The optimization problem for determining the optimal hyper-plane in this FSVM framework is formulated as (Wang et al., 2022)

\begin{matrix} objective : \max {1 / 2 {‖ w ‖}^{2} + C \sum_{i = 1}^{n} s_{i} ξ_{i}} \\ constrains : y_{i} (w^{T} z_{i} + b) \geq 1 - ξ_{i}, ξ_{i} \geq 0 = 1, \dots, n \end{matrix}

(15)

Here, C is a regularization parameter, and $ξ_{i}$ are slack variables allowing for classification errors. The term $s_{i} ξ_{i}$ represents the weighted error for each training point. To solve this optimization problem, we define the Lagrangian as

\begin{matrix} L (w, b, ξ, α, β) = 1 / 2 ‖ w ‖^{2} + C \sum_{i = 1}^{n} s_{i} ξ_{i} - \sum_{i = 1}^{n} α_{i} (y_{i} (w^{T} z_{i} + b) \\ - 1 + ξ_{i}) - \sum_{i = 1}^{n} β_{i} ξ_{i} \end{matrix}

(16)

The parameters must satisfy the following conditions to find the saddle point of the Lagrangian

\begin{matrix} \frac{dL}{dw} = w - \sum_{i = 1}^{n} α_{i} y_{i} z_{i} = 0, \frac{dL}{db} = - \sum_{i = 1}^{n} α_{i} y_{i} = 0, \frac{dL}{d ξ_{i}} \\ = s_{i} C - α_{i} - β_{i} = 0 \end{matrix}

(17)

To solve the optimization problem, we need to maximize the objective function $W (α)$ , which is defined as

\begin{matrix} objective : \max W (α) = \sum_{i = 1}^{n} α_{i} - 1 / 2 \sum_{i, j = 1}^{n} y_{i} y_{j} α_{i} α_{j} K (x_{i}, x_{j})), \\ constrains : \sum_{i = 1}^{n} y_{i} α_{i} = 0, 0 \leq α_{i} \leq s_{i} C, i = 1, \dots, n \end{matrix}

(18)

The Kuhn-Tucker conditions for the optimization problem are given by

\begin{matrix} {\bar{α}}_{i} [y_{i} ({\bar{w}}^{T} x_{i} + \bar{b}) - 1 + {\bar{ξ}}_{i}] = 0, i = 1, \dots, n \\ (s_{i} C - {\bar{α}}_{i}) {\bar{ξ}}_{i} = 0, i = 1, \dots, n \end{matrix}

(19)

Here, ${\bar{α}}_{i}, {\bar{ξ}}_{i} \geq 0$ . A point $y_{i}$ with ${\bar{α}}_{i} > 0$ is called a support vector. There are two types of support vectors: those with $0 < α_{i} < C$ that lie on the margin, and those with $α_{i} = C$ that are misclassified. In FSVM, the same value of $α_{i}$ can correspond to different types of support vectors due to the fuzzy membership factor $s_{i}$ .

Dependence on fuzzy membership

In standard SVM, the parameter C controls the balance between maximizing the margin and minimizing misclassification errors. A larger C reduces misclassifications but narrows the margin, while a smaller C increases the margin but allows for more misclassifications. In FSVM, adjusting the fuzzy membership values $s_{i}$ allows control over the importance of each training point. A smaller $s_{i}$ makes the corresponding point $x_{i}$ less influential in the training process. While SVM has a single free parameter C, FSVM introduces as many free parameters as there are training points, allowing for finer control over the classification process.

Generating fuzzy memberships

To assign appropriate fuzzy memberships for a given problem, follow these steps: First, set the lower bound for fuzzy memberships. Then, identify the main characteristic of the dataset and link this characteristic to the fuzzy memberships.

For example, in a sequential learning problem, start by establishing a lower bound $σ > 0$ for the fuzzy memberships. Consider time as the key characteristic and define the fuzzy membership $s_{i}$ as a function of time $t_{i}$

s_{i} = f (t_{i})

(20)

where $t_{1} < \dots < t_{k}$ represents the time points at which data points are introduced into the system. Assign the highest importance to the last point $x_{k}$ by setting $s_{k} = f (t_{1}) = 1$ and the lowest importance to the first point $x_{1}$ , with $s_{k} = f (t_{1}) = σ$ . If you want the fuzzy membership to follow a linear function of time, you can use

s_{i} = f (t_{i}) = a t_{i} + b

(21)

By applying boundary conditions, you can determine

a = \frac{1 - σ}{t_{k} - t_{1}}, b = σ - a t_{1}

(22)

For a quadratic membership function, you can use

s_{i} = f (t_{i}) = a (t_{i} - b)^{2} + c

(23)

By applying the boundary conditions, we can derive the following equation for fuzzy membership

s_{i} = f (t_{i}) = (1 - σ) (\frac{t_{i} - t_{1}}{t_{n} - t_{1}})^{2}, + σ

(24)

Problem formulation

Replay attack mechanism

The mechanism of replay attack is clearly shown in Figure 1 (Ilyin, 2025):

As shown in above figure, a replay attack usually follows a three-phase protocol:

Capturing: Here, the assailant ensnares the data while it is being relayed. This data could take several forms, for instance, login credentials, particulars of a transaction, or any other classified information.

Registering: The snared data is enumerated and hoarded by the miscreant. This data is typically ciphered, making it a herculean task to sniff out the attack in this phase.

Reiteration: The inventoried data is re-broadcast afterward. The system that receives the data assumes it to be a valid request culminating in unsanctioned actions.

As the protocol explains, this attack causes data from previous system times to be intentionally applied to the system in the form of online data; as if the attacker had intentionally delayed sending data to the process. For this reason, replay attack can be formulated mathematically as a system-tolerable delay (Franzè et al., 2019; Huang et al., 2020).

UNMJCPS model description

Let the system be modeled by a state vector $x (t) \in R^{n}$ representing the continuous physical state of the system, and let $θ \in 1, 2, \dots, N$ be the discrete mode of the system at time t, governed by a Markov process.

The evolution of the system state depends on the current mode $θ (t)$ , and it can be described by a nonlinear differential equation

\overset{\cdot}{x} (t) = A (t, θ (t)) x (t, θ (t)) + B (t, θ (t)) u (t, θ (t)) + f (x (t), θ (t), t)

(25)

$x (t) \in R^{n}$ : Continuous state of the physical system at time t.

$u (t) \in R^{m}$ : Control input provided by the cyber layer at time t.

$θ \in 1, 2, \dots, N$ : Discrete mode of the system, following a Markov process.

$A (t) = A + Δ A (t), B (t) = B + Δ B (t)$ , where $Δ A (t), Δ B (t)$ , are real valued time-varying matrices represent uncertainties of system satisfy:

$[\begin{matrix} Δ A (t) & Δ B (t) \end{matrix}] = HF (t) [\begin{matrix} E_{1} & E_{2} \end{matrix}], F^{T} (t) F (t) \leq I, H, E_{1}, E_{2}$ are known matrices.

$f (\cdot)$ : A nonlinear function describing the system’s dynamics in mode $θ (t)$ and satisfy Lipschitz constraint as $f^{T} f \leq L^{2} x^{T} x$ , where L has unknown value.

The switching between different system modes is modeled by a discrete-time Markov process $θ (t)$ with a transition probability matrix $P = [p_{ij}]$ , where

p_{ij} = P (t + Δ t) = j | θ (t) = i)

(26)

$p_{ij}$ : The probability of transitioning from mode i to mode j over a time interval $Δ t$ .

$θ \in 1, 2, \dots, N$ Represents the mode of operation of the system at time t.

The process satisfies the Markov property

\begin{matrix} p_{ij} = P (θ (t + Δ t) = j | θ (t) = i, θ (t - 1), . . .) = P (θ (t + 1) = j | θ (i) = i) \end{matrix}

(27)

The proposed control input is as

u (t) = K_{θ (t)} x (t) - {\overset{⌢}{Γ}}_{θ (t)} x (t), \overset{\cdot}{\overset{⌢}{Γ}} = ‖ x {(t)}^{T} QBx (t) ‖

(28)

$K_{θ (t)}$ : is the gain of controller in mode $θ (t)$ and Q is defined in Lyapunov function in the sequel. It should be noted that second term of the proposed controller has adaptive mechanism for estimating unknown nonlinearity of system.

Replay attack model

As reported in Huang et al. (2020), Franzè et al. (2019), replay attack is a type of cyber-attack that a valid data transmission is maliciously or fraudulently repeated or delayed. Therefore, similar to mentioned papers, in this article, replay attack is considered as delay imposed to input controller as

\tilde{u} (t) = K_{θ (t)} x (t - d (t)) - Γ_{θ (t)} x (t - d (t)), \overset{\cdot}{Γ} = ‖ x {(t)}^{T} QBx (t - d (t)) ‖

(29)

where $d (t)$ is the time shift of the stored states. The states replayed correspond to the time range $[d_{1} \leq d (t) \leq d_{2}]$ , causing the system to receive delayed states rather than the current ones.

The behavior of the nonlinear MJCPS states, as specified in (25), when subjected to a replay attack can be characterized as

\begin{matrix} \overset{\cdot}{x} (t) = A (t, θ (t)) x (t, θ (t)) + B (t, θ (t)) \tilde{u} (t, θ (t)) + f (x (t), θ (t), t) \\ \overset{\cdot}{x} (t) = A (t, θ (t)) x (t, θ (t)) + B (t, θ (t)) u (t - d (t), θ (t)) + f (x (t), θ (t), t) \end{matrix}

(30)

The goal is to design a control law $u (t)$ that stabilizes the system despite the replay attack.

Resilient controller design

The fuzzy SVM controller is a hybrid approach that combines fuzzy logic and SVM-based control strategies to handle uncertainties and nonlinearities which is applied to the NMJCPS in this paper. In each mode $θ (t)$ , the control input is designed to stabilize the system while accounting for the fuzzy nature of the uncertainties and nonlinear dynamics.

Let the FSVM control law in mode $θ (t)$ be denoted as

u (t) = K_{θ (t)} x (t) - Γ_{θ (t)} x (t) = \sum_{i = 1}^{L} μ_{i} (x (t)) u_{i}

(31)

where:

$μ_{i} (x (t))$ are the fuzzy membership functions that weigh the SVM control decisions,

$u_{i}$ are the SVM-based control actions designed for each fuzzy rule,

L is the number of fuzzy rules.

The SVM controller generates a decision boundary that separates secure and unsecure system as (Figure 2):

while the fuzzy logic system in combination of SVM handles the nonlinearities and uncertainties by blending different control actions.

In this work, to compensate the delay introduced by the replay attack, the control input $u (t)$ must take the delay $u (t)$ into account. The FSVM control law is adjusted to handle the delayed control input $u (t - d (t))$

\begin{matrix} u (t - d (t)) = K_{θ (t)} x (t - d (t)) - Γ_{θ (t)} x (t - d (t)) \\ = \sum_{i = 1}^{L} μ_{i} (x (t - d (t))) u_{i} \end{matrix}

(32)

The FSVM classifier is trained to output control actions based on both the current and delayed states of the system, minimizing the impact of the replay attack on system performance.

In this section, resiliency of UNMJCPS by proposed controller in (32) is proved. The following theorem shows the main result:

Theorem 1: Reciliency in (30) is achieved, if there exist matrices $Q_{θ} > 0, R_{θ} > 0$ , and matrices ${U_{θ}}_{i} (i = 1, 2, \dots, 9)$ , $Z_{θ}, N_{θ}, J_{θ}, H_{θ}$ with appropriate dimension such that the LMI in (42) is feasible. Then, the closed-loop system (30) is resilient, and the gain of controller (32) is $K_{θ} = {Z_{θ}}^{- 1} N_{θ}$ .

Proof. To prove stability, a Lyapunov function for the system analysis is defined. The key idea is to show that the Lyapunov function decreases over time, ensuring that the system remains stable. Let $V (x (t), θ (t))$ be a Lyapunov function candidate that depends on both the system state $x (t)$ and the Markov mode $θ (t)$ . For each mode $θ (t)$ , we define

\begin{matrix} V (x (t), θ (t)) = x^{T} (t) Q_{θ (t)} x (t) \\ + (d_{2} - d_{1}) \int_{t - d_{1}}^{t} e^{σ (s - t)} {\overset{\cdot}{x}}^{T} (s) R_{θ (t)} \overset{\cdot}{x} (s) ds + \frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2}, \end{matrix}

(33)

where $Q_{θ (t)}, R_{θ (t)} \in R^{n}$ are positive definite matrices for each mode $θ (t)$ . The function $V (x (t), θ (t))$ is positive definite if $Q_{θ (t)}, R_{θ (t)}$ are positive definite, that is, $V (x (t), θ (t)) > 0$ for all $x (t)$ .

$V (x (t), θ (t))$ appropriately measures the “energy” of the system, and we now aim to show that it decreases over time under the FSVM control. Therefore, the time derivative of the Lyapunov function for each θ is calculated

\begin{matrix} {\overset{\cdot}{V}}_{θ} (t) = 2 x^{T} Q_{θ} \overset{\cdot}{x} (t) - \int_{t - d_{1}}^{t} e^{σ (s - t)} {\overset{\cdot}{x}}^{T} (t) R_{θ} \overset{\cdot}{x} (s) ds \leq 2 x^{T} Q_{θ} \\ (A + H E_{1} + B + H E_{2} + f (x, u (t - d (t), t)) \\ + {\overset{\cdot}{x}}^{T} (t) R_{θ} \overset{\cdot}{x} (s) - (d_{2} - d (t)) e^{- σ d_{1}} {\overset{\cdot}{x}}^{T} (t - d_{1}) R_{θ} \overset{\cdot}{x} (s - d_{1}) \\ - σ (d_{2} - d (t)) \int_{t - d_{1}}^{t} e^{σ (s - t)} {\overset{\cdot}{x}}^{T} (t) R_{θ} \overset{\cdot}{x} (s) ds, \\ \leq x^{T} (Q_{θ}) x + x^{T} Q_{θ} x (t - d (t)) + (d - d (t)) {\overset{\cdot}{x}}^{T} (t) R_{θ} \overset{\cdot}{x} (s) \\ - (d_{2} - d (t)) e^{- σ d_{1}} {\overset{\cdot}{x}}^{T} (t - d_{1}) R_{θ} \overset{\cdot}{x} (s - d_{1}) - σ (d_{2} \\ - d (t)) \int_{t - d_{1}}^{t} e^{σ (s - t)} {\overset{\cdot}{x}}^{T} (t) R_{θ} \overset{\cdot}{x} (s) ds \\ - \overset{⌢}{Γ} x^{T} (t) QBx (t - d (t)) + Γ ‖ x^{T} (t) QBx (t - d (t)) ‖ \\ - (Γ - \overset{⌢}{Γ}) ‖ x^{T} (t) QBx (t - d (t)) ‖ \end{matrix}

(34)

and utilizing Jesens inequality, we have

\begin{matrix} - e^{- σ d_{1}} \int_{t - d_{1}}^{t} {\overset{\cdot}{x}}^{T} (t) R_{θ} \overset{\cdot}{x} (s) ds \leq - \frac{e^{- σ d_{1}}}{d_{1} - d (t)} η^{T} (t) {e_{1}}^{T} R_{θ} e_{1} η (t) \\ - \frac{e^{- σ d_{1}}}{d (t)} η^{T} (t) {e_{2}}^{T} R_{θ} e_{2} η (t) \end{matrix}

(35)

where:

\begin{matrix} η^{T} (t) = [\begin{matrix} x^{T} (t) & x^{T} (t - d_{1}) & x^{T} (t - d (t)) \end{matrix}], e_{1} = [\begin{matrix} 0 & - I & I \end{matrix}], \\ e_{2} = [\begin{matrix} I & 0 & - I \end{matrix}] \end{matrix}

for a matrix $U_{θ} = [\begin{matrix} {U_{1}}_{θ} & {U_{2}}_{θ} & U_{3 θ} \\ U_{4 θ} & U_{5} θ & U_{6 θ} \\ U_{7 θ} & U_{8 θ} & U_{9 θ} \end{matrix}]$ Lemma 1, Lemma 3 guarantees that

\begin{matrix} - e^{- σ d_{1}} \int_{t - d_{1}}^{t} {\overset{\cdot}{ℵ}}^{T} (t) R_{θ} \overset{\cdot}{ℵ} (s) ds \leq \\ - η^{T} (t) [\begin{matrix} - ϕ R_{θ} & {U_{θ}}^{T} & ϕ R_{θ} - {U_{θ}}^{T} \\ * & - ϕ R_{θ} & ϕ R_{θ} - U_{θ} \\ * & * & - 2 ϕ R_{θ} + He (U θ) \end{matrix}] η (t) \end{matrix}

(36)

where

\begin{matrix} - e^{- σ d_{1}} \int_{t - d_{1}}^{t} {\overset{\cdot}{ℵ}}^{T} (t) R \overset{\cdot}{ℵ} (s) ds \leq \\ - η^{T} (t) [\begin{matrix} - ϕ R_{θ} & {U_{θ}}^{T} & ϕ R_{θ} - {U_{θ}}^{T} \\ * & - ϕ R_{θ} & ϕ R_{θ} - U_{θ} \\ * & * & - 2 ϕ R_{θ} + He (U_{θ}) \end{matrix}] η (t) \end{matrix}

(37)

$ϕ = \frac{e^{- σ d_{1}}}{(d_{2} - d_{1})}$ . with σ is a positive real number.

substituting (35–37) in (34) and using Lemma 2 in Moon et al. (2010) yields

\begin{matrix} \overset{\cdot}{V} (x (t)) + σ V (x (t)) \leq η^{T} (t) (Π_{θ} + {Ξ_{θ}}^{T} {R_{θ}}^{- 1} Ξ_{θ}) η (t) \\ + 2 \overset{⌢}{Γ} \overset{\cdot}{\overset{⌢}{Γ}} + σ (\frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2}) \end{matrix}

(38)

where

\begin{matrix} Π_{θ} = [\begin{matrix} {Π_{θ}}_{11} & {U_{θ}}^{T} & {Π_{θ}}_{13} \\ * & - ϕ R & ϕ R_{θ} - U_{θ} + Q^{T} H \\ * & * & - 2 ϕ R_{θ} + He (U_{θ}) + {E_{2}}^{T} E_{2} \end{matrix}], \\ {Π_{θ}}_{11} = He ((Q_{θ}) + {E_{1}}^{T} E_{1} - ϕ R_{θ} + σ Q_{θ}, \\ {Π_{θ}}_{13} = Q_{θ} + ϕ R_{θ} - {U_{θ}}^{T} + {E_{1}}^{T} E_{2}, \\ Ξ_{θ} = [\begin{matrix} \sqrt{(d_{2} - d_{1})} R_{θ} & 0 & \sqrt{{(d_{2} - d_{1})}_{1}} R_{θ} \end{matrix}] \end{matrix}

(39)

applying Shur-complement to (39) $Π_{θ} < 0$ is equal to

[\begin{matrix} Π_{θ 11} & {U_{θ}}^{T} & {Π_{θ}}_{13} & \sqrt{(d_{2} - d_{1}) (} R_{θ})^{T} \\ * & - ϕ R_{θ} & ϕ R_{θ} - U_{θ} & 0 \\ * & * & - 2 ϕ R_{θ} + He (U_{θ}) & \sqrt{{(d_{2} - d_{1})}_{1}} {(R_{θ})}^{T} \\ * & * & * & - R_{θ} \end{matrix}] < 0

(40)

On the other hand, based on Lemma 2, the following inequality holds

\begin{matrix} [\begin{matrix} I_{3} \\ 0_{3 \times 3} \end{matrix}] J_{θ} [\begin{matrix} I_{3} \\ 0_{3 \times 3} \end{matrix}] T_{θ} + [\begin{matrix} 0_{3 \times 1} \\ θ_{θ_{n} 23} \end{matrix}] Δ_{θ 34} {J_{θ}}^{- 1} {Δ_{θ 34}}^{T} {[\begin{matrix} 0_{3 \times 1} \\ θ_{23} \end{matrix}]}^{T} \\ \geq He ([\begin{matrix} I_{3} \\ 0_{3 \times 3} \end{matrix}] Δ_{θ 34} {Z_{θ}}^{- 1} N_{θ} [\begin{matrix} 0_{1 \times 3} & - I & I \\ - I \end{matrix}]) \\ Δ_{θ 34} = [\begin{matrix} Q_{θ} - Z_{θ} \\ 0 \\ 0 \\ 0 \\ \sqrt{(d_{2} - d_{1})} (Q_{θ} - Z_{θ}) \\ 0 \\ 0 \\ \sqrt{(d_{2} - d_{1})} (Q_{θ} - Z_{θ}) \end{matrix}], \\ Δ_{θ 23} = {[\begin{matrix} - Z θ^{- 1} N_{θ} & Z θ^{- 1} N θ & {Z_{θ_{n}}}^{- 1} N_{θ} - {Z_{θ}}^{- 1} N_{θ} \end{matrix}]}^{T} \end{matrix}

(41)

and finally applying Lemma 3 and Shur-complement to (41), we reach at

[\begin{matrix} Δ_{θ 11} + J_{θ} & Δ_{θ 12} & 0 & 0 \\ 0 & Δ_{θ 22} & Δ_{θ 23} & 0 \\ 0 & 0 & He (- Z_{θ}) & Δ_{θ 34} \\ 0 & 0 & 0 & - J_{θ} \end{matrix}] < 0

(42)

Where

\begin{matrix} Δ_{θ 11} \\ = [\begin{matrix} {- \bar{Δ}}_{θ 11} & 0 & 0 & {U_{θ 1}}^{T} & {U_{θ 4}}^{T} & {U_{θ}}^{T} & \sqrt{(d_{2} - d_{1})} {(R_{θ})}^{T} & 0 & 0 \\ * & {- \bar{Δ}}_{θ 11} & 0 & {U_{θ}}^{T} & {U_{θ}}^{T} & {U_{θ}}^{T} & 0 & 0 & 0 \\ * & * & {- \bar{Δ}}_{θ 11} & {U_{θ}}^{T} & {U_{θ}}^{T} & {U_{θ}}^{T} & 0 & \sqrt{(d_{2} - d_{1})} {(R_{θ})}^{T} & 0 \\ * & * & * & - ϕ Q_{θ} & 0 & 0 & 0 & 0 & \sqrt{(d_{2} - d_{1})} {(R_{θ})}^{T} \\ * & * & * & * & - ϕ Q_{θ} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - ϕ Q_{θ} & 0 & 0 & 0 \\ * & * & * & * & * & * & - R_{θ} & 0 & 0 \\ * & * & * & * & * & * & * & - R_{θ} & 0 \\ * & * & * & * & * & * & * & * & - R_{θ} \end{matrix}] \end{matrix}

ϕ = \frac{e^{- σ d_{1}}}{(d_{2} - d_{1})},

{\bar{Δ}}_{θ 11} = He (Q_{θ}) - ϕ R_{θ} - σ Q_{θ},

\begin{matrix} {Δ_{θ}}_{12} \\ = [\begin{matrix} - N_{θ} & N_{θ} - {U_{θ}}_{1}^{T} + ϕ Q_{θ} & - N_{θ} - {U_{θ}}_{4}^{T} & - N_{θ} - {U_{θ}}_{7}^{T} & ϕ Q_{θ} - {U_{θ}}_{1} & 0 \\ 0 & - {U_{θ}}_{2}^{T} & - {U_{θ}}_{5}^{T} + ϕ Q_{θ} & - {U_{θ}}_{8}^{T} + ϕ Q_{θ} & 0 & ϕ R_{θ} - {U_{θ}}_{5} \\ 0 & - {U_{θ}}_{3}^{T} & - {U_{θ_{n}}}_{6} & - {U_{θ}}_{9} & ϕ Q_{θ} - {U_{θ}}_{4} & 0 \\ 0 & ϕ R_{θ} - {U_{θ}}_{1} & - U_{2} & - {U_{θ}}_{3} & 0 & ϕ R_{θ} - {U_{θ}}_{7} \\ 0 & - {U_{θ}}_{4} & ϕ R_{θ_{n}} - {U_{θ_{n}}}_{5} & - {U_{θ}}_{6} & ϕ Q_{θ} - {U_{θ}}_{6} & 0 \\ 0 & ϕ R_{θ} - {U_{θ}}_{7} & ϕ R - U_{8} & - {U_{θ}}_{9} & 0 & ϕ R_{θ} - {U_{θ}}_{8} \\ - \sqrt{(d_{2} - d_{1})} N_{θ} & - \sqrt{(d_{2} - d_{1})} N_{θ} & - \sqrt{(d_{2} - d_{1})} N_{θ} & - \sqrt{(d_{2} - d_{1})} N_{θ} & - \sqrt{(d_{2} - d_{1})} N_{θ} & - \sqrt{(d_{2} - d_{1})} N_{θ} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}

Δ_{θ 22} = [\begin{matrix} - Π_{θ} & 0 & 0 \\ * & - 2 ϕ R_{θ} + He ({U_{θ}}_{1} + {U_{θ}}_{2} + {U_{θ}}_{3}) & {U_{4}}^{T} + {U_{θ}}_{5}^{T} + {U_{θ 6}}^{T} \\ * & * & - 2 ϕ Q_{θ} + He ({U_{θ}}_{7} + {U_{θ}}_{8} + {U_{θ}}_{9}) \end{matrix}],

{Δ_{θ}}_{23} = [\begin{matrix} - {(N_{θ})}^{T} \\ {(N_{θ})}^{T} \\ - {(N_{θ})}^{T} \end{matrix}], Π_{θ} = [\begin{matrix} Q_{θ} & 0 & 0 & {U_{θ}}_{1}^{T} & {U_{θ}}_{4}^{T} & {U_{θ}}_{7}^{T} \\ * & Q_{θ} & 0 & {U_{θ}}_{2}^{T} & {U_{θ}}_{5}^{T} & {U_{θ}}_{8}^{T} \\ * & * & Q_{θ} & {U_{θ}}_{3}^{T} & {U_{θ}}_{6}^{T} & {U_{θ}}_{9}^{T} \\ * & * & * & R_{θ} & 0 & 0 \\ * & * & * & * & R_{θ} & 0 \\ * & * & * & * & * & R_{θ} \end{matrix}],

Therefore, if LMI in (42) is satisfied, by positive selected Lyapunov function in (33), we have $\overset{\cdot}{V} (x, θ) \leq - σ ‖ x ‖ + 2 \overset{⌢}{Γ} \overset{\cdot}{\overset{⌢}{Γ}} + σ (\frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2})$ .

Until now, the stabilization of system in each sub-system (it means in each θ Markov jumping) has been proved. In order to prove the stabilization of total system, Generalized Itô Formula is applied in sequel.

By defining expectancy of a function as “E”, we have $E (V (x, θ) > 0, E (\overset{\cdot}{V} (x, θ) \leq - σ E (‖ x ‖) + E (2 \overset{⌢}{Γ} \overset{\cdot}{\overset{⌢}{Γ}} + σ (\frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2}))$ . By introducing the Generalized Itô Formula, we have

\begin{matrix} E [V (x, θ)] - V (x (0), θ_{0}) = \int_{0}^{t} E [\overset{\cdot}{V} (x, θ)] ds \leq \\ \int_{0}^{t} - σ E [x^{T} (s) x (s)] + E (2 \overset{⌢}{Γ} \overset{\cdot}{\overset{⌢}{Γ}} + (\frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2})) ds \end{matrix}

(43)

Such that

\begin{matrix} \int_{0}^{t} σ E [x^{T} (s) x (s)] - E (2 \overset{⌢}{Γ} \overset{\cdot}{\overset{⌢}{Γ}} + (\frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2})) ds \\ < V (x (0), θ_{0}) - E [V (x, θ)] \\ \int_{0}^{t} E [x^{T} (s) x (s)] ds < \frac{1}{σ} (V (x (0), θ_{0}) + E (2 \overset{⌢}{Γ} \overset{\cdot}{\overset{⌢}{Γ}} + (\frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2})) \end{matrix}

(44)

That means $\lim_{t \to \infty} E [x^{T} (t) x (t)] \to 0$ , and therefore, (44) means that the system is mean square random exponential stable with rate less than σ.

Remark 1: In this section, the regulation problem with assuming set point = 0 has been solved. Tracking problem of output to reference input r(t) also can be solved similar to above procedure by replacing x(t) to $e (t) = r (t) - y (t)$

Remark 2: As per conditions obtained in (44), $M = \frac{1}{σ} (V (x (0), θ_{0}) + E (2 \overset{⌢}{Γ} \overset{\cdot}{\overset{⌢}{Γ}} + (\frac{{(Γ - \overset{⌢}{Γ})}^{2}}{2}))$ can be considered as a system secure margin (Figure 2) used in the proposed algorithms (Figures 3 and 4).

In the end of this section, the summary of this work has been presented as flowcharts in Figure 3 (attack detection) and Figure 4 (resilient control):

Remark 3. As mentioned previously, the contribution of this paper is novelty in formulation, assumptions and control method. In Table 1, assumption and method of this work in comparison with other similar papers have been summarized:

Table 1.

Comparison this work with other similar papers.

Reference	System model	Attack type	Method
Jiyang Wang et al. (2025b)	Discrete fuzzy MJCPS	DoS	SMC
Deteng Wang et al. (2025a)	Discrete MJCPS	DoS and deception	Event-triggering controller
Ye et al. (2023)	MJCPS	DoS	Event-triggered-based H∞ control
Cao et al. (2024)	fuzzy MJCPS	FDI	Adaptive neural H∞ control
Xu et al. (2023)	Discrete MJCPS	deception	Asynchronous Triggering Control
Zeng et al. (2022)	MJCPS	DoS	L∞ control
Gao et al. (2023)	fuzzy MJCPS	DoS and deception	Adaptive Neural Event-Triggered Control
Wang et al. (2024)	MJCPS	replay	H∞ control
Zhang et al. (2025)	Delayed MJCPS	deception	Robust feedback
Yao et al. (2024)	Discrete MJCPS	DoS and deception	Switching-Like Adaptive Event-Triggered Control
Liu et al. (2023)	NMJCPS	FDI	Adaptive SMC
Gu et al. (2025)	fuzzy MJCPS	mismatched quantization (FDI)	Event-triggered H∞
This paper	NMJCPS	replay	Adaptive FSVM

As per Table 1, until now, methods proposed in this area for MJCPS are robust (SMC, H∞), adaptive, event-triggered, and NN. Advantages of this work (adaptive FSVM) against such methods is explained in sequel:

Adaptive FSVM is an online method, while robust mechanism is usually considered as an offline method. Also, it excels at dealing with time-varying or poorly known parameters by dynamically adjusting controller parameters, while robust control focuses on maintaining stability and performance despite a fixed set of uncertainties. FSVM can achieve better performance in highly variable systems, but robust control offers guaranteed stability within defined uncertainty bounds.

While event-triggered control has its merits, the adaptive FSVM approach offers a more comprehensive solution for CPS. The adaptive FSVMs with their integration, high availability, scalability, and automated operations provide a more robust and efficient way than a purely event-triggered approach. In general, time-triggered systems excel in predictability and determinism due to their fixed schedule, making them suitable for safety-critical applications.

Adaptive FSVMs offer several advantages over traditional adaptive control methods, particularly in handling uncertainty and noise in complex systems. FSVMs excel in situations with limited data, nonlinearity, and high dimensionality, providing better generalization and anti-interference capabilities. They can reduce the influence of outliers and noise through fuzzy membership functions, leading to more robust and accurate classifications.

SVMs are often preferred over neural networks (NN) for their robustness to overfitting, efficiency in training, and ability to handle high-dimensional data, especially with smaller datasets. Also, FSVMs offer several advantages over traditional NNs, particularly in scenarios with smaller datasets, higher dimensionality, and the need for interpretable results. FSVMs excel in handling noisy or uncertain data due to their fuzzy nature and can provide more robust solutions than NNs, especially when dealing with limited data.

Simulation

In this section, a numerical case study of a single-link robotic arm (Figure 5) along with a simulation is presented to validate the effectiveness of the obtained results.

The mathematical model for the single-link arm, as defined in Tan et al. (2024) is

\overset{\cdot\cdot}{θ} (t) = - \frac{gl}{J} (1 / 2 m + m_{d}) \sin (θ (t)) - \frac{H (t)}{J} \overset{\cdot}{θ} (t) + \frac{1}{J} u (t)

(45)

where $θ (t), u (t)$ represent the angular position and control input, respectively. J, $H (t)$ , g, l and $m_{d}$ denote the moment of inertia, viscous friction coefficient, gravitational acceleration, arm length, and payload mass, respectively. The parameter values are set as $l = 0.5 m, g = 9.8 m / s^{2}, m = 0.12 kg, m_{d} = 0.1, 0.43 kg, J = 2.22, 3.33 Nm$ . The value of $H (t)$ is varied within 10% of the nominal value $H_{0} = 2$ .

Let $x_{1} (t) = θ (t)$ and $x_{2} (t) = \overset{\cdot}{θ} (t)$ . The linearized system model is described as (Tan et al., 2024)

\begin{array}{l} \overset{\cdot}{x} (t) = [\begin{matrix} 0 & 1 \\ - \frac{g h}{J} (1 / 2 m + m_{d}) & - \frac{H (t)}{J} \end{matrix}] x (t) \\ + [\begin{matrix} 0 \\ - \frac{1}{J} \end{matrix}] u (t), y (t) = Cx (t) + Du (t) \end{array}

The parameters of the model are given as:

\begin{matrix} A_{1} = [\begin{matrix} 0.5 & 0.1 \\ - 0.2 & 0.3 \end{matrix}], A_{2} = [\begin{matrix} 0.4 & 0.2 \\ - 0.1 & 0.4 \end{matrix}], B = [\begin{matrix} 0.1 \\ 0.2 \end{matrix}], C = [\begin{matrix} 1 & 0 \end{matrix}], \\ D = 0 \end{matrix}

The mode transitions over time represent the switching between two dynamic modes of the system, determined by a Markov chain. This phenomenon is illustrated in Figure 6. This helps visualize the stochastic behavior of the system as it randomly switches between these modes, which can affect its performance and behavior. This plot shows a series of discrete mode transitions, with the system switching between modes 1 and 2 at regular intervals. The duration spent in each mode appears to be roughly equal. The plot suggests a periodic switching behavior or a Markov process governing the mode transitions. However, a more definitive interpretation would require additional context about the system and its dynamics.

The system output with replay attack plot is shown in Figure 7 which visualizes the behavior of the system’s output under both normal and attack conditions. It compares the original system output (without the attack) to the output that is affected by the replay attack. The blue line represents the system’s output when it is operating normally, without any interference and the red dashed line represents the system’s output when it is under attack. The replay attack replaces the actual output with previously recorded values. It is noteworthy that the attack was applied at the 50th second and lasted for 20 seconds.

By comparing two lines, the impact of the replay attack on the system’s behavior can be observed. It can be seen how the attack deviates the output from its normal trajectory. The plot reveals that the replay attack introduces noticeable deviations from the normal output, suggesting its effectiveness in disrupting the system’s behavior.

The state trajectory (State 1 and 2) with replay attack plots visualizes the evolution of the system’s internal states (State 1 and State 2) under both normal and attack conditions. These figures are shown in Figures 8 and 9, respectively. By comparing the normal and attacked state trajectories, the impact of the replay attack on the system’s internal behavior can be seen. If the attack is successful, the state trajectories will deviate significantly from their normal paths, indicating a compromised system. It is observed that by applying FSVM control, the attack effect is eliminated, and the system states under normal and attack conditions are approximately equivalent, showing no deviation.

The primary purpose of the SVM in this context is to differentiate between normal operational data and data affected by a replay attack. By training on labeled data (normal operation vs attack), the SVM can learn the distinguishing features of each class. The SVM model is trained with the training data composed of outputs from normal operation and attack scenarios.

After training, the SVM model is used to predict anomalies in the output data during the replay attack: The predictions are based on the attack data, which includes manipulated output values during the attack phase. The SVM predicts whether each output value belongs to normal operation or indicates an anomaly.

In the prediction phase, the predictions from the FSVM are initially in the form of ${- 1 . 1}$ , indicating the two classes (normal and attack). They are normalized to $[0 . 1]$ for easier interpretation and further processing in the fuzzy logic control system. Integration with Fuzzy Logic: The output of the SVM predictions is then fed into the fuzzy logic controller, which uses this information to adjust control actions. If an anomaly is detected (indicated by the SVM), the fuzzy logic system can trigger appropriate responses to mitigate the effects of the attack.

The role of control action using fuzzy logic in this context is to provide a resilient control strategy that can mitigate the effects of the replay attack. This control action is provided in Figure 10. The plot shows a series of spikes in the control action, indicating that the controller is actively adjusting its output to counteract the effects of the replay attack. The specific shape and timing of the spikes would depend on the fuzzy logic rules and membership functions used in the controller’s design. This illustrates how the control action changes in response to the system’s state and the detection of anomalies. The plot may show a significant increase in control action values during the attack, indicating a stronger response to mitigate the detected anomaly.

The tracking error with replay attack serves as a critical metric for evaluating the system’s performance and resilience against replay attacks. This metric is shown in Figure 11.

After the attack ends, the tracking error may gradually decrease as the system recovers from the attack’s effects. However, the extent of recovery depends on the system’s resilience and the severity of the attack.

This plot is valuable for assessing the impact of the replay attack on the system’s performance and evaluating the effectiveness of the implemented detection and resilience mechanisms. A significant increase in tracking error during the attack indicates that the system’s performance is compromised, while a successful recovery suggests that the resilience mechanisms are working as intended.

The State 2 vs State 1 plot serves several important roles in visualizing and analyzing the system’s behavior during both normal operation and under a replay attack. This plot is shown in Figure 12.

Plotting State 2 vs State 1 shows the convergence of the curve to the system equilibrium point $(0.0)$ .

In the end of this section and in order to show advantages of FSVM against SVM, we have applied SVM without fuzzy logic to system (45). Table 2 show the advantages of results when fuzzy logic is combined with SVM:

Table 2.

SVM vs fuzzy SVM results.

Parameter	Nonlinearity of system γ	Maximum delay $d_{2}$	Settling time (sec)
SVM	6.24	3.58	90
FSVM	8.73	4.62	80

According to Table 2, in this example, by applying FSVM method, the results in some parameters such as nonlinear term strength, upper bound of the delay imposed by the attacker, and settling time were improved by 40%, 29%, and 12% compared to the SVM method, respectively.

Conclusion

In this paper, a novel control strategy using a Robust Adaptive FSVM-based framework was proposed to enhance the detection and resilience of uncertain nonlinear Markov jump cyber-physical systems (UNMJCPS) under replay attacks. Through the integration of adaptive mechanism with FSVM, a robust detection mechanism capable of handling the stochastic and unknown nonlinear nature of UNMJCPS was developed, while ensuring system stability and performance during and after an attack. The Markov jump model captured the system’s mode transitions effectively, and the FSVM classifier provided accurate anomaly detection, allowing the system to mitigate the effects of replay attacks in real time. Also, adaptive mechanism was designed to eliminate the effect of the unknown nonlinear term of the system. Simulation results demonstrated the efficacy of the proposed approach.

Future work direction

While the proposed framework has been effective as a new subject in this area, there are some points that require further exploration to enhance the capabilities and adapt to future challenges and can improve results in this work:

Integration of robust control that has good results in system with external disturbance (Shen et al., 2021) can be considered as next future work in this subject.

In a control system when states are not directly measurable or when sensor data is noisy or unreliable, an observer design is considered to estimate the internal states of system (Shen et al., 2022). Adding observer design to the results of this research can be considered as a topic for further study in this area.

The upper and lower bound of delay imposed to the system (because of attack) is assumed to be known that is conservative assumption. Considering other assumptions such as stochastic delay or unbounded delay, can help to improve results obtained in this work.

Footnotes

ORCID iD

Maryam Fattahi

Author contributions

All study conception and designs, material preparation, data collection and analysis of this work were performed by the author.

Funding

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

Declaration of conflicting interests

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

Data availability statement

Data sharing is not applicable to this article, as no data sets were generated or analyzed during the current study.

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