Quantization-Based Event-Triggered H_∞ Consensus for Discrete-Time Markov Jump Fractional-Order Multiagent Systems with DoS Attacks

Abstract

This paper investigates the $H_{\infty }$ consensus problem of discrete-time Markov jump fractional-order multiagent systems (DTMJFOMASs) under denial-of-service (DoS) attacks. By applying the short-memory principle, we can obtain discrete-time Markov jump multiagent systems with partially unknown probabilities. A novel quantized event-triggering mechanism (QETM), based on a mode-dependent logarithmic quantizer, is proposed to enhance transmission efficiency among multiagents. A distributed controller with quantized output is developed. Sufficient conditions are provided to ensure the system achieves $H_{\infty }$ consensus through Lyapunov stability theory. Finally, two examples are given to verify the effectiveness of the proposed model.

1. Introduction

Multiagent systems (MASs) have been extensively utilized in multi-robot formation control [1], multiple single-link robotic arm systems (SLRASs) [2], and smart grids [3]. The consensus or synchronization problem has always been a central issue in MASs, aiming to achieve agreement among multiagents through communication. The issue of consensus in MASs has been extensively explored, including $H_{\infty }$ consensus [4,5], event-triggered consensus [6,7,8], distributed consensus [9], finite/fixed-time consensus [10,11], and so on.

It is worth mentioning that the research on MASs assumes that system parameters and structures are deterministic. However, in practical systems, system parameters and structures may change because of unexpected factors (external disturbances, hardware failures, etc.) [12,13]. To resolve the above situation, the Markov jump process is introduced into MASs, realizing uncertain jumps through stochastic transition probabilities. This has been studied by many scholars in various ways [14,15,16,17,18,19,20]. The finite-time leader-following consensus issue in MASs with Markov switching parameters has been studied in [14]. Additionally, the problem of leader-following consensus in semi-Markov jump multiagent systems was examined in [15]. In [16,17], the authors addressed the event-triggered consensus problem for Markov jump multiagent systems (MJMASs). Considering quantized multi-channel transmission, Huo et al. [18] explored the output feedback consensus control strategy to examine the $H_{\infty }$ consensus problem in MJMASs. The synchronization problem of heterogeneous MJMASs was studied in [19] using a collaborative output quadratic controller. In [19,20], the $H_{\infty }$ consensus issue of the MJMASs was discussed with incomplete transition probabilities. Given the analysis above, MJMASs with partially unknown transition probabilities have not been extensively studied. Thus, this drives the research in this paper.

Furthermore, we explore fractional-order multiagent systems (FOMASs). FOMASs are considered to be more reliable, flexible, and accurate in modeling the systems than integer-order MASs. Fractional-order systems are widely used for dynamical systems with memory or hereditary features [21,22,23]. The paper addresses the consensus challenges in FOMASs (DTFOMASs). The containment control problem for DTFOMASs with time delays was discussed in [24]. Ref. [23] focused on the $H_{\infty }$ consensus problem of DTFOMASs with finite-dimensional memory states. According to review [25], we consider a class of systems that combines MJMASs and FOMASs, referred to as DTMJFOMASs.

High-frequency communication among multiagents leads to severe congestion in limited channels. Signal quantization is a prevalent control strategy in digital communication for optimizing network communication resources. In practical applications, the event-triggering mechanisms (ETMs) [6,7,8,11,26,27,28,29] are instrumental in lowering communication burdens among multiagents and conserving network resources. Building upon this, MAS consensus has been studied by combining quantization strategies and ETMs. The application of an event-triggered pinning control was discussed in [30] to solve the containment consensus problem in MASs with quantized communication. Under the framework of quantized event-triggered control, Zhang et al. [31] examined the secure consensus problem for linear MASs. In [32], a data-driven event-triggered control algorithm was introduced for nonlinear MASs utilizing uniform quantization in the encoding–decoding scheme. By integrating ETM and a quantized control technique, Ref. [33] addressed the distributed adaptive optimization problem of nonstrict feedback FOMASs with uncertainty. However, the general quantizers [30,31,32,33,34,35] are typically determined by fixed threshold parameters. This contributes to the decline in system performance. Hence, resembling Markov modes, a quantization strategy with dynamic switching is investigated in this study.

The consensus issue in open MASs poses a threat that cannot be ignored. Denial-of-service (DoS) attacks manifest as network assaults that deplete bandwidth, overload servers, or deplete system resources [36]. Thus, how to enhance the robustness of system control is a challenging issue. Ref. [27] investigated DoS attacks governed by a Markov process and explored a model predictive control approach to enhance system robustness. To address network systems with DoS attacks, an event-triggered cognitive controller was introduced in [37]. In [38], a mode-dependent $H_{\infty }$ consensus approach was developed to tackle the leader-following consensus problem affected by DoS attacks. In general, there are two types of stochastic processes for random DoS attacks: the Markov jump process [27,37] and the Bernoulli distribution process [38]. Until now, the impact of DoS attacks with a Bernoulli distribution process on DTMJMASs has not been adequately investigated. This motivates us to undertake further research.

Building upon the aforementioned discussions, this paper investigates the $H_{\infty }$ consensus problem of DTMJFOMASs with distributed controllers. A mode-dependent QETM is introduced to mitigate communication overhead. The contributions of this paper are articulated as follows:

• Compared to MJMASs and FOMASs, we address the more generalized $H_{\infty }$ consensus problem for DTMJFOMASs, which takes into account incomplete probabilistic Markov processes and external disturbances.
• A mode-dependent distributed controller with quantized inputs is developed, and DoS attacks obeying a Bernoulli distribution are addressed to enhance the robustness of the system.
• Considering the high-frequency communication between MASs, a mode-dependent approach to quantization is introduced. Compared with the traditional triggering strategy, the QETM has both a lower triggering frequency and meets the system performance requirements.

The remaining structure of the paper is outlined as follows: In Section 2, some foundational knowledge is provided. Section 3 introduces DTMJFOMASs and designs QETM. $H_{\infty }$ performance is analyzed in Section 4. Section 5 validates the effectiveness of the proposed model. Lastly, Section 6 summarizes this paper.

Notations: Some symbols in this article are as follows: ${\mathbb{Z}}^{+}$ represents a positive real number. I is an identity matrix. ${\mathbb{R}}^n$ denotes an n-dimensional vector. Pr and acot represent the Prob and arccot function, respectively. $\parallel ·\parallel$ and ⊗ represent the two norm and the multiplication cross, respectively. Finally, $1_n$ is an n-dimensional vector with all ones in it, i.e., ${[1,\dots ,1]}^T\in {\mathbb{R}}^n$.

2. Preliminaries

Definition 1

([39]). The Grünwald–Letnikov fractional derivative of $f\left(t\right)$ is outlined in

$${}_a{}^{G}D_t^{a}f\left(t\right)=\underset{h\to 0}{lim}{h}^{-\alpha }\sum _{\varpi =0}^{\left[\frac{t-a}{h}\right]}{(-1)}^{\varpi }\left[\begin{array}{c}\alpha \\ \varpi \end{array}\right] f(t-\varpi h),$$

(1)

where $\alpha \in {\mathbb{Z}}^{+}$, h is the sampling interval.

$$\left[\begin{array}{c}\alpha \\ \varpi \end{array}\right]=\left\{\begin{array}{cc}1\hfill & \varpi =0,\hfill \\ \frac{\alpha (\alpha -1)\cdots (\alpha -\varpi +1)}{\varpi !}\hfill & \varpi =1,2,3,\cdots .\hfill \end{array}\right.$$

Let $c_{\varpi }\left(\alpha \right)={(-1)}^{\varpi }\left[\begin{array}{c}\alpha \\ \varpi \end{array}\right]$. Obviously, $|c_{\varpi }\left(\alpha \right)|\le \frac{{\alpha }^j}{\varpi !}$, $c_{\varpi }\left(\alpha \right)$ is the absolutely summable sequence.

Based on the works in [23], the discrete-time bounded form of (1) can be expressed as

$$D_t^{\alpha }f\left(t\right)\approx {\mathsf{\Delta }}_h^{\alpha }f\left(kh\right)=h^{-\alpha }\sum _{\varpi =0}^k{c}_{\varpi }\left(\alpha \right)f\left((k-\varpi )h\right).$$

(2)

Lemma 1

([40]). Given appropriately dimensioned matrices $S_{11}$, $S_{12}$, and $S_{22}$, the following matrix inequality is satisfied:

$$\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{12}^T& S_{22}\end{array}\right]<0$$

if and only if

$$\begin{array}{c}\hfill S_{11}<0,S_{22}-S_{12}^T{S}_{11}^{-1}{S}_{12}<0,\\ \hfill S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^T<0.\end{array}$$

Lemma 2

([41]). Assuming appropriately dimensioned matrices X, Y, and diagonal matrix M exist with $MM\le I$, the following condition holds for any $\epsilon >0$:

$$XMY+{\left(XMY\right)}^T\le \epsilon X{X}^T+{\epsilon }^{-1}{Y}^{T}Y.$$

Lemma 3

([16]). If there be real matrices A, B, M, and N such that the following conditions hold for any $ϵ>0$:

$$\left[\begin{array}{cc}M& A+ϵB^T\\ \ast & -ϵ(N+N^T)\end{array}\right]<0,$$

then we have

$$M+A{N}^{-1}B+B^T{X}^{-T}{A}^T<0.$$

3. Materials and Methods

3.1. Problem Formulation

Consider a class of DTMJFOMASs with n identical agents:

$$\left\{\begin{array}{c}{\mathsf{\Delta }}^{\alpha }{x}_i(k+1)=A_{\varphi \left(k\right)}{x}_i\left(k\right)+B_{\varphi \left(k\right)}{u}_i\left(k\right)+D_{\varphi \left(k\right)}{\omega }_i\left(k\right)\hfill \\ y_i\left(k\right)=C_{\varphi \left(k\right)}{x}_i\left(k\right)\hfill \\ z_i\left(k\right)=E_{\varphi \left(k\right)}(x_i\left(k\right)-\frac{1}{n}{\displaystyle \sum _{j=1}^n}{x}_j\left(k\right))\hfill \end{array}\right.,$$

(3)

where $x_i\left(k\right)\in {\mathbb{R}}^{n_x}$ is the state of the ith agent; $y_i\left(k\right)\in {\mathbb{R}}^{n_y}$ and $z_i\left(k\right)\in {\mathbb{R}}^{n_z}$ are the measured output and controlled output, respectively, of the ith agent; ${\omega }_i\left(k\right)\in {\mathbb{R}}^{n_{\omega }}$ is the disturbance input of the ith agent; and $u_i\left(k\right)\in {\mathbb{R}}^{n_u}$ is the control input of the ith agent. The matrices A, $B_{\varphi \left(k\right)}$, $C_{\varphi \left(k\right)}$, $D_{\varphi \left(k\right)}$, $E_{\varphi \left(k\right)}$ are the known real matrices. $\varphi \in \mathcal{M}=\{1,\cdots ,\mathfrak{m}\}$ is a Markov chain with the transition probability matrix (TPM) $\mathsf{\Pi }=\left\{{\pi }_{\varsigma p}\right\}$:

$${\pi }_{\varsigma p}=Pr\{\varphi (k+1)=p|\varphi \left(k\right)=\varsigma \},$$

where ${\sum }_{p=1}^{\mathfrak{m}}{\pi }_{\varsigma p}=1$ and ${\pi }_{\varsigma p}\in [0,1]$. Within the TPM, there exist unspecified elements, illustrated by the 3rd-order matrix $\mathsf{\Pi }$:

$$\mathsf{\Pi }=\left[\begin{array}{ccc}{\pi }_{11}& \ast & \ast \\ \ast & {\pi }_{22}& \ast \\ \ast & \ast & {\pi }_{33}\end{array}\right],$$

where “*” indicates an unknown element.

According to (2), the state variable can be articulated as

$${\mathsf{\Delta }}_{k+1}^{{\alpha }_i}{x}_i(k+1)=h^{-\alpha }\sum _{\varpi =0}^{k+1}{c}_{\varpi }\left({\alpha }_i\right)x_i(k+1-\varpi )$$

(4)

Hence, $x_i(k+1)$ can be reformulated as

$$x_i(k+1)=(h^{\alpha }A+\alpha I_n)x_i\left(k\right)+h^{\alpha }{B}_{\varphi \left(k\right)}{u}_i\left(k\right)+h^{\alpha }{D}_{\varphi \left(k\right)}{\omega }_i\left(k\right)+\sum _{v=1}^L{c}_v\left(\alpha \right)x(k-v).$$

(5)

Let $\varphi \left(k\right)=\varsigma$ and $x_i\left(k\right)={\left[x_i^T\left(k\right),x_i^T(k-1),\cdots ,x_i^T(k-L)\right]}^T$, then Equation (3) can be extended to

$$\left\{\begin{array}{c}{x}_i(k+1)={\mathcal{A}}_{\varsigma }{x}_i\left(k\right)+{\mathcal{B}}_{\varsigma }{u}_i\left(k\right)+{\mathcal{D}}_{\varsigma }{\omega }_i\left(k\right)\hfill \\ y_i\left(k\right)={\mathcal{C}}_{\varsigma }{x}_i\left(k\right)\hfill \\ z_i\left(k\right)={\mathcal{E}}_{\varsigma }(x_i\left(k\right)-\frac{1}{n}{\displaystyle \sum _{j=1}^n}{x}_j\left(k\right))\hfill \end{array}\right.,$$

(6)

where

$$\begin{array}{c}{\mathcal{A}}_{\varsigma }=\left[\begin{array}{cccc}{h}^{\alpha }{A}_{\varsigma }+\alpha I_n& c_1{I}_n& \cdots & c_L{I}_n\\ I_n& 0& \cdots & 0\\ 0& I_n& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0\end{array}\right],{\mathcal{B}}_{\varsigma }=\left[\begin{array}{c}{h}^{\alpha }{B}_{\varsigma }\\ 0\\ ⋮\\ 0\end{array}\right],\\ {\mathcal{C}}_{\varsigma }^T=\left[\begin{array}{c}{C}_{\varsigma }\\ 0\\ ⋮\\ 0\end{array}\right],{\mathcal{D}}_{\varsigma }=\left[\begin{array}{c}{h}^{\alpha }{D}_{\varsigma }\\ 0\\ ⋮\\ 0\end{array}\right],{\mathcal{E}}_{\varsigma }^T=\left[\begin{array}{c}{E}_{\varsigma }\\ 0\\ ⋮\\ 0\end{array}\right].\end{array}$$

Assumption 1.

The pair ($\mathcal{A}$, $\mathcal{B}$) is stabilizable.

Assumption 2.

The pair ($\mathcal{A}$, $\mathcal{C}$) is observable.

Remark 1.

According to Theorem 8.19 in [42], it is evident that if $h^{\alpha }{A}_{\varsigma }+\alpha \le 1$, then DTMJFOMASs (6) is stabilizable. That is, Assumption 1 is equivalent to the condition $h^{\alpha }{A}_{\varsigma }+\alpha \le 1$. As per [43], it is known that (A, C) being observable in (3) is equivalent to Assumption 2.

3.2. Quantized Event-Triggered Mechanism

A mode-dependent logarithmic quantizer can be outlined in

$$q_{\psi \left(k\right)}^i\left(v^i\right)=\left\{\begin{array}{cc}{\nu }_{\psi \left(k\right)}^{ij},\hfill & \frac{{\nu }_{\psi \left(k\right)}^{ij}}{1+{\delta }_{\psi \left(k\right)}^i}\le v^i\le \frac{{\nu }_{\psi \left(k\right)}^{ij}}{1-{\delta }_{\psi \left(k\right)}^i}\hfill \\ 0,\hfill & v^i=0\hfill \\ -q_{\psi \left(k\right)}^i(-v^i),\hfill & v^i<0\hfill \end{array}\right.,$$

(7)

where ${\delta }_{\psi \left(k\right)}=[(1-{\rho }_{\psi \left(k\right)})/(1+{\rho }_{\psi \left(k\right)})]$, $0<{\delta }_{\psi \left(k\right)}<1$, $0<{\rho }_{\psi \left(k\right)}<1$. $\psi \in \mathcal{N}=\{1,\cdots ,\mathfrak{N}\}$ is a Markov chain with the TPM $\mathsf{\Theta }=\left\{{\theta }_{\varrho q}\right\}$:

$${\theta }_{\varrho q}=Pr\{\psi (k+1)=q|\psi \left(k\right)=\varrho \},$$

where ${\sum }_{q=1}^{\mathfrak{n}}{\theta }_{\varrho q}=1$ and ${\theta }_{\varrho q}\in [0,1]$.

Let $\psi \left(k\right)=\varrho$. $Q\left(v\right)={[q_{\varrho }^1\left(v_1\right),\cdots ,q_{\varrho }^s\left(v_s\right)]}^T$ be defined as the set of logarithmic quantization levels. As per [44], $q_{\varrho }^i\left(v^i\right)$ is a sector with bounds; then there is

$$Q_{\varrho }\left(y\left(k\right)\right)=(1+{\mathcal{H}}_{\varrho }\left(k\right))y\left(k\right),$$

where ${\mathcal{H}}_{\varrho }={[{\mathcal{H}}_{\varrho }^1\left(k\right),\cdots ,{\mathcal{H}}_{\varrho }^i\left(k\right)]}^T$ and $|{\mathcal{H}}_{\varrho }^i\left(k\right)|\le {\delta }_{\varrho }^i<1$.

To mitigate communication overhead, this study examines the logarithmic quantizer into the ETM, called QETM. For each agent, the trigger instant by the following condition:

$$\begin{array}{c}{k}_{d+1}^i={\displaystyle \underset{k\in \mathbb{N}}{min}}\{k>k_d^i\mid \sigma \left(k\right){\eta }_i^{\mathrm{T}}\left(k\right){\mathsf{\Omega }}_1{\eta }_i\left(k\right)-e_i^{\mathrm{T}}\left(k\right){\mathsf{\Omega }}_2{e}_i\left(k\right)\le 0\}\end{array},$$

(8)

where $\sigma \left(k\right)={\varrho }_1+({\varrho }_2-{\varrho }_1)\frac{2}{\pi }\mathrm{acot}({\epsilon }_0\parallel \mathfrak{y}\left(k\right){\parallel }^2)(0<{\varrho }_1<{\varrho }_2<1,{\epsilon }_0>0)$ is a threshold parameter, $k_d^i$ denotes the last trigger instant of agent i, ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$ represent the weighting matrices. $\eta \left(k\right)=Q\left(y\right(k\left)\right)$, and the measurement error is determined as follows

$$e\left(k\right)=\eta \left(k\right)-\eta \left(k_d^i\right)$$

Remark 2.

A mode-dependent QETM is introduced in this paper, demonstrating enhanced generalization compared to traditional triggering mechanisms.

1. 

A quantizer with dynamic switching features is more able to emphasize the robustness of the system, while traditional quantizers [30,31,32,33,34] lack the capability to achieve system stability through dynamic adjustments.

2. 

If the quantizer $q_{\psi \left(k\right)}^i\left(v^i\right)=v^i$, then the QETM can degenerate into the dynamic event-triggered mechanism [7,11,26].

3. 

If the quantizer $q_{\psi \left(k\right)}^i\left(v^i\right)=v^i$ and $\sigma \left(k\right)$ is a constant, then the QETM can transform into the static event-triggered mechanism [6,27,28].

Remark 3.

In the QETM (8), the adaptive parameter $\sigma \left(k\right)$ is taken into consideration. It is apparent that the dynamic parameter $\sigma \left(k\right)$ changes in real time based on the quantized error $\eta \left(k\right)$.

3.3. Mode-Dependent Distributed Control Protocol

DoS attacks lead to network communication disruptions, which result in packet loss and affect system stability. Under DoS attacks, the distributed control protocol with quantization is considered as follows:

$$u_i\left(k\right)=\left\{\begin{array}{cc}c\beta \left(k\right)K_{\varphi \left(k\right)}\sum _{j=1}^N{g}_{ij}({\eta }_j\left(k\right)-{\eta }_i\left(k\right)),\hfill & i\ne j\hfill \\ 0,\hfill & i=j\hfill \end{array}\right.,$$

(9)

where $c>0$ is a scalar, $K_{\varphi \left(k\right)}$ is a gain matrix, and $\beta \left(k\right)\in [0,1]$ conforms to a Bernoulli distribution, representing stochastic packet loss. If it equals 1, it signifies successful data transmission to the agents; otherwise, it indicates data loss.

$$\left\{\begin{array}{c}\mathrm{Pr}\{\beta \left(k\right)=1\}=\mathbb{E}\left\{\beta \left(k\right)\right\}=\overline{\beta }\\ \mathrm{Pr}\{\beta \left(k\right)=0\}=1-\mathbb{E}\left\{\beta \left(k\right)\right\}=1-\overline{\beta }.\end{array}\right.$$

Remark 4.

The considered network framework primarily illustrates the control process of the ith agent, while the other agents operate similarly to the ith agent, as depicted in Figure 1. The framework mainly consists of the ith agent, a sensor, an event trigger, a controller, a zero-order holder (ZOH), and an actuator. Utilizing a topological structure, the ith agent interacts with other agents, forming a communication network. Given the impact of a DoS attack, the consensus problem of DTMJFOMASs is addressed.

3.4. Model Transformation

Upon substituting control protocol (9) into DTMJFOMASs (6), one obtains

$$\left\{\begin{array}{c}x(k+1)=(I_n\otimes {\mathcal{A}}_{\varsigma })x\left(k\right)+c\overline{\beta }(\mathcal{L}\otimes {\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I_n+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma })x\left(k\right)\hfill \\ -c(\beta \left(k\right)-\overline{\beta })(\mathcal{L}\otimes {\mathcal{B}}_{\varsigma }{K}_{\varsigma })e\left(k\right)+(I_n\otimes {\mathcal{D}}_{\varsigma })\omega \left(k\right)\hfill \\ z\left(k\right)=(\beth \otimes {\mathcal{E}}_{\varsigma })x\left(k\right)\hfill \end{array}\right.,$$

(10)

where $x\left(k\right)=\mathrm{col}\left(x_i^T\left(k\right)\right)$, $y\left(k\right)=\mathrm{col}\left(y_i^T\left(k\right)\right)$, $z\left(k\right)=\mathrm{col}\left(z_i^T\left(k\right)\right)$, $\omega \left(k\right)=\mathrm{col}\left({\omega }_i^T\left(k\right)\right)$, $\beth =I_n-\frac{1}{n}{1}_n{1}_n^T$.

Let $\tilde{x}\left(k\right)=(\beth \otimes I_n)x\left(k\right)$. Based on $\beth \beth =\beth$ and $\beth \mathcal{L}=\mathcal{L}\beth$, the system (10) can be reformulated as

$$\left\{\begin{array}{c}\tilde{x}(k+1)=(I_n\otimes {\mathcal{A}}_{\varsigma })\tilde{x}\left(k\right)+c\overline{\beta }(\mathcal{L}\otimes {\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I_n+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma })\tilde{x}\left(k\right)\hfill \\ -c(\beta \left(k\right)-\overline{\beta })(\mathcal{L}\otimes {\mathcal{B}}_{\varsigma }{K}_{\varsigma })e\left(k\right)+(I_n\otimes {\mathcal{D}}_{\varsigma })\omega \left(k\right)\hfill \\ \tilde{z}\left(k\right)=(\beth \otimes {\mathcal{E}}_{\varsigma })\tilde{x}\left(k\right)\hfill \end{array}\right.$$

(11)

For a given orthogonal matrix $\mathcal{Q}=[{\mathcal{Q}}_1,{\mathcal{Q}}_2]$ with ${\mathcal{Q}}_2=1/\sqrt{n}{1}_n$, such that ${\mathcal{Q}}^T\beth \mathcal{Q}=\mathrm{diag}\{I_{n-1},0\}$ and ${\mathcal{Q}}^T\mathcal{L}\mathcal{Q}=\mathrm{diag}\{{\gamma }_1,\cdots ,{\gamma }_{n-1},0\}$. Then, one has

$$\left\{\begin{array}{c}{\overline{x}}_{\iota }(k+1)=(I_{n-1}\otimes {\mathcal{A}}_{\varsigma }+c\overline{\beta }{\mathsf{\Gamma }}_1){\overline{x}}_{\iota }\left(k\right)-c(\beta \left(k\right)-\overline{\beta }){\mathsf{\Gamma }}_2{\overline{e}}_{\iota }\left(k\right)+(I_{n-1}\otimes {\mathcal{D}}_{\varsigma }){\overline{\omega }}_{\iota }\left(k\right)\hfill \\ {\overline{z}}_{\iota }\left(k\right)=(I_{n-1}\otimes {\mathcal{E}}_{\varsigma }){\overline{x}}_{\iota }\left(k\right)\hfill \end{array}\right.$$

(12)

and

$$\left\{\begin{array}{c}{\overline{x}}_n(k+1)={\mathcal{A}}_{\varsigma }{\overline{x}}_n\left(k\right)+{\mathcal{D}}_{\varsigma }{\overline{\omega }}_n\left(k\right)\hfill \\ {\overline{z}}_n\left(k\right)=0\hfill \end{array}\right.,$$

(13)

where

• $\overline{x}\left(k\right)={[{\overline{x}}_{\iota }^T\left(k\right),{\overline{x}}_n^T\left(k\right)]}^T,\overline{z}\left(k\right)={[{\overline{z}}_{\iota }^T\left(k\right),{\overline{z}}_n^T\left(k\right)]}^T,$
• $\overline{e}\left(k\right)={[{\overline{e}}_{\iota }^T\left(k\right),{\overline{e}}_n^T\left(k\right)]}^T,\overline{\omega }\left(k\right)={[{\overline{\omega }}_{\iota }^T\left(k\right),{\overline{\omega }}_n^T\left(k\right)]}^T,$
• ${\overline{x}}_{\iota }\left(k\right)={[{\overline{x}}_1^T\left(k\right),\cdots ,{\overline{x}}_{n-1}^T\left(k\right)]}^T,{\overline{e}}_{\iota }\left(k\right)={[{\overline{e}}_1^T\left(k\right),\cdots ,{\overline{e}}_{n-1}^T\left(k\right)]}^T,$
• ${\overline{\omega }}_{\iota }\left(k\right)={[{\overline{\omega }}_1^T\left(k\right),\cdots ,{\overline{\omega }}_{n-1}^T\left(k\right)]}^T,{\overline{z}}_{\iota }\left(k\right)={[{\overline{z}}_1^T\left(k\right),\cdots ,{\overline{z}}_{n-1}^T\left(k\right)]}^T,$
• ${\mathsf{\Gamma }}_1=\mathrm{diag}\{{\gamma }_1,\dots ,{\gamma }_{n-1}\}\otimes {\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I_{n-1}+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma },$
• ${\mathsf{\Gamma }}_2=\mathrm{diag}\{{\gamma }_1,\dots ,{\gamma }_{n-1}\}\otimes {\mathcal{B}}_{\varsigma }{K}_{\varsigma }.$

It is evident that with ${\overline{z}}_{\iota }\left(k\right)$ being equivalent to ${\overline{z}}_i\left(k\right)$, the stability of DTMJFOMASs (3) implies the consensus attainment of subsystem (14).

$$\left\{\begin{array}{c}{\widehat{x}}_{\varkappa }(k+1)=({\mathcal{A}}_{\varsigma }+c\overline{\beta }{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma }){\widehat{x}}_{\varkappa }\left(k\right)\hfill \\ -{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }{\widehat{e}}_{\varkappa }\left(k\right)+{\mathcal{D}}_{\varsigma }{\widehat{\omega }}_{\varkappa }\left(k\right)\hfill \\ {\widehat{z}}_{\varkappa }\left(k\right)={\mathcal{E}}_{\varsigma }{\widehat{x}}_{\varkappa }\left(k\right)\hfill \end{array}\right.,$$

(14)

where $\varkappa \in \{1,\cdots ,n-1\}$.

Definition 2

([45]). The DTMJFOMASs (14) with ${\widehat{\omega }}_{\varkappa }\left(k\right)=0$, $\forall \varkappa \in \{1,\cdots ,n-1\}$ are said to be mean-square asymptotic stability if it satisfies:

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\mathbb{E}\{\parallel e_i\left(k\right)-e_j\left(k\right){\parallel \}}^2=0,\forall i,j\in \{1,\dots ,n-1\}.\end{array}$$

(15)

Definition 3

([46]). Considering DTMJFOMASs (14) under DoS attacks and the $H_{\infty }$ consensus performance metrics $\mu >0$ based on distributed control protocols, stochastic stability can be attained if the following conditions are met:

1. 

If $\widehat{\omega }\left(k\right)\equiv 0$, the system consensus is mean-square asymptotic stability, i.e., Definition (2) is met.

2. 

Under zero initial conditions, the following inequality holds for any non-zero $\widehat{\omega }\left(k\right)\in L_2[0,\infty )$:

$$\sum _{k=0}^{\infty }\mathbb{E}\left\{{\widehat{z}}_{\varkappa }^T\left(k\right){\widehat{z}}_{\varkappa }\left(k\right)\right\}<{\mu }^2\sum _{k=0}^{\infty }\mathbb{E}\left\{{\widehat{\omega }}_{\varkappa }^T\left(k\right){\widehat{\omega }}_{\varkappa }\left(k\right)\right\}.$$

(16)

4. Main Results

In this section, firstly, the stochastic stability conditions for all agents are analyzed. Subsequently, the mean-square asymptotic stability and $H_{\infty }$ performance conditions for the system (14) are established. Lastly, the design procedure for the distributed controller (9) with mode-dependent is showcased.

Theorem 1.

Under Assumptions 1 and 2, the DTMJFOMASs (14) with ${\widehat{\omega }}_{\varkappa }\left(k\right)=0$ exhibits stochastic consensus stability. Then, for any scalar $0\le \varkappa \le n-1$, the following condition holds:

$${\mathsf{\Lambda }}_{\varkappa }^{\varsigma \varrho }=\left[\begin{array}{ccc}{\mathsf{\Lambda }}_{11}& 0& {\mathsf{\Lambda }}_{13}\\ \ast & -I& {\mathsf{\Lambda }}_{23}\\ \ast & \ast & {\mathsf{\Lambda }}_{33}\end{array}\right]<0,$$

where

• ${\mathcal{P}}_{\varsigma \varrho }={\sum }_{p=\mathcal{M}}{\sum }_{q=\mathcal{N}}{\pi }_{\varsigma p}{\theta }_{\varrho q}{P}_{pq}$,
• ${\mathsf{\Lambda }}_{11}={\mathcal{C}}_{\varsigma }^T{\mathcal{C}}_{\varsigma }-P_{\varsigma \varrho }$, ${\mathsf{\Lambda }}_{13}={({\mathcal{A}}_{\varsigma }+{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma })}^T$,
• ${\mathsf{\Lambda }}_{23}={\left({\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }\right)}^T$, ${\mathsf{\Lambda }}_{33}={\mathcal{P}}_{\varsigma \varrho }^{-1}$.

Proof. 

To analyze the stochastic consensus stability for the DTMJFOMASs (14) with ${\widehat{\omega }}_{\varkappa }\left(k\right)=0$, one considers the following Lyapunov function as

$$\begin{array}{c}\hfill V_{\varkappa }(k,\varphi \left(k\right),\psi \left(k\right))={\widehat{x}}_{\varkappa }^T\left(k\right)P_{\varphi \left(k\right),\psi \left(k\right)}{\widehat{x}}_{\varkappa }\left(k\right),\end{array}$$

and its expectation manifests as

$$\begin{array}{c}\hfill \mathbb{E}\{▵V_{\varkappa }(k,\varphi \left(k\right),\psi \left(k\right))\}=\mathbb{E}\{V_{\varkappa }(k+1,\varphi (k+1),\psi (k+1))-V_{\varkappa }(k,\varphi \left(k\right),\psi \left(k\right))\}.\end{array}$$

(17)

According to TPMs $\mathsf{\Pi }$ and $\mathsf{\Theta }$, we can obtain

$$\begin{array}{cc}\hfill & Pr\left\{\varphi \right(k+1)=p,\psi (k+1)=q|\varphi \left(k\right)=\varsigma ,\psi \left(k\right)=\varrho \}\hfill \\ \hfill & =Pr\left\{\varphi \right(k+1)=p|\varphi \left(k\right)=\varsigma ,\psi \left(k\right)=\varrho \}\times Pr\{\psi (k+1)=q\left|\varphi \right(k)=\varsigma ,\psi (k)=\varrho \}\hfill \\ \hfill & ={\pi }_{\varsigma p}{\theta }_{\varrho q}.\hfill \end{array}$$

Thus, Equation (17) can be reduced to

$$\begin{array}{cc}\hfill & \mathbb{E}\{▵V_{\varkappa }(k,\varsigma ,\varrho )\}\hfill \\ \hfill & \le \mathbb{E}\{{\widehat{x}}_{\varkappa }^T(k+1){\mathcal{P}}_{\varsigma \varrho }{\widehat{x}}_{\varkappa }\left(k\right)-{\widehat{x}}_{\varkappa }^T\left(k\right)P_{\varsigma ,\varrho }{\widehat{x}}_{\varkappa }\left(k\right)+{\widehat{x}}_{\varkappa }^T\left(k\right){\mathcal{C}}_{\varsigma }^T{\mathcal{C}}_{\varsigma }{\widehat{x}}_{\varkappa }\left(k\right)-{\widehat{e}}_{\varkappa }^T\left(k\right){\widehat{e}}_{\varkappa }\left(k\right)\}\hfill \\ \hfill & ={\xi }_{\varkappa }^T\left(k\right){\mathsf{\Lambda }}_{\varkappa }^{\varsigma \varrho }{\xi }_{\varkappa }\left(k\right),\hfill \end{array}$$

(18)

where ${\xi }_{\varkappa }^T\left(k\right)={[{\widehat{x}}_{\varkappa }^T\left(k\right),{\widehat{e}}_{\varkappa }^T\left(k\right)]}^T$.

Remark 5.

During the solution of Equation (18), a strategic use of an additional term ($g\left(x\right)={\widehat{x}}_{\varkappa }^T\left(k\right){\mathcal{C}}_{\varsigma }^T{\mathcal{C}}_{\varsigma }{\widehat{x}}_{\varkappa }\left(k\right)-{\widehat{e}}_{\varkappa }^T\left(k\right){\widehat{e}}_{\varkappa }\left(k\right)$) was incorporated to simplify the solving process. Clearly, $g\left(x\right)\ge 0$, and the equation is valid.

Using Lemma 1, it is easy to obtain ${\mathsf{\Lambda }}_{\varkappa }^{\varsigma \varrho }<0$. One can obtain

$$\begin{array}{c}\hfill \mathbb{E}\{▵V_{\varkappa }(k,\varsigma ,\varrho )\}\le -\hslash \mathbb{E}\{\parallel {\xi }_{\varkappa }\left(k\right){\parallel }^2\},\end{array}$$

(19)

where $\hslash ={min}_{\varsigma =\mathcal{M},\varrho =\mathcal{N}}\left\{\lambda \left({\mathsf{\Lambda }}_{\varkappa }^{\varsigma \varrho }\right)\right\}(\hslash >0)$. Furthermore, we have

$$\begin{array}{c}\hfill \mathbb{E}\{V_{\varkappa }(n+1,\varsigma ,\varrho )-V_{\varkappa }(0,\varsigma ,\varrho )\}\le -\hslash \sum _{k=0}^n\mathbb{E}\{\parallel {\xi }_{\varkappa }\left(k\right){\parallel }^2\}.\end{array}$$

(20)

When $n\to \infty$, obviously, there is

$$\begin{array}{cc}\hfill \sum _{k=0}^n\mathbb{E}\{\parallel {\xi }_{\varkappa }\left(k\right){\parallel }^2\}& \le \frac{1}{\hslash }(\mathbb{E}\left\{V_{\varkappa }(n+1,\varsigma ,\varrho )\right\}-\mathbb{E}\left\{V_{\varkappa }(0,\varsigma ,\varrho )\right\})\hfill \\ \hfill & \le \frac{1}{\hslash }{V}_{\varkappa }(0,\varsigma ,\varrho )\}<\infty .\hfill \end{array}$$

(21)

This indicates that all agents maintain stochastic consensus, with the proof being completed. □

Theorem 2.

If there exists a matrix $P_{\varsigma \varrho }>0(\varsigma \in \mathcal{M},\varrho \in \mathcal{N})$ that ensures the DTMJFOMASs (14) is stochastic consensus stability with the $H_{\infty }$ performance level μ, then for any scalar $0\le \varkappa \le n-1$, the following condition holds:

$${\mathsf{\Sigma }}_{\varkappa }^{\varsigma \varrho }=\left[\begin{array}{cccc}{\mathsf{\Sigma }}_{11}& 0& 0& {\mathsf{\Lambda }}_{13}\\ \ast & -{\mu }^{2}I& 0& {\mathcal{D}}_{\varsigma }\\ \ast & \ast & -I& {\mathsf{\Lambda }}_{23}\\ \ast & \ast & \ast & -{\mathcal{P}}_{\varsigma \varrho }^{-1}\end{array}\right]<0,$$

(22)

where ${\mathsf{\Sigma }}_{11}={\mathcal{E}}_{\varsigma }^T{\mathcal{E}}_{\varsigma }+{\mathsf{\Lambda }}_{11}$.

Proof. 

To consider the $H_{\infty }$ performance of the system (14), we define

$$\begin{array}{cc}\hfill \mathcal{J}\left(k\right)=& \mathbb{E}\{▵V_{\varkappa }(k,\varsigma ,\varrho )+{\widehat{z}}_{\varkappa }^T\left(k\right){\widehat{z}}_{\varkappa }\left(k\right)-{\mu }^2{\widehat{\omega }}_{\varkappa }^T\left(k\right){\widehat{\omega }}_{\varkappa }\left(k\right)\}\hfill \\ \hfill =& {\widehat{x}}_{\varkappa }^T\left(k\right){[{\mathcal{A}}_{\varsigma }+{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma }]}^T{\mathcal{P}}_{\varsigma \varrho }[{\mathcal{A}}_{\varsigma }+{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma }]{\widehat{x}}_{\varkappa }\left(k\right)\hfill \\ \hfill & +{\widehat{e}}_{\varkappa }^T\left(k\right){\left[{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I+{\mathcal{H}}_{\varrho })\right]}^T{\mathcal{P}}_{\varsigma \varrho }\left[{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{K}_{\varsigma }(I+{\mathcal{H}}_{\varrho })\right]{\widehat{e}}_{\varkappa }\left(k\right)+{\widehat{\omega }}_{\varkappa }^T\left(k\right){\mathcal{D}}_{\varsigma }^T{\mathcal{P}}_{\varsigma \varrho }{\mathcal{D}}_{\varsigma }{\widehat{\omega }}_{\varkappa }\left(k\right)\hfill \\ \hfill & +{\widehat{x}}_{\varkappa }^T\left(k\right){\mathcal{E}}_{\varsigma }^T{\mathcal{E}}_{\varsigma }{\widehat{x}}_{\varkappa }\left(k\right)+{\widehat{x}}_{\varkappa }^T\left(k\right){\mathcal{C}}_{\varsigma }^T{\mathcal{C}}_{\varsigma }{\widehat{x}}_{\varkappa }\left(k\right)-{\widehat{e}}_{\varkappa }^T\left(k\right){\widehat{e}}_{\varkappa }\left(k\right)-{\widehat{x}}_{\varkappa }^T\left(k\right)P_{\varsigma \varrho }{\widehat{x}}_{\varkappa }\left(k\right)-{\mu }^2{\widehat{\omega }}_{\varkappa }^T\left(k\right){\widehat{\omega }}_{\varkappa }\left(k\right)\hfill \\ \hfill =& {\widehat{\xi }}_{\varkappa }^T\left(k\right){\mathsf{\Sigma }}_{\varkappa }^{\varsigma \varrho }{\widehat{\xi }}_{\varkappa }\left(k\right),\hfill \end{array}$$

(23)

where ${\widehat{\xi }}_{\varkappa }^T\left(k\right)={[{\widehat{x}}_{\varkappa }^T\left(k\right),{\widehat{e}}_{\varkappa }^T\left(k\right),{\widehat{\omega }}_{\varkappa }^T\left(k\right)]}^T$.

According to (22), one obtains $\mathcal{J}\left(k\right)<0$. Likewise, we can conclude that

$$\begin{array}{c}\hfill \sum _{k=0}^{\infty }\mathbb{E}\left\{{\widehat{z}}_{\varkappa }^T\left(k\right){\widehat{z}}_{\varkappa }\left(k\right)\right\}<\mathbb{E}\left\{V_{\varkappa }(\infty ,\varsigma ,\varrho )\right\}-\mathbb{E}\left\{V_{\varkappa }(0,\varsigma ,\varrho )\right\}+{\mu }^2\sum _{k=0}^{\infty }\mathbb{E}\left\{{\widehat{\omega }}_{\varkappa }^T\left(k\right){\widehat{\omega }}_{\varkappa }\left(k\right)\right\}.\end{array}$$

(24)

According to (24), $\mathbb{E}\{{\widehat{z}}_{\varkappa }^T\left(k\right){\widehat{z}}_{\varkappa }\left(k\right)-{\mu }^2{\widehat{\omega }}_{\varkappa }^T\left(k\right){\widehat{\omega }}_{\varkappa }\left(k\right)\}<0$. This indicates that the DTMJFOMASs (14) maintain stochastic $H_{\infty }$ consistent stability. This completes the proof. □

Theorem 3.

If there exist matrices $P_{\varsigma \varrho }>0(\varsigma \in \mathcal{M},\varrho \in \mathcal{N})$ and ${\mathcal{Z}}_{\varsigma }$ that ensure that the DTMJFOMASs (14) have stochastic consensus stability with the $H_{\infty }$ performance level μ, then for any scalars ϰ, ε, ${\epsilon }_1$, and ${\epsilon }_2$, the following condition holds:

$${\mathsf{\Xi }}_{\varkappa }^{\varsigma \varrho }=\left[\begin{array}{cccccc}{\mathsf{\Xi }}_{11}& 0& 0& {\mathsf{\Xi }}_{14}& {\mathsf{\Xi }}_{15}& 0\\ \ast & -{\mu }^{2}I& 0& {\mathsf{\Xi }}_{24}& 0& 0\\ \ast & \ast & -I& {\mathsf{\Xi }}_{34}& 0& 0\\ \ast & \ast & \ast & {\mathsf{\Xi }}_{44}& 0& {\mathsf{\Xi }}_{46}\\ \ast & \ast & \ast & \ast & {\mathsf{\Xi }}_{55}& {\mathsf{\Xi }}_{56}\\ \ast & \ast & \ast & \ast & \ast & {\mathsf{\Xi }}_{66}\end{array}\right]<0,$$

(25)

where

• ${\mathsf{\Xi }}_{11}={\mathcal{E}}_{\varsigma }^T{\mathcal{E}}_{\varsigma }-P_{\varsigma \varrho }-({\epsilon }_1+{\epsilon }_2){\mathcal{C}}_{\varsigma }^T{\mathcal{C}}_{\varsigma }$,
• ${\mathsf{\Xi }}_{14}={({\mathcal{Z}}_{\varsigma }{\mathcal{A}}_{\varsigma }+{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{\mathcal{Y}}_{\varsigma }{\mathcal{C}}_{\varsigma })}^T$,
• ${\mathsf{\Xi }}_{15}=\epsilon {\left({\mathcal{Y}}_{\varsigma }{\mathcal{C}}_{\varsigma }\right)}^T-{\gamma }_{\varkappa }({\mathcal{Z}}_{\varsigma }{\mathcal{B}}_{\varsigma }-{\mathcal{B}}_{\varsigma }{\mathcal{X}}_{\varrho })$,
• ${\mathsf{\Xi }}_{24}={\left({\mathcal{Z}}_{\varsigma }{\mathcal{D}}_{\varsigma }\right)}^T$,
• ${\mathsf{\Xi }}_{34}={\left({\gamma }_{\varkappa }{\mathcal{Z}}_{\varsigma }{\mathcal{B}}_{\varsigma }{\mathcal{Y}}_{\varsigma }\right)}^T-{\mathcal{Z}}_{\varsigma }^T$,
• ${\mathsf{\Xi }}_{44}={\mathcal{P}}_{\varsigma \varrho }^{-1}-{\mathcal{Z}}_{\varsigma }-{\mathcal{Z}}_{\varsigma }^T$,
• ${\mathsf{\Xi }}_{46}=[-{\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{\mathcal{Y}}_{\varsigma }0]$,
• ${\mathsf{\Xi }}_{55}=-\epsilon ({\mathcal{X}}_{\varsigma }+{\mathcal{X}}_{\varsigma }^T)$,
• ${\mathsf{\Xi }}_{56}=\left[0\epsilon {\mathcal{Y}}_{\varsigma }\right]$,
• ${\mathsf{\Xi }}_{66}=\mathrm{diag}\{-{\epsilon }_{1}I,{\epsilon }_{2}I\}$.

The controller gain matrix can be obtained according to the following equation:

$$K_{\varsigma }=Y_{\varsigma }^T{\mathcal{Z}}_{\varsigma }^{-T}.$$

(26)

Proof. 

Given the inequality $({\mathcal{P}}_{\varsigma \varrho }-{\mathcal{Z}}_{\varsigma }){\mathcal{P}}_{\varsigma \varrho }^{-1}{({\mathcal{P}}_{\varsigma \varrho }-{\mathcal{Z}}_{\varsigma })}^T\ge 0$, one can deduce that

$$-{\mathcal{Z}}_{\varsigma }{\mathcal{P}}_{\varsigma \varrho }^{-1}{\mathcal{Z}}_{\varsigma }^T\le {\mathcal{P}}_{\varsigma \varrho }-{\mathcal{Z}}_{\varsigma }-{\mathcal{Z}}_{\varsigma }^T,$$

(27)

Leveraging Lemmas 1 and 2, one can derive from Equation (25) that

$$\left[\begin{array}{ccccc}{\tilde{\mathsf{\Xi }}}_{11}& 0& 0& {\tilde{\mathsf{\Xi }}}_{14}& {\tilde{\mathsf{\Xi }}}_{15}\\ \ast & -{\mu }^{2}I& 0& {\mathsf{\Xi }}_{24}& 0\\ \ast & \ast & -I& {\mathsf{\Xi }}_{34}& 0\\ \ast & \ast & \ast & {\tilde{\mathsf{\Xi }}}_{44}& 0\\ \ast & \ast & \ast & \ast & {\mathsf{\Xi }}_{55}\end{array}\right]<0,$$

(28)

where

• ${\tilde{\mathsf{\Xi }}}_{11}={\mathcal{E}}_{\varsigma }^T{\mathcal{E}}_{\varsigma }-P_{\varsigma \varrho }$,
• ${\tilde{\mathsf{\Xi }}}_{14}={\mathsf{\Xi }}_{14}-{\left({\gamma }_{\varkappa }{\mathcal{B}}_{\varsigma }{\mathcal{Y}}_{\varsigma }{\mathcal{C}}_{\varsigma }\right)}^T$,
• ${\tilde{\mathsf{\Xi }}}_{15}=\epsilon {\left({\mathcal{Y}}_{\varsigma }(I+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma }\right)}^T-{\gamma }_{\varkappa }({\mathcal{Z}}_{\varsigma }{\mathcal{B}}_{\varsigma }-{\mathcal{B}}_{\varsigma }{\mathcal{X}}_{\varrho })$,
• ${\tilde{\mathsf{\Xi }}}_{44}=-{\mathcal{Z}}_{\varsigma }{\mathcal{P}}_{\varsigma \varrho }^{-1}{\mathcal{Z}}_{\varsigma }^T$.

Through the application of Lemma 3, it is possible to infer from (28) that

$${\mathsf{\Xi }}_{\varkappa }^{\varsigma \varrho }=\left[\begin{array}{cccc}{\tilde{\mathsf{\Xi }}}_{11}& 0& 0& {\overline{\mathsf{\Xi }}}_{14}\\ \ast & -{\mu }^{2}I& 0& {\mathsf{\Xi }}_{23}\\ \ast & \ast & -I& {\mathsf{\Xi }}_{34}\\ \ast & \ast & \ast & {\tilde{\mathsf{\Xi }}}_{44}\end{array}\right]<0,$$

(29)

where

• ${\overline{\mathsf{\Xi }}}_{14}={\tilde{\mathsf{\Xi }}}_{14}-{\gamma }_{\varkappa }{\left(({\mathcal{Z}}_{\varsigma }{\mathcal{B}}_{\varsigma }-{\mathcal{B}}_{\varsigma }{\mathcal{X}}_{\varrho }){\mathcal{X}}_{\varsigma }^{-1}{\mathcal{Y}}_{\varsigma }(I+{\mathcal{H}}_{\varrho }){\mathcal{C}}_{\varsigma }\right)}^T$.

Following this, pre-multiplying and post-multiplying (29) with $\mathrm{diag}\{I,I,I,{\mathcal{Z}}_{\varsigma }^{-1}\}$ and its transpose lead to the straightforward observation that Equation (22) holds. This indicates that the system (14) achieves consensus at the $H_{\infty }$ performance level $\mu$. The proof is finalized. □

5. Simulation Examples

In this section, the feasibility and practicality of the proposed method are illustrated through a numerical example and a single-link robotic arm demonstration.

5.1. A Numerical Example

Consider the DTMJFOMASs composed of four agents, as depicted in Figure 2. The corresponding Laplacian matrix L is as follows:

$$L=\left[\begin{array}{cccc}1& -1& 0& 0\\ 0& 1& -1& 0\\ -1& 0& 2& 0\\ 0& -1& 0& 1\end{array}\right].$$

The TPMs $\mathsf{\Pi }$ and $\mathsf{\Theta }$ with incomplete transition probabilities in Markov jump modes are as follows:

$$\mathsf{\Pi }=\left[\begin{array}{ccc}0.3& 0.1& 0.6\\ 0.6& \ast & \ast \\ 0.2& \ast & \ast \end{array}\right],\mathsf{\Theta }=\left[\begin{array}{ccc}0.2& 0.6& 0.2\\ 0.5& \ast & \ast \\ \ast & \ast & 0.8\end{array}\right].$$

The matrix parameters are shown in

Table 1. The matrix parameters for three modes.

	A	B	C	D	E
Mode 1	$\left[\begin{array}{cc}0.95& 0.01\\ 0.15& 0.95\end{array}\right]$	$\left[\begin{array}{cc}0.51& -0.1\\ 0.05& 0.15\end{array}\right]$	$\left[\begin{array}{cc}0.2& 0.1\end{array}\right]$	$\left[\begin{array}{c}0.02\\ 0.1\end{array}\right]$	$\left[\begin{array}{cc}0.1& 0.1\end{array}\right]$
Mode 2	$\left[\begin{array}{cc}0.96& 0.02\\ 0.15& 0.8\end{array}\right]$	$\left[\begin{array}{cc}0.01& -0.1\\ 0.05& 0.1\end{array}\right]$	$\left[\begin{array}{cc}0.21& 0.08\end{array}\right]$	$\left[\begin{array}{c}0.01\\ 0.1\end{array}\right]$	$\left[\begin{array}{cc}0.14& -0.1\end{array}\right]$
Mode 3	$\left[\begin{array}{cc}0.95& 0.02\\ 0.1& 0.98\end{array}\right]$	$\left[\begin{array}{cc}0.49& -0.1\\ 0.05& 0.01\end{array}\right]$	$\left[\begin{array}{cc}0.2& 0.11\end{array}\right]$	$\left[\begin{array}{c}0.05\\ 0.1\end{array}\right]$	$\left[\begin{array}{cc}-0.1& 0.14\end{array}\right]$

. The initial state of four agents are set as $x_1\left(0\right)={[1,-1]}^T$, $x_2\left(0\right)={[-5,8]}^T$, $x_3\left(0\right)={[3,-2]}^T$, $x_4\left(0\right)={[2,5]}^T$, and $c=1$. The disturbance input is designated as $\omega \left(k\right)=0.05e^{-0.05k}sin\left(k\right)$. The stochastic packet loss probability is given by $\overline{\beta }=20\%$. The initial values of QETM are established as ${\delta }_{\varrho }=0.55$, $v_{min}=0.01$, ${\varrho }_1=0.1$, ${\varrho }_2=0.99$, and ${\epsilon }_0=0.45$.

By solving LMIs in Theorem 3, we obtain that optimal $H_{\infty }$ performance index $\mu =1.2595$ and controller gains

$$K_1={[1.5250,1.1883]}^T,K_2={[1.5195,0.5143]}^T,K_3={[0.4502,0.1514]}^T.$$

The state trajectory diagrams for four agents are depicted without control input and with control input in Figure 3a,b, respectively. Obviously, the agents without control input fail to converge, whereas the agents under the influence of the distributed controller (9) can realize consensus. This indicates that the distributed control algorithm introduced in the study is genuinely effective.

As depicted in Figure 4, the control output is displayed under the scenario of DoS attacks. Notably, system (14) maintains strong robustness even when confronted with DoS attacks. The evolution of modes $\psi$ and $\varphi$ are presented in Figure 5. By examining the error curve in Figure 6, it is clear that the agents achieve consensus by 43 s. Figure 7 demonstrates the triggering time interval for all agents. The average triggering frequency of multiagents is 7.25%. The comparison between the measurement output $y\left(k\right)$ and the quantized output $\eta \left(k\right)$ is illustrated in Figure 8. The results indicate that the QETM proposed not only saves communication channels but also ensures consensus among multiagents.

5.2. The Single-Link Robotic Arm Systems

We validate the model’s practicability through the SLRASs with six nodes [2,47,48,49,50], with its dynamic equation being formulated as

$$J\ddot{\theta }\left(t\right)=-glM\sin \left(\theta \left(t\right)\right)-R\dot{\theta }\left(t\right)+u\left(t\right)$$

where $\theta \left(t\right)$, J, and M stand for the angle position of the arm, the mass of payload, and the moment of inertia, respectively. The gravity acceleration g, the arm length L, and the payload mass M take the value $9.81{\mathrm{m}/\mathrm{s}}^2,0.5\mathrm{m},\mathrm{and}2\mathrm{N}·\mathrm{m}/\mathrm{s}$, respectively. The dynamic mode of the SLRASs can be expressed as

$$\left\{\begin{array}{c}{x}_{\varkappa }(k+1)={\mathcal{A}}_{\varsigma }{x}_{\varkappa }\left(k\right)+{\mathcal{B}}_{\varsigma }{u}_{\varkappa }\left(k\right)+{\mathcal{D}}_{\varsigma }{\omega }_{\varkappa }\left(k\right)\hfill \\ y_{\varkappa }\left(k\right)={\mathcal{C}}_{\varsigma }{x}_{\varkappa }\left(k\right)\hfill \end{array}\right.$$

where $A_{\varsigma }=\left[\begin{array}{cc}1& T\\ -{\displaystyle \frac{TgLM}{J}}& 1-{\displaystyle \frac{TR}{J}}\end{array}\right]$, $B_{\varsigma }=\left[\begin{array}{cc}0& T\\ 0& -{\displaystyle \frac{TR}{J}}\end{array}\right]$, $D_{\varsigma }\left[\begin{array}{c}0\\ T\end{array}\right]$.

There are three different modes for the parameters J and M: $J_1=1\mathrm{N}·\mathrm{m},M_1=1\mathrm{kg}$; $J_2=2.5\mathrm{N}·\mathrm{m},M_2=2\mathrm{kg}$; $J_3=5\mathrm{N}·\mathrm{m},M_3=4\mathrm{kg}$. The sampling period is $T=0.1\mathrm{s}, f=0.01/\mathsf{\pi }$. The TPMs $\mathsf{\Pi }$ and $\mathsf{\Theta }$ are as follows:

$$\mathsf{\Pi }=\left[\begin{array}{ccc}0.3& 0.2& 0.5\\ \ast & \ast & 0.5\\ 0.4& \ast & \ast \end{array}\right],\mathsf{\Theta }=\left[\begin{array}{ccc}0.1& \ast & \ast \\ 0.4& \ast & \ast \\ \ast & 0.2& \ast \end{array}\right].$$

The directed topology structure of the SLRASs is depicted in Figure 9. The corresponding Laplacian matrix is as follows:

$$L=\left[\begin{array}{cccccc}1& -1& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0\\ 0& -1& 2& -1& 0& 0\\ 0& -1& -1& 3& -1& 0\\ 0& 0& 0& -1& 2& -1\\ 0& 0& -1& 0& 0& 1\end{array}\right].$$

The matrix parameters are defined in

Table 2. The matrix parameters of the SLRASs.

	A	B	C	D	E
Mode 1	$\left[\begin{array}{cc}1& 0.1\\ -4.9505& -1\end{array}\right]$	$\left[\begin{array}{cc}0& 0.1\\ 0& -0.2\end{array}\right]$	$\left[\begin{array}{cc}0& 0.1\end{array}\right]$	$\left[\begin{array}{c}0\\ 0.1\end{array}\right]$	$\left[\begin{array}{cc}0.1& 0.1\end{array}\right]$
Mode 2	$\left[\begin{array}{cc}1& 0.1\\ -3.9240& 0.2\end{array}\right]$	$\left[\begin{array}{cc}0& 0.1\\ 0& -0.8\end{array}\right]$	$\left[\begin{array}{cc}0& 0.04\end{array}\right]$	$\left[\begin{array}{c}0\\ 0.1\end{array}\right]$	$\left[\begin{array}{cc}0.14& -0.1\end{array}\right]$
Mode 3	$\left[\begin{array}{cc}1& 0.1\\ -2.4525& 0.8\end{array}\right]$	$\left[\begin{array}{cc}0& 0.1\\ 0& -0.02\end{array}\right]$	$\left[\begin{array}{cc}0& 0.01\end{array}\right]$	$\left[\begin{array}{c}0\\ 0.1\end{array}\right]$	$\left[\begin{array}{cc}-0.1& 0.14\end{array}\right]$

. The initial state of the SLRASs are configured as $x_1\left(0\right)={[10,-10]}^T$, $x_2\left(0\right)={[-5,8]}^T$, $x_3\left(0\right)={[33,-28]}^T$, $x_4\left(0\right)={[14,35]}^T$, $x_5\left(0\right)={[-35,22]}^T$, $x_6\left(0\right)={[21,16]}^T$, and $c=10.05$. We set the disturbance input as $\omega \left(k\right)=e^{-0.05k}sin\left(0.1\pi k\right)$.

Utilizing Theorem 3, we ascertain optimal $H_{\infty }$ performance index $\mu =1.9523$ and controller gains

$$K_1={[0.0525,0.0218]}^T,K_2={[-0.5200,-0.4143]}^T,K_3={[0.0515,0.0555]}^T.$$

Figure 10 shows the mode $\psi$ and $\varphi$ evolution in the SLRASs. Figure 11, Figure 12, Figure 13 and Figure 14 illustrate the SLRASs consensus simulation under DoS attacks, and it is evident that the state converges to zero after 22 s. In Figure 11 and Figure 12, the control input and output for the SLRASs are presented. The error curve for the SLRASs is displayed in Figure 13. The 3D state trajectory for the SLRASs is depicted in Figure 14.

Table 3. The trigger frequency of all robotic arms.

Trigger Frequency	Robotic Arm 1	Robotic Arm 2	Robotic Arm 3	Robotic Arm 4	Robotic Arm 5	Robotic Arm 6	Average
QETM	18%	23%	21%	19%	24%	23%	21.33%
Method 1 *	23%	24%	24%	26%	21%	26%	24.00%
Method 2 *	38%	35%	40%	33%	30%	33%	34.83%

* Method 1 and Method 2 construct new triggering mechanisms by integrating the logarithmic quantizer in [35] and the uniform quantizer in [31,32] with Equation (8).

illustrates the triggering frequencies and their averages for each robotic arm under QETM, Method 1, and Method 2. Method 1 and Method 2 construct new triggering mechanisms by integrating the logarithmic quantizer in [35] and the uniform quantizer in [31,32] with Equation (8). Overall, the controller presented in this paper maintains consensus under DoS attack and minimizes communication loads efficiently.

6. Conclusions

This paper examines the $H_{\infty }$ consensus control of DTMJFOMASs with DoS attacks and external disturbances, extending the short-memory principle proposed in reference [23] to MJMASs. To conserve bandwidth and minimize triggering frequency, the mode-dependent QETM is employed. According to the designed distributed controller, sufficient conditions are proposed to ensure the system consensus under the given $H_{\infty }$ performance criterion. In the end, the validity and applicability of the proposed model are elucidated through a numeric example and the SLRASs.

It is worth noting that this paper only considers linear time-invariant DTMJFOMASs. Future research will concentrate on addressing the problem of leader–follower consensus in nonlinear DTMJFOMASs. Additionally, our objectives include designing a triggering mechanism that can more effectively conserve communication resources, thereby further optimizing system performance.