Control Design for Fractional Order Leader and Follower Systems with Mixed Time Delays: A Resilience-Based Approach

Abstract

In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weighted digraph. A disturbance term is introduced into the delayed and non-delayed controller part to make it more practical. Our method involves proposing algebraic criteria by utilizing non-delayed and delayed protocols, applying the Razumikhin technique and graph theory respectively. The presented method can well overcome the difficulty resulting from fractional calculus, time delays and fractional derivatives. To demonstrate the validity and effectiveness of our findings, we provide an example at the end of our study.

1. Introduction

In recent years, due to numerous applications, cooperative control for multi-agent systems has gained attention for software systems [1], neural networks [2], intelligent robotics [3] and traffic control [4]. Consensus has drawn a lot of attention as a fundamental and important issue of cooperative control. Both leader-following consensus [5,6,7] and leaderless consensus [8,9,10] have been seriously considered. However, many practical control systems have several leaders, such as satellite-formation control systems [11] and robot cooperation systems [12]. The corresponding consensus control is known as containment control (CC). Designing suitable control protocols to force the followers to converge to the convex hull created by the leaders is the main focus of CC. To solve CC, a variety of protocols are used such as adaptive protocols [13,14], periodic sampling protocols [15,16,17] and observational protocols [18,19,20].

However, many existing studies have paid considerable attention. In recent years, due to unique advantages for modelling some physical phenomena and processes, such as biological systems [21], complex networks [22,23] and rock blasting [24], fractional derivatives have gained increasing interest. In [25,26], the CC of both linear and nonlinear FOMASs is explained. In [27], a novel projection protocol is selected to resolve the CC of linear FOMASs. It is important to note that time delays can occur in many real models and have a significant impact on stability [28,29]. It would be preferable to implement delayed control protocols in order to achieve CC because time delays will arise among the process of agents transmitting data. To solve CC of linear FOMASs, the sampled-data-based technique with time delays is provided [30,31]. Today the basic method for handling CC of linear FOMASs with dynamical leaders [32] or static [33] leaders is the frequency domain method. For fractional-order differential systems, it is generally known that the Lyapunov function technique provides a very practical and efficient way to deal with consensus, CC and stability. However, the fractional calculus’s lack of the semigroup property makes it very challenging to research and analyze the CC of delayed FOMASs. In fact, there are still some important problems that need to be resolved, such as how to deal with the CC of such systems using the Lyapunov function method and how to provide some basic algebraic conditions. This study’s objective is to partially fill this gap.

On the other hand, the geometric extent of communication delays is crucial, and many multi-agent systems cannot be accurately represented by dynamical systems with discrete delays. Instead, distributed delays may better capture the complexity of such lag events. On the basis of the above analysis, resilient base containment control (CC) of fractional-order multi-agent systems (FOMASs) with mixed time delays, where communication topology is a weighted digraph among followers, will be addressed in this paper. Using the fractional Razumikhin method and graph theory, some sufficient criteria are preferred for achieving resilient base CC by using the non-delayed and delayed control protocols. A concluding example is presented to demonstrate the validity and effectiveness of the proposed approach. The major contributions of this study are

(1)

Initially FOMASs with mixed time delays are considered, and a practical and efficient method is developed for achieving resilient base CC.

(2)

The proposed method may handle well the problems resulting from time delays and fractional derivatives.

(3)

Matrix inequalities are used to provide resilient base CC criteria that can be easily verified in practical applications.

2. Preliminaries

First, several essential definitions and lemmas are remembered.

Definition 1

([34]). The β-order derivative for a function $x\left(t\right)$ in the Caputo sense is given by

$$t_0{D}_t^{\beta }x\left(t\right)=\frac{1}{\mathrm{\Gamma }(u-\beta )}{\int }_{t_0}^t\frac{x^u\left(p\right)}{{(t-p)}^{\beta +1-u}}dp,u-1<\beta \le u,$$

(1)

where $\mathrm{\Gamma }\left(y\right)={\int }_0^{+\infty }{p}^{(y-1)}{e}^{-p}dp$.

Definition 2

([35]). If $(1-\zeta )x_1+\zeta x_2\in \mathrm{\Omega }$ for any $x_1,x_2\in \mathrm{\Omega }$ and $\zeta \in [0,1]$, a set $\mathrm{\Omega }\subset R^n$ is said to be convex. Points $x_1,x_2,\dots ,x_N\in R^n$ are contained in the smallest convex hull which is represented by

$$C_0\{x_1,x_2,\dots ,x_N\}=\{\sum _{i=1}^N{\zeta }_i{x}_i|{\zeta }_i\in [0,1],\sum _{i=1}^N{\zeta }_i=1\}$$

Examine a delayed differential equation with β order.

$${}_{t_0}{D}_t^{\beta }x\left(t\right)=f(t,x_t),t\ge t_0,$$

(2)

where $0<\beta \le 1$ and $x_t\left(\theta \right)=x(t+\theta ),\theta \in [-r,0]$. The continuous function $f:R\times R^n\to R^n$ satisfies $f(t,0)=0$.

Lemma 1

([29]). If there are three positive constants $v_1,v_2,v_3$ and a differentiable function $V:R^n\to R$, then the trivial solution of Equation (2) is asymptotically stable such that

$$v_1{\left|\right|x\left|\right|}^2\le V\left(x\right)\le v_2{\left|\right|x\left|\right|}^2,$$

and the β-order derivative fulfils

$${}_{t_0}{D}_t^{\beta }{V\left(x\left(t\right)\right)|}_{\left(2\right)}\le -v_3{\left|\right|x\left|\right|}^2,$$

whenever

$$V\left(x\right(t+\theta \left)\right)\le \lambda V\left(x\right(t\left)\right),\theta \in [-r,0],$$

for some $\lambda >1$.

Lemma 2

([36]). The following relationship holds for any differentiable vector function $x\left(t\right)\in R^n$ and $n\times n$ matrix $P>0$

$${}_{t_0}{D}_t^{\beta }\left(x^T\left(t\right)Px\left(t\right)\right)\le 2x^T\left(t\right)P_{t_0}{D}_t^{\beta }x\left(t\right),0<\beta <1$$

3. Resilient Base Containment Control of FOMAS

To deal with resilient base CC of FOMASs an efficient and easy method is adopted. Certain practical algebraic conditions to ensure resilient base CC are presented by using delayed and non-delayed control protocols with disturbance term using the fractional Razumikhin technique and graph theory, respectively. The problems arising from time delays and fractional derivatives can be effectively overcome by the suggested method.

Initially some necessary graph definitions are recalled.

A weighted digraph $G=(V,E)$ is used to represented the communication topology of a FOMAS, whose vertex set is the set $V=\{1,2,\dots ,N\}$ and $E\subset V\times V$ is the edge set. ${\epsilon }_{ij}=\left(ij\right)\in E$ denotes the ability of agent i to communicate with agent j. The adjacency matrix is denoted by $B={\left(b_{ij}\right)}_{N\times N}$, whose entries are given by $b_{ij}$ > 0 if ${\epsilon }_{ji}\in E$ and $b_{ij}$ = 0 if ${\epsilon }_{ji}\notin E$. Self-links are eliminated in the article, which give $b_{ii}$ = 0.$L={\left(l_{ij}\right)}_{N\times N}$ denotes the Laplacian matrix of G. Its entries are $l_{ii}={\sum }_{j=1,j\ne i}^N{b}_{ij}$ and $l_{ij}=-b_{ij}(i\ne j).$

The FOMAS consist of $N-F$ leaders and F followers, denoted by $\mathcal{R}=\{F+1,\dots ,N\}$ and $\mathcal{F}=\{1,2,\dots ,F\}$ respectively. The ith agent’s dynamics are shown by

$$\left\{\begin{array}{c}{}_{t_0}{D}_t^{\beta }{x}_i\left(t\right)=A{x}_i\left(t\right)+B{x}_i(t-r_1)+{\int }_{-r_2}^{0}C{x}_i(t+p)dp+{\omega }_i\left(t\right),i\in \mathcal{F}\hfill \\ {}_{t_0}{D}_t^{\beta }{x}_i\left(t\right)=A{x}_i\left(t\right)+B{x}_i(t-r_1)+{\int }_{-r_2}^{0}C{x}_i(t+p)dp,i\in \mathcal{R}\hfill \end{array}\right.$$

(3)

where 0 < $\beta$, < 1 ${\omega }_i\in R^n$ and $x_i\in R^n$ indicate the ith agent’s state feedback protocol and state vector. Constant matrices $A,B,C\in R^{n\times n}$ and time delays $r_1$ > 0, $r_2$ > 0.

The next assumption is used to gain the major conclusion.

(H). At least one leader must provide information to each follower, and the leader may not receive any data from other agents.

If (H) is true, then Laplacian matrix L can be expressed in the following form.

$$L=\left(\begin{array}{cc}{L}_F& L_R\\ {}^0(N-F)\times F& {}^0(N-F)\times (N-F)\end{array}\right)$$

where $L_F\in R^{F\times F}$ and $L_R\in R^{F\times (N-F)}$.

Lemma 3

([37]). Under $\left(H\right),L_F$ is a nonsingular N-matrix, -$L_F^{-1}{L}_R$ is a non-negative matrix and its row sums are equal to 1.

The delayed and non-delayed state feedback protocols are used respectively in the following. By using graph theory and the fractional Razumikhin approach, we propose some practical algebraic conditions to ensure resilient base CC of FOMAS (3).

3.1. Case-I

Resilient base CC of FOMAS (3) under a non-delayed control protocol.

We designed the following non-delayed control protocol ${\omega }_i$.

$${\omega }_i\left(t\right)=K\sum _{j=1}^N{b}_{ij}(x_j\left(t\right)-x_i\left(t\right))+\Delta {\omega }_i\left(t\right),i\in \mathcal{F},$$

(4)

where $K\in R^{n\times n}$ is the gain matrix and $\Delta {\omega }_i\left(t\right)$ is the disturbance term.

Substituting Equation (4) into Equation (3), we can obtain

$$\left\{\begin{array}{c}{}_{t_0}{D}_t^{\beta }{x}_i\left(t\right)=A{x}_i\left(t\right)+B{x}_i(t-r_1)+{\int }_{-r_2}^{0}C{x}_i(t+p)dp+K\sum _{j=1}^N{a}_{ij}(x_j\left(t\right)-x_i\left(t\right))+\Delta {\omega }_i\left(t\right),i\in \mathcal{F}\hfill \\ {}_{t_0}{D}_t^{\beta }{x}_i\left(t\right)=A{x}_i\left(t\right)+B{x}_i(t-r_1)+{\int }_{-r_2}^{0}C{x}_i(t+p)dp,i\in \mathcal{R}\hfill \end{array}\right.$$

(5)

which the help of the Kronecker product, Equation (5) can be written as

$$\left\{\begin{array}{c}{}_{t_0}{D}_t^{\beta }{x}_F\left(t\right)=(I_F\otimes A)x_F\left(t\right)+(I_F\otimes B)x_F(t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)x_F(t+p)dp\hfill \\ -(L_F\otimes K)x_F\left(t\right)-(L_R\otimes K)x_R\left(t\right)+\Delta \omega \left(t\right),\hfill \\ {}_{t_0}{D}_t^{\beta }{x}_R\left(t\right)=(I_{N-F}\otimes A)x_R\left(t\right)+(I_{N-F}\otimes B)x_R(t-r_1)+{\int }_{-r_2}^0(I_{N-F}\otimes C)x_R(t+p)dp,\hfill \end{array}\right.$$

(6)

where $x_R\left(t\right)={[x_{F+1}^T\left(t\right),\dots ,y_N^T\left(t\right)]}^T$ and $x_F\left(t\right)={[x_1^T\left(t\right),x_2^T\left(t\right),\dots ,x_F^T\left(t\right)]}^T$.

Let error state $\alpha \left(t\right)=x_F\left(t\right)-[(-L_F^{-1}{L}_R)\otimes I_n]x_R\left(t\right)=x_F\left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)$. Then its $\beta$-order derivative is given by

$$\begin{array}{ccc}\hfill {}_{t_0}{D}_t^{\beta }\alpha \left(t\right)& =& {}_{t_0}{D}_t^{\beta }[x_F\left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)]\hfill \\ \hfill & =& t_0{D}_t^{\beta }{x}_F\left(t\right)+t_0{D}_t^{\beta }[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)]\hfill \\ \hfill & =& {}_{t_0}{D}_t^{\beta }{x}_F\left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_{t_0}{D}_t^{\beta }{x}_R\left(t\right)\hfill \\ \hfill & =& (I_F\otimes A)x_F\left(t\right)+(I_F\otimes B)x_F(t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)x_F(t+p)dp\hfill \\ \hfill & & -(L_F\otimes K)x_F\left(t\right)-(L_R\otimes K)x_R\left(t\right)+\Delta \omega \left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n][(I_{N-F}\otimes A)x_R\left(t\right)\hfill \\ & & +(I_{N-F}\otimes B)x_R(t-r_1)+{\int }_{-r_2}^0(I_{N-F}\otimes C)x_R(t+p)dp]\hfill \\ \hfill & =& (I_F\otimes A)[(x_F\left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)]+(I_F\otimes B)[x_F(t-r_1)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R(t-r_1)]\hfill \\ \hfill & & +\Delta \omega \left(t\right)+{\int }_{-r_2}^0(I_F\otimes C)[x_F(t+p)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R(t+p)]dp\hfill \\ \hfill & & -(L_F\otimes K)[x_F\left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)]\hfill \\ \hfill & =& (I_F\otimes A)\alpha \left(t\right)+(I_F\otimes B)\alpha (t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)\alpha (t+p)dp-(L_F\otimes K)\alpha \left(t\right)+\Delta \omega \left(t\right)\hfill \\ \hfill & =& (I_F\otimes A-L_F\otimes K)\alpha \left(t\right)+(I_F\otimes B)\alpha (t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)\alpha (t+p)dp+\Delta \omega \left(t\right).\hfill \end{array}$$

(7)

Theorem 1.

Under (H), if there exist three scalars ${\eta }_i$ > 0 (i = 1, 2, 3) and a matrix P > 0 then resilient base CC of FOMAS (3) with protocol (4) is achieved such that

$$\left[\begin{array}{cc}{\varphi }_{11}& I_F\otimes \left(PB\right)\\ I_F\otimes \left(B^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]<0,$$

(8)

$$\left[\begin{array}{cc}({\eta }_3-{\eta }_2)(I_F\otimes P)& I_F\otimes \left(PC\right)\\ I_F\otimes \left(C^{T}P\right)& -{\eta }_3(I_F\otimes P)\end{array}\right]<0.$$

(9)

where ${\varphi }_{11}=I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(I_F\otimes (P+P^T)\Delta \omega \left(t\right))]-L_F\otimes \left(PK\right)-L_F^T\otimes \left(K^{T}P\right)$.

Proof. 

Select a Lyapunov function

$$V\left(t\right)={\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right).$$

It follows from Equation (7) and Lemma (2) that

$$\begin{array}{ccc}\hfill {}_{t_0}{D}_t^{\beta }V\left(t\right)& \le & 2{\alpha }^T\left(t\right){(I_F\otimes P)}_{t_0}{D}_t^{\beta }\alpha \left(t\right)\hfill \\ & =& 2{\alpha }^T\left(t\right)(I_F\otimes P)[(I_F\otimes A-L_F\otimes K)\alpha \left(t\right)+(I_F\otimes B)\alpha (t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)\alpha (t+p)dp+\Delta \omega \left(t\right)]\hfill \\ & =& 2{\alpha }^T\left(t\right)[(I_F\otimes P)(I_F\otimes A)-(I_F\otimes P)(L_F\otimes K)]\alpha \left(t\right)+2{\alpha }^T(I_F\otimes P)(I_F\otimes B)\alpha (t-r_1)\hfill \\ & & +2{\alpha }^T\left(t\right){\int }_{-r_2}^0(I_F\otimes P)(I_F\otimes C)\alpha (t+p)dp+2{\alpha }^T\left(t\right)(I_F\otimes P)\Delta \omega \left(t\right)\hfill \\ & =& 2{\alpha }^T\left(t\right)[I_F\otimes \left(PA\right)-L_F\otimes \left(PK\right)]\alpha \left(t\right)+2{\alpha }^T\left(t\right)[I_F\otimes \left(PB\right)]\alpha (t-r_1)\hfill \\ & & +2{\alpha }^T\left(t\right){\int }_{-r_2}^0[I_F\otimes \left(PC\right)]\alpha (t+p)dp+2{\alpha }^T\left(t\right)(I_F\otimes P)\Delta \omega \left(t\right)\hfill \end{array}$$

whenever $\alpha \left(t\right)$ fulfills

$$V\left(\alpha (t+\theta )\right)<\lambda V\left(\alpha \left(t\right)\right),-max(r_1,r_2)\le \theta \le 0,$$

$$0<\lambda V\left(\alpha \right(t\left)\right)-V\left(\alpha \right(t+\theta )$$

for some $\lambda$ > 1, or equivalently

$$\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\alpha }^T(t+\theta )(I_F\otimes P)\alpha (t+\theta )>0,-max(r_1,r_2)\le \theta \le 0$$

Thus we have for any ${\eta }_1>0,{\eta }_2>0,{\eta }_3>0$,

$$\begin{array}{ccc}\hfill {}_{t_0}{D}_t^{\beta }V\left(t\right)& \le & {\alpha }^T\left(t\right)[I_F\otimes (PA+A^{T}P)-L_F\otimes \left(PK\right)-L_F^T\otimes \left(K^{T}P\right)]\alpha \left(t\right)\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes \left(PB\right)]\alpha (t-r_1)+{\alpha }^T(t-r_1)[I_F\otimes \left(B^{T}P\right)]\alpha \left(t\right)\hfill \\ & & +{\eta }_1[\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\alpha }^T(t-r_1)(I_F\otimes P)\alpha (t-r_1)]\hfill \\ & & +{\int }_{-r_2}^0[{\alpha }^T\left(t\right)[I_F\otimes \left(PC\right)]\alpha (t+p)+{\alpha }^T(t+p)[I_F\otimes \left(C^{T}P\right)]\alpha \left(t\right)]dp\hfill \\ & & +{\eta }_2[r_2{\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\int }_{-r_2}^0{\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)dp]\hfill \\ & & +{\int }_{-r_2}^0{\eta }_3[\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\alpha }^T\left(t\right)(t+p)(I_F\otimes P)\alpha (t+p)]dp\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes (P+P^T)]\Delta \omega \left(t\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {}_{t_0}{D}_t^{\beta }V\left(t\right)& =& {\alpha }^T\left(t\right)[I_F\otimes (PA+A^{T}P)-L_F\otimes \left(PK\right)-L_F^T\otimes \left(K^{T}P\right)]\alpha \left(t\right)\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes \left(PB\right)]\alpha (t-r_1)+{\alpha }^T(t-r_1)[I_F\otimes \left(B^{T}P\right)]\alpha \left(t\right)\hfill \\ & & +{\eta }_1\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\eta }_1{\alpha }^T(t-r_1)(I_F\otimes P)\alpha (t-r_1)\hfill \\ & & +{\int }_{-r_2}^0{\alpha }^T\left(t\right)[I_F\otimes \left(PC\right)]\alpha (t+p)dp+{\int }_{-r_2}^0{\alpha }^T\left(t\right)(t+p)[I_F\otimes \left(C^{T}P\right)]\alpha \left(t\right)dp\hfill \\ & & +{\eta }_2{r}_2{\alpha }^T\left(t\right)(I_N\otimes P)\alpha \left(t\right)-{\eta }_2{\int }_{-{\tau }_2}^0{\alpha }^T(I_N\otimes P)\alpha \left(t\right)ds\hfill \\ & & +{\int }_{-r_2}^0{\eta }_3\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\int }_{-r_2}^0{\eta }_3{\alpha }^T(t+p)(I_F\otimes P)\alpha (t+p)dp\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes (P+P^T)]\Delta \omega \left(t\right)\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\alpha }^T\left(t\right)[I_F\otimes (PA+A^{T}P+{\eta }_1\lambda P+{\eta }_2{r}_{2}H)-L_F\otimes \left(PK\right)-L_F^T\otimes \left(K^{T}P\right)\hfill \\ & & +{\alpha }^{-1}\left(t\right)[(I_F\otimes (P+P^T)]\Delta \omega \left(t\right)]\alpha \left(t\right)\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes \left(PB\right)]\alpha (t-r_1)+{\alpha }^T(t-r_1)[I_F\otimes \left(B^{T}P\right)]\alpha \left(t\right)\hfill \\ & & -{\eta }_1{\alpha }^T(t-r_1)(I_F\otimes P)\alpha (t-r_1)]\hfill \\ & & +{\int }_{-r_2}^0[({\eta }_3\lambda -{\eta }_2){\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)+{\alpha }^T\left(t\right)[I_F\otimes \left(PC\right)]\alpha (t+p)\hfill \\ & & +{\alpha }^T(t+p)[I_F\otimes \left(C^{T}P\right)]\alpha \left(t\right)-{\eta }_3{\alpha }^T(t+p)(I_F\otimes P)\alpha (t+p)]dp\hfill \\ & =& \begin{array}{c}[{\alpha }^T\left(t\right),{\alpha }^T(t-r_1)]\end{array}\left[\begin{array}{cc}{\varphi }_{11}& I_F\otimes \left(PB\right)\\ I_F\otimes \left(B^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]\begin{array}{c}{[\alpha \left(t\right),\alpha (t-r_1)]}^T\end{array}\hfill \\ & & +\begin{array}{c}{\int }_{-r_2}^0[{\alpha }^T\left(t\right),{\alpha }^T(t+p)]\end{array}\left[\begin{array}{cc}({\eta }_3\lambda -{\eta }_2)(I_F\otimes P)& I_F\otimes \left(PC\right)\\ I_F\otimes \left(C^{T}P\right)& -{\eta }_3(I_F\otimes P)\end{array}\right]\begin{array}{c}{[\alpha \left(t\right),\alpha (t+p)]}^{T}dp,\end{array}\hfill \end{array}$$

where ${\varphi }_{11}=I_F\otimes (PA+A^{T}P+{\eta }_1\lambda P+{\eta }_2{r}_{2}P)-L_F\otimes \left(PK\right)-L_F^T\otimes \left(K^{T}P\right)+{\alpha }^{-1}\left(t\right)[I_F\otimes (P+P^T)]\Delta \omega \left(t\right)$.

Relationships (8) and (9) imply that for an adequately small ${\lambda }_1>0,\lambda ={\lambda }_1+1$,

$$\left[\begin{array}{cc}{\varphi }_{11}& I_F\otimes \left(PB\right)\\ I_F\otimes \left(B^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]<0,$$

(10)

$$\left[\begin{array}{cc}({\eta }_3\lambda -{\eta }_2)(I_F\otimes P)& I_F\otimes \left(PC\right)\\ I_F\otimes \left(C^{T}P\right)& -{\eta }_3(I_F\otimes P)\end{array}\right]<0,$$

(11)

which indicates that ${}_{t_0}{D}_t^{\beta }V\left(t\right)$< 0. This implies Equation (7) is asymptotically stable by Lemma (1). Hence, resilient base CC of FOMAS (3) with protocol (4) can be achieved. □

3.2. Case-II

Resilient base CC of FOMAS (3) under a delayed control protocol.

We designed the following delayed control protocol ${\omega }_i$

$${\omega }_i\left(t\right)=K_1\sum _{j=1}^N{b}_{ij}(x_j(t-r_3)-x_i(t-r_3))+\Delta {\omega }_i\left(t\right),i\in \mathcal{F},$$

(12)

where $K_1\in R^{n\times n}$ is the gain matrix.

Substituting Equation (12) into Equation (3), we can obtain

$$\left\{\begin{array}{c}{}_{t_0}{D}_t^{\beta }{x}_i\left(t\right)=A{x}_i\left(t\right)+B{x}_i(t-r_1)+{\int }_{-r_2}^{0}C{x}_i(t+p)dp+K_1\sum _{j=1}^N{b}_{ij}(x_j(t-r_3)-x_i(t-r_3))\hfill \\ +\Delta {\omega }_i\left(t\right),i\in \mathcal{F}\hfill \\ {}_{t_0}{D}_t^{\beta }{x}_i\left(t\right)=A{x}_i\left(t\right)+B{x}_i(t-r_1)+{\int }_{-r_2}^{0}C{x}_i(t+p)dp,i\in \mathcal{R}\hfill \end{array}\right.$$

Under protocol (12) Equation (3) can be rewritten as

$$\left\{\begin{array}{c}{}_{t_0}{D}_t^{\beta }{x}_F\left(t\right)=(I_F\otimes A)x_F\left(t\right)+(I_F\otimes B)x_F(t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)x_F(t+p)dp\hfill \\ -(L_F\otimes K_1)x_F(t-r_3)-(L_R\otimes K_1)x_R(t-r_3)+\Delta \omega \left(t\right),\hfill \\ {}_{t_0}{D}_t^{\beta }{x}_R\left(t\right)=(I_{N-F}\otimes A)x_R\left(t\right)+(I_{N-F}\otimes B)x_R(t-r_1)+{\int }_{-r_2}^0(I_{N-F}\otimes C)x_R(t+p)dp.\hfill \end{array}\right.$$

Then the $\mu$-order derivative of $\alpha \left(t\right)$ is given by

$$\begin{array}{ccc}\hfill {}_{t_0}{D}_t^{\beta }\alpha \left(t\right)& =& {}_{t_0}{D}_t^{\beta }[x_F\left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)]\hfill \\ \hfill & =& t_0{D}_t^{\beta }{x}_F\left(t\right)+t_0{D}_t^{\beta }[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)\hfill \\ \hfill & =& {}_{t_0}{D}_t^{\beta }{x}_F\left(t\right)+{[\left(L_F^{-1}{L}_R\right)\otimes I_n]}_{t_0}{D}_t^{\beta }{x}_R\left(t\right)\hfill \\ \hfill & =& (I_F\otimes A)x_F\left(t\right)+(I_F\otimes B)x_F(t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)x_F(t+p)dp\hfill \\ \hfill & & -(L_F\otimes K_1)x_F(t-r_3)-(L_R\otimes K_1)x_R(t-r_3)+\Delta \omega \left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n][(I_{N-F}\otimes A)x_R\left(t\right)\hfill \\ & & +(I_{N-F}\otimes B)x_R(t-r_1)+{\int }_{-r_2}^0(I_{N-F}\otimes C)x_R(t+p)dp]\hfill \\ \hfill & =& (I_F\otimes A)[x_F\left(t\right)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R\left(t\right)]+(I_F\otimes B)[x_F(t-r_1)\hfill \\ \hfill & & +[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R(t-r_1)]+\Delta \omega \left(t\right)\hfill \\ \hfill & & +{\int }_{-r_2}^0(I_F\otimes C)[x_F(t+p)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R(t+p)]dp\hfill \\ \hfill & & -(L_F\otimes K_1)[x_F(t-r_3)+[\left(L_F^{-1}{L}_R\right)\otimes I_n]x_R(t-r_3)]\hfill \\ \hfill & =& (I_F\otimes A)\alpha \left(t\right)+(I_F\otimes B)\alpha (t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)\alpha (t+p)dp-(L_F\otimes K_1)\alpha (t-r_3)+\Delta \omega \left(t\right)\hfill \\ \hfill & =& (I_F\otimes A)\alpha \left(t\right)-(L_F\otimes K_1)\alpha (t-r_3)+(I_F\otimes B)\alpha (t-r_1)+{\int }_{-r_2}^0(I_F\otimes C)\alpha (t+p)dp+\Delta \omega \left(t\right).\hfill \end{array}$$

(13)

Theorem 2.

Under (H), if there exist two matrices P > 0, Q > 0 and scalers ${\eta }_i$ > 0 (i = 1, 2, 3) then resilient base CC of FOMAS (3) with protocol (12) is attain as

$$\left[\begin{array}{cc}{I}_F\otimes (PA+A^{T}P+{\eta }_{1}P+Q& I_F\otimes \left(PB\right)\\ +{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right))\\ I_F\otimes \left(B^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]<0,$$

(14)

$$\left[\begin{array}{cc}{I}_F\otimes [({\eta }_3{r}_2+{\eta }_2)P-Q]& -L_F\otimes \left(P{K}_1\right)\\ -L_F^T\otimes \left(K_1^{T}P\right)& -{\eta }_2(I_F\otimes P)\end{array}\right]<0,$$

(15)

$$\left[\begin{array}{cc}({\eta }_4-{\eta }_3)(I_F\otimes P)& I_F\otimes \left(PC\right)\\ I_F\otimes \left(C^{T}P\right)& -{\eta }_4(I_F\otimes P)\end{array}\right]<0.$$

(16)

Proof. 

A Lyapunov function is chosen

$$V\left(t\right)={\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right).$$

From Lemma (2) and Equation (13), we have

$$\begin{array}{ccc}\hfill {}_{t_0}{D}_t^{\beta }V\left(t\right)& \le & 2{\alpha }^T\left(t\right){(I_F\otimes P)}_{t_0}{D}_t^{\beta }\alpha \left(t\right)\hfill \\ & =& 2{\alpha }^T\left(t\right)(I_F\otimes P)[(I_F\otimes A)\alpha \left(t\right)-(L_F\otimes K_1)\alpha (t-r_3)+(I_F\otimes B)\alpha (t-r_1)\hfill \\ & & +{\int }_{-r_2}^0(I_F\otimes C)\alpha (t+p)dp+\Delta \omega \left(t\right)]\hfill \\ & =& 2{\alpha }^T\left(t\right)\left[(I_F\otimes P)(I_F\otimes A)\right]\alpha \left(t\right)-2{\alpha }^T\left(t\right)\left[(I_F\otimes P)(L_F\otimes K_1)\right]\alpha (t-r_3)\hfill \\ & & +2{\alpha }^T\left(t\right)(I_F\otimes P)(I_F\otimes B)\alpha (t-r_1)\hfill \\ & & +2{\alpha }^T\left(t\right){\int }_{-r_2}^0(I_F\otimes P)(I_F\otimes C)\alpha (t+p)dp+2{\alpha }^T\left(t\right)(I_F\otimes P)\Delta \omega \left(t\right)\hfill \\ & =& 2{\alpha }^T\left(t\right)[I_F\otimes \left(PA\right)]\alpha \left(t\right)-2{\alpha }^T\left(t\right)[L_F\otimes \left(P{K}_1\right)]\alpha (t-r_3)+2{\alpha }^T\left(t\right)[I_F\otimes \left(PB\right)]\alpha (t-r_1)\hfill \\ & & +2{\alpha }^T\left(t\right){\int }_{-r_2}^0[I_F\otimes \left(PC\right)]\alpha (t+p)dp+2{\alpha }^T\left(t\right)(I_F\otimes P)\Delta \omega \left(t\right)\hfill \end{array}$$

whenever $\alpha \left(t\right)$ fulfills

$$\begin{array}{ccc}\hfill V\left(\alpha \right(t+\theta \left)\right)& <& \lambda V\left(\alpha \left(t\right)\right),-max(r_1,r_2,r_3)\le \theta \le 0\hfill \\ \hfill \lambda V\left(\alpha \right(t\left)\right)-V\left(\alpha \right(t+\theta )& >& 0\hfill \end{array}$$

for some $\lambda$ > 1,or equivalently

$$\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\alpha }^T(t+\theta )(I_F\otimes P)\alpha (t+\theta )>0,-max(r_1,r_2,r_3,)\le \theta \le 0$$

Thus we have for any ${\eta }_i>0(i=1,2,3,4)$,

$$\begin{array}{ccc}\hfill {}_{t_0}{D}_t^{\beta }V\left(t\right)& \le & {\alpha }^T\left(t\right)[I_F\otimes (PA+A^{T}P)]\alpha \left(t\right)+{\alpha }^T\left(t\right)[I_F\otimes \left(PB\right)]\alpha (t-r_1)+{\alpha }^T(t-r_1)[I_F\otimes \left(B^{T}P\right)]\alpha \left(t\right)\hfill \\ & & -{\alpha }^T\left(t\right)[L_F\otimes \left(P{K}_1\right)]\alpha (t-r_3)-{\alpha }^T(t-r_3)[L_F^T\otimes \left(K_1^{T}P\right)]\alpha \left(t\right)\hfill \\ & & +{\int }_{-r_2}^0[{\alpha }^T\left(t\right)[I_F\otimes \left(PC\right)]\alpha (t+p)+{\alpha }^T(t+p)[I_F\otimes \left(C^{T}P\right)]\alpha \left(t\right)]dp\hfill \\ & & +{\eta }_1[\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\alpha }^T(t-r_1)(I_F\otimes P)\alpha (t-r_1)]\hfill \\ & & +{\eta }_2[\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\alpha }^T(t-r_3)(I_F\otimes P)\alpha (t-r_3)]\hfill \\ & & +{\eta }_3[r_2{\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\int }_{-r_2}^0{\alpha }^T(I_F\otimes P)\alpha \left(t\right)dp]\hfill \\ & & +{\int }_{-r_2}^0{\eta }_4[\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\alpha }^T(t+p)(I_F\otimes P)\alpha (t+p)]dp\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes (P+P^T)]\Delta \omega \left(t\right)+{\alpha }^T\left(t\right)[I_F\otimes Q]\alpha \left(t\right)-{\alpha }^T\left(t\right)[I_F\otimes Q]\alpha \left(t\right)\hfill \\ & =& {\alpha }^T\left(t\right)[I_F\otimes (PA+A^{T}P)]\alpha \left(t\right)+{\alpha }^T\left(t\right)[I_F\otimes \left(PB\right)]\alpha (t-r_1)+{\alpha }^T(t-r_1)(I_F\otimes \left(B^{T}P\right))]\alpha \left(t\right)\hfill \\ & & -{\alpha }^T\left(t\right)[L_F\otimes \left(P{K}_1\right)]\alpha (t-r_3)-{\alpha }^T(t-r_3)[L_F^T\otimes \left(K_1^{T}P\right)]\alpha \left(t\right)\hfill \\ & & +{\int }_{-r_2}^0{\alpha }^T\left(t\right)[I_F\otimes \left(PC\right)]\alpha (t+p)dp+{\int }_{-r_2}^0{\alpha }^T(t+p)[I_F\otimes \left(C^{T}P\right)]\alpha \left(t\right)dp\hfill \\ & & +{\eta }_1\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\eta }_1{\alpha }^T(t-r_1)(I_F\otimes P)\alpha (t-r_1)\hfill \\ & & +{\eta }_2\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\eta }_2{\alpha }^T(t-r_3)(I_F\otimes P)\alpha (t-r_3)\hfill \\ & & +{\eta }_3{r}_2{\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\int }_{-r_2}^0{\eta }_3{\alpha }^T(I_F\otimes P)\alpha \left(t\right)dp\hfill \\ & & +{\int }_{-r_2}^0{\eta }_4\lambda {\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)-{\int }_{-r_2}^0{\eta }_4{\alpha }^T(t+p)(I_F\otimes P)\alpha (t+p)dp\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes (P+P^T)]\Delta \omega \left(t\right)+{\alpha }^T\left(t\right)[I_F\otimes Q]\alpha \left(t\right)-{\alpha }^T\left(t\right)[I_F\otimes Q]\alpha \left(t\right)\hfill \\ & =& {\alpha }^T\left(t\right)[I_F\otimes (PA+A^{T}P+{\eta }_1\lambda P+Q+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]\alpha \left(t\right)\hfill \\ & & +{\alpha }^T\left(t\right)[(I_F\otimes \left(PB\right)]\alpha (t-r_1)+{\alpha }^T(t-r_1)[I_F\otimes \left(B^{T}P\right)]\alpha \left(t\right)\hfill \\ & & -{\eta }_1{\alpha }^T(t-r_1)(I_F\otimes P)\alpha (t-r_1)+{\alpha }^T\left(t\right)[I_F\otimes ({\eta }_3{r}_{2}P+{\eta }_2\lambda P-Q)]\alpha \left(t\right)\hfill \\ & & -{\alpha }^T\left(t\right)[L_F\otimes \left(P{K}_1\right)]\alpha (t-r_3)-{\alpha }^T(t-r_3)[L_F^T\otimes \left(K_1^{T}P\right)]\alpha \left(t\right)-{\eta }_2{\alpha }^T(t-r_3)(I_F\otimes P)\alpha (t-r_3)\hfill \\ & & +{\int }_{-r_2}^0[({\eta }_4\lambda -{\eta }_3){\alpha }^T\left(t\right)(I_F\otimes P)\alpha \left(t\right)+{\alpha }^T\left(t\right)[I_F\otimes \left(PC\right)]\alpha (t+p)\hfill \\ & & +{\alpha }^T(t+p)[I_F\otimes \left(C^{T}P\right)]\alpha \left(t\right)-{\eta }_4{\alpha }^T(t+p)(I_F\otimes P)\alpha (t+p)]dp.\hfill \\ & =& {\mathrm{\Pi }}_{r_1}^T\left[\begin{array}{cc}{I}_F\otimes [PA+A^{T}P+{\eta }_1\lambda P+Q& I_F\otimes \left(PB\right)\\ +{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]\\ I_F\otimes \left(B^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]{\mathrm{\Pi }}_{r_1}\hfill \\ & & +{\mathrm{\Pi }}_{r_3}^T\left[\begin{array}{cc}{I}_F\otimes [({\eta }_3{r}_2+{\eta }_2\lambda )P-Q]& -L_F\otimes \left(P{K}_1\right)\\ -L_F^T\otimes \left(K_1^{T}P\right)& -{\eta }_2(I_F\otimes P)\end{array}\right]{\mathrm{\Pi }}_{r_3}\hfill \\ & & +{\int }_{-r_2}^0{\mathrm{\Pi }}_p^T\left[\begin{array}{cc}({\eta }_4\lambda -{\eta }_3)(I_F\otimes P)& I_F\otimes \left(PC\right)\\ I_F\otimes \left(C^{T}P\right)& -{\eta }_4(I_F\otimes P)\end{array}\right]{\mathrm{\Pi }}_{p}dp,\hfill \end{array}$$

where ${\mathrm{\Pi }}_{r_1}={[{\alpha }^T\left(t\right),{\alpha }^T(t-r_1)]}^T,{\mathrm{\Pi }}_{r_3}={[{\alpha }^T\left(t\right),{\alpha }^T(t-r_3)]}^T$ and ${\mathrm{\Pi }}_p={[{\alpha }^T\left(t\right),{\alpha }^T(t+p)]}^T$. Relationships (14)–(16) imply that, for an adequately small ${\lambda }_1>0,\lambda ={\lambda }_1+1$,

$$\left[\begin{array}{cc}{I}_F\otimes [PA+A^{T}P+{\eta }_1\lambda P+Q& I_F\otimes \left(PB\right)\\ +{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]\\ I_F\otimes \left(B^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]<0,$$

(17)

$$\left[\begin{array}{cc}{I}_F\otimes [({\eta }_3{r}_2+{\eta }_2\lambda )P-Q]& -L_F\otimes \left(P{K}_1\right)\\ -L_F^T\otimes \left(K_1^{T}P\right)& -{\eta }_2(I_F\otimes P)\end{array}\right]<0,$$

(18)

$$\left[\begin{array}{cc}({\eta }_4\lambda -{\eta }_3)(I_F\otimes P)& I_F\otimes \left(PC\right)\\ I_F\otimes \left(C^{T}P\right)& -{\eta }_4(I_F\otimes P)\end{array}\right]<0,$$

(19)

which indicate that ${}_{t_0}{D}_t^{\beta }V\left(t\right)$ < 0, this demonstrates that from Lemma (1) Equation (13) is asymptotically stable. As a result, resilient base CC of FOMAS (3) with protocol (12) is possible. In particular, if $r_1=r_2=r_3=r$, one obtains a simpler criterion. □

Corollary 1.

Under $\left(H\right)$, If both a matrix P > 0 and a scalar ${\eta }_i$ > 0 (i = 1,2,3) exist, then the resilient base CC of FOMAS (3) with protocol (12) is obtained.

$$\left[\begin{array}{cc}{I}_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P& I_F\otimes \left(PB\right)-L_F\otimes \left(P{K}_1\right)\\ +{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]\\ I_F\otimes \left(B^{T}P\right)-L_F^T\otimes \left(K_1^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]<0,$$

(20)

$$\left[\begin{array}{cc}({\eta }_3-{\eta }_2)(I_F\otimes P)& I_F\otimes \left(PC\right)\\ I_F\otimes \left(C^{T}P\right)& -{\eta }_3(I_F\otimes P)\end{array}\right]<0.$$

(21)

If C = 0, FOMAS (3) is reduced to

$${}_{t_0}{D}_t^{\beta }{x}_i\left(t\right)=A{x}_i\left(t\right)+B{x}_i(t-r_1)+\omega \left(t\right)+\Delta {\omega }_i\left(t\right),i=1,2,\dots ,N.$$

(22)

The following conclusion can be reached according to the proof of the previous theorem.

Theorem 3.

Under (H), if there exist constants η > 0 and a matrix P > 0, then resilient base CC of FOMAS (22) with protocol (4) is obtained.

$$\left[\begin{array}{cc}{I}_F\otimes [PA+A^{T}P+\eta P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]& I_F\otimes \left(PB\right)\\ -L_F\otimes \left(PK\right)-L_F^T\otimes \left(K^{T}P\right)& \\ I_N\otimes \left(B^{T}P\right)& -\eta (I_N\otimes P)\end{array}\right]<0.$$

(23)

Theorem 4.

Under $\left(H\right)$, if there are two matrices Q > 0, P > 0 and a scalar η > 0, then resilient base CC of FOMAS (22) with protocol (12) is obtained.

$$\left[\begin{array}{cc}{I}_F\otimes [PA+A^{T}P+\eta P+Q& I_F\otimes \left(PB\right)\\ +{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]\\ I_F\otimes \left(B^{T}P\right)& -\eta (I_F\otimes P)\end{array}\right]<0,$$

(24)

$$\left[\begin{array}{cc}{I}_F\otimes (\eta P-Q)& -L_F\otimes \left(P{K}_1\right)\\ -L_F^T\otimes \left(K_1^{T}P\right)& -\eta (I_F\otimes P)\end{array}\right]<0.$$

(25)

In particular, if $r_1=r_2=r_3=r$, one can obtain a simpler criterion.

Corollary 2.

Under $\left(H\right)$, if there are constants η > 0 and a matrix P > 0 then resilient base CC of FOMAS (22) with protocol (12) is obtained.

$$\left[\begin{array}{cc}{I}_F\otimes [PA+A^{T}P+\eta P& I_F\otimes \left(PB\right)-L_F\otimes \left(P{K}_1\right)\\ +{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]& \\ I_F\otimes \left(B^{T}P\right)-L_F^T\otimes \left(K_1^{T}P\right)& -{\eta }_1(I_F\otimes P)\end{array}\right]<0.$$

(26)

Remark 1.

This study offers a more practical and efficient method to resolve resilient base CC of delayed FOMASs in comparison to the frequency domain method used in [25,32,33]. Additionally, the suggested approach works well for fractional-order systems with different kinds of delays.

4. Examples

Two examples are used in this section to clarify the effectiveness of the result achieved.

Example 1.

Consider FOMAS (3) with five followers and two leaders describe as Figure 1, where $A,B,C\in R^{n\times n}$ are constant matrices.

$$A=\left[\begin{array}{cc}-1& 0\\ 0.1& -3\end{array}\right]$$

$$B=\left[\begin{array}{cc}-2& 0\\ -0.1& -1\end{array}\right]$$

$$C=\left[\begin{array}{cc}0.25& -0.5\\ -1.3& 0\end{array}\right]$$

According to Figure 1 one has

$$L=\left[\begin{array}{cc}{L}_F& L_R\\ 0& 0\end{array}\right]$$

$$L=\left[\begin{array}{ccccccc}1.01& -1& 0& 0& 0& -0.01& 0\\ 0& 2& 0& 0& -1.1& -0.2& -0.7\\ 0& -0.5& 1.7& 0& -0.3& 0& -0.9\\ 0& -0.03& 0& 0.03& 0& 0& 0\\ 0& 0& 0& -2& 2& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

According to the Schur lemma, association (8) and (9) are equivalent to the inequalities

$$\begin{array}{ccc}\hfill [I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)](-{\eta }_1(I_F\otimes P))-(I_F\otimes \left(PB\right))(I_F\otimes B^{T}P)& <& 0\hfill \\ \hfill I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)](-{\eta }_1(I_F\otimes P))& <& [I_F\otimes \left(PB{B}^{T}P\right)]\hfill \\ \hfill I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)& <& \frac{-1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)]\hfill \\ \hfill I_N\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)+\frac{1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)]& <& 0\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \left[({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))\right]-(I_F\otimes \left(PC\right))(I_F\otimes \left(C^{T}P\right))& <& 0\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))& <& [I_F\otimes \left(PC{C}^{T}P\right)]\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)& <& \frac{1}{-{\eta }_3}[I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right)]\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)+\frac{1}{{\eta }_3}[I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right)]& <& 0\hfill \end{array}$$

(28)

Choose $P=\left[\begin{array}{cc}0.1& 1\\ 1& 0.1\end{array}\right],{\eta }_1=2,{\eta }_2=3,{\eta }_3=2and{r}_2=0.3.Then,K=\left[\begin{array}{cc}0.01& -1\\ -0.5& -0.01\end{array}\right]$ can be chosen to satisfy (31) and (32), where $K\in R^{n\times n}isthegainmatrix$ and P is the $n\times n$ matrix.

From Equation (32) we have

$$\left[\begin{array}{cc}-0.1118897306& -0.7830639731\\ -0.7830639731& -0.1493063973\end{array}\right]<0$$

Thus, Theorem (1) is used to achieve CC of FOMAS (3). If we assume that $\beta =0.6$ and $r_1=0.4$, Figure 2 displays the error ${\alpha }_i\left(t\right)$.

On contrary, the Schur lemma demonstrates that mutual relations between (20) and (21) are equivalent to the inequalities

$$\begin{array}{ccc}\hfill I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)](-{\eta }_1(I_F\otimes P))& & \\ \hfill -(I_F\otimes \left(PB\right)-L_F\otimes \left(P{K}_1\right))(I_F\otimes \left(B^{T}P\right)-L_F^T\otimes \left(K_1^{T}P\right))& <& 0\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)](-{\eta }_1(I_F\otimes P))\hfill \\ \hfill & <& (I_F\otimes \left(PB\right)-L_F\otimes \left(P{K}_1\right))(I_F\otimes \left(B^{T}P\right)-L_F^T\otimes \left(K_1^{T}P\right))\hfill \\ \hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)](-{\eta }_1(I_F\otimes P))\hfill \\ \hfill & <& [I_F\otimes \left(PB{B}^{T}P\right)-L_F^T\otimes \left(PB{K}_1^{T}P\right)-L_F\otimes \left(P{K}_1{B}^{T}P\right)\hfill \\ \hfill & & +L_F{L}_F^T\otimes \left(P{K}_1{K}_1^{T}P\right)]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]\hfill \\ \hfill & <& \frac{-1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)-L_F^T\otimes \left(PB{P}^{-1}{K}_1^{T}P\right)-L_F\otimes \left(P{K}_1{P}^{-1}{B}^{T}P\right)\hfill \\ \hfill & & +L_F{L}_F^T\otimes \left(P{K}_1{P}^{-1}{K}_1^{T}P\right)]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]+\frac{1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)\hfill \\ \hfill & & -L_F^T\otimes \left(PB{P}^{-1}{K}_1^{T}P\right)-L_F\otimes \left(P{K}_1{P}^{-1}{B}^{T}P\right)+L_F{L}_F^T\otimes \left(P{K}_1{P}^{-1}{K}_1^{T}P\right)]\hfill \\ \hfill & <& 0\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)+\frac{1}{{\eta }_1}\left(PB{P}^{-1}{B}^{T}P\right)]& & \\ \hfill +\frac{1}{{\eta }_1}[\left(L_F{L}_F^T\right)\otimes \left(P{K}_1{P}^{-1}{K}_1^{T}P\right)& & \\ \hfill -L_F\otimes \left(P{K}_1{P}^{-1}{B}^{T}P\right)-L_F^T\otimes \left(PB{P}^{-1}{K}_1^{T}P\right)]& <& 0\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))-(I_F\otimes \left(PC\right))(I_F\otimes \left(C^{T}P\right))& <& 0\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))-(I_F\otimes \left(PC{C}^{T}P\right))& <& 0\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))& <& (I_F\otimes \left(PC{C}^{T}P\right))\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)& <& \frac{-1}{{\eta }_3}(I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right))\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)+\frac{1}{{\eta }_3}(I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right))& <& 0\hfill \\ \hfill I_F\otimes [({\eta }_3-{\eta }_2)P+\frac{1}{{\eta }_3}\left(PC{P}^{-1}{C}^{T}P\right)]& <& 0.\hfill \end{array}$$

(30)

Choose $P=\left[\begin{array}{cc}1& 0.1\\ 0.1& 1\end{array}\right],{\eta }_1=2,{\eta }_2=3,{\eta }_3=2andr=0.3.Then,K_1=\left[\begin{array}{cc}-0.3& -0.3\\ 0& -0.3\end{array}\right]$ can be chosen to satisfy (33) and (34), where $K_1\in R^{n\times n}$ and P is an $n\times n$ matrix.

From Equation (34) we have

$$\left[\begin{array}{cc}-0.5976190476& -0.8445238095\\ -0.8445238095& -0.2754047619\end{array}\right]<0$$

Thus, Corollary (1) is used to achieve resilient base CC of FOMAS (3). Taking $\beta =0.6$, Figure 3 shows the error ${\alpha }_i\left(t\right)$.

Example 2.

We examine FOMAS (3) with five followers and two leaders, as shown in Figure 1, where $A,B,C\in R^{n\times n}$ are constant matrices.

$$A=\left[\begin{array}{cc}-6& 0\\ 4& -7\end{array}\right]$$

$$B=\left[\begin{array}{cc}2& -0.3\\ 2& -0.2\end{array}\right]$$

$$C=\left[\begin{array}{cc}-0.26& -0.16\\ 0.6& -0.6\end{array}\right]$$

According to Figure 4, one has

$$L=\left[\begin{array}{cc}{L}_F& L_R\\ 0& 0\end{array}\right]$$

$$L=\left[\begin{array}{ccccccc}2.6& -0.2& 0& 0& -1.3& -1.1& 0\\ 0& 2.9& -1& 0& 0& -0.7& -1.2\\ 0& 0& 0.3& 0& 0& 0& -0.3\\ 0& -0.4& -1.5& 1.9& 0& 0& 0\\ 0& -0.9& 0& -0& 0.9& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

The Schur lemma implies that relationship (8) and (9) are equivalent to the inequalities

$$\begin{array}{ccc}\hfill [I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)](-{\eta }_1(I_F\otimes P))-(I_F\otimes \left(PB\right))(I_F\otimes B^{T}P)& <& 0\hfill \\ \hfill [I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)](-{\eta }_1(I_F\otimes P))& <& [I_F\otimes \left(PB{B}^{T}P\right)]\hfill \\ \hfill I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)& <& \frac{-1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)]\hfill \\ \hfill I_N\otimes [PA+A^{T}P+({\eta }_1+{\eta }_2{r}_2)P+{\alpha }^{-1}(P+P^T)\Delta \omega \left(t\right)]-L_F\otimes \left(PK\right)& & \\ \hfill -L_F^T\otimes \left(K^{T}P\right)+\frac{1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)]& <& 0\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill [({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P)]-(I_F\otimes \left(PC\right))(I_F\otimes \left(C^{T}P\right))& <& 0\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))& <& [I_F\otimes \left(PC{C}^{T}P\right)]\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)& <& \frac{1}{-{\eta }_3}[I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right)]\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)+\frac{1}{{\eta }_3}[I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right)]& <& 0\hfill \end{array}$$

(32)

Choose $P=\left[\begin{array}{cc}2& 0.2\\ 0.2& 2\end{array}\right],{\eta }_1=1.6,{\eta }_2=2.1,{\eta }_3=1.1and{r}_2=0.5.Then,K=\left[\begin{array}{cc}-0.2& 0.2\\ -0.4& -0.2\end{array}\right]$ can be chosen to satisfy (31) and (32), where $K\in R^{n\times n}$ is the gain matrix and P is $n\times n$ matrix. From Equation (32) we have

$$\left[\begin{array}{cc}-1.8391919192& -0.157519192\\ -0.157519194& -0.4249519198\end{array}\right]<0$$

Thus, Theorem (1) is used to achieve CC of FOMAS (3). If we assume that $\beta =0.6$ and $r_1=0.4$, Figure 5 displays the error ${\alpha }_i\left(t\right)$.

Conversely, the Schur lemma shows that relations (20) and (21) are equivalent to the inequalities

$$\begin{array}{ccc}\hfill I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)](-{\eta }_1(I_F\otimes P))& & \\ \hfill -(I_F\otimes \left(PB\right)-L_F\otimes \left(P{K}_1\right))(I_F\otimes \left(B^{T}P\right)-L_F^T\otimes \left(K_1^{T}P\right))& <& 0\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)](-{\eta }_1(I_F\otimes P))\hfill \\ \hfill & <& (I_F\otimes \left(PB\right)-L_F\otimes \left(P{K}_1\right))(I_F\otimes \left(B^{T}P\right)-L_F^T\otimes \left(K_1^{T}P\right))\hfill \\ \hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)](-{\eta }_1(I_F\otimes P))\hfill \\ \hfill & <& [I_F\otimes \left(PB{B}^{T}P\right)-L_F^T\otimes \left(PB{K}_1^{T}P\right)-L_F\otimes \left(P{K}_1{B}^{T}P\right)\hfill \\ \hfill & & +L_F{L}_F^T\otimes \left(P{K}_1{K}_1^{T}P\right)]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]\hfill \\ \hfill & <& \frac{-1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)-L_F^T\otimes \left(PB{P}^{-1}{K}_1^{T}P\right)-L_F\otimes \left(P{K}_1{P}^{-1}{B}^{T}P\right)\hfill \\ \hfill & & +L_F{L}_F^T\otimes \left(P{K}_1{P}^{-1}{K}_1^{T}P\right)]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F& \otimes & [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)]+\frac{1}{{\eta }_1}[I_F\otimes \left(PB{P}^{-1}{B}^{T}P\right)\hfill \\ \hfill & & -L_F^T\otimes \left(PB{P}^{-1}{K}_1^{T}P\right)-L_F\otimes \left(P{K}_1{P}^{-1}{B}^{T}P\right)+L_F{L}_F^T\otimes \left(P{K}_1{P}^{-1}{K}_1^{T}P\right)]\hfill \\ \hfill & <& 0\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_F\otimes [PA+A^{T}P+({\eta }_1+{\eta }_{2}r)P+{\alpha }^{-1}\left(t\right)(P+P^T)\Delta \omega \left(t\right)+\frac{1}{{\eta }_1}\left(PB{P}^{-1}{B}^{T}P\right)]& & \\ \hfill +\frac{1}{{\eta }_1}[\left(L_F{L}_F^T\right)\otimes \left(P{K}_1{P}^{-1}{K}_1^{T}P\right)& & \\ \hfill -L_F\otimes \left(P{K}_1{P}^{-1}{B}^{T}P\right)-L_F^T\otimes \left(PB{P}^{-1}{K}_1^{T}P\right)]& <& 0\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))-(I_F\otimes \left(PC\right))(I_F\otimes \left(C^{T}P\right))& <& 0\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))-(I_F\otimes \left(PC{C}^{T}P\right))& <& 0\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)(-{\eta }_3(I_F\otimes P))& <& (I_F\otimes \left(PC{C}^{T}P\right))\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)& <& \frac{-1}{{\eta }_3}(I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right))\hfill \\ \hfill ({\eta }_3-{\eta }_2)(I_F\otimes P)+\frac{1}{{\eta }_3}(I_F\otimes \left(PC{P}^{-1}{C}^{T}P\right))& <& 0\hfill \\ \hfill I_F\otimes [({\eta }_3-{\eta }_2)P+\frac{1}{{\eta }_3}\left(PC{P}^{-1}{C}^{T}P\right)]& <& 0.\hfill \end{array}$$

(34)

Choose $P=\left[\begin{array}{cc}3& 0.2\\ 0.2& 0.9\end{array}\right],{\eta }_1=3,{\eta }_2=4,{\eta }_3=3andr=0.5.Then,K_1=\left[\begin{array}{cc}-0.4& -0.4\\ 0& -0.4\end{array}\right]$ can be chosen to satisfy (33) and (34), where $K_1\in R^{n\times n}$ is the gain matrix and P is the $n\times n$ matrix. From Equation (34) we have

$$\left[\begin{array}{cc}-2.8353834587& -0.1094255639\\ -0.1094255639& -0.7361483703\end{array}\right]<0$$

Thus, Corollary (1) is used to achieve resilient base CC of FOMAS (3). Taking $\beta =0.8$, Figure 6 shows the errors ${\alpha }_i\left(t\right)$.

5. Conclusions

We have provided an easy and effective way to look into the resilient base CC of fractional order multi-agent system with mixed time delays. In order to achieve resilient base CC, both delayed and non-delayed control protocols are used. By applying graph theory, the fractional Razumikhin technique and the Lyapunov function technique, certain useful algebraic criteria have been proposed to ensure resilient base CC. Delayed and non delayed control protocol with disturbance term have been designed to solve resilient base containment control. An additional numerical example demonstrates the reliability of the results presented and it is easy to verify the criteria using matrix inequalities. Our approach offers a reliable and easy method to overcome the problems resulting from fractional derivatives and time delays. Our upcoming research will concentrate on CC of single delayed fractional order multi-agent systems that are nonlinear.