Observer-Based Robust Containment Control of Multi-Agent Systems with Structural Uncertainties and Multiple Delays

Abstract

In this paper, a distributed output-feedback control law is proposed to explore the containment control problem for uncertain multi-agent systems with nonlinear dynamics. First, unlike the existing control schemes, an improved observer-based robust containment controller without utilizing the observation of leaders is developed, where the inaccuracy of the model structure is considered. Second, different from previous studies that considered a single transmission delay, both parameter uncertainties and multiple time delays are investigated over a directed graph. Notably, the considered delays are nonuniform: the state delay is related to the nonlinearities, and the transmission delay is taken into account in the controller design. In addition, making use of the delay-product-type quadratic approach, sufficient conditions are rigorously derived to realize containment control, and it is proved that the containment error converges to zero. Finally, a simulation model is constructed in MATLAB R2021a platform to validate the effectiveness of the algorithm.

1. Introduction

Over the previous 20 years, consensus problems of multi-agent systems (MASs), encompassing synchronization, swarming, flocking, consensus tracking, and containment, have been an active research area in various fields, such as unmanned aerial vehicles [1,2], urban traffic management [3,4], energy efficiency and energy integration [5,6], attitude of astrovehicle alignment [7], optimization [8,9], and so on. Numerous studies have investigated the consensus and containment problems of MASs. In [10], the secure consensus is analyzed via a fully distributed robust consensus protocol in the presence of Denial-of-Service attacks under Markovian switching graphs. For heterogeneous dynamics, the consensus tracking of a dynamic leader is solved by a data-driven consensus protocol [11]. Moreover, for containment and formation–containment problems of multiple leaders, several remarkable achievements have been reported [12,13,14,15,16]. It is worth noting that the cited papers [10,11,12,13,14,15,16] focus on particular MASs. However, parameter uncertainties [17] lead to system performance degradation or even loss of stability. Thus, it is of great significance to further explore consensus problems for MASs subject to structural uncertainties.

The robust consensus of uncertain first-order and second-order MASs is discussed by developing parameter-dependent Lyapunov functions in [18]. Depending on the relative states among neighboring agents, the distributed $H_{\infty }$ containment problem of second-order MASs is solved in [19]. Different from low-order dynamics [18,19], many consensus results of high-order dynamics with structural uncertainties have been reported. Specifically, for linear dynamics, an adaptive control algorithm is adopted to address the consensus problem in the presence of heterogeneous matching uncertainties [20]. Based on a neural network approximation method, the quasi-containment and asymptotic containment results for uncertain dynamics are, respectively, shown in [21]. By using a state feedback control protocol, the $H_2$ consensus is studied in [22] subject to parametric uncertainties and measurement noises. Further, in [23], a fixed-time control algorithm is developed to discuss the robustness and convergence speed for uncertain robotic systems. Also, for uncertain heterogeneous singular MASs, the adaptive fault-tolerant consensus tracking problem is investigated in [24] via a proportional and differential control algorithm. Moreover, the containment reports of heterogeneous MASs with uncertain parameters can be found in [25,26,27,28].

The above results only focus on consensus problems with parameter uncertainties, yet the time delays in the network communication are ignored. However, delays inevitably arise in practical complex environments. Regarding an observer-based adaptive algorithm, the consensus tracking problem with state time delay is addressed in [29]. By introducing a unified D-type learning algorithm, the results of a consensus tracking problem with state time delay are obtained in [30]. In the presence of transmission delay, ref. [31] is concerned with the bipartite containment control issue, where an event-based protocol with a state-dependent trigger function is developed. When subject to multiple delays, a great number of achievements are presented [32,33,34,35,36,37]. Particularly, a data-based adaptive dynamic programming algorithm is constructed. Further, optimal consensus tracking with multiple delays is discussed in [33]. Taking into account both unknown state and input delays, a truncated prediction feedback algorithm is introduced to cope with the output-feedback control problem of interconnected MASs [36]. Further, aiming at heterogeneous multi-human MASs, ref. [37] considers the containment problem subject to transmission delays as well as human reaction delays. Moreover, in the study of fractional MASs with mixed delays, the bipartite containment control problem is addressed via a delayed control algorithm [38].

Indeed, some results for uncertain MASs with delays that have been presented attract research attention, such as weighted consensus [39], output consensus tracking control [40], and robust containment control [41]. However, only limited results have been obtained for MASs with different delays and parameter uncertainties, which motivates us to undertake the present study. Inspired by the above discussion, this paper focuses on the robust containment control of high-order MASs with structural uncertainties and multiple delays.

The three main contributions of the paper are as follows: (1) Without taking the observations of leaders into account, an observer-based output-feedback control protocol is proposed over the directed graph, where the design of the feedback-gain matrices is carried out independently. (2) For dynamics considering structural uncertainties, in comparison with leaderless consensus [18,22] and consensus tracking control [20,24] in the absence of delays, the proposed control protocol aims to deal with the robust containment problems of multi-leader MASs with multiple delays. (3) Our designed protocol not only handles parameter uncertainties but also state delay and transmission delay, in contrast with existing results considering delays that ignore structural uncertainties [29,30,31,32,33,34,35,36].

2. Preliminaries and Problem Formulation

2.1. Preliminaries and Notations

Let $\mathcal{G}=\left.\left\{\mathcal{V},\mathcal{E},\mathcal{A}\right.\right\}$ represent the interaction graph where $\mathcal{G}$ has N nodes and $\mathcal{E}=\left.\left\{(k,j):k,j\in \mathcal{V}\right.\right\}$ represent edges. Define the adjacency matrix $\mathcal{A}=\left[a_{kj}\right]\in R^{N\times N}$ for $a_{kj}=1$ if $(k,j)\in \mathcal{E}$ and $a_{kj}=0$ otherwise. Further, denote $L=\left[l_{kj}\right]\in R^{N\times N}$ the Laplacian matrix for $l_{kj}=-a_{kj}$ with $k\ne j$ and $l_{kk}={\sum }_{j=1,j\ne k}^N{a}_{kj}$ [42]. Moreover, $\mathrm{He}\left(X\right)=X+X^T$, $\mathrm{diag}\{z_1,z_2,\dots ,z_n\}$ represents a diagonal matrix, $\mathrm{col}(z_1,z_2,\dots ,z_n)$ denotes a column vector, and $X\otimes Y$ stands for the Kronecker product. Further, the convex hull $Co(\Pi )$ is defined by $Co(\Pi )=\left\{{\sum }_{k=1}^n{\varpi }_k{\pi }_k\left|{\pi }_k\in R,{\varpi }_k\ge 0,\right.{\sum }_{k=1}^n{\varpi }_k=1\right\}$ with the set $\Pi =\{{\pi }_1,{\pi }_2,\dots ,{\pi }_n\}$.

2.2. Problem Formulation

The system consists of M followers with $k\in S_F=\{1,2,\dots ,M\}$ and $N-M$ leaders with $k\in S_L=\{M+1,M+2,\dots ,N\}$. Consider the MASs with the following uncertain dynamics:

$$\begin{array}{c}\hfill \begin{array}{c}{\dot{x}}_k\left(t\right)=(A+\Delta A\left(t\right))x_k\left(t\right)+(B+\Delta B\left(t\right))u_k\left(t\right)+Dg\left(x_k(t-{\tau }_1)\right),\hfill \\ y_k\left(t\right)=C{x}_k\left(t\right),k\in \mathcal{V},\hfill \end{array}\end{array}$$

(1)

where $A\in R^{n\times n},B\in R^{n\times m},D\in R^{n\times n},C\in R^{q\times n}$. $x_k\left(t\right)\in R^n$, $y_k\left(t\right)\in R^q$, and $u_k\left(t\right)\in R^m$ represent the state, measurement output, and control input of the system. $g\left(x_k(t-{\tau }_1)\right)$ denotes the inner nonlinear function, and the known constant ${\tau }_1\ge 0$ stands for the state delay. Note that A, B, C, and D are known constant matrices. $\Delta A\left(t\right)\in R^{n\times n}$ and $\Delta B\left(t\right)\in R^{n\times m}$ are the unknown uncertainties satisfying

$$\begin{array}{c}\hfill [\Delta A\left(t\right)\Delta B\left(t\right)]=EH\left(t\right)\left[F_1{F}_2\right],\end{array}$$

where $E\in R^{n\times n}$, $F_1\in R^{n\times n}$, and $F_2\in R^{n\times m}$ are known constant matrices and $H\left(t\right)\in R^{n\times n}$ is an unknown time-varying matrix satisfying

$$\begin{array}{c}\hfill H^T\left(t\right)H\left(t\right)\le I.\end{array}$$

In addition, note that $u_k\left(t\right)=0$ for $k\in S_L$. Then, L can be expressed as

$$\begin{array}{c}\hfill L=\left[\begin{array}{cc}{L}_{11}& L_{12}\\ 0& 0\end{array}\right],\end{array}$$

where $L_{11}\in R^{M\times M}$ and $L_{12}\in R^{M\times (N-M)}$, and it yields $L_{11}{1}_M+L_{12}{1}_{N-M}=0$; thus, it derives $-{L_{11}}^{-1}{L}_{12}{1}_{N-M}=1_M$, and $-{L_{11}}^{-1}{L}_{12}$ is nonnegative.

Remark 1.

It should be pointed out that the state delays inherent in the system are frequently overlooked. In [29,30], the consensus tracking problems with state time delay are discussed, while the structural uncertainties are not addressed. Thus, considering both the state delays and uncertain dynamics, the model (1) is paid attention to in the presence of multiple leaders.

Definition 1

([25]). The containment control of MAS (1) with uncertain parameters $\Delta A\left(t\right)$ and $\Delta B\left(t\right)$ is achieved when $\underset{t\to \infty }{lim}\mathrm{dist}(x_k,Co(x_{M+1},\dots ,x_N))=0,k\in S_F$ holds.

Assumption 1.

For each follower, there exists at least one leader that has a directed path to that follower.

Lemma 1

([43]). There exist a matrix $\Delta >0$ and a constant $\zeta >0$ such that

$$\begin{array}{c}\hfill {\int }_{t-\zeta }^t{\phi }^T\left(s\right)\Delta \phi \left(s\right)ds\ge {\displaystyle \frac{1}{\zeta }}{\left({\int }_{t-\zeta }^t\phi \left(s\right)ds\right)}^T\Delta \left({\int }_{t-\zeta }^t\phi \left(s\right)ds\right).\end{array}$$

Lemma 2

([13]). For matrices $Z=Z^T$, X, and Y with compatible dimensions,

$$\begin{array}{c}\hfill Z+X\Phi \left(t\right)Y+Y^T{\Phi }^T\left(t\right)X^T<0\end{array}$$

satisfies ${\Phi }^T\left(t\right)\Phi \left(t\right)\le I$, and then there exists a constant $\delta >0$ such that

$$\begin{array}{c}\hfill Z+\delta X{X}^T+{\delta }^{-1}{Y}^{T}Y<0.\end{array}$$

3. Robust Containment Control Design Condition

In order to enforce the agents to research containment control, a robust transmission delay controller for follower k is presented

$$\begin{array}{cc}\hfill {\dot{\varphi }}_k\left(t\right)=& (A+\Delta A\left(t\right)){\varphi }_k\left(t\right)+(B+\Delta B\left(t\right))G_1\sum _{j\in S_F}{a}_{kj}({\varphi }_k(t-{\tau }_2)-{\varphi }_j(t-{\tau }_2))\hfill \\ \hfill & +G_2\left(\sum _{j\in S_F\cup S_L}{a}_{kj}(y_k(t-{\tau }_2)-y_j(t-{\tau }_2))-C{\varphi }_k\left(t\right)\right),\hfill \\ \hfill u_k\left(t\right)=& G_1{\varphi }_k\left(t\right),\hfill \end{array}$$

(2)

where ${\varphi }_k\left(t\right)$ stands for the state estimation, $G_1$ and $G_2$ are feedback-gain matrices, and the known constant ${\tau }_2\ge 0$ represents the transmission delay.

Remark 2.

Unlike the protocols that only focus on structural uncertainties [18,19,20,21,22,23,24], by making use of the measurement output information, our proposed protocol (2) is employed to cope with the structural uncertainties, as well as the multiple delays.

Let ${\epsilon }_k\left(t\right)={\sum }_{j\in S_F\cup S_L}{a}_{kj}\left(x_k\left(t\right)-x_j\left(t\right)\right),k\in S_F$, and $\epsilon \left(t\right)=\mathrm{col}({\epsilon }_1\left(t\right),\dots ,{\epsilon }_M\left(t\right))$. Then, it has

$$\begin{array}{c}\hfill \epsilon \left(t\right)=(L_{11}\otimes I_n)x_F\left(t\right)+(L_{12}\otimes I_n)x_L\left(t\right),\end{array}$$

(3)

where $x_F\left(t\right)=\mathrm{col}(x_1\left(t\right),\dots ,x_M\left(t\right))$, $x_L\left(t\right)=\mathrm{col}(x_{M+1}\left(t\right),\dots ,x_N\left(t\right))$.

Let ${\rm Y}\left(x(t-{\tau }_1)\right)=(L_{11}\otimes D)[{{\rm Y}}_1\left(x(t-{\tau }_1)\right)+(L_{11}^{-1}{L}_{12}\otimes I_n){{\rm Y}}_2\left(x(t-{\tau }_1)\right)]$, ${{\rm Y}}_1\left(x(t-{\tau }_1)\right)=\mathrm{col}(g\left(x_1(t-{\tau }_1)\right),\dots ,g\left(x_M(t-{\tau }_1)\right))$, ${{\rm Y}}_2\left(x(t-{\tau }_1)\right)=\mathrm{col}(g\left(x_{M+1}(t-{\tau }_1)\right),\dots ,f\left(x_N(t-{\tau }_1)\right))$. It follows from (1)–(3) that

$$\begin{array}{c}\hfill \dot{\epsilon }\left(t\right)=(I_M\otimes (A+\Delta A\left(t\right)))\epsilon \left(t\right)+(L_{11}\otimes (B+\Delta B\left(t\right))G_1){\varphi }_F\left(t\right)+{\rm Y}\left(x(t-{\tau }_1)\right),\end{array}$$

(4)

where ${\varphi }_F\left(t\right)=\mathrm{col}({\varphi }_1\left(t\right),\dots ,{\varphi }_M\left(t\right))$.

Let ${\tilde{\varphi }}_k\left(t\right)={\sum }_{j\in S_F\cup S_L}{a}_{kj}\left(x_k\left(t\right)-x_j\left(t\right)\right)-{\varphi }_k\left(t\right)$ denote the containment estimation error, and it has ${\varphi }_F\left(t\right)=\epsilon \left(t\right)-{\tilde{\varphi }}_F\left(t\right)$, where ${\tilde{\varphi }}_F\left(t\right)=\mathrm{col}({\tilde{\varphi }}_1\left(t\right),\dots ,{\tilde{\varphi }}_M\left(t\right))$. It follows from (4) that

$$\begin{array}{cc}\hfill \dot{\epsilon }\left(t\right)=& (I_M\otimes (A+\Delta A\left(t\right))+L_{11}\otimes (B+\Delta B\left(t\right))G_1)\epsilon \left(t\right)\hfill \\ \hfill & -(L_{11}\otimes (B+\Delta B\left(t\right))G_1){\tilde{\varphi }}_F\left(t\right)+{\rm Y}\left(x(t-{\tau }_1)\right).\hfill \end{array}$$

(5)

And, based on (2), it has

$$\begin{array}{cc}\hfill {\dot{\varphi }}_F\left(t\right)=& (I_M\otimes (A+\Delta A\left(t\right)-G_{2}C)){\varphi }_F\left(t\right)+(L_3\otimes (B+\Delta B\left(t\right))G_1){\varphi }_F(t-{\tau }_2)\hfill \\ \hfill & +(I_M\otimes G_{2}C)\epsilon (t-{\tau }_2),\hfill \end{array}$$

(6)

where

$$\begin{array}{c}\hfill L_3=\left[\begin{array}{cccc}{\sum }_{j\in S_F}{a}_{1j}& -a_{12}& \dots & -a_{1M}\\ ⋮& \ddots & \dots & \dots \\ ⋮& \dots & \ddots & \dots \\ -a_{M1}& -a_{M2}& \dots & {\sum }_{j\in S_F}{a}_{Mj}\end{array}\right].\end{array}$$

Further, it yields

$$\begin{array}{cc}\hfill {\dot{\tilde{\varphi }}}_F\left(t\right)=& \dot{\epsilon }\left(t\right)-{\dot{\varphi }}_F\left(t\right)\hfill \\ \hfill =& (I_M\otimes G_{2}C+L_{11}\otimes (B+\Delta B\left(t\right))G_1)\epsilon \left(t\right)+{\rm Y}\left(x(t-{\tau }_1)\right)\hfill \\ \hfill & +(I_M\otimes (A+\Delta A\left(t\right)-G_{2}C)-L_{11}\otimes (B+\Delta B\left(t\right))G_1){\tilde{\varphi }}_F\left(t\right)\hfill \\ \hfill & +(L_3\otimes (B+\Delta B\left(t\right))G_1){\tilde{\varphi }}_F(t-{\tau }_2)-(L_3\otimes (B+\Delta B\left(t\right))G_1\hfill \\ \hfill & +I_M\otimes G_{2}C)\epsilon (t-{\tau }_2).\hfill \end{array}$$

(7)

Thus, from (5) and (7), it yields

$$\begin{array}{c}\hfill \dot{\beta }\left(t\right)=R_1\left(t\right)\beta \left(t\right)+R_2\left(t\right)\beta (t-{\tau }_2)+\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right),\end{array}$$

(8)

where

$$\begin{array}{c}\hfill \begin{array}{c}\beta \left(t\right)=\left[\begin{array}{c}\epsilon \left(t\right)\\ {\tilde{\varphi }}_F\left(t\right)\end{array}\right],\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)=\left[\begin{array}{c}{\rm Y}\left(x(t-{\tau }_1)\right)\hfill \\ {\rm Y}\left(x(t-{\tau }_1)\right)\hfill \end{array}\right],\mathrm{Z}\left(t\right)=L_{11}\otimes (B+\Delta B\left(t\right))G_1,\hfill \\ R_1\left(t\right)=\left[\begin{array}{cc}{I}_M\otimes (A+\Delta A\left(t\right))+\mathrm{Z}\left(t\right)& -\mathrm{Z}\left(t\right)\\ \mathrm{Z}\left(t\right)+I_M\otimes G_{2}C& I_M\otimes (A+\Delta A\left(t\right)-G_{2}C)-\mathrm{Z}\left(t\right)\end{array}\right],\hfill \\ R_2\left(t\right)=\left[\begin{array}{cc}0& 0\\ -L_3\otimes (B+\Delta B\left(t\right))G_1-I_M\otimes G_{2}C& L_3\otimes (B+\Delta B\left(t\right))G_1\end{array}\right].\hfill \end{array}\end{array}$$

In order to facilitate reading, the following symbols are given:

$$\begin{array}{c}\hfill \begin{array}{c}{\stackrel{⌢}{\Gamma }}_6={\stackrel{⌣}{\Gamma }}_7,{\stackrel{⌢}{\Gamma }}_7={\stackrel{⌢}{\Gamma }}_2,{\stackrel{⌢}{\Gamma }}_8={\stackrel{⌢}{\Gamma }}_5,{\overline{\Lambda }}_2=L_3\otimes F_2{G}_1,{\Lambda }_2=L_{11}\otimes F_2{G}_1,\hfill \\ {\overline{\Gamma }}_4={\overline{\Gamma }}_3,{\widehat{\Theta }}_1=L_{11}^T\otimes V_1^T{B}^T,{\widehat{\Theta }}_2=I_M\otimes C^T{V}_2^T,{\Lambda }_1=I_M\otimes P_{1}E,\hfill \\ {\tilde{\Theta }}_1=L_3\otimes B{V}_1,{\Theta }_2=I_M\otimes A^T{P}_1,{\Theta }_4=I_M\otimes S_1,{\widehat{\Theta }}_4=I_M\otimes S_2,\hfill \\ {\overline{\Lambda }}_1=I_M\otimes F_1,{\Theta }_3={\Theta }_1+{\Theta }_4,{\widehat{\Theta }}_3={\Theta }_1+{\widehat{\Theta }}_4,{\Theta }_1=I_M\otimes P_1,\hfill \\ {\Theta }_{11}={\Theta }_4+{\widehat{\Theta }}_4-2{\Theta }_1+\mathrm{He}({\widehat{\Theta }}_1+{\Theta }_2),{\Theta }_{12}=-{\widehat{\Theta }}_1^T+{\widehat{\Theta }}_1+{\widehat{\Theta }}_2,\hfill \\ {\Theta }_{22}={\Theta }_4+{\widehat{\Theta }}_4-2{\Theta }_1+\mathrm{He}({\Theta }_2-{\widehat{\Theta }}_2-{\widehat{\Theta }}_1),{\Theta }_{25}=-{\tilde{\Theta }}_1-{\widehat{\Theta }}_2^T,\hfill \\ {\Theta }_{26}={\Theta }_1+{\tilde{\Theta }}_1,{\tilde{\Theta }}_{11}=\mu ({\Theta }_2+{\widehat{\Theta }}_1)+{\Theta }_1,{\tilde{\Theta }}_{12}=\mu ({\widehat{\Theta }}_2+{\widehat{\Theta }}_1),\hfill \\ {\tilde{\Theta }}_{21}=-\mu {\widehat{\Theta }}_1,{\tilde{\Theta }}_{22}=\mu ({\Theta }_2-{\widehat{\Theta }}_2-{\widehat{\Theta }}_1)+{\Theta }_1,{\tilde{\Theta }}_{13}={\tau }_1({\Theta }_2+{\widehat{\Theta }}_1),\hfill \\ {\tilde{\Theta }}_{14}={\tau }_1({\widehat{\Theta }}_1+{\widehat{\Theta }}_2),{\tilde{\Theta }}_{23}=-{\tau }_1{\widehat{\Theta }}_1,{\tilde{\Theta }}_{24}={\tau }_1({\Theta }_2-{\widehat{\Theta }}_2-{\widehat{\Theta }}_1),\hfill \\ {\tilde{\Theta }}_{15}={\tau }_2({\Theta }_2+{\widehat{\Theta }}_1),{\tilde{\Theta }}_{16}={\tau }_2({\widehat{\Theta }}_2+{\widehat{\Theta }}_1),{\stackrel{⌣}{\Theta }}_{44}=\mu {\stackrel{⌣}{\Theta }}_{12},\hfill \\ {\tilde{\Theta }}_{25}=-{\tau }_2{\widehat{\Theta }}_1,{\tilde{\Theta }}_{26}=-{\tau }_2({\Theta }_2-{\widehat{\Theta }}_2-{\widehat{\Theta }}_1),{\stackrel{⌣}{\Theta }}_{66}={\stackrel{⌣}{\Theta }}_{55}=-{\stackrel{⌣}{\Theta }}_{12}.\hfill \end{array}\end{array}$$

Theorem 1.

Suppose that Assumption 1 holds. For given parameters ${\tau }_1>0$, ${\tau }_2>0$ and $\mu ={\tau }_1^2+{\tau }_2^2$, the distributed robust containment control of MAS (1) with uncertain parameters can be achieved by (2) if there exist matrices $P_1>0$, $S_1>0$, $S_2>0$, $V_1>0$, $V_2>0$, P and scalars ${\delta }_j$, ${\stackrel{⌢}{\delta }}_j$, $j=1,2,\dots ,8$ and the following linear matrix inequality (LMI) holds:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\Theta & \Gamma \\ \ast & -\Lambda \end{array}\right]<0,\end{array}$$

(9)

with

$$\begin{array}{c}\hfill \begin{array}{c}\Gamma =[{\overline{\Gamma }}_1,\dots ,{\overline{\Gamma }}_8,{\widehat{\Gamma }}_1,\dots ,{\widehat{\Gamma }}_8,{\delta }_1{\stackrel{⌣}{\Gamma }}_1,\dots ,{\delta }_8{\stackrel{⌣}{\Gamma }}_8,{\stackrel{⌢}{\delta }}_1{\stackrel{⌢}{\Gamma }}_1,\dots ,{\stackrel{⌢}{\delta }}_8{\stackrel{⌢}{\Gamma }}_8],\hfill \\ \Lambda =\mathrm{diag}\{{\delta }_1,\dots ,{\delta }_8,{\stackrel{⌢}{\delta }}_1,\dots ,{\stackrel{⌢}{\delta }}_8,{\delta }_1,\dots ,{\delta }_8,{\stackrel{⌢}{\delta }}_1,\dots ,{\stackrel{⌢}{\delta }}_8\},\hfill \\ \Theta =\left[\begin{array}{cccccc}{\stackrel{⌣}{\Theta }}_{11}& {\stackrel{⌣}{\Theta }}_{12}& {\stackrel{⌣}{\Theta }}_{13}& {\stackrel{⌣}{\Theta }}_{14}& {\stackrel{⌣}{\Theta }}_{15}& {\stackrel{⌣}{\Theta }}_{16}\\ \ast & {\stackrel{⌣}{\Theta }}_{22}& 0& 0& 0& 0\\ \ast & \ast & {\stackrel{⌣}{\Theta }}_{33}& {\stackrel{⌣}{\Theta }}_{34}& {\stackrel{⌣}{\Theta }}_{35}& {\stackrel{⌣}{\Theta }}_{36}\\ \ast & \ast & \ast & {\stackrel{⌣}{\Theta }}_{44}& 0& 0\\ \ast & \ast & \ast & \ast & {\stackrel{⌣}{\Theta }}_{55}& 0\\ \ast & \ast & \ast & \ast & \ast & {\stackrel{⌣}{\Theta }}_{66}\end{array}\right],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}{\stackrel{⌣}{\Theta }}_{11}=\left[\begin{array}{cc}{\Theta }_{11}& {\Theta }_{12}\\ \ast & {\Theta }_{22}\end{array}\right],{\stackrel{⌣}{\Theta }}_{12}=\left[\begin{array}{cc}{\Theta }_1& 0\\ \ast & {\Theta }_1\end{array}\right],{\stackrel{⌣}{\Theta }}_{13}=\left[\begin{array}{cc}{\Theta }_1& 0\\ {\Theta }_{25}& {\Theta }_{26}\end{array}\right],\hfill \\ {\stackrel{⌣}{\Theta }}_{14}=\left[\begin{array}{cc}{\tilde{\Theta }}_{11}& {\tilde{\Theta }}_{12}\\ {\tilde{\Theta }}_{21}& {\tilde{\Theta }}_{22}\end{array}\right],{\stackrel{⌣}{\Theta }}_{15}=\left[\begin{array}{cc}{\tilde{\Theta }}_{13}& {\tilde{\Theta }}_{14}\\ {\tilde{\Theta }}_{23}& {\tilde{\Theta }}_{24}\end{array}\right],{\stackrel{⌣}{\Theta }}_{16}=\left[\begin{array}{cc}{\tilde{\Theta }}_{15}& {\tilde{\Theta }}_{16}\\ {\tilde{\Theta }}_{25}& {\tilde{\Theta }}_{26}\end{array}\right],\hfill \\ {\stackrel{⌣}{\Theta }}_{22}=\left[\begin{array}{cc}-{\Theta }_3& 0\\ \ast & -{\Theta }_3\end{array}\right],{\stackrel{⌣}{\Theta }}_{33}=\left[\begin{array}{cc}-{\widehat{\Theta }}_3& 0\\ \ast & -{\widehat{\Theta }}_3\end{array}\right],{\stackrel{⌣}{\Theta }}_{34}=\left[\begin{array}{cc}0& {\tilde{\Theta }}_{52}\\ 0& {\tilde{\Theta }}_{62}\end{array}\right],\hfill \\ {\stackrel{⌣}{\Theta }}_{35}=\left[\begin{array}{cc}0& {\tilde{\Theta }}_{54}\\ 0& {\tilde{\Theta }}_{64}\end{array}\right],{\stackrel{⌣}{\Theta }}_{36}=\left[\begin{array}{cc}0& {\tilde{\Theta }}_{56}\\ 0& {\tilde{\Theta }}_{66}\end{array}\right],\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{c}{\overline{\Gamma }}_1=\mathrm{col}\left(\begin{array}{cccccccccccc}{\Lambda }_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌣}{\Gamma }}_1=\left[\begin{array}{cccccccccccc}{\overline{\Lambda }}_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ {\overline{\Gamma }}_2=\mathrm{col}\left(\begin{array}{cccccccccccc}{\Lambda }_1& {\Lambda }_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌣}{\Gamma }}_2=\left[\begin{array}{cccccccccccc}{\Lambda }_2& -{\Lambda }_2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ {\overline{\Gamma }}_3=\mathrm{col}\left(\begin{array}{cccccccccccc}0& {\Lambda }_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌣}{\Gamma }}_3=\left[\begin{array}{cccccccccccc}0& {\overline{\Lambda }}_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ {\stackrel{⌣}{\Gamma }}_4=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& -{\overline{\Lambda }}_2& {\overline{\Lambda }}_2& 0& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ {\overline{\Gamma }}_5=\mathrm{col}\left(\begin{array}{cccccccccccc}\mu {\overline{\Lambda }}_1^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌣}{\Gamma }}_5=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& {\Lambda }_1^T& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ {\overline{\Gamma }}_6=\mathrm{col}\left(\begin{array}{cccccccccccc}\mu {\Lambda }_2^T& -\mu {\Lambda }_2^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌣}{\Gamma }}_6=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& {\Lambda }_1^T& {\Lambda }_1^T& 0& 0& 0& 0\end{array}\right],\hfill \\ {\overline{\Gamma }}_7=\mathrm{col}\left(\begin{array}{cccccccccccc}0& \mu {\overline{\Lambda }}_1^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌣}{\Gamma }}_7=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& {\Lambda }_1^T& 0& 0& 0& 0\end{array}\right],\hfill \\ {\overline{\Gamma }}_8=\mathrm{col}\left(\begin{array}{cccccccccccc}{\tau }_1{\overline{\Lambda }}_1^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌣}{\Gamma }}_8=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& {\Lambda }_1^T& 0& 0& 0\end{array}\right],\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}{\widehat{\Gamma }}_1=\mathrm{col}\left(\begin{array}{cccccccccccc}{\tau }_1{\overline{\Lambda }}_2^T& -{\tau }_1{\overline{\Lambda }}_2^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌢}{\Gamma }}_1=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& {\Lambda }_1^T& {\Lambda }_1^T& 0& 0\end{array}\right],\hfill \\ {\widehat{\Gamma }}_2=\mathrm{col}\left(\begin{array}{cccccccccccc}0& {\tau }_1{\overline{\Lambda }}_1^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌢}{\Gamma }}_2=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& {\Lambda }_1^T& 0& 0\end{array}\right],\hfill \\ {\widehat{\Gamma }}_3=\mathrm{col}\left(\begin{array}{cccccccccccc}{\tau }_2{\overline{\Lambda }}_1^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌢}{\Gamma }}_3=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {\Lambda }_1^T& 0\end{array}\right],\hfill \\ {\widehat{\Gamma }}_4=\mathrm{col}\left(\begin{array}{cccccccccccc}{\tau }_2{\Lambda }_2^T& -{\tau }_2{\Lambda }_2^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌢}{\Gamma }}_4=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {\Lambda }_1^T& {\Lambda }_1^T\end{array}\right],\hfill \\ {\widehat{\Gamma }}_5=\mathrm{col}\left(\begin{array}{cccccccccccc}0& {\tau }_2{\overline{\Lambda }}_1^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\stackrel{⌢}{\Gamma }}_5=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {\Lambda }_1^T\end{array}\right],\hfill \\ {\widehat{\Gamma }}_6=\mathrm{col}\left(\begin{array}{cccccccccccc}0& 0& 0& 0& -\mu {\overline{\Lambda }}_2^T& \mu {\overline{\Lambda }}_2^T& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\widehat{\Gamma }}_7=\mathrm{col}\left(\begin{array}{cccccccccccc}0& 0& 0& 0& -{\tau }_1{\overline{\Lambda }}_2^T& {\tau }_1{\overline{\Lambda }}_2^T& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ {\widehat{\Gamma }}_8=\mathrm{col}\left(\begin{array}{cccccccccccc}0& 0& 0& 0& -{\tau }_2{\overline{\Lambda }}_2^T& {\tau }_2{\overline{\Lambda }}_2^T& 0& 0& 0& 0& 0& 0\end{array}\right).\hfill \end{array}\end{array}$$

Moreover, the gain matrices are calculated by $BP=P_{1}B$, $G_1=P^{-1}{V}_1$ and $G_2=P_1^{-1}{V}_2$.

Proof. 

Choose the following Lyapunov–Krasovskii functional (LKF) candidate for MAS (1):

$$\begin{array}{c}\hfill V\left(t\right)=\sum _{p=1}^4{V}_p\left(t\right),\end{array}$$

(10)

where

$$\begin{array}{cc}\hfill V_1\left(t\right)=& {\beta }^T\left(t\right)\overline{P}\beta \left(t\right),\hfill \\ \hfill V_2\left(t\right)=& {\int }_{t-{\tau }_1}^t{\beta }^T\left(s\right)\overline{S}\beta \left(s\right)ds+{\int }_{t-{\tau }_2}^t{\beta }^T\left(s\right)\tilde{S}\beta \left(s\right)ds,\hfill \\ \hfill V_3\left(t\right)=& {\tau }_1{\int }_{-{\tau }_1}^0{\int }_{t+\theta }^t{\dot{\beta }}^T\left(s\right)\overline{P}\dot{\beta }\left(s\right)dsd\theta ,\hfill \\ \hfill V_4\left(t\right)=& {\tau }_2{\int }_{-{\tau }_2}^0{\int }_{t+\theta }^t{\dot{\beta }}^T\left(s\right)\overline{P}\dot{\beta }\left(s\right)dsd\theta ,\hfill \end{array}$$

with

$$\begin{array}{c}\hfill \overline{P}=\left[\begin{array}{cc}{\Theta }_1& 0\\ 0& {\Theta }_1\end{array}\right],\overline{S}=\left[\begin{array}{cc}{\Theta }_4& 0\\ 0& {\Theta }_4\end{array}\right],\tilde{S}=\left[\begin{array}{cc}{\widehat{\Theta }}_4& 0\\ 0& {\widehat{\Theta }}_4\end{array}\right],\end{array}$$

and ${\Theta }_1$, ${\Theta }_4$, ${\widehat{\Theta }}_4$ are defined in Theorem 1.

Calculating the time derivative of $V\left(t\right)$ along the trajectories of (8) yields

$$\begin{array}{c}\hfill {\dot{V}}_1\left(t\right)={\beta }^T\left(t\right)\mathrm{He}\left(\overline{P}{R}_1\left(t\right)\right)\beta \left(t\right)+\mathrm{He}\left({\beta }^T\left(t\right)\overline{P}{R}_2\left(t\right)\beta (t-{\tau }_2)\right)+\mathrm{He}\left({\beta }^T\left(t\right)\overline{P}\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)\right),\end{array}$$

(11)

and

$$\begin{array}{c}\hfill {\dot{V}}_2\left(t\right)={\beta }^T\left(t\right)\overline{S}\beta \left(t\right)-{\beta }^T(t-{\tau }_1)\overline{S}\beta (t-{\tau }_1)+{\beta }^T\left(t\right)\tilde{S}\beta \left(t\right)-{\beta }^T(t-{\tau }_2)\tilde{S}\beta (t-{\tau }_2).\end{array}$$

(12)

Also, we have

$$\begin{array}{cc}\hfill {\dot{V}}_3\left(t\right)=& {\tau }_1^2{\dot{\beta }}^T\left(t\right)\overline{P}\dot{\beta }\left(t\right)-{\tau }_1{\int }_{t-{\tau }_1}^t{\dot{\beta }}^T\left(s\right)\overline{P}\dot{\beta }\left(s\right)ds\hfill \\ \hfill =& {\tau }_1^2[{\beta }^T\left(t\right)R_1^T\left(t\right)\overline{P}{R}_1\left(t\right)\beta \left(t\right)+\mathrm{He}\left({\beta }^T\left(t\right)R_1^T\left(t\right)\overline{P}{R}_2\left(t\right)\beta (t-{\tau }_2)\right)\hfill \\ \hfill & +{\beta }^T(t-{\tau }_2)R_2^T\left(t\right)\overline{P}{R}_2\left(t\right)\beta (t-{\tau }_2)+\mathrm{He}\left({\beta }^T\left(t\right)R_1^T\left(t\right)\overline{P}\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)\right)\hfill \\ \hfill & +\mathrm{He}\left({\beta }^T(t-{\tau }_2)R_2^T\left(t\right)\overline{P}\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)\right)+{\widetilde{{\rm Y}}}^T\left(x(t-{\tau }_1)\right)\overline{P}\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)]\hfill \\ \hfill & -{\tau }_1{\int }_{t-{\tau }_1}^t{\dot{\beta }}^T\left(s\right)\overline{P}\dot{\beta }\left(s\right)ds,\hfill \end{array}$$

(13)

by Lemma 1, it yields

$$\begin{array}{c}\hfill -{\tau }_1{\int }_{t-{\tau }_1}^t{\dot{\beta }}^T\left(s\right)\overline{P}\dot{\beta }\left(s\right)ds\le -{(\beta \left(t\right)-\beta (t-{\tau }_1))}^T\overline{P}(\beta \left(t\right)-\beta (t-{\tau }_1)).\end{array}$$

Adopting the same method in taking the derivative of $V_3\left(t\right)$ yields

$$\begin{array}{cc}\hfill {\dot{V}}_4\left(t\right)\le & {\tau }_2^2[{\beta }^T\left(t\right)R_1^T\left(t\right)\overline{P}{R}_1\left(t\right)\beta \left(t\right)+\mathrm{He}\left({\beta }^T\left(t\right)R_1^T\left(t\right)\overline{P}{R}_2\left(t\right)\beta (t-{\tau }_2)\right)\hfill \\ \hfill & +{\beta }^T(t-{\tau }_2)R_2^T\left(t\right)\overline{P}{R}_2\left(t\right)\beta (t-{\tau }_2)+\mathrm{He}\left({\beta }^T\left(t\right)R_1^T\left(t\right)\overline{P}\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)\right)\hfill \\ \hfill & +\mathrm{He}\left({\beta }^T(t-{\tau }_2)R_2^T\left(t\right)\overline{P}\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)\right)+{\widetilde{{\rm Y}}}^T\left(x(t-{\tau }_1)\right)\overline{P}\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right)]\hfill \\ \hfill & -{(\beta \left(t\right)-\beta (t-{\tau }_2))}^T\overline{P}(\beta \left(t\right)-\beta (t-{\tau }_2)).\hfill \end{array}$$

(14)

Let $\xi \left(t\right)=\mathrm{col}(\beta \left(t\right),\beta (t-{\tau }_1),\beta (t-{\tau }_2),\widetilde{{\rm Y}}\left(x(t-{\tau }_1)\right))$. From Theorem 1, it is obtained that $V_1=P{G}_1$ and $V_2=P_1{G}_2$. Further, combining with (11)–(14), it has

$$\begin{array}{c}\hfill \dot{V}\left(t\right)\le {\xi }^T\left(t\right)\tilde{\Psi }\left(t\right)\xi \left(t\right),\end{array}$$

(15)

where

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{\Psi }\left(t\right)=\left[\begin{array}{cccc}{\tilde{\theta }}_{11}\left(t\right)& \overline{P}& {\tilde{\theta }}_{13}\left(t\right)& {\tilde{\theta }}_{14}\left(t\right)\\ \ast & -\overline{S}-\overline{P}& 0& 0\\ \ast & \ast & {\tilde{\theta }}_{33}\left(t\right)& {\tilde{\theta }}_{34}\left(t\right)\\ \ast & \ast & \ast & {\tilde{\theta }}_{44}\left(t\right)\end{array}\right],\hfill \\ {\tilde{\theta }}_{11}\left(t\right)=\mathrm{He}(R_1^T\left(t\right)\overline{P})+\overline{S}+\tilde{S}-2\overline{P}+\mu R_1^T\left(t\right)\overline{P}{R}_1\left(t\right),\hfill \\ {\tilde{\theta }}_{13}\left(t\right)=\overline{P}+\overline{P}{R}_2\left(t\right)+\mu R_1^T\left(t\right)\overline{P}{R}_2\left(t\right),{\tilde{\theta }}_{14}\left(t\right)=\overline{P}+\mu R_1^T\left(t\right)\overline{P},\hfill \\ {\tilde{\theta }}_{34}\left(t\right)=\mu R_2^T\left(t\right)\overline{P},{\tilde{\theta }}_{33}\left(t\right)=-\tilde{S}-\overline{P}+\mu R_2^T\left(t\right)\overline{P}{R}_2\left(t\right),{\tilde{\theta }}_{44}\left(t\right)=\mu \overline{P}.\hfill \end{array}\end{array}$$

For $\tilde{\Psi }\left(t\right)<0$, depending on Schur Complement Lemma, (15) turns into

$$\begin{array}{c}\hfill \widehat{\Psi }\left(t\right)=\left[\begin{array}{cccccc}{\widehat{\theta }}_{11}\left(t\right)& \overline{P}& \overline{P}+\overline{P}{R}_2\left(t\right)& {\tilde{\theta }}_{14}\left(t\right)& {\widehat{\theta }}_{15}\left(t\right)& {\widehat{\theta }}_{16}\left(t\right)\\ \ast & -\overline{S}-\overline{P}& 0& 0& 0& 0\\ \ast & \ast & -\tilde{S}-\overline{P}& {\tilde{\theta }}_{34}\left(t\right)& {\widehat{\theta }}_{35}\left(t\right)& {\widehat{\theta }}_{36}\left(t\right)\\ \ast & \ast & \ast & {\tilde{\theta }}_{44}\left(t\right)& 0& 0\\ \ast & \ast & \ast & \ast & -\overline{P}& 0\\ \ast & \ast & \ast & \ast & \ast & -\overline{P}\end{array}\right]<0,\end{array}$$

(16)

where

$$\begin{array}{c}\hfill \begin{array}{c}{\widehat{\theta }}_{11}\left(t\right)=\mathrm{He}(R_1^T\left(t\right)\overline{P})+\overline{S}+\tilde{S}-2\overline{P},{\widehat{\theta }}_{15}\left(t\right)={\tau }_1{R}_1^T\left(t\right)\overline{P},\hfill \\ {\widehat{\theta }}_{35}\left(t\right)={\tau }_1{R}_2^T\left(t\right)\overline{P},{\widehat{\theta }}_{16}\left(t\right)={\tau }_2{R}_1^T\left(t\right)\overline{P},{\widehat{\theta }}_{36}\left(t\right)={\tau }_2{R}_2^T\left(t\right)\overline{P},\hfill \\ {\vartheta }_{11}=I_M\otimes P_1(A+\Delta A\left(t\right))+L_{11}\otimes P_1(B+\Delta B\left(t\right))G_1,\hfill \\ {\vartheta }_{22}=I_M\otimes P_1(A+\Delta A\left(t\right)-G_{2}C)-L_{11}\otimes P_1(B+\Delta B\left(t\right))G_1,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}\overline{P}{R}_1\left(t\right)=\left[\begin{array}{cc}{\vartheta }_{11}& -L_{11}\otimes P_1(B+\Delta B\left(t\right))G_1\\ L_{11}\otimes P_1(B+\Delta B\left(t\right))G_1+I_M\otimes P_1{G}_{2}C,& {\vartheta }_{22}\end{array}\right],\hfill \\ \overline{P}{R}_2\left(t\right)=\left[\begin{array}{cc}0& 0\\ -L_3\otimes P_1(B+\Delta B\left(t\right))G_1-I_M\otimes P_1{G}_{2}C& L_3\otimes P_1(B+\Delta B\left(t\right))G_1\end{array}\right].\hfill \end{array}\end{array}$$

Further, substituting $[\Delta A\left(t\right)\Delta B\left(t\right)]=EH\left(t\right)\left[F_1{F}_2\right]$ into (16), it yields

$$\begin{array}{c}\hfill \Theta +\Delta \Psi \left(t\right)<0,\end{array}$$

(17)

and Θ can be found in (9), where

$$\begin{array}{cc}\hfill \Delta \Psi \left(t\right)=& \mathrm{He}({\overline{\Gamma }}_{1}H\left(t\right){\stackrel{⌣}{\Gamma }}_1+{\overline{\Gamma }}_{2}H\left(t\right){\stackrel{⌣}{\Gamma }}_2+{\overline{\Gamma }}_{3}H\left(t\right){\stackrel{⌣}{\Gamma }}_3+{\overline{\Gamma }}_{4}H\left(t\right){\stackrel{⌣}{\Gamma }}_4)\hfill \\ \hfill & +\mathrm{He}({\overline{\Gamma }}_5{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_5+{\overline{\Gamma }}_6{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_6+{\overline{\Gamma }}_7{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_7+{\overline{\Gamma }}_8{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_8)\hfill \\ \hfill & +\mathrm{He}({\widehat{\Gamma }}_1{H}^T\left(t\right){\stackrel{⌢}{\Gamma }}_1+{\overline{\Gamma }}_2{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_2+{\overline{\Gamma }}_3{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_3+{\overline{\Gamma }}_4{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_4)\hfill \\ \hfill & +\mathrm{He}({\widehat{\Gamma }}_5{H}^T\left(t\right){\stackrel{⌢}{\Gamma }}_5+{\overline{\Gamma }}_6{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_6+{\overline{\Gamma }}_7{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_7+{\overline{\Gamma }}_8{H}^T\left(t\right){\stackrel{⌣}{\Gamma }}_8).\hfill \end{array}$$

Making use of Lemma 2, there exist scalars ${\delta }_i>0,{\delta }_i^{-1}>0,{\stackrel{⌢}{\delta }}_i>0,{\stackrel{⌢}{\delta }}_i^{-1}>0,i=1,2,\dots 8$; then, (17) can be changed to

$$\begin{array}{cc}\hfill \Theta & +{\delta }_1{\overline{\Gamma }}_1{\overline{\Gamma }}_1^T+{\delta }_1^{-1}{\stackrel{⌣}{\Gamma }}_1^T{\stackrel{⌣}{\Gamma }}_1+{\delta }_2{\overline{\Gamma }}_2{\overline{\Gamma }}_2^T+{\delta }_2^{-1}{\stackrel{⌣}{\Gamma }}_2^T{\stackrel{⌣}{\Gamma }}_2+{\delta }_3{\overline{\Gamma }}_3{\overline{\Gamma }}_3^T+{\delta }_3^{-1}{\stackrel{⌣}{\Gamma }}_3^T{\stackrel{⌣}{\Gamma }}_3\hfill \\ \hfill & +{\delta }_4{\overline{\Gamma }}_4{\overline{\Gamma }}_4^T+{\delta }_4^{-1}{\stackrel{⌣}{\Gamma }}_4^T{\stackrel{⌣}{\Gamma }}_4+{\delta }_5{\overline{\Gamma }}_5{\overline{\Gamma }}_5^T+{\delta }_5^{-1}{\stackrel{⌣}{\Gamma }}_5^T{\stackrel{⌣}{\Gamma }}_5+{\delta }_6{\overline{\Gamma }}_6{\overline{\Gamma }}_6^T+{\delta }_6^{-1}{\stackrel{⌣}{\Gamma }}_6^T{\stackrel{⌣}{\Gamma }}_6\hfill \\ \hfill & +{\delta }_7{\overline{\Gamma }}_7{\overline{\Gamma }}_7^T+{\delta }_7^{-1}{\stackrel{⌣}{\Gamma }}_7^T{\stackrel{⌣}{\Gamma }}_7+{\delta }_8{\overline{\Gamma }}_8{\overline{\Gamma }}_8^T+{\delta }_8^{-1}{\stackrel{⌣}{\Gamma }}_8^T{\stackrel{⌣}{\Gamma }}_8+{\stackrel{⌢}{\delta }}_1{\widehat{\Gamma }}_1{\widehat{\Gamma }}_1^T+{\stackrel{⌢}{\delta }}_1^{-1}{\stackrel{⌢}{\Gamma }}_1^T{\stackrel{⌢}{\Gamma }}_1\hfill \\ \hfill & +{\stackrel{⌢}{\delta }}_2{\widehat{\Gamma }}_2{\widehat{\Gamma }}_2^T+{\stackrel{⌢}{\delta }}_2^{-1}{\stackrel{⌢}{\Gamma }}_2^T{\stackrel{⌢}{\Gamma }}_2+{\stackrel{⌢}{\delta }}_3{\widehat{\Gamma }}_3{\widehat{\Gamma }}_3^T+{\stackrel{⌢}{\delta }}_3^{-1}{\stackrel{⌢}{\Gamma }}_3^T{\stackrel{⌢}{\Gamma }}_3+{\stackrel{⌢}{\delta }}_4{\widehat{\Gamma }}_4{\widehat{\Gamma }}_4^T+{\stackrel{⌢}{\delta }}_4^{-1}{\stackrel{⌢}{\Gamma }}_4^T{\stackrel{⌢}{\Gamma }}_4\hfill \\ \hfill & +{\stackrel{⌢}{\delta }}_5{\widehat{\Gamma }}_5{\widehat{\Gamma }}_5^T+{\stackrel{⌢}{\delta }}_5^{-1}{\stackrel{⌢}{\Gamma }}_5^T{\stackrel{⌢}{\Gamma }}_5+{\stackrel{⌢}{\delta }}_6{\widehat{\Gamma }}_6{\widehat{\Gamma }}_6^T+{\stackrel{⌢}{\delta }}_6^{-1}{\stackrel{⌢}{\Gamma }}_6^T{\stackrel{⌢}{\Gamma }}_6+{\stackrel{⌢}{\delta }}_7{\widehat{\Gamma }}_7{\widehat{\Gamma }}_7^T+{\stackrel{⌢}{\delta }}_7^{-1}{\stackrel{⌢}{\Gamma }}_7^T{\stackrel{⌢}{\Gamma }}_7\hfill \\ \hfill & +{\stackrel{⌢}{\delta }}_8{\widehat{\Gamma }}_8{\widehat{\Gamma }}_8^T+{\stackrel{⌢}{\delta }}_8^{-1}{\stackrel{⌢}{\Gamma }}_8^T{\stackrel{⌢}{\Gamma }}_8<0.\hfill \end{array}$$

(18)

Clearly, the inequality (18) is guaranteed by LMI (9). Hence, $\dot{V}\left(t\right)<0$ is induced. This completes the proof. □

Remark 3.

In [32,33,34,35,36,37], the consensus and containment problems for certain MASs with multiple delays are researched. For uncertain MASs with one leader, only the transmission delay is handled in [40]. Although the containment problem concerned with structural uncertainties is dealt with in [41] considering multiple delays, the communication graph is undirected.

Remark 4.

It is noteworthy that the study in [12] only focuses on certain dynamics and transmission delay. In the case of uncertain dynamics, the multiple delays involving both state delay and transmission delay are investigated in this paper. Furthermore, compared with [29,30] considering single state delay, the multiple delays make the construction of LKFs and solving of LMIs more difficult. Especially, the dimension of matrix Θ in (9) significantly surpasses that in [12]. In addition, the computational complexity of solving LMIs will increase accordingly as the number of agents increases.

4. Simulation Results

The directed connection graph is shown in Figure 1, and Assumption 1 holds. From Figure 1, the nodes 1–8 stand for followers and 9–11 are the leaders.

The matrices $L_{11}$ and $L_{12}$ are expressed as

$$\begin{array}{c}\hfill L_{11}=\left[\begin{array}{cccccccc}2& 0& 0& 0& 0& 0& 0& -1\\ -1& 3& 0& 0& 0& 0& -1& 0\\ 0& -1& 3& 0& 0& 0& 0& 0\\ 0& 0& -1& 2& 0& 0& 0& 0\\ 0& 0& 0& -1& 2& 0& 0& 0\\ 0& 0& -1& 0& -1& 2& 0& 0\\ 0& -1& 0& 0& 0& -1& 2& 0\\ -1& 0& 0& 0& 0& 0& -1& 2\end{array}\right],\end{array}$$

and

$$\begin{array}{c}\hfill L_{12}=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& -1& -1\\ 0& 0& -1\\ 0& 0& -1\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

Based on (1), we set

$$\begin{array}{c}\hfill \begin{array}{c}{x}_k=\left[\begin{array}{c}{x}_{k1}\\ x_{k2}\end{array}\right],A=\left[\begin{array}{cc}-1& 1.6\\ -1.6& 0\end{array}\right],B=\left[\begin{array}{c}1\\ 2\end{array}\right],C=\left[\begin{array}{cc}1& -1\end{array}\right],\hfill \\ D=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right],g\left(x_k(t-{\tau }_1)\right)=\left[\begin{array}{c}0.5sin\left(x_{k1}(t-0.2)\right)\\ 0.5sin\left(x_{k2}(t-0.2)\right)\end{array}\right].\hfill \end{array}\end{array}$$

Before solving the LMI (9), we set that

$$\begin{array}{c}\hfill \begin{array}{c}E=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right],F_1=\left[\begin{array}{cc}0.05& 0\\ 0& 0.05\end{array}\right],F_2=\left[\begin{array}{c}0.05\\ 0.05\end{array}\right],{\tau }_1=0.2,{\tau }_2=0.35,\hfill \\ ({\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4,{\delta }_5,{\delta }_6,{\delta }_7,{\delta }_8)=(1.09,1.08,1.10,1.13,1.12,1.13,1.14,1.15),\hfill \\ ({\stackrel{⌢}{\delta }}_1,{\stackrel{⌢}{\delta }}_2,{\stackrel{⌢}{\delta }}_3,{\stackrel{⌢}{\delta }}_4,{\stackrel{⌢}{\delta }}_5,{\stackrel{⌢}{\delta }}_6,{\stackrel{⌢}{\delta }}_7,{\stackrel{⌢}{\delta }}_8)=(1.12,1.05,1.11,1.15,1.14,1.07,1.20,1.16).\hfill \end{array}\end{array}$$

Further, applying the LMI techniques to (9), we obtain that

$$\begin{array}{c}\hfill \begin{array}{c}{P}_1=\left[\begin{array}{cc}0.6396& -0.0952\\ -0.0952& 0.7825\end{array}\right],S_1=\left[\begin{array}{cc}0.3521& 0.0385\\ 0.0385& 0.1364\end{array}\right],\hfill \\ V_1=\left[\begin{array}{cc}0.0161& -0.0318\end{array}\right],V_2={\left[\begin{array}{cc}-0.0159& -0.2652\end{array}\right]}^T,\hfill \end{array}\end{array}$$

and the gain matrices $G_1$ and $G_2$ are shown as

$$\begin{array}{c}\hfill G_1=\left[\begin{array}{cc}0.0194& -0.0383\end{array}\right],G_2={\left[\begin{array}{cc}-0.0768& -0.3482\end{array}\right]}^T.\end{array}$$

We set initial states as $x_1={[-6,3]}^T$, $x_2={[-4,8]}^T$, $x_3={[0,-7]}^T$, $x_4={[8,4]}^T$, $x_5={[6,-8]}^T$, $x_6={[2,-1]}^T$, $x_7={[-4,5]}^T$, $x_8={[7,-6]}^T$, $x_9={[2,-8]}^T$, $x_{10}={[-8,-7]}^T$, $x_{11}={[8,3]}^T$ and let $H\left(t\right)=\left[\begin{array}{cc}0.1\times sin\left(t\right)& 0\\ 0& 0.1\end{array}\right]$, ${\tau }_1=0.2$ and ${\tau }_2=0.35$. Figure 2 and Figure 3 display the trajectories of containment errors and observer errors for followers, respectively. Obviously, when $T=30$ s, ${\tilde{\varphi }}_{Fk}\to 0$ and ${\epsilon }_k\to 0$. Furthermore, the trajectories of containment error for followers when ${\tau }_1=0.2$ and ${\tau }_2=0.1$ and when ${\tau }_1=0.2$ and ${\tau }_2=0.75$ are, respectively, presented in Figure 4 and Figure 5. It can be seen that the convergence speed of the error curve becomes slower when the transmission delay ${\tau }_2$ becomes larger. In addition, the trajectories of containment error for followers in [12] are shown in Figure 6. Then, it follows that the convergence speed of the error trajectories in Figure 5 becomes slower than that in Figure 6 because of multiple time delays.

Moreover, in Figure 7, the small black circles denote leaders and the colorful diamonds denote followers, and the snapshots of the agents are depicted at moments 0 s, 10 s, 20 s, and 30 s, respectively. Clearly, the colorful diamonds enter into the convex hull surrounded by the black circles when $t\to \infty$, and then the agents achieve robust containment with uncertain parameters.

5. Conclusions

This article focused on the robust cooperative containment problem of uncertain MASs with nonlinearities in the presence of multiple time delays. In order to address model uncertainties and transmission delay, an observer-based control protocol was employed without exploiting any knowledge of the leaders. Furthermore, solutions originating from the LMI equation were obtained, relying on developing novel quadratic-type LKFs. Based on Lyapunov theory, the stability of the observer dynamics and containment-error time-delay systems was proven. Moreover, under the framework of the developed containment strategy, the followers asymptotically converged to a convex hull created by the leaders. In the near future, it will be interesting to extend the current results into the time-varying group formation–containment control of MASs with heterogeneous uncertainties and multiple varying delays over switching directed graphs [15,16].