2. Department of Electrical and Computer Engineering at University of Alberta, Edmonton, Alberta T6R1Z6, Canada
During the last decade, cooperative control of multiagent systems has been extensively studied due to its everincreasing applications in many fields such as formation control, sensor network, flocking, synchronization of coupled oscillators and robot position synchronization [1][6]. Leaderfollowing consensus problem as one of the essential problems in cooperative control has also received considerable attention in recent years. For example, Meng et al. [7] provided some analysis results for both leaderless and leaderfollowing consensus algorithms for firstorder and secondorder multiagent systems in presence of communication and input delays. An LMI approach for designing a leaderfollowing consensus protocol was also introduced for general linear multiagent systems in presence of timedelay by Ding et al. [8] in which the leader has the same dynamics as the followers. Furthermore, for timevarying delayed firstorder multiagent systems with a static leader, leaderfollowing consensus analysis was studied in [9] and some sufficient conditions were obtained to guarantee the convergence of the agents to the leader. A leaderfollowing consensus problem for secondorder multiagent systems with communication timevarying delays was also investigated in [10]. For systems with large delay sequences, leaderfollowing consensus problem of discretetime multiagent systems has been addressed in [11]. Moreover, Cao et al. [12] studied the distributed containment control problem of mobile autonomous agents with multiple stationary or dynamic leaders under both fixed and switching directed network topologies. For uncertain multiagent systems, Liu and Huang [13] addressed a leaderfollowing consensus problem for a class of uncertain higherorder nonlinear multiagent systems in presence of external disturbances. Recently, a leaderfollowing consensus problem has been addressed in [14] for a class of uncertain multiagent systems with actuator faults.
In many practical leaderfollowing multiagent systems, the control objective is to follow a combination of the leader states as an output. This is analogous to the case that the follower agents have a virtual leader which means that there is an exosystem acting as a leader. This problem is called leaderfollowing output regulation that has attracted attention in recent years [15][18]. Therefore, the problem of leaderfollowing output consensus can be viewed as the special case of cooperative output regulation problem. In [15], an algorithm for designing a distributed control law is provided for a robust cooperative control of multiagent system with distributed output regulation. Yan and Huang [16] addressed the analysis of a cooperative output regulation for linear timedelay multiagent systems. Moreover, a cooperative output regulation analysis of heterogeneous multiagent systems is carried out in [17]. To the best of our knowledge, very few works have focused on output regulation problem of multiagent systems with timedelay. Recently, a descriptor approach to robust leaderfollowing output regulation design of multiagent systems was given in [19] in the presence of both uncertainty and constant transmission delay. The presented method in this study works for both stable and unstable agents. Furthermore, Yan and Huang [18] addressed the cooperative robust output regulation problem for a discretetime linear timedelay multi agent systems with stable agents. Since, the presented output regulation methods in the literature for timedelay multiagent systems cannot handle the timevarying delays, we are motivated to investigate a robust leaderfollowing output regulation for uncertain general linear multiagent systems with stationary and ramptype dynamic leaders as well as both stable and unstable agents in presence of timevarying delay. The contributions of this paper are summarized as follows:
1) A new regulation protocol is proposed in this paper that works for both stationary and ramptype dynamic leaders.
2) Applying this protocol, a timedelay closedloop system of retarded type is obtained that is simpler to deal with than the closedloop system of neutral type presented in [19].
3) Using direct LyapunovKrasovskii method, robust analysis and design conditions for the leaderfollowing output regulation of uncertain multiagent systems with transmission timevarying delay are given in terms of linear matrix inequalities. The derived LMI conditions have smaller dimensions than the ones obtained in [19].
4) As mentioned, in this paper, a timevarying transmission delay is assumed in leaderfollowing multiagent system whereas the studies in [18] and [19] are based on a constant timedelay. Furthermore, our method can deal with both stable and unstable agents.
This paper is organized as follows. Problem formulation and preliminary results as well as a new regulation protocol are given in Section Ⅱ. In Section Ⅲ, The robust stability analysis conditions are provided in terms of certain linear matrix inequalities. Moreover, the design conditions are given in Section Ⅳ. Some illustrative examples are provided in Section Ⅴ to show the effectiveness of the proposed methods. Finally, the concluding remarks are given in Section Ⅵ.
Notations :
Let
Assume
Consider a group of N uncertain
$\begin{eqnarray} \dot{x}_{i} (t)& =& (A+\Delta A(t))x_{i} (t)+(B+\Delta B(t))u_{i} (t) \nonumber\\ y_{i} (t)& =& Cx_{i} (t), \, \, \, \, \, \, i=1, \, 2, \ldots, N \end{eqnarray} $  (1) 
where
$ \begin{equation} \Delta A(t)=D_{a} F_{a} (t)E_{a} , \, \, \, \, \, \, \, \, \, \Delta B(t)=D_{b} F_{b} (t)E_{b} \end{equation} $  (2) 
where
$ \begin{equation} F_{a} ^{T} (t)F_{a} (t)\le I, \, \, \, \, \, \, \, \, \, F_{b} ^{T} (t)F_{b} (t)\le I\, \, \, \, \, \, \, \forall t \end{equation} $  (3) 
and
In this paper we consider the design of an appropriate feedback control
Definition 1: For any initial condition
$ \begin{eqnarray*} \lim\limits_{t\to \infty } \left\ y_{i} (t)y_{0} (t)\right\ =0, \quad i=1, \, \ldots, \, N. \end{eqnarray*} $ 
The control objective of the leaderfollowing output regulation of a multiagent system is to follow the output of a real or virtual leader. This output plays the role of a setpoint for the followers. Moreover, in many practical problems, the leader has its own feedback independent of the followers in order to obtain the appropriate setpoint in its output for the followers. Therefore, we find that in a general case, the model of the followers and the leader can be nonidentical. On the other hand, in practice, many of the desired outputs of the agents can be approximated by stationary or ramp signals. Hence, we focus on designing leaderfollowing output regulation of the uncertain multiagent systems for stationary and ramptype dynamic leaders in this paper. These leaders are labeled as
$ \begin{equation} y_{0} (t):=\left\{\begin{array}{c} {a_{0} , \, \, \, \, \, \, \, \, t\ge 0} \\ {0, \, \, \, \, \, \, \, \, \, \, t<0} \end{array}\right. \end{equation} $  (4) 
$ \begin{equation} \label{GrindEQ__5_} y_{0} (t):=\left\{\begin{array}{c} {r_{0} t, \, \, \, \, \, \, \, t\ge 0} \\ {0, \, \, \, \, \, \, \, \, \, \, t<0} \end{array}\right. \end{equation} $  (5) 
where
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Fig. 1 Network topology of the directed multiagent system. 
Now, we state the following assumptions that are needed throughout the paper.
Assumption 1: The graph
Assumption 2: The pair (
Assumption 3: The digraph
To deal with the leaders (4) and (5) in the network of multiagent system presented in Fig. 1, we propose a new consensus protocol for system (1) in presence of transmission delay shown in a general form that works for both stationary and dynamic leaders (4) and (5) as
$ \begin{equation} u_{i} (t)=u_{iC} (t)+u_{iT} (t), \, \, \, \, \, \, i=1, \, 2, \ldots, N \end{equation} $  (6) 
in which
$ \begin{eqnarray*} u_{iC} (t)& =& K\sum _{j=1}^{N}a_{ij} (y_{i} (t\tau (t))y_{j} (t\tau (t))) \\ && +K_{I} (\sum _{j=1}^{N}a_{ij} \int _{0}^{t\tau (t)}(y_{i} (\alpha )y_{j} (\alpha ))d\alpha)\\ && +K_{II} (\sum _{j=1}^{N}a_{ij} \int _{0}^{t\tau (t)}\int _{0}^{\beta }(y_{i} (\alpha )y_{j} (\alpha ))d\alpha d\beta)\\ u_{iT} (t)& =& m_{i} K'_{i} \, (y_{i} (t)y_{0} (t\tau _{r} (t))) \\ && +m_{i} K'_{Ii} \, \int _{0}^{t}y_{i} (\alpha )d\alpha m_{i} K'_{Ii} \, \int _{0}^{t\tau _{r} (t)}y_{0} (\alpha )d\alpha \\ && +m_{i} K'_{IIi}\int _{0}^{t}\int _{0}^{\beta }y_{i} (\alpha )d\alpha d\beta\\ && m_{i} K'_{IIi}\int _{0}^{t\tau _{r} (t)}\int _{0}^{\beta }y_{0} (\alpha )d\alpha d\beta \end{eqnarray*} $ 
with
As seen in (6), the proposed regulation protocol is composed of two parts: consensus
$ \begin{eqnarray} \dot{x}(t)& =& A_{\Delta } x_{i} (t)\nonumber\\[2mm] && +B_{\Delta } KC\sum _{j=1}^{N}a_{ij} (x_{i} (t\tau (t))x_{j} (t\tau (t))) \nonumber \\[2mm] && +B_{\Delta } K_{I} C(\sum _{j=1}^{N}a_{ij} \int _{0}^{t\tau (t)}(x_{i} (\alpha )x_{j} (\alpha ))d\alpha ) \nonumber\\[2mm] && +B_{\Delta } K_{II} C(\sum _{j=1}^{N}a_{ij} \int _{0}^{t\tau (t)} \int _{0}^{\beta }(x_{i} (\alpha )  x_{j} (\alpha ))d\alpha d\beta) \nonumber\\[2mm] && +m_{i} B_{\Delta } K'_{i} Cx_{i} (t)+m_{i} B_{\Delta } K'_{Ii} \int _{0}^{t}Cx_{i} (\alpha )d\alpha \nonumber \\[2mm] && +m_{i} B_{\Delta } K'_{IIi} \int _{0}^{t}\int _{0}^{\beta }Cx_{i} (\alpha )d\alpha d\beta \nonumber\\[2mm] && m_{i} B_{\Delta } K'y_{0} (t  \tau _{r} (t))  m_{i} B_{\Delta } K'_{Ii} \int _{0}^{t\tau _{r} (t)} y_{0} (\alpha )d\alpha \nonumber\\[2mm] && m_{i} B_{\Delta } K'_{IIi} \int _{0}^{t\tau _{r} (t)}\int _{0}^{\beta }y_{0} (\alpha )d\alpha d\beta \end{eqnarray} $  (7) 
with
$ \begin{eqnarray} \dot{x}(t)& =& (I\otimes A_{\Delta } )x(t)+(L_{s} \otimes (B_{\Delta } KC))\, x(t\tau (t))\nonumber\\ && +(L_{s} \otimes (B_{\Delta } K_{I} C))\int _{0}^{t\tau (t)}x(\alpha )d\alpha \nonumber\\ && +(L_{s} \otimes (B_{\Delta } K_{II} C))\int _{0}^{t\tau (t)}\int _{0}^{\beta }x(\alpha )d\alpha d\beta \nonumber\\ && +(M\otimes B_{\Delta } )K'(I\otimes C)\, x(t) \nonumber\\ && +(M\otimes B_{\Delta } )K'_{I} (I\otimes C)\int _{0}^{t}x(\alpha )d\alpha \nonumber\\ && +(M\otimes B_{\Delta } )K'_{II} (I\otimes C)\int _{0}^{t}\int _{0}^{\beta }x(\alpha )d\alpha d\beta \nonumber\\ && (M\otimes B_{\Delta } )K'(I\otimes C)\, y_{0} (t\tau _{r} (t)) \nonumber\\ && (M\otimes B_{\Delta } )K'_{I} (I\otimes C)\int _{0}^{t\tau _{r} (t)}y_{0} (\alpha )d\alpha \nonumber\\ && (M\otimes B_{\Delta } )K'_{II} (I \otimes C) \int _{0}^{t\tau _{r} (t)} \int_{0}^{\beta} y_{0} (\alpha)d\alpha d\beta \end{eqnarray} $  (8) 
Remark 1: It is wellknown that a ProportionalIntegral (PⅠ) or ProportionalIntegralDouble integral (PⅡ
$ \begin{eqnarray} \dot{x}(t)& =& (I\otimes A_{\Delta } )x(t)+(L_{s} \otimes (B_{\Delta } KC))\, x(t\tau (t))\nonumber\\ && +(L_{s} \otimes (B_{\Delta } K_{I} C))\int _{0}^{t\tau (t)}x(\alpha )d\alpha \nonumber\\ && +(L_{s} \otimes (B_{\Delta } K_{II} C))\int _{0}^{t\tau (t)}\int _{0}^{\beta }x(\alpha )d\alpha d\beta \nonumber\\ && +(M\otimes B_{\Delta } )K'(I\otimes C)\, x(t) \nonumber\\ && +(M\otimes B_{\Delta } )K'_{I} (I\otimes C)\int _{0}^{t}x(\alpha )d\alpha \nonumber\\ && +(M\otimes B_{\Delta } )K'_{II} (I\otimes C)\int _{0}^{t}\int _{0}^{\beta }x(\alpha )d\alpha d\beta. \end{eqnarray} $  (9) 
Remark 2: The simple and double integrator statements in (6) play important roles in achieving a general condition for the robust leaderfollowing output regulation of the uncertain multiagent systems with general linear dynamics in presence of each of the leaders (4) and (5). In other words, these integrators provide output tracking of a closedloop multiagent system for stationary or ramptype dynamic leader that can prevalently occur in practice.
Now, we consider (9) and define new variables
$ \begin{eqnarray} \dot{\eta }(t)& =& \xi (t)\nonumber\\ \dot{\xi }(t)& =& x(t)\nonumber\\ \dot{x}(t)& =& (I\otimes A_{\Delta } )x(t)+(L_{s} \otimes B_{\Delta } KC)\, x(t\tau (t))\nonumber\\ && +(L_{s} \otimes B_{\Delta } K_{I} C)\xi (t\tau (t)) \nonumber\\ && +(L_{s} \otimes B_{\Delta } K_{II} C)\eta (t\tau (t)) \nonumber\\ && +(M\otimes B_{\Delta } )K'(I\otimes C)\, x(t) \nonumber\\ && +(M\otimes B_{\Delta } )K'_{I} (I\otimes C)\, \xi (t)\nonumber\\ && +(M\otimes B_{\Delta } )K'_{II} (I\otimes C)\eta (t) \end{eqnarray} $  (10) 
or
$ \begin{equation} \dot{\bar{x}}(t)=\bar{A}\bar{x}(t)+\bar{B}\bar{x}(t\tau (t)) \end{equation} $  (11) 
in which
$ \xi (t)=\left[\begin{array}{c} {\xi _{1} (t)} \\ {\vdots } \\ {\xi _{N} (t)} \end{array}\right], \eta (t)=\left[\begin{array}{c} {\eta _{1} (t)} \\ {\vdots } \\ {\eta _{N} (t)} \end{array}\right] $ 
$ \begin{eqnarray*} \bar{A}& =& \left[\begin{array}{ccc} {0} & {I}&{0} \\ {0}&{0}&{I} \\{\bar{A}_{3\Delta } }&{\bar{A}_{2\Delta } }&{I\otimes A_{\Delta } +\bar{A}_{1\Delta } \, } \end{array}\right]\\ \bar{B}& =& \left[ \begin{array}{ccc} {0}&{0}&{0} \\ {0}&{0}&{0} \\ {(L_{s} \otimes B_{\Delta } K_{II} C)} & {(L_{s} \otimes B_{\Delta } K_{I} C)} & {(L_{s} \otimes B_{\Delta } KC)} \end{array} \right]\\ K'& =& {\text {diag}}\{K'_{11} , \ldots , K'_{1N}\}, \, \, K'_{I} ={\text{diag}\{K'_{21} , \ldots , K_{2N}\}}\\ K'_{II} & =& {\text {diag}}\{K'_{31} , \ldots , K'_{3N}\}, M={\text {diag}}\{m_{1}, \ldots , m_{N}\} \end{eqnarray*} $ 
where
$ \begin{eqnarray*} \bar{A}_{1\Delta } & =& (M\otimes B_{\Delta } )K'(I\otimes C), \bar{A}_{2\Delta } = (M\otimes B_{\Delta } )K'_{I} (I\otimes C)\\ \bar{A}_{3\Delta } & =& (M\otimes B_{\Delta } )K'_{II} (I\otimes C). \end{eqnarray*} $ 
Now, we present the following lemmas which will be used in the main results of the paper.
Lemma 1 [20]: Let
$ \begin{equation} LF(t)H+H^{T} F^{T} (t)L^{T} \le \frac{1}{\sigma ^{2} } LL^{T} +\sigma ^{2} H^{T} H. \end{equation} $  (12) 
Lemma 2 [21]: If
$ \begin{equation} SW^{1} S\le (S^{T} +SW). \end{equation} $  (13) 
For the sake of brevity, the proofs of these two lemmas are omitted.
Ⅲ. ROBUST STABILITY OF LEADERFOLLOWING OUTPUT REGULATIONIn this section, we concentrate on deriving a robust stability analysis criterion for leaderfollowing output regulation of uncertain timedelay multiagent system (10). Defining
$ \begin{eqnarray*} &&A_{1} = (M\otimes B)K'(I\otimes C), \, \, A_{1\Delta } = (M\otimes \Delta B(t))K'(I\otimes C)\\ &&A_{2} = (M\otimes B)K'_{I} (I\otimes C), \, \, A_{2\Delta } = (M\otimes \Delta B(t))K'_{I} (I\otimes C)\\ &&A_{3} = (M\otimes B)K'_{II} (I\otimes C), \, \, A_{3\Delta } = (M\otimes \Delta B(t))K'_{II} (I\otimes C)\\ &&B_{1} = BKC, B_{1\Delta } =\Delta B(t)KC \\ &&B_{2} = BK_{I} C, B_{2\Delta } = \Delta B(t)K_{I} C\\ &&B_{3} = BK_{II} C ~and~ B_{3\Delta } =\Delta B(t)K_{II} C \end{eqnarray*} $ 
in (10), we will have
$ \begin{eqnarray} \dot{\eta }(t)& =& \xi (t)\nonumber\\ \dot{\xi }(t)& =& x(t)\nonumber\\ \dot{x}(t)& =& (I\otimes (A+\Delta A))x(t)\nonumber\\ && +(L_{s} \otimes (B_{1} +B_{1\Delta } ))\, x(t\tau (t)) \nonumber\\ && +(L_{s} \otimes (B_{2} +B_{2\Delta } ))\xi (t\tau (t)) \nonumber\\ && +(L_{s} \otimes (B_{3} +B_{3\Delta } ))\eta (t\tau (t))+(A_{1} +A_{1\Delta } )\, x(t) \nonumber\\ && +(A_{2} +A_{2\Delta } )\xi (t)+(A_{3} +A_{3\Delta } )\eta (t). \end{eqnarray} $  (14) 
A LyapunovKrasovskii functional for system (14) has the form
$ \begin{equation} V=V_{1} +V_{2} +V_{3} \end{equation} $  (15) 
where
$ \begin{eqnarray} &&V_{1} =\bar{x}^{T} \bar{P}\bar{x}\nonumber\\ &&V_{2} =2\int _{\tau (t)}^{0}\int _{t+\beta }^{t}{\dot{\bar{x}}}^{T} (\alpha )Z{\bar{\dot{x}}}(\alpha )d\alpha \, d\beta\nonumber\\ &&V_{3} =\int _{t\tau (t)}^{t}\bar{x}^{T} (\alpha )Q\bar{x}(\alpha )d\alpha \end{eqnarray} $  (16) 
in which
Theorem 1: Under Assumptions 13, for given timedelay
$ \begin{equation} \left[\begin{array}{cc} {\Sigma _{11} }&{\Sigma _{12} } \\ {*} & {\Sigma _{22} } \end{array}\right]<0 \end{equation} $  (17) 
$ \begin{equation} \left[\begin{array}{cc} {X}&{Y} \\ {Y^{T} }&{Z} \end{array}\right]>0 \end{equation} $  (18) 
in which
$ \begin{eqnarray*} \Sigma _{11} & =& \left[\begin{array}{cccc} {\phi _{n} }&{Y+\bar{P}\, \, \Psi }&{0}&{\bar{\tau }\, \, \Omega Z} \\ {*}&{(1\lambda )Q}&{0}&{\bar{\tau }\, \Psi ^{T} Z} \\ {*} & {*}&{(\frac{1\lambda }{\bar{\tau }} )Z}&{0} \\ {*}&{*} & {*} & {\bar{\tau }Z} \end{array}\right]\\ \Sigma _{12} & =& \left[\begin{array}{cccc} {L_{a} } & {L_{b} } & {H_{a}^{T} }&{H_{b}^{T} } \end{array}\right]\\ \Sigma _{22} & =& \left[\begin{array}{cccc} {\sigma _{1}^{2} I}&{0}&{0}&{0} \\ {0}&{\sigma _{2}^{2} I}&{0} & {0} \\ {0}&{0}&{\frac{1}{\sigma _{1}^{2} } I}&{0} \\ {0} & {0}&{0} & {\frac{1}{\sigma _{2}^{2} } I} \end{array}\right]\\ \phi _{n} & =& \Omega ^{T} \bar{P}+\bar{P}\, \Omega +\bar{\tau }X+Y+Y^{T} +Q \end{eqnarray*} $ 
with
$ \begin{eqnarray*} \Omega & =& \left[\begin{array}{ccc} {0} & {I}&{0} \\ {0}&{0}&{I} \\ {A_{3} }&{A_{2} }&{(I\otimes A)+A_{1} } \end{array}\right]\\[4mm] \Psi & =& \left[\begin{array}{ccc} {0}&{0}&{0} \\ {0}&{0}&{0} \\ {L_{s} \otimes B_{3} }&{L_{s} \otimes B_{2} }&{L_{s} \otimes B_{1} } \end{array}\right]\\[4mm] L_{a} & =& \left[ \begin{array}{c} {\bar{P}\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{a} } \end{array} \right]} \\ {0} \\ {0} \\ {\bar{\tau }Z\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{a} } \end{array} \right]} \end{array} \right], \, \, \, \, L_{b} =\left[ \begin{array}{c} {\bar{P}\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{b} } \end{array} \right]} \\ {0} \\ {0} \\ {\bar{\tau }Z\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{b} } \end{array} \right]} \end{array} \right] \\[4mm] H_{b}^{T}& =& \left[\begin{array}{c} {\left[\begin{array}{c} {((M\otimes E_{b} )K'_{II} (I\otimes C))^{T} } \\[2mm] {((M\otimes E_{b} )K'_{I} (I\otimes C))^{T} } \\[2mm] {((M\otimes E_{b} )K'(I\otimes C))^{T} } \end{array}\right]} \\[4mm] {\left[\begin{array}{c} {(L_{s} \otimes E_{b} K_{II} C)^{T} } \\[2mm] {(L_{s} \otimes E_{b} K_{I} C)^{T} } \\[2mm] {(L_{s} \otimes E_{b} KC)^{T} } \end{array}\right]} \\[2mm] {0} \\ {0} \end{array}\right] \\[4mm] H_{a} & =& \left[\begin{array}{cccc} {\left[\begin{array}{ccc} {0}&{0}&{I\otimes E_{a} } \end{array}\right]}&{0}&{0}&{0} \end{array}\right]. \end{eqnarray*} $ 
Proof: Using LyapunovKrasovskii functional in (15) and (16), the delaydependent sufficient conditions for the stability of the delayed multiagent system (14) are presented. For sake of brevity, the detailed proof of this theorem is given in Appendix.
Remark 3: Theorem 1 provides robust stability analysis conditions for uncertain multiagent system (14). Feasibility of the set of LMI conditions (17) and (18) guarantees the robust leaderfollowing output regulation of the uncertain timedelay multiagent system (14) with known statespace matrices. The significant advantage of this theorem is to give a set of stability analysis LMI conditions that are valid for both leaders (4) and (5).
Remark 4: It is worthwhile mentioning that the proposed leaderfollowing output regulation protocol can be further used for low frequency sinusoidal leaders or the leaders that can be approximated by a combination of a number of sinusoidal signals. The tracking quality for high frequency sinusoidal leaders can be improved by incorporating a design criterion for widening the bandwidth of the closedloop system.
Remark 5: In nontimedelay case, all the rows and columns containing
In this section, using the analysis conditions provided in the previous section as well as the preliminaries given in the Section Ⅱ, design conditions for the proposed leaderfollowing output regulation control are given. These conditions are presented by the following theorem.
Theorem 2: Consider the multiagent system (1) and the control law (6) with timevarying transmission delay
$ \begin{equation} \left[\begin{array}{cc} {\Theta _{11} }&{\Theta _{12} } \\ {*} & {\Theta _{22} } \end{array}\right]<0 \end{equation} $  (19) 
$ \begin{equation} \left[\begin{array}{cc} {T_{1} }&{N} \\ {N^{T} }&{2LR} \end{array}\right]>0 \end{equation} $  (20) 
in which
$ \begin{eqnarray*} \Theta _{11} & =& \\[2mm] && \left[ \begin{array}{cccc} {\phi _{11} } & {N+\phi _{12} } & {0} & {\bar{\tau }\bar{L}\left[\begin{array}{ccc} {0} & {I} & {0} \\ {0} & {0} & {I} \\ {0} & {0} & {(I\otimes A)} \end{array}\right]^{T} + \bar{\tau }\phi _{13}^{T} } \\ {*} & {(1\lambda )T_{2} } & {0} & {\bar{\tau }\phi _{12}^{T} } \\ {*} & {*} & {(\frac{1\lambda }{\bar{\tau }} )R} & {0} \\ {*} & {*} & {*} & {\bar{\tau }R} \end{array} \right]\\[2mm] \phi _{11} & =& \bar{L}\left[\begin{array}{ccc} {0}&{I}&{0} \\ {0}&{0}&{I} \\ {0}&{0}&{(I\otimes A)} \end{array}\right]^{T} +\left[\begin{array}{ccc} {0}&{I}&{0} \\ {0}&{0}&{I} \\ {0} & {0}&{(I\otimes A)} \end{array}\right]\bar{L}\\[2mm] && +\phi _{13} +\phi_{13}^{T} +\bar{\tau }T_{1} +N+N^{T}+T_{2} \\ \phi _{12} & =& \left[ \begin{array}{ccc} {0} & {0} & {0} \\ {0} & {0} & {0} \\ {L_{s} \otimes BK_{II} CL_{1} } & {L_{s} \otimes BK_{I} CL_{1} } & {L_{s} \otimes BKCL_{1} } \end{array} \right] E\\[2mm] \phi _{13} & =& \left[ \begin{array}{ccc} {0} & {0}&{0} \\ {0}&{0} & {0} \\ {A'_{3} }&{A'_{2} }&{A'_{1} } \end{array} \right] E\\[2mm] \Theta _{12} & =& \left[ \begin{array}{cccc} {\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{a} } \end{array} \right]} & {\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{b} } \end{array} \right]} & {\bar{L}\left[ \begin{array}{c} {0} \\ {0} \\ {(I\otimes E_{a} )^{T} } \end{array} \right]} & {\phi _{14} } \\ {0} & {0} & {0} & {\phi _{24} } \\ {0} & {0} & {0} & {0} \\ {\bar{\tau }\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{a} } \end{array} \right]} & {\bar{\tau }\left[ \begin{array}{c} {0} \\ {0} \\ {I\otimes D_{b} } \end{array} \right]} & {0} & {0} \end{array} \right]\\[2mm] &&\Theta _{22} =\left[\begin{array}{cccc} {\sigma _{1}^{2} I}&{0}&{0}&{0} \\ {0}&{\sigma _{2}^{2} I}&{0} & {0} \\ {0}&{0}&{\frac{1}{\sigma _{1}^{2} } I}&{0} \\ {0} & {0}&{0} & {\frac{1}{\sigma _{2}^{2} } I} \end{array}\right]\\[2mm] &&\phi _{14} =E\left[\begin{array}{c} {\begin{array}{l} {((M\otimes E_{b} )K'_{II} (I\otimes CL_{1} ))^{T} } \\ {((M\otimes E_{b} )K'_{I} (I\otimes CL_{1} ))^{T} } \end{array}} \\ {((M\otimes E_{b} )K'(I\otimes CL_{1} ))^{T} } \end{array}\right]\\[2mm] &&\phi _{24} =E\left[\begin{array}{c} {\begin{array}{l} {(L_{s} \otimes E_{b} K_{II} CL_{1} )^{T} } \\ {(L_{s} \otimes E_{b} K_{I} CL_{1} )^{T} } \end{array}} \\ {(L_{s} \otimes E_{b} KCL_{1} )^{T} } \end{array}\right] \end{eqnarray*} $ 
and
$ \begin{eqnarray*} &&L=\left[ \begin{array}{ccccccccccccc} {\hat{L}} & {0} & {0} \\ {0} & {\hat{L}} & {0} \\ {0} & {0} & {\hat{L}} \end{array} \right], E=\left[ \begin{array}{ccc} {\alpha _{1} I_{N} } & {\varepsilon I} & {\varepsilon I} \\ {\varepsilon I} & {\alpha _{2} I_{N} } & {\varepsilon I} \\ {\varepsilon I} & {\varepsilon I} & {\alpha _{3} I_{N} } \end{array} \right], \hat{L}=I_{N} \otimes L_{1}\\ &&\begin{array}{l} {K'={\text{diag}}\{K'_{1} , \ldots , K'_{N}\}, \, \, K'_{I} ={\text{diag}}\{K'_{I1} , \ldots , K_{IN}\}\, \, } \\ K'_{II} ={\text{diag}}\{K'_{II1} , \ldots , K'_{IIN}\} \end{array}\\ &&A'_{1}=(M\otimes B)K'(I\otimes CL_{1} ), A'_{2} =(M\otimes B)K'_{I} (I\otimes CL_{1} ) \\ &&A'_{3} =(M\otimes B)K'_{II} (I\otimes CL_{1}). \end{eqnarray*} $ 
Proof: See Appendix.
Remark 6: As we mentioned earlier, Theorem 1 gives a general set of matrix inequality conditions for stability analysis and design of leaderfollowing output regulation of multiagent system (1) with regulation protocol (8). Solving the presented conditions in (19) and (20), we obtain the appropriate solution for the regulation protocol (8) that works for each of the leaders (4) and (5). In a special case that the stationary leader is merely considered, one can set
$ \begin{eqnarray} u_{i} (t)& = &K\sum _{j=1}^{N}a_{ij} (y_{i} (t\tau (t))y_{j} (t\tau (t))) \nonumber\\ && +K_{I} (\sum _{j=1}^{N}a_{ij} \int _{0}^{t\tau (t)}(y_{i} (\alpha )y_{j} (\alpha ))d\alpha)\nonumber\\ && +m_{i} K'_{i} \, (y_{i} (t)y_{0} (t)) \nonumber\\ && +m_{i} K'_{Ii} \, \int _{0}^{t}(y_{i} (\alpha )y_{0} (\alpha ))d\alpha. \end{eqnarray} $  (21) 
Then, the following corollary is obtained.
Corollary 1: Under Assumptions 13, the leaderfollowing output regulation is asymptotically achieved for the multiagent system (1) with
$ \begin{equation} \left[\begin{array}{cc} {\Theta _{11} }&{\Theta _{12} } \\ {*} & {\Theta _{22} } \end{array}\right]<0 \end{equation} $  (22) 
$ \begin{equation} \label{GrindEQ__23_} \left[\begin{array}{cc} {M}&{N} \\ {N^{T} }&{2LR} \end{array}\right]>0 \end{equation} $  (23) 
in which
$ \begin{eqnarray*} &&\Theta _{11} = \\ &&\left[ \begin{array}{cccc} {\phi }&{N+\Omega _{1} }&{0}&{\bar{\tau }\bar{L}\left[ \begin{array}{cc} {0}&{I} \\ {0}&{(I\otimes A)} \end{array} \right]^{T} +\bar{\tau }\, \Omega _{2}^{T} } \\ {*}&{(1\lambda )Q}&{0}&{\bar{\tau }\, \Omega _{1}^{T} } \\ {*}&{*}&{(\frac{1\lambda }{\bar{\tau }} )R}&{0} \\ {*} & {*}&{*}&{\bar{\tau }R} \end{array} \right]\\ &&\Theta _{12} =\left[ \begin{array}{cccc} {\left[ \begin{array}{c} {0} \\ {I\otimes D_{a} } \end{array} \right]} & {\left[ \begin{array}{c} {0} \\ {I\otimes D_{b} } \end{array} \right]} & {\bar{L}\left[ \begin{array}{c} {0} \\ {(I\otimes E_{a} )^{T} } \end{array} \right]} & {\Omega _{3} } \\ {0}&{0}&{0} & {\Omega _{4} } \\ {0}&{0}&{0} & {0} \\ {\bar{\tau }\left[ \begin{array}{c} {0} \\ {I\otimes D_{a} } \end{array} \right]}&{\bar{\tau }\left[ \begin{array}{c} {0} \\ {I\otimes D_{b} } \end{array} \right]}&{0} & {0} \end{array} \right]\\ &&\Theta _{22} ={\text{diag}}\{\sigma _{1}^{2} I, \, \, \sigma _{2}^{2} I, \, \, \frac{1}{\sigma _{1}^{2} } I, \, \, \frac{1}{\sigma _{2}^{2} } I\}\\ &&~~~\phi =\bar{L}\left[\begin{array}{cc} {0}&{I} \\ {0}&{(I\otimes A)} \end{array}\right]^{T} +\left[\begin{array}{cc} {0}&{I} \\ {0}&{(I\otimes A)} \end{array}\right]\bar{L}+\Omega _{2} +\Omega _{2}^{T} \\ &&~~~~~+\bar{\tau }T_{1} +N+N^{T} +T_{2} \end{eqnarray*} $ 
and
$ \begin{eqnarray*} &&\Omega _{1} =\left[\begin{array}{cc} {0}&{0} \\ {(L_{s} \otimes BK_{I} CL_{1} )}&{(L_{s} \otimes BKCL_{1} )} \end{array}\right]E\\ &&\Omega _{2} =\left[\begin{array}{cc} {0}&{0} \\ {(M\otimes BK'_{I} CL_{1} )}&{(M\otimes BK'CL_{1} )} \end{array}\right]E\\ &&\Omega _{3} =E\left[\begin{array}{c} {(M\otimes E_{b} K'_{I} CL_{1} )^{T} } \\ {(M\otimes E_{b} K'CL_{1} )^{T} } \end{array}\right]\\ &&\Omega _{4} =E\left[\begin{array}{c} {(L_{s} \otimes E_{b} K_{I} CL_{1} )^{T} } \\ {(L_{s} \otimes E_{b} KCL_{1} )^{T} } \end{array}\right]\\ &&L=\left[\begin{array}{cc} {\hat{L}}&{0} \\ {0} & {\hat{L}} \end{array}\right], \, \, E=\left[\begin{array}{cc} {I_{N} } & {\varepsilon I_{N} } \\ {\varepsilon I_{N} }&{I_{N} } \end{array}\right], \, \, \, \, \hat{L}=I_{N} \otimes L_{1}. \end{eqnarray*} $ 
Proof: The proof is omitted since it can be directly established using Theorem 2.
Remark 7: As seen in the leaderfollowing conditions presented in Theorem 2 and Corollary 1, the matrix inequalities (19) and (22) are bilinear (BMI). Although BMIs are categorized in NPhard problems, there exist practically effective algorithms for BMI solutions [22], [23]. Furthermore, the PENBMI solver in MATLAB environment can be used to solve BMIs [24]. It is noteworthy that for the firstorder systems as a special case, the obtained BMI condition in (19) and (22) turn into LMI that are much easier to deal with as compared to BMIs. The details are given as follows.
A. Special Case: FirstOrder SystemsConsider the following firstorder system:
$ \begin{eqnarray} \dot{x}_{i} (t)& = &(a+\Delta a(t))x_{i} (t)+(b+\Delta b)u_{i} (t)\, \, \, \, \, \, i=1, \ldots, N \nonumber\\ y_{i} (t)& = &cx_{i} (t) \end{eqnarray} $  (24) 
where
In this section, two examples are given to illustrate the effectiveness of the theoretical results.
A. Example 1In this example, we consider two multiagent systems Ⅰ and Ⅱ with six uncertain firstorder agents as follows:
$ \begin{eqnarray} \dot{x}_{i} (t)& = &(a+D_{a} F_{a} (t)E_{a} )x_{i} (t) \nonumber\\ && +(b+D_{b} F_{b} (t)E_{b} )u_{i} (t)\nonumber\\ y_{i} (t)& = &cx_{i}(t) (i=1, 2, 3) \end{eqnarray} $  (25) 
The agents in system Ⅰ are stable in which
$ \begin{eqnarray*} &&K=0.8, K_{I} = 0.93, K_{II} = 0.29\\[1mm] &&K'={\text{ diag}}\{1.004, \, \, 0, \, \, 1.004, \, \, 0, \, \, 1.004, \, \, 0\}\\[1mm] &&K'_{I}={\text {diag}}\{1.6, \, \, 0, \, \, 1.6, \, \, 0, \, \, 1.6, \, \, 0\}\\[1mm] &&K'_{II} ={\text {diag}}\{0.59, \, \, 0, \, \, 0.59, \, \, 0, \, \, 0.59, \, \, 0\} \end{eqnarray*} $ 
and the multiagent system Ⅱ as
$ \begin{eqnarray*} &&K = 0.8, K_{I} = 0.93, K_{II} = 0.29\\ &&K'={\text {diag}}\{4.3, \, \, 0, \, \, 4.3, \, \, 0, \, \, 4.3, \, \, 0\}\nonumber\\ &&K'_{I}={\text {diag}}\{4.5, \, \, 0, \, \, 4.5, \, \, 0, \, \, 4.5, \, \, 0\}\nonumber\\ &&K'_{II} ={\text {diag}}\{1.7, \, \, 0, \, \, 1.7, \, \, 0, \, \, 1.7, \, \, 0\}. \end{eqnarray*} $ 
Two set points ramp and sinusoidal are applied to the agent 1 and 2 as the state of the leader. Figs. 25 display the simulation results of the multiagent systems Ⅰ and Ⅱ. The initial conditions for the agents 1, 2,
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Fig. 2 Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅰ. 
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Fig. 3 Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅰ. 
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Fig. 4 Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅱ. 
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Fig. 5 Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅱ. 
As seen in Figs. 2 and 3, a leaderfollowing output regulation is achieved for multiagent system Ⅰ and the agents' outputs follow a ramp path as well as sinusoidal path using the regulation protocol (6). Moreover, the unstable multiagent system Ⅱ has been stabilized and the leaderfollowing output regulation is obtained as seen in Figs. 4 and 5. These simulation results show that the proposed method can be used for the leaderfollowing output regulation problems with hyperbolic leaders as well. Furthermore, as expected, the presented method provides leaderfollowing output regulation for multiagent systems in which the followers do not have necessarily the same dynamics as their leader.
B. Example 2In this example, we address the problem of the robust stability analysis of the leaderfollowing output regulation for multiagent systems investigated in Section Ⅲ. To this aim, we consider equation (1) with the following statespace matrices of an unstable nominal multiagent system:
$ \begin{equation} A=\left[\begin{array}{cc} {1}&{1} \\ {10}&{0} \end{array}\right], \, \, \, \, B=\left[\begin{array}{cc} {2}&{0} \\ {1}&{4} \end{array}\right], \, \, \, \, C=\left[\begin{array}{cc} {1}&{0} \\ {0}&{1} \end{array}\right]. \end{equation} $  (26) 
The communication topology and the leader adjacency matrices are represented as
$ \begin{eqnarray} K_{II} & = &\left[ \begin{array}{cc} {2} & {0.3} \\ {0.65} & {0.47} \end{array} \right], \ K'_{i} =\left[ \begin{array}{cc} {52.53} & {5.99} \\ {23.16} & {8.7} \end{array} \right] \nonumber\\ K'_{Ii} & = &\left[ \begin{array}{cc} {71.52} & {9.07} \\ {25.05} & {13.51} \end{array} \right], K'_{IIi} =\left[ \begin{array}{cc} {17.21} & {2.3} \\ {6.17} & {2.99} \end{array} \right] \nonumber\\ K& = &\left[ \begin{array}{cc} {6.4} & {0.75} \\ {3} & {1.4} \end{array} \right], K_{I} =\left[ \begin{array}{cc} {8.8} & {1.2} \\ {2.7} & {2.1} \end{array} \right], i=1, 2, 3. \end{eqnarray} $  (27) 
Now, Theorem 1 can be applied for the stability analysis of the closedloop system in presence of delay. For this purpose, we use the LMI Toolbox in MATLAB and solve the LMI conditions (17) and (18) with
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Fig. 6 First controlled outputs of the follower agents (solid) and output of the leader (dashed). 
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Fig. 7 Second controlled outputs of the follower agents (solid) and output of the leader (dashed). 
Now, we perform some simulations in order to examine the robust stability of the closedloop system in the presence of system parameter variations. Since the stability of the closedloop system is affected by the uncertainty in timedelay, gain and poles of the openloop system, we investigate the performance of the closedloop system in the presence of these uncertainties. To this aim, we set
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Fig. 8 Stability region in presence of 
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Fig. 9 Stability region in presence of 
The problem of leaderfollowing output regulation analysis and design of uncertain general linear multiagent systems with transmission delay has been presented in this paper. The proposed method can be used for both stable and unstable follower agents under a directed graph. Many of the leader outputs can be approximated by the stationary and ramp signals. Therefore, two stationary and ramptype dynamic leaders have been investigated in this paper that cover adequate variety of the leaders. To this aim, we proposed a new regulation protocol for the closedloop system. The analysis conditions have been presented in terms of certain LMIs in which the provided results for design purposes are bilinear. It was shown that for firstorder systems as special case, the presented results are turned into LMI. Finally, we presented two analysis and design examples for the leaderfollowing output regulation of timedelay uncertain multiagent systems. We showed that our proposed method effectively meets the quality requirements of the leaderfollowing output regulation for both stationary and dynamic leaders. Moreover, we showed that this method can be further used in presence of lowfrequency sinusoidal leaders. To achieve superior leaderfollowing output regulation for highfrequency sinusoidal leaders, the bandwidth of the closedloop system should be widened. To this aim, an appropriate performance objective is required to be added to the stability criterion that can be considered in the future works.
APPENDIXProof of Theorem 1: Differentiating
$ \begin{eqnarray} \dot{V}_{1} & =& 2{\dot{\bar{x}}}^{T} (t)\bar{P}\bar{x}(t)=\bar{x}^{T} (\bar{A}^{T} \bar{P}+\bar{P}\bar{A})\bar{x} \nonumber\\ && +2\bar{x}^{T} (t)\bar{P}\bar{B}\bar{x}(t\tau (t)) \nonumber\\ & \le& \bar{x}^{T} (t)(\bar{A}^{T} \bar{P}+\bar{P}\bar{A})\bar{x}(t)+\bar{\tau }\bar{x}(t)X\bar{x}(t) \nonumber\\ && +\bar{x}^{T} (t)(Y+Y^{T} )\bar{x}(t) \nonumber\\ && 2\bar{x}^{T} (t)(Y\bar{P}\bar{B})\bar{x}(t\tau (t))\nonumber\\ && +\int _{t\tau (t)}^{t}{\dot{\bar{x}}}^{T} \, (\alpha )Z{\dot{\bar{x}}}\, (\alpha) d\alpha \end{eqnarray} $  (28) 
with
$ \begin{eqnarray} \left[\begin{array}{cc} {X}&{Y} \\ {Y^{T} }&{Z} \end{array}\right]>0 \end{eqnarray} $  (29) 
where
$ \begin{array}{l} {\bar{A}=} \\ {\left[ \begin{array}{cc} {0}&{I} \\ {(M\otimes (A_{2} +A_{2\Delta } ))}&{(I\otimes (A + \Delta A)) + (M\otimes (A_{1} + A_{1\Delta } ))} \end{array} \right]} \end{array} $ 
and
$ \bar{B}=\left[\begin{array}{cc} {0}&{0} \\ {L_{s} \otimes (B_{2} +B_{2\Delta } )}&{L_{s} \otimes (B_{1} +B_{1\Delta } )} \end{array}\right]. $ 
Also, the timederivative of
$ \begin{eqnarray} \dot{V}_{2} & \le & \bar{\tau }\bar{x}^{T} (t)\bar{A}^{T} Z\bar{A}\bar{x}(t) \nonumber\\ && +\bar{\tau }\bar{x}^{T} (t\tau (t))\bar{B}^{T} Z\bar{B}\bar{x}(t\tau (t)) \nonumber\\ && +2\bar{\tau }\bar{x}(t)\bar{A}Z\bar{B}\bar{x}(t\tau (t)) \nonumber\\ & & \int _{t\tau (t)}^{t}{\dot{\bar{x}}}^{T} (\alpha )Z{\dot{\bar{x}}}(\alpha )d\alpha \nonumber\\ & & (\frac{1\lambda }{\bar{\tau }} )(\int _{t\tau (t)}^{t}{\dot{\bar{x}}}^{T} (\alpha )d\alpha)Z(\int _{t\tau (t)}^{t}{\dot{\bar{x}}}(\alpha )d\alpha) \end{eqnarray} $  (30) 
$ \begin{eqnarray} \dot{V}_{3} \le \bar{x}^{T} (t)Q\bar{x}(t)(1\lambda )\bar{x}^{T} (t\tau (t))Q\bar{x}(t\tau (t)) \end{eqnarray} $  (31) 
Since
$ \begin{eqnarray} \dot{V}&\le& \bar{x}^{T} (t)(\bar{A}^{T} \bar{P}+\bar{P}\bar{A})\bar{x}(t)+\bar{\tau }\bar{x}^{T} (t)X\bar{x}(t) \nonumber\\ && +\bar{x}^{T} (t)(Y+Y^{T} )\bar{x}(t) \nonumber\\ && 2\bar{x}^{T} (t)(Y\bar{P}\bar{B})\bar{x}(t\tau (t)) \nonumber\\ && +\bar{\tau }\bar{x}^{T} (t)\bar{A}^{T} Z\bar{A}\bar{x}(t) \nonumber\\ && +\bar{\tau }\bar{x}^{T} (t\tau (t))\bar{B}^{T} Z\bar{B}\bar{x}(t\tau (t))\nonumber\\ && +2\bar{\tau }\bar{x}^{T} (t)\bar{A}^{T} Z\bar{B}\bar{x}(t\tau (t)) \nonumber\\ && +\bar{\tau }\bar{x}^{T} (t)X\bar{x}(t)\, +\bar{x}^{T} (t)Q\bar{x}(t) \nonumber\\ && (1\lambda )\bar{x}^{T} (t\tau (t))Q\bar{x}(t\tau (t)) \nonumber\\ && (\frac{1\lambda }{\bar{\tau }} )(\int _{t\tau (t)}^{t}{\dot{\bar{x}}}^{T} (\alpha )d\alpha)Z(\int _{t\tau (t)}^{t}{\dot{\bar{x}}}(\alpha )d\alpha ) \nonumber\\ &=&\xi ^{T} \Theta \xi \end{eqnarray} $  (32) 
in which
$ \xi =col\left[\begin{array}{ccc} {\bar{x}(t)} & {\bar{x}(t\tau (t))}&{\int _{t\tau (t)}^{t}{\dot{\bar{x}}} (\alpha )d\alpha } \end{array}\right]. $ 
If the condition
$ \begin{eqnarray} \left[\begin{array}{cc} {\Upsilon _{11} }&{\Upsilon _{12} } \\ {*}&{\Upsilon _{22} } \end{array}\right]<0 \end{eqnarray} $  (33) 
in which
$ \begin{eqnarray*} && \Upsilon _{12} = \\ && \left[ \begin{array}{c} {\bar{\tau }\left[ \begin{array}{ccc} {0}&{I}&{0} \\ {0}&{0}&{I} \\ {(A_{3} + A_{3\Delta } )}&{(A_{2} + A_{2\Delta } )} & {(I\otimes A) + (A_{1} + A_{1\Delta } )} \end{array} \right] ^{T} Z} \\ {} \\ {\begin{array}{l} {\bar{\tau }\, \Omega _{1}^{T} Z} \\ {} \\ {0} \end{array}} \end{array} \right]\\ \end{eqnarray*} $ 
$ \begin{eqnarray*} && \Upsilon _{11} =\left[ \begin{array}{ccc} {\phi }&{Y+\bar{P}\Omega _{1} }&{0} \\ {(Y+\bar{P}\Omega _{1} )^{T} }&{(1\lambda )Q}&{0} \\ {0}&{0}&{(\frac{1\lambda }{\bar{\tau }})Z} \end{array} \right]\\ && \Upsilon _{22} =\bar{\tau }Z\\ && \phi =\left[ \begin{array}{ccc} {0}&{I}&{0} \\ {0}&{0}&{I} \\ {(A_{3} + A_{3\Delta } )}&{(A_{2} + A_{2\Delta } )}&{(I\otimes A) + (A_{1} + A_{1\Delta } )} \end{array} \right]^{T} \bar{P} \\ && +\bar{P}\left[ \begin{array}{ccc} {0}&{I}&{0} \\ {0}&{0}&{I} \\ {(A_{3} + A_{3\Delta } )}&{(A_{2} + A_{2\Delta } )}&{(I\otimes A) + (A_{1} + A_{1\Delta } )} \end{array} \right]\\ && +\bar{\tau }X+Y+Y^{T} +Q \end{eqnarray*} $ 
where
$ \begin{array}{l} {\Omega _{1} =} \\ {\left[ \begin{array}{ccc} {0}&{0}&{0} \\ {0}&{0}&{0} \\ {L_{s} \otimes (B_{3} + B_{3\Delta } )}&{L_{s} \otimes (B_{2} + B_{2\Delta } )}&{L_{s} \otimes (B_{1} + B_{1\Delta } )} \end{array} \right].} \end{array} $ 
Partitioning the nominal and uncertain parts in (33) and considering the definitions in (2), we have
$ \begin{eqnarray} \Pi_{n} & +& L_{a} (I\otimes F_{a} (t))H_{a} +H_{a}^{T} (I\otimes F_{a} (t))^{T} L_{a}^{T} \nonumber\\ & +& L_{b} (I\otimes F_{b} (t))H_{b} +H_{b}^{T} (I\otimes F_{b} (t))^{T} L_{b}^{T} <0 \end{eqnarray} $  (34) 
where
$ \begin{eqnarray*} && H_{a} =\left[\begin{array}{cccc} {\left[\begin{array}{ccc} {0}&{0}&{I\otimes E_{a} } \end{array}\right]}&{0}&{0}&{0} \end{array}\right]\\ && L_{a} =\left[\begin{array}{c} {\bar{P}\left[\begin{array}{c} {0} \\ {0} \\ {I\otimes D_{a} } \end{array}\right]} \\ {0} \\ {0} \\ {\bar{\tau }Z\left[\begin{array}{c} {0} \\ {0} \\ {I\otimes D_{a} } \end{array}\right]} \end{array}\right]\, \, \, \, L_{b} =\left[\begin{array}{c} {\bar{P}\left[\begin{array}{c} {0} \\ {0} \\ {I\otimes D_{b} } \end{array}\right]} \\ {0} \\ {0} \\ {\bar{\tau }Z\left[\begin{array}{c} {0} \\ {0} \\ {I\otimes D_{b} } \end{array}\right]} \end{array}\right]\\ && H_{b}^{T} =\left[\begin{array}{c} {\left[\begin{array}{c} {((M\otimes E_{b} )K'_{II} (I\otimes C))^{T} } \\ {((M\otimes E_{b} )K'_{I} (I\otimes C))^{T} } \\ {((M\otimes E_{b} )K'(I\otimes C))^{T} } \end{array}\right]} \\ {\left[\begin{array}{c} {(L_{s} \otimes E_{b} K_{II} C)^{T} } \\ {(L_{s} \otimes E_{b} K_{I} C)^{T} } \\ {(L_{s} \otimes E_{b} KC)^{T} } \end{array}\right]} \\ {0} \\ {0} \end{array}\right]\\ && \Pi _{n} =\left[\begin{array}{cccc} {\phi _{n} } & {Y+\bar{P}\, \, \Psi }&{0}&{\bar{\tau }\, \, \Omega Z} \\ {*} & {(1\lambda )Q}&{0}&{\bar{\tau }\, \Psi ^{T} Z} \\ {*}&{*} & {(\frac{1\lambda }{\bar{\tau }} )Z}&{0} \\ {*}&{*}&{*} & {\bar{\tau }Z} \end{array}\right]. \end{eqnarray*} $ 
Since the conditions in (3) are satisfied, it can be easily shown that
$ \begin{eqnarray} (I\otimes F_{a} (t))^{T} (I\otimes F_{a} (t))\le I \nonumber\\ (I\otimes F_{b} (t))^{T} (I\otimes F_{b} (t))\le I, \, \, \, \, \, \, \, \forall t. \end{eqnarray} $  (35) 
Therefore, using Lemma 1, the inequality (34) is written as
$ \begin{eqnarray} \Pi _{n} &&+\frac{1}{\sigma _{1}^{2} } L_{a} L_{a}^{T} + \sigma _{1}^{2} H_{a}^{T} H_{a} +\frac{1}{\sigma _{2}^{2} } L_{b} L_{b}^{T} \nonumber \\ &&+~\sigma _{2}^{2} H_{b}^{T} H_{b} <0. \end{eqnarray} $  (36) 
Applying Schur complement and considering (29), the matrix inequalities (17) and (18) are obtained.
Proof of Theorem 2: Pre and post multiplying the matrix inequality (17) by diag
in which
$ \begin{eqnarray*} && L=\left[\begin{array}{ccc} {\hat{L}}&{0}&{0} \\ {0}&{\hat{L}}&{0} \\ {0}&{0}&{\hat{L}} \end{array}\right], \, \, E=\left[\begin{array}{ccc} {\alpha _{1} I_{N} }&{\varepsilon I}&{\varepsilon I} \\ {\varepsilon I} & {\alpha _{2} I_{N} }&{\varepsilon I} \\ {\varepsilon I} & {\varepsilon I}&{\alpha _{3} I_{N} } \end{array}\right]\\ && \hat{L}=I_{N} \otimes L_{1} \end{eqnarray*} $ 
the matrix inequality (22) is obtained. Furthermore, Pre and post multiplying the matrix inequality (29) by diag
[1]  W. Ren, R. Beard, and E. Atkins, "Information consensus in multivehicle cooperative control, " IEEE Contr. Syst. Mag., vol. 27, no. 2, pp. 7182, Apr. 2007. http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=4140748 
[2]  H. Yang, M. Staroswiecki, B. Jiang, and J. Liu, "Fault tolerant cooperative control for a class of nonlinear multiagent systems, " Syst. Contr. Lett., vol. 60, no. 4, pp. 271277, Apr. 2011. http://www.sciencedirect.com/science/article/pii/S0167691111000259 
[3]  J. Yan, X. Yang, C. L. Chen, X. Y. Luo, and X. P. Guan, "Bilateral teleoperation of multiple agents with formation control, " IEEE/CAA J. of Autom. Sinica, vol. 1, no. 2, pp. 141148, Apr. 2014. http://ieeexplore.ieee.org/document/7004543/ 
[4]  R. OlfatiSaber and R. M. Murray, "Consensus problems in networks of agents with switching topology and timedelays, " IEEE Trans. Autom. Contr., vol. 49, no. 9, pp. 15201533, Sep. 2004. http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=1333204 
[5]  M. S. Mahmoud and G. D. Khan, "LMI consensus condition for discretetime multiagent systems, " IEEE/CAA J. of Autom. Sinica, vol. 5, no. 2, pp. 509513, Mar. 2018. http://ieeexplore.ieee.org/document/7738322/ 
[6]  Z. K. Li, G. H. Wen, Z. S. Duan, and W. Ren, "Designing fully distributed consensus protocols for linear multiagent systems with directed graphs, " IEEE Trans. Autom. Contr., vol. 60, no. 4, pp. 11521157, Apr. 2015. http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=6881684 
[7]  Z. Y. Meng, W. Ren, Y. C. Cao, and Z. You, "Leaderless and leaderfollowing consensus with communication and input delays under a directed network topology, " IEEE Trans. Syst. Man, Cybernet. Part B (Cybernet. ), vol. 41, no. 1, pp. 7588, Feb. 2011. http://ieeexplore.ieee.org/document/5456144 
[8]  L. Ding, Q. L. Han, and G. Guo, "Networkbased leaderfollowing consensus for distributed multiagent systems, " Automatica, vol. 49, no. 7, pp. 22812286, July. 2013. http://www.sciencedirect.com/science/article/pii/S0005109813002331 
[9]  Z. Y. Ye, Y. G. Chen, and H. Zhang, "Leaderfollowing consensus of multiagent systems with timevarying delays via impulsive control, " Math. Probl. Eng., vol. 2014, Article No. 240503, Mar. 2014. https://www.researchgate.net/publication/286361965_LeaderFollowing_Consensus_of_Multiagent_Systems_with_TimeVarying_Delays_via_Impulsive_Control 
[10]  H. Xia, T. Z. Huang, JL. Shao, and J. Y. Yu, "Formation control of secondorder multiagent systems with timevarying delays, " Math. Probl. Eng., vol. 2014, Article No. 764580, Jan. 2014. http://dx.doi.org/10.1155/2014/764580 
[11]  H. W. Liu, H. R. Karimi, S. L. Du, W. G. Xia, and C. Q. Zhong, "Leaderfollowing consensus of discretetime multiagent systems with timevarying delay based on large delay theory, " Inf. Sci., vol. 417, pp. 236246, Nov. 2017. http://www.sciencedirect.com/science/article/pii/S0020025516313299 
[12]  Y. C. Cao, W. Ren, and M. Egerstedt, "Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks, " Automatica, vol. 48, no. 8, pp. 15861597, Aug. 2012. http://dl.acm.org/citation.cfm?id=2343432 
[13]  W. Liu and J. Huang, "Adaptive leaderfollowing consensus for a class of highorder nonlinear multiagent systems with directed switching networks, " Automatica, vol. 79, pp. 8492, May 2017. http://www.sciencedirect.com/science/article/pii/S0005109817300687 
[14]  X. Z. Jin, Z. Zhao, and Y. G. He, "Insensitive leaderfollowing consensus for a class of uncertain multiagent systems against actuator faults, " Neurocomputing, vol. 272, pp. 189196, Jan. 2018. http://www.researchgate.net/publication/318201437_Insensitive_Leaderfollowing_Consensus_for_a_Class_of_Uncertain_Multiagent_Systems_Against_Actuator_Faults 
[15]  L. Yu and J. Wang, "Robust cooperative control for multiagent systems via distributed output regulation, " Systems & Control Letters, vol. 62, no. 11, pp. 10491056, Nov. 2013. http://www.sciencedirect.com/science/article/pii/S0167691113001771 
[16]  Y. M. Yan and J. Huang, "Cooperative output regulation of discretetime linear timedelay multiagent systems, " IET Contr. Theory Appl., vol. 10, no. 16, pp. 20192026, Oct. 2016. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7725848 
[17]  W. F. Hu and L. Liu, "Cooperative output regulation of heterogeneous linear multiagent systems by eventtriggered control, " IEEE Trans. Cybern., vol. 47, no. 1, pp. 105116, Jan. 2017. http://ieeexplore.ieee.org/document/7378921/ 
[18]  Y. M. Yan and J. Huang, "Cooperative robust output regulation problem for discretetime linear timedelay multiagent systems, " Int. J. Robust Nonlin. Contr., vol. 28, no. 3, pp. 10351048, Feb. 2018. 
[19]  A. Shariati and M. Tavakoli, "A descriptor approach to robust leaderfollowing output consensus of uncertain multiagent systems with delay, " IEEE Trans. Autom. Contr., vol. 62, no. 10, pp. 53105317, Oct. 2017. http://ieeexplore.ieee.org/document/7792586/ 
[20]  Y. Y. Wang, L. H. Xie, and C. E. de Souza., "Robust control of a class of uncertain nonlinear systems, " Syst. Contr. Lett., vol. 19, no. 2, pp. 139149, Aug. 1992. http://dl.acm.org/citation.cfm?id=139259 
[21]  L. A. Mozelli and R. M. Palhares, "Less conservative H_{∞} fuzzy control for discretetime takagisugeno systems, " Math. Probl. Eng., vol. 2011, Article No. 361640, 2011. 
[22]  H. D. Tuan, P. Apkarian, and Y. Nakashima, "A new Lagrangian dual global optimization algorithm for solving bilinear matrix inequalities, Int. Robust Nonlin J.". Contr. , vol.10, no.7, pp.561–578, 2000. 
[23]  Y. Y. Cao, J. Lam, and Y. X. Sun, "Static output feedback stabilization: an ILMI approach, " Automatica, vol. 34, no. 12, pp. 16411645, Dec. 1998. http://www.sciencedirect.com/science/article/pii/S0005109898800216 
[24]  M. Kocvara and M. Stingl, TOMLAB/PENBMI solver (MATLAB Toolbox), PENOPT GbR, 2005. 
[25]  Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee, "Delaydependent robust stabilization of uncertain statedelayed systems". Int. Contr J. , vol.74, no.14, pp.1447–1455, 2001. DOI:10.1080/00207170110067116 