In recent years, distributed coordination of multiagent systems (MASs) has received great attention from many researchers due to its broad applications on MASs in many areas including formation control [1], [2], flocking [3], [4], distributed filtering [5], [6], synchronization of coupled chaotic oscillators [7][9]. Consensus is an essential problem of distributed coordination of MASs, which is to make each agent agree on some common values of interest through feedback of local information from neighboring agents.
The theoretical framework for posing and solving the consensus problem for MASs was first introduced in [10][12]. Their work mostly focused on the firstorder and secondorder consensus in MASs. Furthermore, the consensus problem of MASs has obtained a tremendous surge of interest and extensive development. These works can be generally divided into two categories depending on whether the agent models are continuoustime or discretetime. The union of interaction topologies must contain a spanning tree if MASs are expected to achieve consensus asymptotically [13]. A framework of highdimensional state space for the consensus problems of MASs was studied in [14], and then the consensus problems of high order or more general linear MASs models were discussed in [15][17]. The consensus problem of discretetime MASs (DMASs) based on general linear models was investigated in [18], [19]. The leaderfollowing consensus problem of DMASs based on general linear models was studied in [20]. The robust guaranteed cost consensus problem of general linear DMASs models with parameter uncertainties and timevarying delays was investigated in [21].
With the development of the research, the
Motivated by the above, in this paper, the robust
The rest of the paper is organized as follows. The problem formulation is presented in Section 2. In Section 3, the robust
Notations: A matrix or a vector is said to be positive (respectively, nonnegative) if all of its entries are positive (respectively, nonnegative). A square matrix is called Schur stable if all of its eigenvalues lie in the open unit disk. Let
A highorder MAS can be described as a linear system, which has been presented in [15], and thus, consider a highorder identical DMAS consisting of
$ \begin{cases} {{\pmb x}_i}(k + 1) = (A + \Delta A){{\pmb x}_i}(k) \\ \qquad +~ (B + \Delta B){{\pmb u}_i}(k)+ {B_\omega }{{\pmb\omega} _{i, x}}(k)\\[1mm] {{{\pmb z}}_i}(k) = C{{{\pmb x}}_i}(k) \end{cases} $  (1) 
where
$ \begin{align} [\begin{array}{*{20}{c}} {\Delta A}&{\Delta B} \end{array}] = DF[\begin{array}{*{20}{c}} {{E_1}}&{{E_2}} \end{array}] \end{align} $  (2) 
where
$ {F^T}F \le I. $  (3) 
The parameter uncertainties
$ {{\pmb u}_i}(k) = K\sum\limits_{j \in {N_i}} {{a_{ij}}({{\pmb x}_j}(k)  {{\pmb x}_i}(k))} $  (4) 
where
Definition 1: For a given gain matrix
Let
$ \begin{align} \begin{cases} {\pmb{x}}(k + 1) = ({I_N} \otimes (A + \Delta A) \\ \qquad  ~L \otimes (B + \Delta B)K){\pmb{x}}(k) + ({I_N} \otimes {B_{ {\omega }}}){{\pmb{\omega }}_x}(k)\\ {\pmb{z}}(k) = ({I_N} \otimes C){\pmb{x}}(k). \end{cases} \end{align} $  (5) 
The suboptimal robust
1) with
2) with
In order to analyze the robust
Lemma 1 [29] : Let
Moreover, let
$ U = \left[{\begin{array}{*{20}{c}} {\dfrac{1}{{\sqrt N }}}&{\dfrac{{{\bf{1}}_{N 1}^T}}{{\sqrt N }}}\\[3mm] {\dfrac{{{\bf{1}}_{N 1}^T}}{{\sqrt N }}}&{\bar U} \end{array}} \right] $ 
such that
Theorem 1: For a given
$ \begin{align} \begin{cases} {{\tilde{\pmb x}}_i}(k + 1) = (A + \Delta A  {\lambda _i}(B + \Delta B)K){\tilde{\pmb x}_i}(k) \\ \qquad +~ {B_{\omega}}{{\tilde{\pmb\omega}}_{i, { x}}}(k)\\[0.5mm] {{\tilde {\pmb z}}_i}(k) = C{{{\tilde{\pmb x}}}_i}(k), ~~~i = 1, 2, \ldots, N. \end{cases} \end{align} $  (6) 
Proof: Let
$ U = \left[{\begin{array}{*{20}{c}} {\dfrac{1}{{\sqrt N }}}&{\dfrac{{{\bf{1}}_{N 1}^T}}{{\sqrt N }}}\\[3mm] {\dfrac{{{\bf{1}}_{N 1}^T}}{{\sqrt N }}}&{\bar U} \end{array}} \right] $ 
such that
$ \begin{align} {\tilde{\pmb x}}(k) & = \left( {{U^{T}} \otimes {I_d}} \right){\pmb x}(k) = {\left[{\tilde{\pmb x}}_c^T(k), {\tilde{\pmb x}}_r^T(k)\right]^T}\nonumber\\ &= {\left[{\tilde{\pmb x}}_1^T(k), {\tilde{\pmb x}}_2^T(k), \ldots , {\tilde{\pmb x}}_N^T(k)\right]^T}. \end{align} $  (7) 
Then, system (5) can be rewritten in terms of
$ \begin{align} \begin{cases} {\tilde {\pmb x}}(k + 1) = ({I_N} \otimes (A + \Delta A) \\ \qquad~\Lambda \otimes (B + \Delta B)K){\tilde{\pmb x}}(k) + ({U^T} \otimes {B_{ {\omega }}}){{\pmb{\omega }}_x}(k)\\[1mm] {{\pmb z}}(k) = (U \otimes C){\tilde{\pmb x}}(k). \end{cases} \end{align} $  (8) 
Moreover, reformulate the disturbance variable
$ \begin{align} {\tilde{\pmb \omega }_x}(k) &= \left( {{U^{T}}} \otimes {I_{{m_2}}} \right){{\pmb{\omega }}_x}(k)\nonumber\\ & = {\left[\tilde \omega _{1, x}^T(k), \tilde \omega _{2, x}^T(k), \ldots, \tilde \omega _{N, x}^T(k)\right]^T} \end{align} $  (9) 
$ \tilde {\pmb z}(k) = \left( {{U^{T}}} \otimes {I_l} \right){\pmb z}(k) = {\left[\tilde z_1^T(k), \tilde z_2^T(k), \ldots, \tilde z_N^T(k)\right]^T}. $  (10) 
Subsequently, substituting (9) and (10) into (8) gives
$ \begin{cases} {\tilde{\pmb x}}(k + 1) = ({I_N} \otimes (A + \Delta A) \\ \qquad ~\Lambda \otimes (B + \Delta B)K)\tilde x(k) + ({I_N} \otimes {B_{\pmb{\omega }}}){{{\tilde{\pmb \omega }}}_x}(k)\\[1mm] {\tilde{\pmb z}}(k) = ({I_N} \otimes C){\tilde{\pmb x}}(k). \end{cases} $  (11) 
Note that (11) is composed of
$ \begin{align} {T_{{{\tilde \omega \tilde z}}}} &= {\rm diag}({T_{{{{ \tilde{ \omega }}}_1}{{{\tilde{ z}}}_1}}}, {T_{{{{\tilde { \omega }}}_2}{{{\tilde{ z}}}_2}}}, \ldots, {T_{{{{ \tilde{ \omega }}}_N}{{{\tilde{ z}}}_N}}})\nonumber\\ &= \left( {{U^{T}}} \otimes {I_l} \right){T_{{ {\omega z}}}}(U \otimes {I_{{m_2}}}) \end{align} $  (12) 
which implies that
$ {\left\ {{T_{{ {\tilde \omega \tilde z}}}}} \right\_\infty } = \mathop {\max }\limits_{i = 2, 3, \ldots, N} {\left\ {{T_{{{{\tilde{ \omega }}}_i}{{{\tilde{ z}}}_i}}}} \right\_\infty } = {\left\ {{T_{{ {\omega z}}}}} \right\_\infty }. $  (13) 
In addition, it is worth mentioning that,
$ \begin{align*} {\tilde{\pmb x}}(k)& = {\left[{\tilde{\pmb x}}_c^T(k), {\tilde{\pmb x}}_r^T(k)\right]^T} = {\left[{\tilde{\pmb x}}_c^T(k), {\tilde{\pmb x}}_{r, 2}^T(k), \ldots, {\tilde{\pmb x}}_{r, N}^T(k)\right]^T}\\ {{\tilde {\pmb \omega }}_x}(k)& = {\left[\tilde {\pmb \omega } _{c, x}^T(k), \tilde {\pmb \omega } _{r, x}^T(k)\right]^T}\\ & = {\left[\tilde {\pmb \omega } _{c, x}^T(k), \tilde {\pmb \omega } _{r, 2x}^T(k), \ldots, \tilde {\pmb \omega } _{r, Nx}^T(k)\right]^T} \\ {\tilde {\pmb z}}(k) &= {\left[{\tilde {\pmb z}}_c^T(k), {\tilde {\pmb z}}_r^T(k)\right]^T} = {\left[{\tilde {\pmb z}}_c^T(k), {\tilde {\pmb z}}_{r, 2}^T(k), \ldots, {\tilde {\pmb z}}_{r, N}^T(k)\right]^T}. \end{align*} $ 
By Lemma 1, the discretetime system (11) also can be rewritten as the following
$ \begin{cases} {{\tilde {\pmb x}}_c}(k + 1) = (A + \Delta A){{\tilde {\pmb x}}_c}(k) + {B_\omega }{{{\tilde{\pmb \omega }} }_{c{\rm{, }}{ x}}}(k)\\ {{{\tilde{\pmb z}}}_c}(k) = C{{{\tilde{\pmb x}}}_c}(k) \end{cases} $  (14) 
$ \begin{cases} {{\tilde {\pmb x}}_{r, i}}(k + 1) = (A + \Delta A  {\lambda _i}(B + \Delta B)K){{{\tilde {\pmb x}}}_{r, i}}(k) \\ \qquad + ~{B_\omega }{{ {\tilde {\pmb \omega}}}_{i, x}}(k)\\[1mm] {{{\tilde{\pmb z}}}_{r, i}}(k) = C{{{{\tilde {\pmb x}}}}_{r, i}}(k), ~~~{i = 2, 3, \ldots, N.} \end{cases} $  (15) 
Obviously, if subsystems (15) are asymptotically stable, then DMASs (5) reach consensus. Subsystem (14) determines the final consensus value of DMASs (5), and the details of it will be discussed below.
Remark 1: The robust
Lemma 2: Given the pair
$ \begin{align} &{\bar A_{{\lambda _i}}^{T}}P{{\bar A}_{{\lambda _i}}}  P + P{C^{\mathop{T}\nolimits} }CP + {\gamma ^{  2}}{B_\omega }B_{ {\omega }^{T}} < 0\notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad i = 1, 2, \ldots, N \end{align} $  (16) 
admits a symmetric positive definite solution
Proof: Given
$ {\bar A_{{\lambda _i}}^{T}}P{\bar A_{{\lambda _i}}  P \le  {\gamma ^{  2}}{B_\omega }B_{\pmb{\omega }}^{T}} < 0, ~~~i = 1, 2, \ldots, N $  (17) 
(complying with the previous assumptions) that
$ {H_{{\lambda _i}}}(s) = C{(sI  {\bar A_{{\lambda _i}}})^{  1}}(B + \Delta B){\kern 1pt}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1, 2, \ldots, N. $  (18) 
Defining
$ \begin{align} CP{C^{T}}&  {{\bar L}_{{\lambda _i}}}(s)  {{\bar L}_{{\lambda _i}}} {({s^{  1}})^{\mathop{T}\nolimits} } + {{\bar L}_{{\lambda _i}}}(s){{\bar L}_{{\lambda _i}}} {({s^{  1}})^{\mathop{ T}\nolimits} }\notag\\ & + {\bar A_{{\lambda _i}}^{T}}P{{\bar A}_{{\lambda _i}}}  P <  {\gamma ^{  2}}{B_\omega }B_{\pmb{\omega }}^{T} \le 0 . \end{align} $  (19) 
which, after completing squares, becomes
$ \begin{align} &{H_{{\lambda _i}}}(s){H_{{\lambda _i}}}{({s^{  1}})^{T}} \le {\gamma ^2}I  {\gamma ^2}CP{C^{T}}\notag\\ & \qquad \, {\gamma ^2}[I{{\bar L}_{{\lambda _i}}}(s)] {[I{{\bar L}_{{\lambda _i}}}({s^{1}})]^{T}} < {\gamma ^2}I \end{align} $  (20) 
meaning that
Remark 2: In Lemma 2, a sufficient condition is given to guarantee DMASs (5) achieving robust consensus with a
To cope with the uncertain matrices
Lemma 4 [30]: Given matrices
$ Y + DFE + {E^T}{F^T}{D^T} < 0 $  (21) 
for all
$ Y + \varepsilon D{D^T} + {\varepsilon ^{  1}}{E^T}E < 0. $  (22) 
Lemma 5 (Schur complement) [31]: The linear matrix inequality
$ \left( {\begin{array}{*{20}{l}} {Q(x)}&{S(x)}\\ {S{{(x)}^T}}&{R(x)} \end{array}} \right) > 0 $ 
where
1)
2)
Theorem 2: Consider DMASs (1) with a fixed, undirected and connected communication topology
$ \begin{align} &\left[{\begin{array}{*{20}{c}} { X + \varepsilon D{D^{T}}}&{AX {\lambda _i}BW}\\ {{{(AX {\lambda _i}BW)}^{T}}}&{  X + {C^{\mathop{T}\nolimits} }C}\\ 0&{{E_1}X  {\lambda _i}{E_2}W}\\ 0&{B_{ {\omega }}^{T}X} \end{array}} \right.\notag\\[2mm] &\qquad\qquad\qquad \left. {\begin{array}{*{20}{c}} 0&0\\ {{{\left( {{E_1}X {\lambda _i}{E_2}W} \right)}^{T}}}&{X{B_\omega }}\\ { \varepsilon {I_d}}&0\\ 0&{ {\gamma ^2}{I_{{m_2}}}} \end{array}} \right] < 0 \end{align} $  (23) 
where
Proof: By Lemma 5, matrix inequality (16) is equivalent to
$ \begin{align} \left[{\begin{array}{*{20}{c}} {{P^{1}}}&{{{\bar A}_{{\lambda _i}}}}\\ {\bar A_{{\lambda _i}}^{T}}&{P + P{C^{\mathop{T}\nolimits} }CP + {\gamma ^{  2}}{B_\omega }B_{ {\omega }}^{T}} \end{array}} \right] < 0. \end{align} $  (24) 
Moreover, the above inequality can be rewritten as
$ \begin{align} &\left[{\begin{array}{*{20}{c}} {{P^{1}}}&{{A_{{\lambda _i}}}}\\ {A_{{\lambda _i}}^{T}}&{P + P{C^{\mathop{T}\nolimits} }CP + {\gamma ^{  2}}{B_\omega }B_{ {\omega }}^{T}} \end{array}} \right]\notag\\ &\quad\ + \left[{\begin{array}{*{20}{c}} D\\ 0 \end{array}} \right]F\left[{\begin{array}{*{20}{c}} 0&{{E_{{\lambda _i}}}} \end{array}} \right] + {\left[{\begin{array}{*{20}{c}} 0&{{E_{{\lambda _i}}}} \end{array}} \right]^{T}}F{\left[{\begin{array}{*{20}{c}} D\\ 0 \end{array}} \right]^{T}} < 0. \end{align} $  (25) 
It follows from the Lemma 4 that (25) can be expressed as
$ \begin{align} &{\small\left[{\begin{array}{*{20}{c}} { {P^{ 1}} + \varepsilon D{D^{T}}}&{{A_{{\lambda _i}}}}\\[2mm] {A_{{\lambda _i}}^{T}}& P + P{C^{\mathop{T}\nolimits} }CP {\gamma ^{ 2}}{B_\omega }B_{{\omega }}^{T} + {\varepsilon ^{  1}}E_{{\lambda _i}}^{T}{E_{{\lambda _i}}} \end{array}} \right]\notag}\\[2mm] &\ \ < 0. \end{align} $  (26) 
Through Lemma 5 again, matrix inequality (26) is equivalent to
$ \begin{align} &{ \left[{\begin{array}{*{20}{c}} { {P^{ 1}} + \varepsilon D{D^{T}}}&{{A_{{\lambda _i}}}}&0\\[2mm] {A_{{\lambda _i}}^{T}}&  P + P{C^{\mathop{T}\nolimits} }CP + {\gamma ^{  2}}{B_\omega }B_{{\omega }}^{T}&{E_{{\lambda _i}}^{T}}\\[4mm] 0&{{E_{{\lambda _i}}}}&{ \varepsilon I} \end{array}} \right]\notag}\\[2mm] &\ \ < 0. \end{align} $  (27) 
Preand postmultiplying both sides of (27) by
$ \left[{\begin{array}{*{20}{c}} {{I_d}}&0&0\\ 0&{{P^{1}}}&0\\ 0&0&{{I_d}} \end{array}} \right] $ 
letting
Remark 3: In Theorem 2, it can be noted that the NMI (16) is transformed to a LMI condition (23). Subsequently, highorder DMASs (1) with the distributed consensus protocol (4) achieve robust consensus with a
Remark 4: From Theorem 2, we can also get that, the communication disturbances have effects on the performance of the control object, such as switching interaction topologies. In [32], [33], the timevarying formation tracking problems for secondorder MASs with switching interaction topologies were studied. Switching topologies include two cases. One is that every interaction topology of MASs has a spanning tree; another is jointcontained spanning tree topologies. It should be mentioned that, this approach can be easily extended to the first case, and more details can be seen in our work [34]. There have been some difficulties to the jointcontained spanning tree case. We will consider it in the future.
Theorem 3: With
$ \begin{align} &\mathop {\lim }\limits_{k \to \infty } \left( {\pmb c}(k)  {{\bf{1}}_N} \otimes \left( \bigg( {\frac{1}{N}{{(A + \Delta A)}^k}\sum\limits_{i = 1}^N {{{\pmb x}_i}(0)} } \bigg)\right.\right.\notag\\ &\qquad \left.\left.+ \sum\limits_{i{\rm{ = 0}}}^{k  1} {\sum\limits_{l{\rm{ = 0}}}^{l = k  i  1} {\sum\limits_{j = 1}^{j = N} {\left( {\frac{1}{N}{{(A + \Delta A)}^i}{B_{{\omega}} }{{\pmb {\omega}} _{{ x}, j}}(l)} \right)} } } \right) \right) = 0. \end{align} $  (28) 
Proof: Let
$ \begin{align*} {{\pmb x}_C}(0) &= {{\bar {\pmb u}}_1} \otimes \left(({\pmb e}_1^T \otimes {I_d}) * ({U^T} \otimes {I_d}){\pmb x}(0)\right) \nonumber\\ & = {{\bar {\pmb u}}_1} \otimes \left(({\pmb e}_1^T{U^T} \otimes {I_d}){\pmb x}(0)\right)\nonumber\\ & = {{\bar {\pmb u}}_1} \otimes \left((\frac{1}{{\sqrt N }}{\bf{1}}_N^T \otimes {I_d}){\pmb x}(0)\right)\notag\\ & = {\bf{1}}_N^T \otimes \left(\frac{1}{N}\sum\limits_{i = 1}^N {{{\pmb x}_i}(0)} \right). \end{align*} $ 
Likewise, let
$ \begin{align*} {{\pmb \omega} _{C, {\pmb x}}}(k)& = {{\bar {\pmb u}}_1} \otimes \left(({\pmb e}_1^T \otimes {I_d}) * ({U^T} \otimes {I_d}){{\pmb {\omega}} _{ x}}(k)\right)\\ &= {{\bar {\pmb u}}_1} \otimes \left(({\pmb e}_1^T{U^T} \otimes {I_d}){{\pmb \omega} _{ x}}(k)\right)\\ &= {{\bar {\pmb u}}_1} \otimes \left((\frac{1}{{\sqrt N }}{\bf{1}}_N^T \otimes {I_d}){{\pmb \omega} _{ x}}(k)\right) \\&= {\bf{1}}_N^T \otimes \left(\frac{1}{N}\sum\limits_{i = 1}^N {{{\pmb \omega} _{{ x}, i}}(k)} \right). \end{align*} $ 
Hence, we have
$ \begin{align*} {{\pmb x}_C}(k) =&\ {(A + \Delta A)^k}{{\pmb x}_C}(0)\\ & \, + {{\bf{1}}_N} \otimes \sum\limits_{i = 0}^{k  1} {\left( {{{(A + \Delta A)}^i}{B_\omega }{{\tilde {\pmb \omega} }_{c{\rm{, }}x}}(k  i  1)} \right)} \\ = & \ {{\bf{1}}_N} \otimes \left( \left( {\frac{1}{N}{{(A + \Delta A)}^k}\sum\limits_{i = 1}^N {{{\pmb x}_i}(0)} } \right)\right. \\ & \left.\, +\, \sum\limits_{i{\rm{ = 0}}}^{k  1} {\sum\limits_{l{\rm{ = 0}}}^{l = k  i  1} {\left( {{{(A + \Delta A)}^i}{B_\omega }{{\pmb \omega} _{C, x}}(l)} \right)} } \right)\\ =&\ {{\bf{1}}_N} \otimes \left( \left( {\frac{1}{N}{{(A + \Delta A)}^k}\sum\limits_{i = 1}^N {{{\pmb x}_i}(0)} } \right)\right.\\ & \left.\, +\, \sum\limits_{i = 0}^{k  1} {\sum\limits_{l{= 0}}^{l = k  i  1} {\sum\limits_{j = 1}^{j = N} {\left( {\frac{1}{N}{{(A + \Delta A)}^i}{B_\omega }{{\pmb \omega} _{{ x}, j}}(l)} \right)} } } \right) \end{align*} $ 
then the final consensus value
Corollary 1: With
$ \mathop {\lim }\limits_{k \to \infty } \left( {{\pmb c}(k)  {{\bf{1}}_N} \otimes \left( {\frac{1}{N}{{(A + \Delta A)}^k}\sum\limits_{i = 1}^N {{{\pmb x}_i}(0)} } \right)} \right) = 0. $  (29) 
Proof: This proof can be easily obtained from the proof of Theorem 3.
Remark 5: With
Remark 6: It should be pointed out that, in [26][28], by recursive linear matrix inequalities (RLMIs) techniques, the robust
In this section, a numerical example is given to illustrate the effectiveness of the proposed theoretical results. We apply the above proposed consensus protocol (4) to achieve state alignment among 8 agents. The dynamics of them are described by (1), where
$ \begin{align*} &A = \left[{\begin{array}{*{20}{c}} \dfrac{\sqrt{2}}{2}&\dfrac{\sqrt{2}}{2}&0\\[1.1mm] \dfrac{\sqrt{2}}{2}&\dfrac{\sqrt{2}}{2}&0\\[1.1mm] 0&0&1 \end{array}} \right], ~~~B = \left[{\begin{array}{*{20}{c}} {0.2}\\ {0.4}\\ 1 \end{array}} \right]\\ &D = \left[{\begin{array}{*{20}{c}} {0.1}&0&0\\ 0&{0.2}&0\\ 0&0&{0.3} \end{array}} \right], ~~~ {E_1} = \left[{\begin{array}{*{20}{c}} {0.1}&{0.3}&0\\ {0.2}&{0.4}&0\\ 0&0&1 \end{array}} \right]\\ &{E_2} = \left[{\begin{array}{*{20}{c}} {0.2}\\ \begin{array}{l} 0.1\\ 0.3 \end{array} \end{array}} \right], ~~~ F = \left[{\begin{array}{*{20}{c}} {{r_1}}&0&0\\ 0&{{r_2}}&0\\ 0&0&{{r_3}} \end{array}} \right]\end{align*} $ 
and
$ {{{\pmb x}}_i}(k + 1) = (A + DF{E_1}){{{\pmb x}}_i}(k) + (B + DF{E_2}){{{\pmb u}}_i}(k). $  (30) 
We apply the consensus protocol (4) to achieve consensus among the above those 8 agents under a fixed topology
Assume that the initial state values of the all agents
$ \begin{align*} &K = [{\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.0983}}}&{ {\rm{0}}{\rm{.0884}}}&{{\rm{0}}{\rm{.2803}}} \end{array}}]\\ &X = \left[{\begin{array}{*{20}{r}} {{\rm{0}}{\rm{.8086}}}&{{\rm{0}}{\rm{.0443}}}&{{\rm{0}}{\rm{.2130}}}\\ {{\rm{0}}{\rm{.0443}}}&{{\rm{0}}{\rm{.6593}}}&{{\rm{0}}{\rm{.0279}}}\\ {{\rm{0}}{\rm{.2130}}}&{ {\rm{0}}{\rm{.0279}}}&{{\rm{1}}{\rm{.1160}}} \end{array}} \right], ~~~\varepsilon = 4.2304\\ &W = [{\begin{array}{*{20}{c}} {0.1353}&{0.0617}&{0.3362} \end{array}}]\end{align*} $ 
In Figs. 24, the simulation results are given. The state trajectories of uncertain DMASs (1) with and without external disturbances are shown in Figs. 24(b) and (a) , respectively. Final consensus values
From Figs. 24(a), it can be seen that the state trajectories of DMASs (1) with
$ \begin{align*} &{{\bf{1}}_N} \otimes \left( {(A + DF{E_1})^k}\left(\dfrac{{1}}{N}\sum\limits_{i = 1}^N {{{\pmb x}_i}(0)} \right)\right.\\ &\qquad\left. + \, \sum\limits_{i = 0}^{k  1} \sum\limits_{j = 1}^{j = N} \sum\limits_{l = 0}^{l = k  i  1} {\left({1 \ N} {{(A + \Delta A)}^i}{B_\omega }{{\pmb \omega} _{{\pmb x}, j}}(l) \right)} \right)\end{align*} $ 
which is in accordance with Theorem 3. By Definition 1, it is clear that DMASs (1) achieves robust consensus with protocol (4). Therefore, the correctness and validity of proposed protocols and theorems are demonstrated.
5 ConclusionsThe robust
Let a weighted digraph (or directed graph)
The node indexes belong to a finite index set
$ {\Delta _{ij}} = \begin{cases} 0, &{i \ne j}\\ {{{\deg }_{{\rm{out}}}}({v_i}), }&{i = j.} \end{cases} $ 
The graph Laplacian matrix associated with the digraph
Given matrices
$ P \otimes Q = [{{p_{ij}}Q}] \in \mathbb{R}{^{mp \times nq}} $ 
in [36]. For matrices
1)
2)
3)
4)
5) Suppose that
6) If
7) If
8) Suppose that
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