2. Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 3378570, Japan
There has been great interest in cooperative control of multiagent systems, where all agents are connected by a network (described by a graph), and they communicate to neighbor agents for necessary information such that the whole group can achieve a collective behavior. The specification for collective behavior includes flocks and swarms, sensor fusion, random networks, synchronization of coupled oscillators, formation control of multi robots, optimizationbased cooperative control, etc. For more detailed information, see the cornerstone paper [1], the survey paper [2], the book [3], [4] and the references therein.
One important control problem in cooperative control is the consensus problem, which involves reaching an agreement regarding a certain quantity of interest that depends on the states of all agents. There are many important papers which have made great contributions in consensus problems for selforganizing networked systems [5][8]. For networked secondorder integrators, [9] presented a necessary and sufficient condition for achieving consensus, where it is shown that both real and imaginary parts of the eigenvalues of the Laplacian matrix of the corresponding network play key roles. Recently, [10] presented a necessary and sufficient condition for consensus among secondorder controllable canonical multiagent systems. Both [9] and [10] dealt with communication delay in the discussion. The consensus problem of thirdorder nonlinear multiagent systems with a fixed communication topology was discussed, and a consensus algorithm was proposed with several sufficient conditions in [11]. For consensus control of higher order linear systems, a sufficient and necessary condition was obtained in [12] where the local control input included two parts: a feedback controller and the interactions from the neighbors. The interconnection graph considered in [12] is undirected, and thus, the discussion can not be extended to directed interconnection cases. Moreover, an approach of achieving consensus for more general linear agents in the framework of matrix inequalities and stabilization was proposed in [13], and the extension to the consensus problem for networked nonholonomic systems was dealt with in [14]. For multiagent systems with switching interconnection graphs, [15] adopted a combination of connected and disconnected graphs to achieve desired consensus, where the basic idea was inspired by switched systems analysis.
The main purpose of this paper is to extend the results in [9] and [10] to third order agents networked by directed interconnection graphs, that is, to provide a necessary and sufficient condition to design a consensus algorithm for a group of agents described by third order integrators. The motivation of dealing with third order agents is not only from theoretical interest but also from the fact that there are many real systems which are described by third order differential equations; for example, the location of bouncing robots, road vehicles with random terrain, and aircrafts in the wind [16], [17]. A typical example is a massspringdamper system with jerk term [16], [17] whose dynamics is described by
As in the literature including [9] and [10], we assume here that the control input of each agent is constructed based on the weighted difference between its states and those of its neighbor agents, in a decentralized manner, and aim at proposing an algorithm for computing the weighting coefficients in the control input. The problem is reduced to design a thirdorder Hurwitz polynomial with real or complex coefficients. It is noted that the direct computation method in [9] and the frequency domain test approach in [10] can not be applied directly and effectively to design the thirdorder polynomial with complex coefficients. For that purpose, we propose to adopt the generalized Hurwitz condition [18] in our design, and obtain a necessary and sufficient condition for the consensus algorithm design. Since the obtained condition is both necessary and sufficient, we provide a kind of parametrization for all the weighting coefficients (feedback gains) achieving consensus. Moreover, the condition turns out to be a natural extension to second order consensus, and is reasonable and practical due to its comparatively decreased computation burden. The discussion is extended to the case where communication delay exists in the control input. With the same control gains as in the case without delay, we aim to establish a tight upper bound for the delay as a necessary and sufficient condition such that consensus is maintained when involving communication delay. It is noted that the method of discussing delay in the Hurwitz polynomial is encouraged and revised from that in [10]. Moreover, although the main idea and approach are different, we would like to mention that there have been many references discussing communication delays in consensus control of networked agents [19][23].
The remainder of this paper is organized as follows. In Section Ⅱ we give some preliminary information about graph theory and provide two lemmas for the benefit of discussion later. Section Ⅲ is devoted to describe the system of networked agents with the control input protocol including three design parameters, and to reduce the control problem to designing Hurwitz polynomials with complex coefficients. Then, Section Ⅳ gives a detailed discussion on the design of thirdorder Hurwitz polynomials together with some possible simplification and generalization. Furthermore, the extension to the case of control input with communication delay among agents is made in Section Ⅴ, where a tight upper bound is established for the delay such that the desired consensus is maintained when the real communication delay is smaller than the bound. Two numerical examples are given in Sections Ⅳ and Ⅴ to show effectiveness of the results. Finally, Section Ⅵ concludes the paper.
Ⅱ. PRELIMINARIESLet us first review some basic definitions for graphs and consensus in network of multiagents. The interconnection of a family of agents can be represented by using a directed graph (or digraph)
A directed path is a sequence of ordered edges, and a digraph is called strongly connected if there is a directed path from every node to every other node. A directed tree is a digraph, where every node, except the root, has exactly one parent. A spanning tree of a digraph is a directed tree formed by graph edges that connect all the nodes of the graph. We say that a digraph has (or contains) a spanning tree if there exists a spanning tree that is a subset of the graph. It is known that the condition that a digraph has a spanning tree is equivalent to the case that there exists a node having a directed path to all other nodes.
The Laplacian of a graph is defined as
$ \begin{eqnarray*} l_{ij}= \begin{cases} 1, & j\in \mathcal{N}_i \\ \mathcal{N}_i, & j=i \\ 0, & \mbox{otherwise} \end{cases} \end{eqnarray*} $ 
and
The next two lemmas will be used in the following sections.
Lemma 1 [25]: Suppose
$ \begin{eqnarray} W=\left[\begin{array}{cc} A & B \\ C & D \end{array}\right]. \end{eqnarray} $ 
1) If
2) If
3) If
4) If
Lemma 2 [25]: Let
1)
2)
Consider a group of networked third order agents whose dynamics are described by
$ \begin{equation} y_i^{(3)}(t) = u_i(t)\, , \quad i=1, 2, \ldots, N \end{equation} $  (1) 
where
Define
$ \begin{eqnarray} x_i &=& \left[\begin{array}{c} x_i^1 \\ x_i^2 \\ x_i^3 \end{array}\right] \equiv \left[\begin{array}{c} y_i \\ y_i^{(1)} \\ y_i^{(2)} \end{array}\right] \\ A_i &=& \left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\, , \quad b_i= \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \end{eqnarray} $ 
and rewrite the system as
$ \begin{eqnarray} x_i^{(1)}(t) = A_i x_i(t) + b_i u_i(t)\, , \quad i=1, 2, \ldots, N\, . \end{eqnarray} $  (2) 
As in the literature, we assume that the available information for the
$ \begin{eqnarray} x_ix_j = \left[\begin{array}{c} x_i^1x_j^1 \\ x_i^2x_j^2 \\ x_i^3x_j^3 \end{array}\right] \to 0\, , \quad \forall i \ne j. \end{eqnarray} $  (3) 
Note that the consensus problem is different from the stabilization one in the sense that convergence to an equilibrium point such as the origin is not desired. In real systems, we usually do not want the system state to stop but rather, to track each other in a flexible manner. Therefore, even if the pair
Similarly to the literature such as [9] and [10], we adopt the control input (consensus protocol) given by
$ \begin{eqnarray} u_i &=& \displaystyle{\sum\limits_{j\in \mathcal{N}_i}} \left[\gamma_1(x_j^1x_i^1)+\gamma_{2} (x_j^2x_i^2) +\gamma_{3} (x_j^3x_i^3)\right] \\ &=& \displaystyle{\sum\limits_{j\in \mathcal{N}_i} \sum\limits_{k=1}^3 \gamma_k (x_j^kx_i^k)} \end{eqnarray} $  (4) 
where
Remark 1: It is noted that the formulation of the third order integrator model in (1) is reasonable in two aspects. First, there are real examples that are described by third order integrators especially in electronic circuits. For example, consider a parallel circuit composed of an inductor (
Secondly, in the case that the dynamical equation of each agent is given by the more general form
$ \begin{eqnarray} y_i^{(3)}(t) + \beta_1 y_i^{(2)}(t) + \beta_{2} y_i^{(1)}(t) + \beta_3 y_i (t)= u_i(t) \end{eqnarray} $ 
since it is reasonable to assume that each agent can obtain all states of itself, we define
$ \begin{eqnarray} u_i &=& \displaystyle{\sum\limits_{j\in \mathcal{N}_i} \left[\gamma_1(x_j^1x_i^1)+\gamma_{2} (x_j^2x_i^2)+\gamma_{3} (x_j^3x_i^3)\right]} \\ && + \beta_1 x_i^{3} + \beta_{2} x_i^2 + \beta_3 x_i^1 \end{eqnarray} $  (5) 
with the same design parameters
Remark 2: Although in this paper we focus our attention on third order agents for notation simplicity, as can be seen later, it is easy to extend the discussion to higher order integrators described by
$ \begin{eqnarray*} y_i^{(n)}(t) = u_i(t)\, , \quad i=1, 2, \ldots, N\, . \end{eqnarray*} $ 
In this case, the matrices
$ \begin{eqnarray*} A_i = \left[\begin{array}{ccc} 0 & I_{n1} \\ 0 & 0 \end{array}\right]\, , \quad b_i= \left[\begin{array}{c} 0_{(n1)\times 1} \\ 1 \end{array}\right] \end{eqnarray*}s $ 
and the control input is
The closedloop system composed of the system (1) (or the system (2)) and the controller (4) is
$ \begin{eqnarray} \dot{\tilde{x}} = \left(\Theta\otimes I_m\right) \tilde{x} \end{eqnarray} $  (6) 
where
$ \begin{eqnarray} \tilde{x}= \left[\begin{array}{c} x^1 \\ x^2\\ x^3 \end{array}\right]\, , \quad x^j = \left[\begin{array}{c} x_1^j \\ x_2^j \\ \vdots \\ x_N^j \end{array}\right]\, , \quad j=1, 2, 3 \end{eqnarray} $ 
and
$ \begin{eqnarray} \Theta = \left[\begin{array}{ccc} 0 & I_N & 0 \\ 0 & 0 & I_N \\ \gamma_1 \mathcal{L} & \gamma_2 \mathcal{L} & \gamma_{3} \mathcal{L} \end{array}\right]. \end{eqnarray} $ 
According to Lemma 2, we obtain that the characteristic polynomial is
$ \left\lambda I \left(\Theta\otimes I_m\right) \right= \left(\lambda I \Theta)\otimes I_m \right= \left(\lambda I \Theta) \right^m\, . $ 
Then, using the same technique as in [4], [9] and [10], the following fact can be proved easily.
Lemma 3: Consensus is achieved in (1) with (4) if and only if
According to Lemma 3, our design problem is to choose
Lemma 4: The characteristic equation of
$ \begin{equation} \left\lambda I \Theta\right = \left\lambda^3 I_N+ (\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2 ) \mathcal{L}\right\, . \end{equation} $  (7) 
Furthermore, if the eigenvalues of
$ \begin{equation} \left\lambda I \Theta\right = \displaystyle{\prod\limits_{i=1}^N} \left( \lambda^3 + \left(\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2\right) \mu_i \right) \, . \end{equation} $  (8) 
Proof: Using the definition of matrix eigenvalues, the characteristic equation of
$ \begin{eqnarray} \left\lambda I \Theta\right = \left\begin{array}{ccc} \lambda I_{N} & I_{N} & 0 \\ 0 & \lambda I_{N}&I_{N} \\ \gamma_{1}\mathcal{L}& \gamma_{2}\mathcal{L}& \lambda I_{N} + \gamma_{3}\mathcal{L} \end{array}\right\, . \end{eqnarray} $ 
In the case of
$ \begin{eqnarray} \left\lambda I \Theta\right &=& \left\begin{array}{ccc} 0 & I_{N} & 0 \\ 0 & 0 & I_{N} \\ \gamma_{1}\mathcal{L} & \gamma_{2}\mathcal{L} & \gamma_{3}\mathcal{L} \end{array}\right \\ &=& \left\begin{array}{ccc} I_{N} & 0 & 0\\ 0 & I_{N} & 0\\ \gamma_{2}\mathcal{L} & \gamma_{3}\mathcal{L} &\gamma_{1}\mathcal{L} \end{array}\right = \gamma_1 \mathcal{L} \end{eqnarray} $ 
which is consistent with (7) with
In the case of
$ \begin{eqnarray} \left\lambda I \Theta\right &=& \left\begin{array}{ccc} \lambda I_{N} & I_{N} & 0 \\ 0 & \lambda I_{N}&I_{N} \\ \gamma_{1}\mathcal{L} & \gamma_{2}\mathcal{L} & \lambda I_{N} + \gamma_{3}\mathcal{L} \end{array}\right \\ &=& \left \lambda I_{N} \right \times \left \left[\begin{array}{cc} \lambda I_{N} & I_{N} \\ \gamma_{2}\mathcal{L} + \frac{\gamma_1}{\lambda} \mathcal{L}& \lambda I_{N} + \gamma_{3}\mathcal{L}\end{array}\right] \right \end{eqnarray} $ 
and, by 3) or 4) of Lemma 1,
$ \begin{eqnarray} \left\lambda I \Theta\right &=& \left \lambda I_{N} \right \times \left \lambda^2 I_{N} + \gamma_3\lambda \mathcal{L} + \gamma_2 \mathcal{L} + \frac{\gamma_1}{\lambda} \mathcal{L} \right \\ &=& \left\lambda^3 I_N+ (\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2 )\mathcal{L}\right \, . \end{eqnarray} $ 
This completes the proof of (7).
Next, when the eigenvalues of
$ \begin{eqnarray} \left\zeta I_N+ \mathcal{L}\right = \prod\limits_{i=1}^N (\zeta+\mu_i)\, , \end{eqnarray} $ 
and thus
$ \begin{eqnarray} {\left\lambda I \Theta\right} \\ &=& \left( \gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2 \right)^N \left\frac{\lambda^3}{\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2} I_N+ \mathcal{L}\right \\ &=& \left( \gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2 \right)^N \displaystyle{\prod\limits_{i=1}^N} \left( \frac{\lambda^3}{\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2} +\mu_i \right) \\ &=& \displaystyle{\prod\limits_{i=1}^N} \left( \lambda^3 + \left(\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2\right) \mu_i \right)\, . \end{eqnarray} $ 
It is observed from the above that
$ \begin{eqnarray} p_{\mu}(\lambda) = \lambda^3 + (\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2)\mu \end{eqnarray} $ 
have negative real parts for all
Remark 3: Although the approaches in [9] and [10] are efficient in analyzing second order Hurwitz polynomials, they can not be used to design the third order polynomial
In this section, we discuss how to design the parameters
Notice that when
$ \begin{eqnarray*} p_{\mu_i}(\lambda) p_{\bar{\mu}_i}(\lambda) &=& \left( \lambda^3 + \left(\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2\right) \mu_i \right) \\ && \times \left( \lambda^3 + \left(\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2\right) \bar{\mu}_i \right) \end{eqnarray*} $ 
is a polynomial with real coefficients. Then, based on the discussion in Section Ⅲ, we have the following result.
Lemma 5: Suppose
For
$ \begin{eqnarray} p_{\mu_i}(\lambda) = \lambda^3 + (\gamma_3 a_i) \lambda^2 + (\gamma_2 a_i) \lambda + \gamma_1 a_i\, . \end{eqnarray} $ 
The Hurwitz matrix of
$ \begin{equation} H_1=\left[\begin{array}{ccc} \gamma_3 a_i & \gamma_1 a_i & 0 \\ 1 & \gamma_2 a_i & 0 \\ 0 & \gamma_3 a_i & \gamma_1 a_i \end{array}\right] \end{equation} $  (9) 
and thus, the necessary and sufficient condition for the zeros of
$ \begin{eqnarray} \gamma_3 a_i > 0\, , \quad \left\begin{array}{cc} \gamma_3 a_i & \gamma_1 a_i \\ 1 & \gamma_2 a_i \end{array}\right>0 \\ \left\begin{array}{ccc} \gamma_3 a_i & \gamma_1 a_i & 0 \\ 1 & \gamma_2 a_i & 0 \\ 0 & \gamma_3 a_i & \gamma_1 a_i \end{array}\right>0\, . \end{eqnarray} $ 
Solving these inequalities, we reach
$ \begin{equation} \gamma_1 >0, \quad \gamma_3>0, \quad \gamma_2> \frac{\gamma_1}{\gamma_3 a_i} \end{equation} $  (10) 
which gives a simple and explicit condition for the parameters.
However, for
To overcome the computational difficulty mentioned in the above subsection, we choose to consider the zeros of each
$ \begin{eqnarray} \bar{p}_{\sigma_j}(\lambda) = p_{\bar{\sigma}_j}(\lambda)\, , \end{eqnarray} $ 
zeros of
Lemma 6: Suppose
To use the above lemma for consensus parameters design, we need the condition under which a polynomial with complex coefficients is Hurwitz.
Lemma 7 [18]: The polynomial
$ \begin{eqnarray} \begin{array}{c} p(z)=z^n+\alpha_1 z^{n1}+\alpha_2 z^{n2}+ \cdots+\alpha_n\, , \\ (\alpha_k=p_k+\sqrt{1}q_k, \ k=1, \ldots, n) \end{array} \end{eqnarray} $ 
has all its zeros in the left halfplane if and only if the next determinants are all positive.
$ \begin{eqnarray} \begin{array}{l} \Delta_1=p_1\, , \quad \Delta_2=\left\begin{array}{ccc} p_1 & p_3 & q_2 \\ 1 & p_2 & q_1 \\ \hline 0 & q_2 & p_1 \end{array}\right \\ \Delta_3=\left\begin{array}{ccccc} p_1 & p_3 & p_5 & q_2 & q_4 \\ 1 & p_2 & p_4 & q_1 & q_3 \\ 0 & p_1 & p_3 & 0 & q_2 \\ \hline 0 & q_2 & q_4 & p_1 & p_3 \\ 0 & q_1 & q_3 & 1 & p_2 \end{array}\right \end{array} \end{eqnarray} $ 
$ \begin{array}{l} \Delta_k=\left\begin{array}{ccccccccc} p_1 & p_3 & p_5 & \ldots & p_{2k1} & q_2 & q_4 & \ldots & q_{2k2} \\[0mm] 1 & p_2 & p_4 & \ldots & p_{2k2} & q_1 & q_3 & \ldots & q_{2k3} \\[0mm] & & & \ldots & & & & \ldots & \\[0mm] 0 & & & \ldots & p_k & 0 & & \ldots & q_{k1} \\[0mm] \hline 0 & q_2 & q_4 & \ldots & q_{2k2} & p_1 & p_3 & \ldots & p_{2k3} \\[0mm] 0 & q_1 & q_3 & \ldots & q_{2k3} & 1 & p_2 & \ldots & p_{2k4} \\[0mm] & & & \ldots & & & & \ldots & \\[0mm] 0 & & & \ldots & q_k & 0 & & \ldots & p_{k1} \end{array}\right \\ k=2,3, \ldots, n\,\, (p_r=q_r=0\ \mbox{for}\ r>n). \end{array} $ 
Using the above lemma for
$ \begin{array}{l} {p_{\sigma_j}(\lambda)} \\ &=& \lambda^3 +(\gamma_1+\gamma_2 \lambda + \gamma_3 \lambda^2)(b_j+\sqrt{1}c_j) \\ &=& \lambda^3 +(\gamma_3b_j+\sqrt{1}\, \gamma_3 c_j) \lambda^2 + (\gamma_2 b_j + \sqrt{1}\, \gamma_2 c_j) \lambda \\ && + (\gamma_1 b_j + \sqrt{1}\, \gamma_1 c_j) \, , \end{array} $ 
where
$ \Delta_{1j} = \gamma_3b_j>0 $  (11) 
$ \begin{eqnarray} \Delta_{2j} &=& \left\begin{array}{rrr} \gamma_3b_j & \gamma_1b_j & \gamma_2c_j \\ 1 & \gamma_2b_j & \gamma_3c_j \\ \hline 0 & \gamma_2c_j & \gamma_3b_j \end{array}\right >0 \end{eqnarray} $  (12) 
$ \begin{eqnarray} \Delta_{3j} &=& \left\begin{array}{ccccc} \gamma_3b_j & \gamma_1b_j & 0 & \gamma_2c_j & 0 \\ 1 & \gamma_2b_j & 0 & \gamma_3c_j & \gamma_1c_j \\ 0 & \gamma_3b_j & \gamma_1b_j & 0 & \gamma_2c_j \\ \hline 0 & \gamma_2c_j & 0 & \gamma_3b_j & \gamma_1b_j \\ 0 & \gamma_3c_j & \gamma_1c_j & 1 & \gamma_2b_j \end{array}\right>0\, . \\ \end{eqnarray} $  (13) 
Since
$ \begin{equation} \Delta_{2j} = \gamma_2\gamma_3^2 b_j(b_j^2+c_j^2)\gamma_1 \gamma_3 b_j^2\gamma_2^2 c_j^2 > 0 \, . \end{equation} $  (14) 
Next, we try to simplify the computation of
$ \begin{eqnarray*} \Delta_{3j} = \left\begin{array}{ccccc} \gamma_3b_j & \gamma_1b_j & \gamma_2c_j & 0 & 0 \\ 1 & \gamma_2b_j & \gamma_3c_j & 0 & \gamma_1c_j \\ 0 & \gamma_2c_j & \gamma_3b_j & 0 & \gamma_1b_j \\ \hline 0 & \gamma_3b_j & 0 & \gamma_1b_j & \gamma_2c_j \\ 0 & \gamma_3c_j & 1 & \gamma_1c_j & \gamma_2b_j \end{array}\right. \end{eqnarray*} $ 
Notice that the leading principal minor of order
$ \begin{array}{l} &&{ \left[\begin{array}{ccc} \gamma_3b_j & \gamma_1b_j & \gamma_2c_j \\ 1 & \gamma_2b_j & \gamma_3c_j \\ 0 & \gamma_2c_j & \gamma_3b_j \end{array}\right]^{1} = \frac{1}{\Delta_{2j}} } \\ && \times\left[\begin{array}{ccc} \gamma_2\gamma_3(b_j^2+c_j^2) & \gamma_1\gamma_3b_j^2\gamma_2^2c_j^2 & (\gamma_2^2\gamma_1\gamma_3)b_jc_j \\ \gamma_3b_j & \gamma_3^2b_j^2 & \gamma_3^2b_jc_j\gamma_2c_j \\ \gamma_2c_j & \gamma_2\gamma_3b_jc_j & \gamma_2\gamma_3b_j^2\gamma_1b_j \end{array}\right]\, . \end{array} $ 
Then, using 1) of Lemma 1, we obtain after some manipulation that
$ \begin{array}{l} {\Delta _{3j}} = {\mkern 1mu} {\Delta _{2j}}\\ \;\;\;\;\;\;\;\;\;\; \times \left {\left[ {\begin{array}{*{20}{c}} {{\gamma _1}{b_j}}&{  {\gamma _2}{c_j}}\\ {{\gamma _1}{c_j}}&{{\gamma _2}{b_j}} \end{array}} \right]  \left[ {\begin{array}{*{20}{c}} 0&{{\gamma _3}{b_j}}&0\\ 0&{{\gamma _3}{c_j}}&1 \end{array}} \right]} \right.\\ \;\;\;\;\;\;\;\;\;\; \times \left. {{{\left[ {\begin{array}{*{20}{c}} {{\gamma _3}{b_j}}&{{\gamma _1}{b_j}}&{  {\gamma _2}{c_j}}\\ 1&{{\gamma _2}{b_j}}&{  {\gamma _3}{c_j}}\\ 0&{{\gamma _2}{c_j}}&{{\gamma _3}{b_j}} \end{array}} \right]}^{  1}}\left[ {\begin{array}{*{20}{c}} 0&0\\ 0&{  {\gamma _1}c}\\ 0&{{\gamma _1}b} \end{array}} \right]} \right\\ \;\;\;\;\; = {\Delta _{2j}} \times \left {\left[ {\begin{array}{*{20}{c}} {{\gamma _1}{b_j}}&{  {\gamma _2}{c_j}}\\ {{\gamma _1}{c_j}}&{{\gamma _2}{b_j}} \end{array}} \right]  \left[ {\begin{array}{*{20}{c}} 0&{\frac{{  {\gamma _1}{\gamma _2}{\gamma _3}b_j^2{c_j}}}{{{\Delta _{2j}}}}}\\ 0&{\frac{{{\gamma _1}{\gamma _2}{\gamma _3}b_j^3  \gamma _1^2b_j^2}}{{{\Delta _{2j}}}}} \end{array}} \right]} \right\\ \;\;\;\;\; = {\Delta _{2j}} \times \left {\begin{array}{*{20}{c}} {{\gamma _1}{b_j}}&{  {\gamma _2}{c_j} + \frac{{{\gamma _1}{\gamma _2}{\gamma _3}b_j^2{c_j}}}{{{\Delta _{2j}}}}}\\ {{\gamma _1}{c_j}}&{{\gamma _2}{b_j}  \frac{{{\gamma _1}{\gamma _2}{\gamma _3}b_j^3  \gamma _1^2b_j^2}}{{{\Delta _{2j}}}}} \end{array}} \right\\ \;\;\;\; = \left {\begin{array}{*{20}{c}} {{\gamma _1}{b_j}}&{  {\gamma _2}{c_j}{\Delta _{2j}} + {\gamma _1}{\gamma _2}{\gamma _3}b_j^2{c_j}}\\ {{\gamma _1}{c_j}}&{{\gamma _2}{b_j}{\Delta _{2j}}  {\gamma _1}{\gamma _2}{\gamma _3}b_j^3 + \gamma _1^2b_j^2} \end{array}} \right\\ \;\;\;\; = {\gamma _1}{\gamma _2}(b_j^2 + c_j^2){\Delta _{2j}} + \gamma _1^2b_j^2[{\gamma _1}{b_j}  {\gamma _2}{\gamma _3}(b_j^2 + c_j^2)]{\mkern 1mu} . \end{array} $ 
It is noted here that
Moreover, the above discussion is also valid for real eigenvalues, i.e.,
$ \Delta_{2j}= \gamma_2\gamma_3^2 b_j^3 \gamma_1\gamma_3 b_j^2 =\gamma_3(\gamma_2\gamma_3b_j\gamma_1)b_j^2>0 $ 
requires that
$ \begin{eqnarray*} \Delta_{3j} &=& \gamma_1\gamma_2b_j^2 \Delta_{2j}+\gamma_1^2b_j^2(\gamma_1b_j\gamma_2\gamma_3b_j^2) \\ &=& \gamma_1\gamma_2b_j^2 \gamma_3(\gamma_2\gamma_3b_j\gamma_1) b_j^2 +\gamma_1^2b_j^2(\gamma_1b_j\gamma_2\gamma_3b_j^2) \\ &=& \gamma_1 b_j^3 (\gamma_2\gamma_3b_j\gamma_1)^2 > 0 \end{eqnarray*} $ 
leads to
We summarize the above discussion in the following theorem, where we treat all the nonzero eigenvalues of
Theorem 1: Suppose that the Laplacian matrix
$ \gamma_3 > 0 $  (15) 
$ \Delta_{2j} = \gamma_2\gamma_3^2 b_j(b_j^2+c_j^2) \gamma_1\gamma_3 b_j^2 \gamma_2^2 c_j^2 > 0 $  (16) 
$ \begin{eqnarray} \Delta_{3j} &=& \gamma_1\gamma_2(b_j^2+c_j^2) \Delta_{2j} \\ && +\gamma_1^2b_j^2[\gamma_1b_j\gamma_2\gamma_3(b_j^2+c_j^2)] > 0 \end{eqnarray} $  (17) 
for all
As mentioned in the above, when the Laplacian matrix only has real eigenvalues (including the case of undirected interconnection among agents), the inequalities (15)(17) shrink into (10). Since the condition (10) is much simpler to implement, we state the following corollary.
Corollary 1: Suppose that the Laplacian matrix
$ \begin{array}{l} \gamma_1 >0, \quad \gamma_3>0, \quad \gamma_2> \frac{\gamma_1}{\gamma_3\min\limits_{2\le j \le N}\{ \mu_j \}} \, . \end{array} $  (18) 
When the Laplacian matrix
Theorem 2: Suppose that the Laplacian matrix
Proof: Consensus in (1) with (4) is achieved if and only if the conditions (15)(17) are satisfied.
First, we obtain from the definition of
$ \begin{equation} \gamma_2\gamma_3b_j(b_j^2+c_j^2)\gamma_1b_j^2=\frac{\Delta_{2j}+\gamma_2^2c_j^2}{\gamma_3}. \end{equation} $  (19) 
Substituting (19) into
$ \begin{eqnarray} \Delta_{3j} &=& \gamma_1\gamma_2(b_j^2+c_j^2) \Delta_{2j}+\gamma_1^2b_j^2[\gamma_1b_j\gamma_2\gamma_3(b_j^2+c_j^2)] \\ &=& \gamma_1\gamma_2(b_j^2+c_j^2) \Delta_{2j} \frac{\gamma_1^2b_j}{\gamma_3} (\Delta_{2j}+\gamma_2^2c_j^2) \\ &=& \frac{\gamma_1 \Delta_{2j}}{\gamma_3 b_j} \left( \gamma_2\gamma_3 b_j (b_j^2+c_j^2)\gamma_1 b_j^2 \right)  \frac{\gamma_1^2 \gamma_2^2}{\gamma_3} b_j c_j^2 \\ &=& \frac{\gamma_1 \Delta_{2j}}{\gamma_3^2 b_j} \left(\Delta_{2j}+\gamma_2^2 c_j^2\right) \frac{\gamma_1^2 \gamma_2^2}{\gamma_3} b_j c_j^2 \\ &=& \frac{\gamma_1 \Delta_{2j}(\Delta_{2j}+\gamma_2^2 c_j^2)}{\gamma_3^2 b_j}  \frac{\gamma_1^2 \gamma_2^2}{\gamma_3} b_j c_j^2 \end{eqnarray} $  (20) 
which is equivalent to
$ \Delta_{3j} + \frac{\gamma_1^2 \gamma_2^2}{\gamma_3} b_j c_j^2 = \gamma_1 \frac{\Delta_{2j}(\Delta_{2j}+\gamma_2^2 c_j^2)}{\gamma_3^2 b_j}. $ 
Since
Next, rewriting (19) into
$ \gamma_2\gamma_3b_j(b_j^2+c_j^2)=\gamma_1b_j^2+\frac{\Delta_{2j}+\gamma_2^2c_j^2}{\gamma_3}>0, $ 
we get
The above theorem shows that the parameters have to be positive. In addition, we shall show that the conditions (16) and (17) may be combined to compute the solution. Since (20) is equivalent to
$ \begin{eqnarray*} \frac{\gamma_3^2 b_j}{\gamma_1}\Delta_{3j} = \Delta_{2j}(\Delta_{2j}+\gamma_2^2 c_j^2)  \gamma_1 \gamma_2^2 \gamma_3 b_j^2 c_j^2\, , \end{eqnarray*} $ 
our problem is reduced to solving
$ \begin{equation} \Delta_{2j}(\Delta_{2j}+\gamma_2^2 c_j^2)  \gamma_1 \gamma_2^2 \gamma_3 b_j^2 c_j^2 > 0 \end{equation} $  (21) 
together with
$ \left( \Delta_{2j}+\frac{\gamma_2^2 c_j^2}{2} \right)^2 > \gamma_1 \gamma_2^2 \gamma_3 b_j^2 c_j^2 + \frac{\gamma_2^4 c_j^4}{4} $ 
and thus
$ \begin{equation} \Delta_{2j}+\frac{\gamma_2^2 c_j^2}{2} > \sqrt{\gamma_1 \gamma_2^2 \gamma_3 b_j^2 c_j^2 + \frac{\gamma_2^4 c_j^4}{4}}\, , \end{equation} $  (22) 
which guarantees
Substituting
Theorem 3: Suppose that the Laplacian matrix
$ \begin{array}{l} \gamma_2\gamma_3^2 b_j(b_j^2+&c_j^2)\gamma_1\gamma_3 b_j^2\frac{\gamma_2^2 c_j^2}{2} \\ & > \sqrt{\gamma_1 \gamma_2^2 \gamma_3 b_j^2 c_j^2 + \frac{\gamma_2^4 c_j^4}{4}} \end{array} $  (23) 
for all
Remark 4: Although (23) is a nonlinear and complicated inequality, we can analyze it efficiently to design the parameters. Firstly, when all
$ \begin{array}{l} \gamma_{2U} = \sup\left\{\gamma_2>0: (23)\ \mbox{holds for all }c_j\neq 0\right\} \\ \gamma_{2L} = \inf\left\{\gamma_2>0: (23)\ \mbox{holds for all }c_j\neq 0\right\}\, , \end{array} $ 
and then choose
There are many practical tools for evaluating
$ \begin{eqnarray*} f_j(\gamma_2) &=& \gamma_2\gamma_3^2 b_j(b_j^2+c_j^2)\gamma_1\gamma_3 b_j^2 \frac{1}{2}\gamma_2^2 c_j^2 \\ &&  \sqrt{\gamma_1 \gamma_2^2 \gamma_3 b_j^2 c_j^2 + \frac{\gamma_2^4 c_j^4}{4}}=0\, , \\ && j=2, \ldots, N \end{eqnarray*} $ 
by using bisection, the Newton method or other numerical algorithms.
D. Numerical ExampleIn the end of this section, we give an example for the consensus algorithm proposed above.
Example 1: Consider 5 agents connected as in Fig. 1, which is a directed graph. Then,
$ \begin{eqnarray*} \mathcal{L} = \left[\begin{array}{ccccc} 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 \\ 1 & 0 & 0 & 0 & 1 \end{array}\right]. \end{eqnarray*} $ 
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Fig. 1 Five agents networked by directed graph (Example 1, 2). 
The eigenvalues of
$ \begin{eqnarray} \gamma_1 >0, \ \gamma_3>0, \ \gamma_2> \frac{\gamma_1}{\gamma_3}\, \end{eqnarray} $  (24) 
and for the complex eigenvalues we have
$ \begin{eqnarray} \Delta_{21} &=& \frac{9}{2}\gamma_2\gamma_3^2\frac{9}{4}\gamma_1\gamma_3\frac{3}{4}\gamma_2^2 \\ &=& \frac{3}{4}\left(6\gamma_2\gamma_3^2\gamma_2^23\gamma_1\gamma_3 \right)>0 \end{eqnarray} $  (25) 
and
$ \begin{eqnarray} \Delta_{31} &=& 3\gamma_1\gamma_2 \Delta_{21} + \frac{9}{4} \gamma_1^2 \left(\frac{3}{2}\gamma_1 3\gamma_2\gamma_3\right) \\ &=& \frac{9}{4} \gamma_1\gamma_2 \left(6\gamma_2\gamma_3^2\gamma_2^23\gamma_1\gamma_3 \right) \\ && + \frac{9}{4} \gamma_1^2 \left(\frac{3}{2}\gamma_1 3\gamma_2\gamma_3\right) \\ &=& \frac{9}{8} \gamma_1 \left[12\gamma_2\gamma_3(\gamma_2\gamma_3\gamma_1)+3 \gamma_1^22\gamma_2^3\right]>0 \end{eqnarray} $  (26) 
On the other hand, the inequality (23) turns out to be
$ \begin{eqnarray*} \frac{9}{2}\gamma_2\gamma_3^2  \frac{9}{4} \gamma_1\gamma_3\frac{3}{8}\gamma_2^2 > \sqrt{\frac{27}{16}\gamma_1\gamma_2^2\gamma_3 + \frac{9}{64}\gamma_2^4} \end{eqnarray*} $ 
or equivalently,
$ \begin{eqnarray} 12\gamma_2\gamma_3^2  6 \gamma_1\gamma_3\gamma_2^2 > \gamma_2 \sqrt{12\gamma_1\gamma_3 + \gamma_2^2}\, . \end{eqnarray} $  (27) 
It is easy to observe that there are infinite solutions to the simultaneous inequalities (24)(26) or (27). For example,
$ \begin{eqnarray*} \gamma_1=1, \ \ \gamma_2=1, \ \ \gamma_3=2 \end{eqnarray*} $ 
is one solution.
Using the consensus algorithm (4) with these parameters, the plots of the systems' trajectories are depicted in Fig. 2, where the initial conditions are
$ \begin{align*} &x_1(0)= \begin{bmatrix}8 & 10 & 1 \end{bmatrix}^T \\ &x_2(0)= \begin{bmatrix}2 & 5 & 5 \end{bmatrix}^T \\ &x_3(0)= \begin{bmatrix} 4 & 5 & 7 \end{bmatrix}^T \\ &x_4(0)= \begin{bmatrix} 10 & 10 & 14 \end{bmatrix}^T \\ &x_5(0)= \begin{bmatrix} 16 & 15 & 20 \end{bmatrix}^T. \end{align*} $ 
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Fig. 2 States of five agents achieving consensus (Example 1, 
It can be observed that consensus has been achieved.
On the contrary, if we choose the parameters
$ \begin{eqnarray*} \gamma_1=1, \ \ \gamma_2=30, \ \ \gamma_3=2 \end{eqnarray*} $ 
then (26) and (27) are not satisfied. With the same initial conditions, the plots of the systems' trajectories are depicted in Fig. 3, and consensus has not been achieved.
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Fig. 3 States of five agents not achieving consensus (Example 1, 
In this section, we assume that the design parameters
$ \begin{eqnarray} u_{i}(t) &=&\displaystyle{\sum\limits_{j\in \mathcal{N}_i}}\left(\gamma_{1}(x_{j}^{1}(t\tau)x_{i}^{1}(t\tau))+ \cdots \right. \\ && \left. +\, \gamma_{3}(x_{j}^{3}(t\tau)x_{i}^{3}(t\tau))\right) \\ &=& \displaystyle{\sum\limits_{j\in \mathcal{N}_i} \sum\limits_{k=1}^3} \gamma_k \left(x_{j}^{k}(t\tau)x_{i}^{k}(t \tau)\right)\, \end{eqnarray} $  (28) 
where
$ \begin{eqnarray*} \dot{\tilde{x}}=(\Theta_{1}\otimes I_{m})\tilde{x}+(\Theta_{2}\otimes I_{m})\tilde{x}(t\tau) \end{eqnarray*} $ 
where
$ \begin{eqnarray*} \Theta_{1}=\left[\begin{array}{cccc}0&I_{N}&0 \\ 0 & 0 &I_{N} \\ 0 & 0 & 0 \end{array}\right]\, , \Theta_{2}=\left[\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & 0 \\ \gamma_{1} \mathcal{L} & \gamma_{2}\mathcal{L} & \gamma_{3}\mathcal{L} \end{array}\right]\, . \end{eqnarray*} $ 
Similarly to the discussion in Lemma 4, the characteristic polynomial of the above system is
$ \begin{array}{l} {\left\lambda I \Theta_{1}\otimes I_{m}  e^{\lambda\tau}(\Theta_{2}\otimes I_{m} )\right} \\ ~~~~~ = \left(\lambda I \Theta_{1}e^{\lambda\tau}\Theta_{2})\otimes I_m \right \\ ~~~~~= \left\lambda I \Theta_{1}e^{\lambda\tau}\Theta_{2}\right^m \end{array} $ 
and thus the characteristic equation is dominated by
$ \begin{array}{l} { \left\lambda I\Theta_{1}e^{\lambda\tau}\Theta_{2}\right} \\ = \left\begin{array}{ccc} \lambda I_{N} & I_{N} & 0 \\ 0 & \lambda I_{N}&I_{N} \\ \gamma_{1}e^{\lambda\tau}\mathcal{L} & \gamma_{2}e^{\lambda\tau}\mathcal{L} & \lambda I_{N} + \gamma_{3}e^{\lambda\tau}\mathcal{L} \end{array}\right \\ = \left\lambda^{3}I_N+(\gamma_{1}+\gamma_{2}\lambda+\gamma_{3}\lambda^{2})e^{\lambda\tau}\mathcal{L}\right \\ =\displaystyle{\prod\limits_{i=1}^{N}}\left(\lambda^{3}+(\gamma_{1}+\gamma_{2}\lambda+\gamma_{3}\lambda^{2})e^{\lambda\tau}\mu_{i}\right) = 0\, . \end{array} $ 
Then, consensus is achieved if and only if the above equation has exactly three zero eigenvalues and all other eigenvalues have negative real parts.
Let
$ \begin{eqnarray*} \left\lambda I\Theta_{1}e^{\lambda\tau}\Theta_{2}\right =\prod\limits_{i=1}^{N}p_{\mu_i}(\lambda, \tau) \, . \end{eqnarray*} $ 
We see from the above that
Next, we seek the condition about the communication delay
For this purpose, we substitute
$ \begin{eqnarray*} \omega_{i}^{3} \sqrt{1}+(\gamma_{1}+\gamma_{2}\omega_{i}\sqrt{1}\gamma_{3}\omega_{i}^{2})e^{\omega_{i}\tau_{i} \sqrt{1}}\mu_{i}=0\, . \end{eqnarray*} $ 
Separating the real and imaginary parts of the above yields
$ \begin{equation} \left\{ \begin{array}{l} A_{i}\sin(\omega_{i}\tau_{i})B_{i}\cos(\omega_{i}\tau_{i})=0 \\ B_{i}\sin(\omega_{i}\tau_{i})+A_{i}\cos(\omega_{i}\tau_{i})=\omega_{i}^{3} \end{array}\right. \end{equation} $  (29) 
where
$ \begin{eqnarray*} \begin{array}{l} A_{i}=\gamma_{2}\omega_{i}\Re(\mu_{i})(\gamma_{3}\omega_{i}^{2}\gamma_{1})\Im(\mu_{i}) \\ B_{i}=\gamma_{2}\omega_{i}\Im(\mu_{i})+(\gamma_{3}\omega_{i}^{2}\gamma_{1})\Re(\mu_{i}). \end{array} \end{eqnarray*} $ 
Then, it is easy to reach
$ \begin{eqnarray*} \sin(\omega_{i}\tau_{i})=\frac{B_{i}\omega_{i}^{3}}{A_{i}^{2}+B_{i}^{2}} \, , \quad \cos(\omega_{i}\tau_{i})=\frac{A_{i}\omega_{i}^{3}}{A_{i}^{2}+B_{i}^{2}} \end{eqnarray*} $ 
where
$ \begin{equation} \tau_{i}=\frac{\arctan\psi_{i}+k_i\pi}{\omega_{i}}\, , \quad \psi_{i} = \frac{B_i}{A_i} \end{equation} $  (30) 
and
Using
$ \begin{eqnarray*} A_i^2 + B_i^2 = \omega_i^6 \end{eqnarray*} $ 
and after simple calculation,
$ \begin{equation} \omega_{i}^{6}\gamma_{3}^{2}\mu_{i}^{2}\omega_{i}^{4}(\gamma_{2}^{2}2\gamma_{1}\gamma_{3})\mu_{i}^{2}\omega_{i}^{2}\gamma_{1}^{2}\mu_{i}^{2}=0\, . \end{equation} $  (31) 
The procedure of calculating the upper bound
1) For
2) Calculate
3) Let
Theorem 4: Suppose that the parameters
Example 2: Consider the multiagent system used in Example 1. The nonzero eigenvalues of
Next, we compute
1) For
$ \begin{eqnarray*} \omega_{2}^{6}4\omega_{2}^{4}+3\omega_{2}^{2}1=0 \end{eqnarray*} $ 
and its positive solution is
2) For
$ \begin{eqnarray*} \omega_{3}^{6}16\omega_{3}^{4}+12\omega_{3}^{2}4=0 \end{eqnarray*} $ 
and its positive solution is
3) For
$ \begin{equation} \omega_{4}^{6}12\omega_{4}^{4}+9\omega_{4}^{2}3=0 \end{equation} $  (32) 
and its positive solution is
4) For
Therefore,
Using the same initial condition as in Example 1, Fig. 4 shows that consensus has been achieved when
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Fig. 4 States of five agents achieving consensus (Example 2, 
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Fig. 5 States of five agents not achieving consensus (Example 2, 
In this paper, we have considered the consensus algorithm for networked third order agents, and have established a necessary and sufficient condition (protocol) for designing the parameters in the control input. The key idea is to reduce the consensus problem into designing a Hurwitz polynomial with complex coefficients. The main result turns out to be a natural extension from second order consensus, and it does not require large computational efforts. Moreover, the discussion has also been extended to the case where a constant communication delay exists in the control inputs. A tight upper bound for the delay has been established as a necessary and sufficient condition under which consensus is maintained when communication delay is involved.
[1]  T. Vicsek, A. Czirók, E. BenJacob, I. Cohen, and O. Shochet, "Novel type of phase transition in a system of selfdriven particles, " Phys. Rev. Lett., vol. 75, no. 6, pp. 12261229, Aug. 1995. http://europepmc.org/abstract/MED/10060237 
[2]  R. OlfatiSaber, J. A. Fax, and R. M. Murray, "Consensus and cooperation in networked multiagent systems, " Proc. IEEE, vol. 95, no. 1, pp. 215233, Jan. 2007. http://ieeexplore.ieee.org/document/4118472/ 
[3]  J. Shamma, Cooperative Control of Distributed MultiAgent Systems. New York: Wiley, 2008. 
[4]  W. Ren and R. Beard, Distributed Consensus in MultiVehicle Cooperative Control: Theory and Applications. London: Springer, 2008. 
[5]  J. A. Fax and R. M. Murray, "Information flow and cooperative control of vehicle formations, " IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 14651476, Sep. 2004. http://www.sciencedirect.com/science/article/pii/S1474667015385219 
[6]  A. Jadbabaie, J. Lin, and A. S. Morse, "Coordination of groups of mobile autonomous agents using nearest neighbor rules, " IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 9881001, Jun. 2003. http://www.ams.org/mathscinetgetitem?mr=1986266 
[7]  L. Moreau, "Stability of multiagent systems with timedependent communication links, " IEEE Trans. Autom. Control, vol. 50, no. 2, pp. 169182, Feb. 2005. https://www.researchgate.net/publication/3032048_Stability_of_multiagent_systems_with_timedependent_communication_links 
[8]  W. Ren and R. W. Beard, "Consensus seeking in multiagent systems under dynamically changing interaction topologies, " IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 655661, May 2005. http://ieeexplore.ieee.org/document/1431045 
[9]  W. W. Yu, G. R. Chen, and M. Cao, "Some necessary and sufficient conditions for secondorder consensus in multiagent dynamical systems, " Automatica, vol. 46, no. 6, pp. 10891095, Jun. 2010. https://www.sciencedirect.com/science/article/pii/S0005109810001251 
[10]  W. Y. Hou, M. Y. Fu, H. S. Zhang, and Z. Z. Wu, "Consensus conditions for general secondorder multiagent systems with communication delay, " Automatica, vol. 75, pp. 293298, Jan. 2017. http://www.sciencedirect.com/science/article/pii/S0005109816303879 
[11]  Y. M. Xin, Y. X. Li, X. Huang, and Z. S. Cheng, "Consensus of thirdorder nonlinear multiagent systems, " Neurocomputing, vol. 159, pp. 8489, Jul. 2015. http://dl.acm.org/citation.cfm?id=2782194 
[12]  W. He and J. Cao, "Consensus control for highorder multiagent systems, " IET Control Theory Appl., vol. 5, no. 1, pp. 231238, Jan. 2011. https://www.mendeley.com/researchpapers/consensuscontrolhighordermultiagentsystems/ 
[13]  G. S. Zhai, S. H. Okuno, J. Imae, and T. Kobayashi, "A matrix inequality based design method for consensus problems in multiagent systems, " Int. J. Appl. Math. Comput. Sci., vol. 19, no. 4, pp. 639646, Dec. 2009. http://dl.acm.org/citation.cfm?id=1721644 
[14]  G. Zhai, J. Takeda, J. Imae, and T. Kobayashi, "Towards consensus in networked nonholonomic systems, " IET Control Theory Appl., vol. 4, no. 10, pp. 22122218, Oct. 2010. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5611741 
[15]  G. S. Zhai and C. Huang, "A note on basic consensus problems in multiagent systems with switching interconnection graphs". Int. J. Control , vol.88, no.3, pp.631–639, 2015. DOI:10.1080/00207179.2014.971431 
[16]  A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control. Switzerland: Springer, 2016. 
[17]  R. K. Upadhyay and S. R. K. Lyengar, Introduction to Mathematical Modeling and Chaotic Dynamics.. London: CRC Press, 2014. 
[18]  E. Frank, "On the zeros of polynomials with complex coefficients". Bull. Am. Math. Soc. , vol.52, no.2, pp.144–157, 1946. DOI:10.1090/S000299041946085262 
[19]  Z. H. Wang, J. J. Xu, and H. S. Zhang, "Consensus seeking for discretetime multiagent systems with communication delay, " IEEE/CAA J. of Autom. Sinica, vol. 2, no. 2, pp. 151157, Apr. 2015. http://www.ieeejas.org/EN/abstract/abstract54.shtml 
[20]  H. Xia, T. Z. Huang, J. L. Shao, and J. Y. Yu, "Group consensus of multiagent systems with communication delays, " Neurocomputing, vol. 171, pp. 16661673, Jan. 2016. http://www.sciencedirect.com/science/article/pii/S0925231215011571 
[21]  X. G. Yang, J. X. Xi, J. Y. Wu, and B. L. Yang, "Consensus transformation for multiagent systems with topology variances and timevarying delays, " Neurocomputing, vol. 168, pp. 10591064, Nov. 2015. http://dl.acm.org/citation.cfm?id=2824619 
[22]  M. Yu, C. Yan, D. M. Xie, and G. M. Xie, "Eventtriggered tracking consensus with packet losses and timevarying delays, " IEEE/CAA J. of Autom. Sinica, vol. 3, no. 2, pp. 165173, Apr. 2016. http://ieeexplore.ieee.org/document/7451104/ 
[23]  B. Zhou and Z. L. Lin, "Consensus of highorder multiagent systems with large input and communication delays, " Automatica, vol. 50, no. 2, pp. 452464, Feb. 2014. http://www.sciencedirect.com/science/article/pii/S0005109813005657 
[24]  B. Mohar, "The Laplacian spectrum of graphs, " Graph Theory, Combinatorics, and Applications, Y. Alavi, G. Chartrand, O. Ollermann, and A. Schwenk, eds. New York: Wiley, 1991, pp. 871898. http://www.researchgate.net/publication/233407318_The_Laplacian_spectrum_of_graphs 
[25]  A. J. Laub, Matrix Analysis for Scientists and Engineers. Philadelphia: SIAM, 2004. 