Weak synchronization in heterogeneous multi-agent systems

Multi-agent systems have been extensively studied over the past 20 years. Initiated by early work such as [7, 5], although the roots can be found in much earlier work [15], it has become an active research area. But the realization that control systems often consist of many components with limited or restricted communication between them was already studied in the area of decentralized control, see e.g. [9, 1]. Applications are for instance systems with many generators connected through a grid or traffic applications such as platoons of cars. The fallacy of early decentralized control is that it often created a specific agent which has a kind of supervisory role while other agents ensure communication to and from this supervisory agent. This approach turned out to be highly sensitive to failures in the network. Multi-agent systems created a different type of structure in these networks where all agents basically have a similar role towards achieving synchronization in the network. However, early work still heavily relied on knowledge of the network.

Later it was established that the protocols designed for a multi-agent systems would work for any network structure satisfying some underlying assumptions such as lower or upper bounds on the spectrum of the Laplacian matrix associated to the graph describing the network structure. This suggested some form of robustness against changes in the network. However, this idea still has two major flaws:

• •

  Firstly, if the network is unknown, then we can never check whether these assumptions are actually satisfied and hence we do not know whether we will achieve synchronization.

• •

  Secondly, changes in the network can have significant effect on the bounds that have been used. It is easily seen that a few links failing might yields a network that fails connectivity properties. But also [12] showed that these lower bounds on the eigenvalues of the Laplacian almost always converge to zero when the network gets large makes these assumptions impossible to guarantee.

In recent years scale-free protocols have been studied, see for instance [3] and references therein. These protocols get rid of all assumptions on the network such as these bounds on the eigenvalues of the Laplacian. However, it still requires that the network is strongly connected or has a direct spanning tree. This actually still inherently has some of the difficulties presented before. How can we check if this connectivity is present in the network? Secondly, what happens in case of a fault in the network that makes the network fail this assumption.

In the basic setup of a multi-agent system, the signals exchanged over the network converge to zero whenever the network synchronizes. So the fact that the network communication dies out over time is a weaker condition than output synchronization. We will refer to this weaker condition as weak synchronization in this paper. We will consider heterogeneous agents in this paper but the concept equally applies to homogeneous networks.

It turns out that if we have a linear scale-free protocol then synchronization implies weak synchronization. But, more importantly, if the network has a directed spanning tree then the converse implication is true: weak synchronization implies classical synchronization.

We can therefore design protocols which achieve weak synchronization for any network without making any kind of assumptions. If the network happens to have a directed spanning tree then we obtain classical synchronization. However, if this is not the case then we describe in detail in this paper what kind of synchronization properties are preserved in the system. For applications this kind of weak synchronization what one would hope for. If the cars in a platoon lose connectivity between two subgroups because their distance has become too large the protocols will still achieve synchronization in both of these groups. If in a power system the connectivity between two subgroups is lost, each of these groups will internally achieve synchronization but, obviously, no global synchronization will be achieved.

To describe the information flow among the agents we associate a weighted graph $\mathcal{G}$ to the communication network. The weighted graph $\mathcal{G}$ is defined by a triple $(\mathcal{V},\mathcal{E},\mathcal{A})$ where $\mathcal{V}=\{1,\ldots,N\}$ is a node set, $\mathcal{E}$ is a set of pairs of nodes indicating connections among nodes, and $\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ is the weighted adjacency matrix with non negative elements $a_{ij}$. Each pair in $\mathcal{E}$ is called an edge, where $a_{ij}>0$ denotes an edge $(j,i)\in\mathcal{E}$ from node $j$ to node $i$ with weight $a_{ij}$. Moreover, $a_{ij}=0$ if there is no edge from node $j$ to node $i$. We assume there are no self-loops, i.e. we have $a_{ii}=0$. A path from node $i_{1}$ to $i_{k}$ is a sequence of nodes $\{i_{1},\ldots,i_{k}\}$ such that $(i_{j},i_{j+1})\in\mathcal{E}$ for $j=1,\ldots,k-1$. A directed tree is a subgraph (subset of nodes and edges) in which every node has exactly one parent node except for one node, called the root, which has no parent node. A directed spanning tree is a subgraph which is a directed tree containing all the nodes of the original graph. If a directed spanning tree exists, the root of this spanning tree has a directed path to every other node in the network [2].

For a weighted graph $\mathcal{G}$, the matrix $L=[\ell_{ij}]$ with

$$\ell_{ij}=\left\{\;\begin{array}[]{cl}\sum_{k=1}^{N}a_{ik},&i=j,\\ -a_{ij},&i\neq j,\end{array}\right.$$

is called the Laplacian matrix associated with the graph $\mathcal{G}$. The Laplacian matrix $L$ has all its eigenvalues in the closed right half plane and at least one eigenvalue at zero associated with right eigenvector 1 [2]. The zero eigenvalues of Laplacian matrix is always semi simple, i.e. its algebraic and geometric multiplicities coincides. Moreover, if the graph contains a directed spanning tree, the Laplacian matrix $L$ has a single eigenvalue at the origin and all other eigenvalues are located in the open right-half complex plane [8].

A directed communication network is said to be strongly connected if it contains a directed path from every node to every other node in the graph. For a given graph $\mathcal{G}$ every maximal (by inclusion) strongly connected subgraph is called a bicomponent of the graph. A bicomponent without any incoming edges is called a basic bicomponent. Every graph has at least one basic bicomponent. A network has one unique basic bicomponent if and only if the network contains a directed spanning tree. In general, every node in a network can be reached by at least one basis bicomponent, see [10, page 7]. In Fig. 1 a directed communication network with its bicomponents is shown. The network in this figure contains 6 bicomponents, 3 basic bicomponents (the blue ones) and 3 non-basic bicomponents (the yellow ones). In Fig. 2 a directed communication network with its bicomponents is shown. The network in this figure contains 4 bicomponents but only one basic bicomponent (the blue one).

In the absence of a directed spanning tree, the Laplacian matrix of the graph has an eigenvalue at the origin with a multiplicity $k$ larger than $1$. This implies that it is a $k$-reducible matrix and the graph has $k$ basic bicomponents. The book [14, Definition 2.19] shows that, after a suitable permutation of the nodes, a Laplacian matrix with $k$ basic bicomponents can be written in the following form:

$$L=\begin{pmatrix}L_{0}&L_{01}&\cdots&\cdots&L_{0k}\\ 0&L_{1}&0&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&&\ddots&L_{k-1}&0\\ 0&\cdots&\cdots&0&L_{k}\end{pmatrix}$$

(1)

where $L_{1},\ldots,L_{k}$ are the Laplacian matrices associated to the $k$ basic bicomponents in our network. These matrices have a simple eigenvalue in $0$ because they are associated with a strongly connected component. On the other hand, $L_{0}$ contains all non-basic bicomponents and is a grounded Laplacian with all eigenvalues in the open right-half plane. After all, if $L_{0}$ would be singular then the network would have an additional basic bicomponent.

In this section, we introduce the concept of weak synchronization for heterogeneous MAS. Consider $N$ heterogeneous agents

$$\begin{array}[]{cl}x_{i}^{+}&=A_{i}x_{i}+B_{i}u_{i},\\ y_{i}&=C_{i}x_{i},\end{array}$$

(2)

where $x_{i}\in\mathbb{R}^{n_{i}}$, $u_{i}\in\mathbb{R}^{m_{i}}$ and $y_{i}\in\mathbb{R}^{p}$ are the state, input, output of agent $i$th for $i=1,\ldots,N$. In the aforementioned presentation, for continuous-time systems, $x_{i}^{+}(t)=\dot{x}_{i}(t)$ with $t\in\mathbb{R}$ while for discrete-time systems, $x_{i}^{+}(t)=x_{i}(t+1)$ with $t\in\mathbb{Z}$.

The communication network provides agent $i$ with the following information which is a linear combination of its own output relative to that of other agents

$$\zeta_{i}=\sum_{j=1}^{N}a_{ij}(y_{i}-y_{j}),$$

(3)

where $a_{ij}\geqslant 0$ and $a_{ii}=0$. The communication topology of the network can be described by a weighted and directed graph $\mathcal{G}$ with nodes corresponding to the agents in the network and the weight of edges given by coefficient $a_{ij}$. In terms of the coefficients of the associated Laplacian matrix $L$, $\zeta_{i}$ can be rewritten as

$$\zeta_{i}=\sum_{j=1}^{N}\ell_{ij}y_{j}.$$

(4)

We denote by $\mathbb{G}^{N}$ the set of all graphs with $N$ nodes. We also introduce a possible additional localized information exchange among agents and their neighbors, i.e. each agent $i\in\{1,\ldots,N\}$ has access to the localized information, denoted by $\hat{\zeta}_{i}$, of the form

$$\hat{\zeta}_{i}=\sum_{j=1}^{N}a_{ij}(\eta_{i}-\eta_{j}),$$

(5)

where $\eta_{i}$ is a variable produced internally by the protocol of agent $i$ and to be defined later. Finally we might have introspective agents which implies that

$$y_{i,m}=C_{i,m}x_{i}$$

(6)

is available to the protocol.

Our protocols are of the form:

$$\begin{array}[]{ccl}\xi_{i}^{+}&=&K_{i}\xi_{i}+L_{i}\zeta_{i}+L_{i,e}\hat{\zeta}_{i}+L_{i,m}y_{i,m}\\ u_{i}&=&M_{i}\xi_{i}\\ \eta_{j}&=&N_{i}\xi_{i}\end{array}$$

(7)

For an agent $i$ which is introspective we might have $L_{i,m}\neq 0$ while for non-introspective agents we have that $L_{i,m}=0$. Similar, for an agent $i$ where extra communication is available of the form (5) we might have $L_{i,e}\neq 0$ while for agents without extra communication we have $L_{i,e}=0$.

In the following, we introduce the concepts of network stability and weak synchronization that is vastly different from output synchronization.

Next, we present two lemmas to explain the difference between these two kind of synchronization

If we achieve weak synchronization we have that

$$\zeta(t)=(I\otimes L)y(t)\rightarrow 0$$

as $t\rightarrow\infty$ where:

$$\zeta=\begin{pmatrix}\zeta_{1}\\ \vdots\\ \zeta_{N}\end{pmatrix},\qquad y=\begin{pmatrix}y_{1}\\ \vdots\\ y_{N}\end{pmatrix},\qquad$$

This implies

$$(I\otimes V)\zeta(t)=\begin{pmatrix}y_{1}(t)-y_{N}(t)\\ \vdots\\ y_{N-1}-y_{N}(t)\end{pmatrix}\rightarrow 0$$

as $t\rightarrow\infty$ which clearly implies output synchronization is achieved.

Next we consider the case that the network doesn’t contain a directed spanning tree. In that case, we have at least two basic bicomponents. By construction the behavior within a basic bicomponent is not influenced by the behavior of the rest of the network. Moreover, within this basic bicomponent we have a strongly connected graph. Hence within a basic bicomponent weak synchronization implies output synchronization. Assume within a basic bicomponent consisting of nodes $k_{1},\ldots,k_{i}$ we have, for certain initial conditions, output synchronization such that

$$y_{k_{a}}(t)-y_{\text{ss}}(t)\rightarrow 0,\qquad y_{\text{ss}}(t)\nrightarrow 0$$

as $t\rightarrow\infty$ for all $a\in 1,\ldots,i$. Take a node $j$ within another basic bicomponent. If we have output synchronization then this would imply:

$$y_{k_{a}}(t)-y_{j}(t)\rightarrow 0$$

as $t\rightarrow\infty$ which is equivalent to

$$y_{j}(t)-y_{\text{ss}}(t)\rightarrow 0$$

(9)

as $t\rightarrow\infty$. But if we multiply all initial conditions in the first basic bicomponent by a factor $2$ we get:

$$y_{k_{a}}(t)-2y_{\text{ss}}(t)\rightarrow 0$$

(10)

as $t\rightarrow\infty$. On the other hand, if we keep the initial conditions for the second basic bicomponent the same then we would still have (9). Clearly, (9) and (10) contradict output synchronization given that $y_{\text{ss}}(t)\nrightarrow 0$ as $t\rightarrow\infty$. Therefore we obtain our result by contradiction.  

In this section, we focus on the behavior of the output of the agents of a MAS for given protocols which achieves weak synchronization. We have following theorem.

Since the matrices $L_{1},\ldots,L_{k}$ have a one-dimensional nullspace while $L_{0}$ is nonsingular, there exists $\{\beta_{1},\ldots\beta_{k}\}\in\mathbb{R}^{M_{0}}$ such that the nullspace of $L$ is given by the image of the following matrix:

$$\begin{pmatrix}\beta_{1}&\beta_{2}&\cdots&\beta_{k}\\ \textbf{1}_{M_{1}}&0&\cdots&0\\ 0&\textbf{1}_{M_{2}}&\ddots&\vdots\\ \vdots&\ddots&\ddots&0\\ 0&\cdots&0&\textbf{1}_{M_{k}}\end{pmatrix}$$

(13)

We define scalars $\beta_{1,i},\ldots,\beta_{M_{0},i}$ such that:

$$\beta_{i}=\begin{pmatrix}\beta_{1,i}\\ \vdots\\ \beta_{M_{0},i}\end{pmatrix}.$$

Since we know $\textbf{1}_{N}$ is an element of the nullspace of $L$ we find that:

$$\sum_{i=1}^{k}\beta_{i}=\textbf{1}_{M_{0}}$$

which yields (12). Using that

$$\beta_{i}=-L_{0}^{-1}L_{0i}\textbf{1}_{M_{i}},$$

we find that all coefficients of $\beta_{i}$ are nonnegative. This follows since the structure of the Laplacian guarantees that all coefficients of $L_{0i}$ are nonpositive while all coefficients of $L_{0i}^{-1}$ are nonnegative. The latter follows from [6, Theorem 4.25].

Assume $\tau^{i}_{1},\ldots\tau^{i}_{M_{i}}$ are the agents contained in basic bicomponent $i$. Then

$$(I\otimes L)\begin{pmatrix}y_{1}(t)\\ \vdots\\ y_{N}(t)\end{pmatrix}\rightarrow 0$$

(which follows from weak synchronization) implies:

$$(I\otimes L_{i})\begin{pmatrix}y_{\tau^{i}_{1}}(t)\\ \vdots\\ y_{\tau^{i}_{M_{i}}}(t)\end{pmatrix}\rightarrow 0$$

and since $L_{i}$ is strongly connected we find:

$$y_{\tau^{i}_{a}}-y_{\tau^{i}_{b}}(t)\rightarrow 0$$

as $t\rightarrow\infty$ which implies the first bullet point of the theorem if we set $y^{i}_{s}=y^{i}_{\tau^{i}_{1}}$.

Next we consider an agent not part of any of the basic bicomponents. Define the following matrix:

$$B=\begin{pmatrix}\beta_{11}M_{1}^{-1}\textbf{1}_{M_{1}}^{\mbox{\tiny T}}&\quad\cdots\quad&\beta_{1k}M_{k}^{-1}\textbf{1}_{M_{k}}^{\mbox{\tiny T}}\\ \vdots&&\vdots\\ \beta_{M_{0}1}M_{1}^{-1}\textbf{1}_{M_{1}}^{\mbox{\tiny T}}&\quad\cdots\quad&\beta_{1k}M_{k}^{-1}\textbf{1}_{M_{k}}^{\mbox{\tiny T}}\end{pmatrix}$$

(14)

Then it is easy to verify from the structure of the kernel of $L$ presented in (13) that

$$\operatorname{ker}L\subset\ker\begin{pmatrix}-I&B\end{pmatrix}.$$

which implies that there exists a matrix $V$ such that

$$VL=\begin{pmatrix}-I&B\end{pmatrix}.$$

(15)

but then weak synchronization implies that

$$(I\otimes VL)y(t)\rightarrow 0$$

Using (14) and (15) this implies:

$$y_{j}(t)-\beta_{j1}M_{1}^{-1}\sum_{j=1}^{M_{1}}y_{\tau^{1}_{j}}(t)-\beta_{j2}M_{2}^{-1}\sum_{j=1}^{M_{2}}y_{\tau^{2}_{j}}(t)-\\ \cdots-\beta_{jk}M_{k}^{-1}\sum_{j=1}^{M_{k}}y_{\tau^{k}_{j}}(t)\rightarrow 0$$

as $t\rightarrow\infty$ for $j=1,\ldots,M_{0}$. Using that $y_{\tau^{i}_{j}}(t)-y^{i}_{s}(t)\rightarrow 0$ established earlier this yields:

$$y_{j}(t)-\beta_{j1}y_{\tau^{1}_{s}}(t)-\beta_{j2}y_{\tau^{2}_{s}}(t)-\cdots-\beta_{jk}y_{\tau^{k}_{s}}(t)\rightarrow 0$$

as $t\rightarrow\infty$ for $j=1,\ldots,M_{0}$ which yields the second bullet point of the theorem.  

Since it is known that many multi-agent systems suffer from scale fragility it is desirable if our protocol (7) for the agent (2) to be scale free. This implies that the protocol (7) for agent $i$ must be designed only based on agent model $i$. This is formally defined below:

Scale free designs are used in the context that the network is not known. But this makes it hard to verify the assumption that the network has a directed spanning tree since verifying this intrinsically requires knowledge of the network. In that sense, the concept of scale-free weak synchronization is more appropriate:

The next objective of this paper is to show that protocols that achieve scale-free output synchronization as defined in Definition 3, also achieve weak synchronization in the absence of a spanning tree due to a fault.

Remains to establish that we obtain scale-free weak synchronization if we know that we have achieved scale-free output synchronization. If we look at the interconnection of (2) and (7) we can write this of the form:

$$\begin{array}[]{ccl}x_{e,i}^{+}&=&\tilde{A}_{i}x_{e,i}+\tilde{B}_{i}\tilde{\zeta}_{i}\\ y_{i}&=&\tilde{C}_{i}x_{e,i}\\ z_{i}&=&\tilde{H}_{i}x_{e,i}\end{array}$$

(16)

with

$$\tilde{\zeta}_{i}=\sum_{j=1}^{N}\ell_{i,j}z_{j}$$

(17)

	$\displaystyle\tilde{A}_{i}$	$\displaystyle=\begin{pmatrix}A_{i}&B_{i}M_{i}\\ L_{i,m}C_{i,m}&K_{i}\end{pmatrix},\tilde{B}_{i}=\begin{pmatrix}0&0\\ L_{i}&L_{i,e}\end{pmatrix},$
	$\displaystyle\tilde{C}_{i}$	$\displaystyle=\begin{pmatrix}C_{i}&0\end{pmatrix},\tilde{H}_{i}=\begin{pmatrix}C_{i}&0\\ 0&N_{i}\end{pmatrix}$

If we define

	$\displaystyle\tilde{A}$	$\displaystyle=\operatorname{diag}\{\tilde{A}_{1},\ldots,\tilde{A}_{N}\},\qquad\tilde{B}=\operatorname{diag}\{\tilde{B}_{1},\ldots,\tilde{B}_{N}\},$
	$\displaystyle\tilde{C}$	$\displaystyle=\operatorname{diag}\{\tilde{C}_{1},\ldots,\tilde{C}_{N}\},\qquad\tilde{H}=\operatorname{diag}\{\tilde{H}_{1},\ldots,\tilde{H}_{N}\}$

and

$$x_{e}=\begin{pmatrix}x_{e,1}\\ \vdots\\ x_{e,N}\end{pmatrix}\qquad y=\begin{pmatrix}y_{1}\\ \vdots\\ y_{N}\end{pmatrix},$$

then we can write the complete system as:

$$\begin{array}[]{ccl}{x}_{e}^{+}&=&[\tilde{A}+\tilde{B}(L\otimes\tilde{H})]x_{e}\\ y&=&\tilde{C}x_{e}\end{array}$$

(18)

We next consider the following $k$ differential equations:

$$\begin{array}[]{ccl}(x^{i}_{e})^{+}&=&[\tilde{A}+\tilde{B}(L\otimes\tilde{H})]x^{i}_{e}\\ y^{i}&=&\tilde{C}x_{e}^{i}\end{array}$$

(19)

for $i=1,\ldots,k$ where

$$x^{i}_{e}=\begin{pmatrix}x^{i}_{e,1}\\ \vdots\\ x^{i}_{e,N}\end{pmatrix}\qquad y^{i}=\begin{pmatrix}y_{1}^{i}\\ \vdots\\ y^{i}_{N}\end{pmatrix}.$$

Each of these systems is kind of connected to one of the basic bicomponents through its initial conditions. In particular, assume agent $j$ is part of a basic bicomponent $i$ then we choose

$$\left\{\;\begin{array}[]{ll}x^{v}_{e,j}(0)=x_{e,j}(0),&\qquad v=i,\\ x^{v}_{e,j}(0)=0,&\qquad v\neq i,\end{array}\right.$$

and hence

$$x^{1}_{e,j}(0)+\cdots+x^{k}_{e,j}(0)=x_{e,j}(0).$$

On the other hand, if agent $j$ is not part of any basic bicomponent, then there is at least one bicomponent $i$ from which agent $j$ can be reached.

Assume that $i$ is the bicomponent with smallest index with this property (which implies $\beta_{j,i}\neq 0$). then we choose

$$\left\{\;\begin{array}[]{ll}x^{v}_{e,j}(0)=x_{e,j}(0),&\qquad v=i,\\ x^{v}_{e,j}(0)=0,&\qquad v\neq i,\end{array}\right.$$

and hence again

$$x^{1}_{e,j}(0)+\cdots+x^{k}_{e,j}(0)=x_{e,j}(0).$$

In other words, we have

$$\begin{pmatrix}x^{1}_{e,1}(0)+\cdots+x^{k}_{e,1}(0)\\ x^{1}_{e,2}(0)+\cdots+x^{k}_{e,2}(0)\\ \vdots\\ x^{1}_{e,N}(0)+\cdots+x^{k}_{e,N}(0)\end{pmatrix}=\begin{pmatrix}x_{e,1}(0)\\ x_{e,2}(0)\\ \vdots\\ x_{e,N}(0)\end{pmatrix}$$

It means that we have

$$x^{1}_{e}(0)+\cdots+x^{k}_{e}(0)=x_{e}(0)$$

with

$$x_{e}(0)=\begin{pmatrix}x_{e,1}(0)\\ x_{e,2}(0)\\ \vdots\\ x_{e,N}(0)\end{pmatrix}$$

Using equation (19) we obtain

$$\begin{array}[]{l}(x^{1}_{e}+\cdots+{x}^{k}_{e})^{+}=[\tilde{A}+\tilde{B}(L\otimes\tilde{H})](x^{1}_{e}+\cdots+x^{k}_{e})\\ (y^{1}+\cdots+y^{n})=\tilde{C}(x^{1}_{e}+\cdots+x^{k}_{e})\end{array}$$

(20)

But from equation (18) and (20) we see that $x^{1}_{e}+\cdots+x^{k}_{e}$ and $x_{e}$ satisfy the same differential equation and have the same initial condition. It implies that

$$x_{e}=x^{1}_{e}+\cdots+x^{k}_{e},\qquad y=y^{1}+\cdots+y^{k}$$

(21)

Next consider $x^{i}_{e}$. Define:

$$\begin{pmatrix}\gamma^{i}_{1}\\ \vdots\\ \gamma^{i}_{N}\end{pmatrix}$$

as the $i$’th column of the matrix (13). This implies

$$\sum_{v=1}^{N}\ell_{j,v}\gamma^{i}_{v}=0$$

(22)

Consider an agent $j$ for which $\gamma^{i}_{j}=0$. Then we note that all terms of the summation are nonpositive since the $\gamma^{i}_{v}$ are nonnegative and the $\ell_{j,v}$ for $v\neq j$ are nonpositive. This implies:

$$\ell_{j,v}\gamma^{i}_{v}=0\quad\text{ for }v=1,\ldots,N.$$

(23)

This yields that an agent $j$ for which $\gamma_{j}=0$ only depends on other agents $v$ (i.e. $\ell_{j,v}\neq 0$) for which $\gamma_{v}=0$. But since all agents for which $\gamma_{v}=0$ satisfy, by construction, $x^{i}_{e,j}(0)=0$ this yield that $x^{i}_{e,j}(t)=0$ for all $t>0$.

We consider all agents $\tau^{i}_{1},\ldots,\tau^{i}_{N_{i}}$ for which the corresponding $\gamma_{v}$ is nonzero, i.e.

$$\gamma_{\tau^{i}_{j}}\neq 0\quad\text{ for }j=1,\ldots,N_{i}$$

(24)

and let $L_{i}$ denote the matrix obtained from $L$ by deleting both columns and rows whose index is not contained in the set $\tau^{i}_{1},\ldots,\tau^{i}_{N_{i}}$. It can be shown that this matrix has rank $N_{i}-1$. After all, we already know

$$\begin{pmatrix}\gamma_{\tau^{i}_{1}}\\ \vdots\\ \gamma_{\tau^{i}_{N_{i}}}\end{pmatrix}$$

(25)

is contained in the null space of $L_{i}$. If, additionally,

$$\begin{pmatrix}\eta_{\tau^{i}_{1}}\\ \vdots\\ \eta_{\tau^{i}_{N_{i}}}\end{pmatrix}$$

is in the null space of $L_{i}$ and linearly independent of (25) then it is easily verified that

$$\eta=\begin{pmatrix}\eta_{1}\\ \vdots\\ \eta_{N}\end{pmatrix}$$

is in the null space of $L$ where we have chosen $\eta_{i}=0$ when $i\not\in\{\tau^{i}_{1},\ldots,\eta_{\tau^{i}_{N_{i}}}\}$. Here we use that (23) implies that

$$\ell_{j,v}\eta_{v}=0\quad\text{ for }v=1,\ldots N.$$

for any $j$ for which $\gamma_{j}=0$ (recall that $\ell_{j,v}\neq 0$ implies $\gamma_{v}\neq 0$). But given the structure of the null space of $L$ given by (13), $L\eta=0$ yields a contradiction.

Note that $L_{i}$ has the structure of a Laplacian except for the zero row sum. However, the matrix:

$$\Gamma=\operatorname{diag}\{\gamma_{\tau^{i}_{1}},\ldots,\gamma_{\tau^{i}_{N_{i}}}\}$$

is invertible and

$$\tilde{L}_{i}=\Gamma^{-1}L_{i}\Gamma$$

is a classical Laplacian matrix with zero row sum. Moreover, it contains a directed spanning tree since its rank is equal to $N_{i}-1$.

We consider agents $\tau^{i}_{1},\ldots,\tau^{i}_{N_{i}}$. Using the above, we obtain that

$$\begin{array}[]{ccl}x_{s,e}^{+}&=&[\tilde{A}_{s}+\tilde{B}_{s}(L_{i}\otimes\tilde{H})]x_{s,e}\\ y_{s}&=&\tilde{C}x_{s,e}\end{array}$$

(26)

where

	$\displaystyle\tilde{A}_{s}$	$\displaystyle=\operatorname{diag}\{\tilde{A}_{\tau^{i}_{1}},\ldots,\tilde{A}_{\tau^{i}_{N_{i}}}\},\qquad\tilde{B}_{s}=\operatorname{diag}\{\tilde{B}_{\tau^{i}_{1}},\ldots,\tilde{B}_{\tau^{i}_{N_{i}}}\},$
	$\displaystyle\tilde{C}_{s}$	$\displaystyle=\operatorname{diag}\{\tilde{C}_{\tau^{i}_{1}},\ldots,\tilde{C}_{\tau^{i}_{N_{i}}}\},\qquad\tilde{H}_{s}=\operatorname{diag}\{\tilde{H}_{\tau^{i}_{1}},\ldots,\tilde{H}_{\tau^{i}_{N_{i}}}\}$

and

$$x_{s,e}=\begin{pmatrix}x^{i}_{e,\tau^{i}_{1}}\\ \vdots\\ x^{i}_{e,\tau^{i}_{N_{i}}}\end{pmatrix}\qquad y_{s}=\begin{pmatrix}y^{i}_{\tau^{i}_{1}}\\ \vdots\\ y^{i}_{\tau^{i}_{N_{i}}}\end{pmatrix}.$$

But then we obtain:

$$\begin{array}[]{ccl}\tilde{x}_{s,e}^{+}&=&[\tilde{A}_{s}+\tilde{B}_{s}(\tilde{L}_{i}\otimes\tilde{H}_{s})]\tilde{x}_{s,e}\\ \tilde{y}_{s}&=&\tilde{C}_{s}\tilde{x}_{s,e}\end{array}$$

(27)

where $\tilde{x}_{s,e}=(\Gamma^{-1}\otimes I)x_{s,e}$ and $\tilde{y}_{s}=(\Gamma^{-1}\otimes I)y_{s}$. Since our protocol achieved scale-free synchronization and the network associated to $\tilde{L}_{i}$ contains a directed spanning tree we obtain output synchronization using Definition 3 and therefore also weak synchronization by Lemma 1, i.e.

$$(\tilde{L}_{i}\otimes I)\tilde{y}_{s}\rightarrow 0$$

as $t\rightarrow\infty$ which implies

$$(L_{i}\otimes I)y_{s}\rightarrow 0$$

Given the way we constructed $y_{s}$, this implies:

$$(L\otimes I)y^{i}\rightarrow 0$$

Since this is true for $i=1,\ldots,k$ we can use (21) to establish:

$$(L\otimes I)y\rightarrow 0$$

In other words, we achieve weak synchronization since the above derivation is valid for all possible initial conditions.  

In this section, we consider a special case of protocol (7): all agent models are introspective and protocols are collaborative. We choose the existing examples in both continuous- and discrete-time presented in [4] and [13].

In this paper we have introduced the concept of weak synchronization for MAS. We have shown that this is the right concept if you have no information available about the network. If we have a directed spanning tree it is equal to the classical concept of output synchronization. However when, due to a fault, the network no longer contains a directed spanning tree then we still achieve the best synchronization properties possible for the given network. We have seen that the protocols guarantee a stable response to these faults: within basic bicomponents we still achieve synchronization and the outputs of the agents not contained in a basic bicomponent converge to a convex combination of the asymptotic behavior achieved in the basic bicomponents. This behavior is completely independent of the specific scale-free protocols being used.

For the heterogeneous MAS that we consider in this paper, the main focus improving the available protocol design methodologies since for heterogeneous networks the current designs are still limited in scope but the concept of weak synchronization introduced in this paper is the correct concept for this protocol design.