# Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol +
## 2 Problem Formulation
### Agents' Dynamics and Coordination Protocol  
Investigate a discrete-time multi-agent system consisting of \(m\) leaders and \(n-m\) followers labelled by \(1,\ldots,m\) and \(m+1,\ldots,n\), respectively. The dynamic model of agent \(i\) is given by  
\[\begin{split} x_{i}(k+1)&=x_{i}(k),i=1,\ldots,m,\\ x_{i}(k+1)&=x_{i}(k)+u_{i}(k),i=m+1,\ldots,n,\end{split} \tag{1}\]  
where \(x_{i}(k)\in R^{p}\) and \(u_{i}(k)\in R^{p}\) are the state and control input of agent \(i\), respectively.  
According to the PI control strategy, we use the following PI coordination control algorithm for the first-order agents,  
\[\begin{split} u_{i}(k)&=\gamma_{1}\sum_{j=1}^{n}a_ {ij}(x_{j}(k)-x_{i}(k))+\gamma_{2}r_{i}(k),\\ r_{i}(k+1)&=r_{i}(k)+\sum_{j=m+1}^{n}a_{ij}(x_{j}( k)-x_{i}(k)),\\ i&=m+1,\ldots,n,\end{split} \tag{2}\]  
where \(a_{ij}\) is the \((i,j)\) entry of the adjacency matrix \(A\), \(r_{i}(k)\) represents the integral term, \(\gamma_{1}\) and \(\gamma_{2}\) are positive gain parameters to be decided for the proportional term and integral term, respectively.  
The states of leaders (1) remain static since the inputs of leaders are always zero, so we only investigate the followers' dynamics here. With the protocol (2), the dynamics of follower \(i\) are rewritten as  
\[\begin{split} x_{i}(k+1)&=x_{i}(k)+\gamma_{1}\sum_{ j=1}^{n}a_{ij}(x_{j}(k)-x_{i}(k))+\gamma_{2}r_{i}(k),\\ r_{i}(k+1)&=r_{i}(k)+\sum_{j=m+1}^{n}a_{ij}(x_{j}( k)-x_{i}(k)),\\ i&=m+1,\ldots,n.\end{split} \tag{3}\]  
In order to analyze the convergence performance of system (3), we introduce two topologies, one of which is named as leader-follower topology composed of the leaders and followers, and the other one is named as follower topology composed of the followers. The Laplacian matrix of leader-follower topology is \(L\) given by  
\[L=\begin{bmatrix}0_{m\times m}&0_{m\times(n-m)}\\ L_{1}&L_{2}\end{bmatrix},\]  
where \(L_{1}\) represents the topology between leaders and followers, and \(L_{2}\) represents the topology among followers. Meanwhile, the Laplacian matrix corresponding to the follower topology is \(L_{F}\) formulated as  
\[L_{F}=L_{2}-D,\]  
where the diagonal matrix \(D\) is defined as  
\[D=diag\{\sum_{j=1}^{m}a_{m+1,j},\ldots,\sum_{j=1}^{m}a_{nj}\}.\]  
Generally, we need the basic assumption on the leader-follower topology of system (3) as follows.  
**Assumption 1**.: _For each of the followers in the leader-follower topology, there is at least one leader that has a directed path to the follower._  
On the basis of Assumption 1, we make further assumptions on the follower topology as follows.  
**Assumption 2**.: _The follower topology has a directed spanning tree._  
**Assumption 3**.: _The edges of the follower topology are bidirectional, i.e., Laplacian matrix \(L_{F}\) is symmetric, and it has a directed spanning tree._  
Then we have the following lemma.  
**Lemma 1** ([38]).: _The Laplacian matrix \(L_{F}\) of the follower topology has a simple eigenvalue \(0\) with the corresponding eigenvector \([1,1,\dots,1]^{T}\) and all the other eigenvalues have positive real parts if and only if the topology satisfies Assumption 2._