# Decentralized Robust Control of a Network of Inverter-Based Distributed Generation Systems
## 2 Methodology
### State Space Representation of the System in the Overlapping Decomposition Framework
#### 2.3.2 Extended State Space Model of the Network of IBDGs Using Overlapping Decomposition  
Consider the network of IBDGs including \(n\) inverters in Equations (3) and (4) with the quasi-adjacency matrix defined in Equation (1). To extend the state space model of the entire network, the overall extended state vector \(\bar{x}\) is defined as follows:  
\[\bar{x}=\left[\begin{array}{cccccc}\bar{x}_{1}^{T}&\bar{x}_{2}^{T}&\cdots& \bar{x}_{n}^{T}\end{array}\right]^{T}, \tag{9}\] \[\bar{x}_{i}=\left[\begin{array}{cccccc}x_{i}^{T}&x_{ik_{1}}^{T}&x _{ik_{2}}^{T}&\cdots&x_{ik}^{T}&\cdots&x_{ik_{alg(i)}}^{T}\end{array}\right]^ {T}\]  
where \(\bar{x}_{i}\) is the extended state vector of inverter \(\#i\) and \(x_{i}\) and \(x_{ik}\) (\(k\in\mathcal{N}(i)=\{k_{1},k_{2},...,k_{deg(i)}\}\)) are defined in Equation (4). Based on (9), the extended state space model of the entire network is provided as follows:  
\[\begin{array}{cccc}\bar{x}=\bar{A}\bar{x}+\bar{B}\bar{u}+\bar{D}\bar{d},\\ \bar{y}=\bar{C}\bar{x},\end{array} \tag{10}\]  
where the matrices \(\bar{A}\), \(\bar{B}\), \(\bar{C}\), and \(\bar{D}\) are expressed as follows:  
\[\bar{A}=\left[\begin{array}{cccc}\bar{A}_{1}&\bar{A}_{12}&\cdots&\bar{A}_{1 n}\\ \bar{A}_{21}&\bar{A}_{2}&\cdots&\bar{A}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \bar{A}_{n1}&\bar{A}_{n2}&\cdots&\bar{A}_{n}\end{array}\right], \tag{11}\] \[\bar{B}=diag(\bar{B}_{1},\bar{B}_{2},\cdots,\bar{B}_{n}),\] \[\bar{C}=diag(\bar{C}_{1},\bar{C}_{2},\cdots,\bar{C}_{n}),\] \[\bar{D}=diag(\bar{D}_{1},\bar{D}_{2},\cdots,\bar{D}_{n})\]  
where \(\bar{A}_{i}\), \(\bar{A}_{ik}\), \(\bar{B}_{i}\), \(\bar{C}_{i}\), and \(\bar{D}_{i}\) are formulated in Equations (12)-(14). The operator \"_diag_\" represents the block-diagonal combination of its arguments.  
\[\bar{A}_{i}=\left[\begin{array}{cccccc}0&-\frac{1}{L_{i}}&0&0&\cdots&0&\cdots &0\\ \frac{1}{C_{i}}&0&-\frac{s_{ik_{1}}}{C_{i}}&-\frac{s_{ik_{2}}}{C_{i}}&\cdots&- \frac{s_{ik_{alg(i)}}}{C_{i}}\\ 0&\frac{s_{ik_{1}}}{L_{ik_{1}}}&-\frac{R_{ik_{1}}}{L_{ik1}}&0&\cdots&0&\cdots&0 \\ 0&\frac{s_{ik_{2}}}{L_{ik_{2}}}&0&-\frac{R_{ik_{2}}}{L_{ik2}}&\cdots&0&\cdots&0 \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\frac{s_{ik}}{L_{ik}}&0&0&\cdots&-\frac{R_{ik_{1}}}{L_{ik_{1}}}&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&\frac{s_{ik_{gs(i)}}}{L_{ik_{gs(i)}}}&0&0&\cdots&0&\cdots&-\frac{R_{ik_{ dg(i)}}}{L_{ik_{dg(i)}}}\end{array}\right] \tag{12}\]\[\bar{A}_{ik}=\left[\begin{array}{ccccc}0&0&0&\cdots&0\\ 0&0&0&\cdots&0\\ 0&0&0&\cdots&0\\ 0&0&0&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&-\frac{s_{ik}}{L_{ik}}&0&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&0\end{array}\right] \tag{13}\]  
\[\bar{B}_{i}=\left[\begin{array}{c}\frac{1}{L_{i}}\\ 0\\ \vdots\\ 0\end{array}\right],\ \bar{C}_{i}=\left[\begin{array}{c}0\\ 1\\ \vdots\\ 0\end{array}\right]^{T},\ \bar{D}_{i}=\left[\begin{array}{c}0\\ -\frac{1}{c_{i}}\\ \vdots\\ 0\end{array}\right] \tag{14}\]  
In the next subsection, the extended matrix \(\bar{A}\) in Equation (11) is used to cluster the network of IBDGs into several subnetworks based on the relative norms of the off-diagonal blocks \(\bar{A}_{ik}\) as compared to the diagonal blocks \(\bar{A}_{i}\) in Equation (12).