## 3 Control and Structure Design
### Structure Design by Sparse Neural Networks  
The structure design involves the placement of sensor, actuator, and communication topology. It is formulated by the neural network weight \(W_{i}\). The weight matrix is separated into column groups, so we can define the radius basis \(\phi\) as  
\[\phi= \left(\begin{array}{cccc}\phi_{1}{}^{T}&\phi_{2}{}^{T}&\cdots&\phi_{n }{}^{T}\end{array}\right)^{T}, \tag{18}\]  
where \(\phi_{i}=\left(\begin{array}{cccc}e^{-\frac{\|x_{i}-x_{d}\|^{2}}{e^{2}}}&e^ {-\frac{\|x_{i}-x_{d}\|^{2}}{e^{2}}}&\cdots&e^{-\frac{\|x_{i}-x_{d}\|^{2}}{e^{2 }}}\end{array}\right)^{T}\), \(x_{cj}\) is a given radius center, \(j\in N^{+}\cap[1,l]\), and \(\sigma\) is a given radius width. According to the definition of \(\phi\), we know that the neural weight is separated into \(n\) column groups. Each group that has \(l\) columns corresponds to the radius basis \(\phi_{i}\) of one area. If the weight matrix \(W_{i}\) is group sparse [28] with respect to these columns, the structure of control system can be figured out. For instance, if the column group \(j\) of \(W_{i}\) corresponding to the gain of \(\phi_{i}\) is zero, the controller of area \(i\) does not need the information from area \(j\), and the communication channel between area \(j\) and \(i\) is also not necessary. If all the controllers do not require the information from area \(j\), any sensor is not necessary for area \(j\). If \(W_{i}=0\), it means that the actuator in area \(i\) is not required. The key is to force \(W_{i}\) to be column group sparse. Thus, a group sparse regulation term is added to the loss function. Thus, the loss function becomes  
\[V_{i}(k)= \frac{1}{2}\left[\left\|B_{i}\hat{a}_{i}(k)-B_{i}u_{di}(k)\right\| ^{2}+\left\|\hat{Q}_{i}(k)\right\|^{2}\right]+ \tag{19}\] \[\frac{1}{2}\left\|\alpha^{N+1}p_{i}(k)+\alpha^{-1}\hat{Q}_{i}(k+1 )-\hat{Q}_{i}(k)\right\|^{2}+ \gamma_{i}\sum\limits_{j=1}^{n}\left\|vec\left(W_{iG(j)}\right)\right\|,\]  
where \(W_{iG(j)}\) denotes the column group \(j\) or the gain of radius basis \(\phi_{j}\). \(\gamma_{i}\) is the regulation weight. The sparse regulated learning iteration is demonstrated in Equation (20), which can be also derived by Equation (17):  
\[\left\{\begin{array}{l}W_{i}(k+1)=W_{i}(k)-\beta_{i}\frac{\partial V_{i}(k)} {\partial W_{i}(k)}\\ \frac{\partial V_{i}(k)}{\partial W_{i}(k)}=\left\{M_{a}^{\mathrm{T}}\left[x_ {i}(k+1)-L_{i}x_{i}(k)\right]+M_{c}^{\mathrm{T}}\left[\alpha^{N+1}p_{i}(k)+ \alpha^{-1}\hat{Q}_{i}(k+1)\right]\right\}\phi(k)^{T}\\ \hskip 14.226378 pt+\gamma_{i}W_{i}(k)D_{i}(k),\end{array}\right. \tag{20}\]where \(D_{i}=diag\left[\begin{array}{cc}\mathbf{1}\Big{\|}vec(W_{iG(1)})\Big{\|}^{-1}& \mathbf{1}\Big{\|}vec(W_{iG(2)})\Big{\|}^{-1}&\cdots&\mathbf{1}\Big{\|}vec(W_{ iG(n)})\Big{\|}^{-1}\end{array}\right]\), and \(\mathbf{1}{=[1,1,...,1]\in\mathbb{R}^{l}}\). \(W_{ij}\) means the \(j\)th weight block in \(W_{i}\), which involves \(\phi_{j}\). Thus, we have  
\[\hat{u}_{di}=W_{i1}\phi_{1}+W_{i2}\phi_{2}+...+W_{in}\phi_{n}. \tag{21}\]  
The stability analysis of iteration in Equation (20) can be taken in a similar way used in [24]. We know that if \(\beta_{i}\) and \(\gamma_{i}\) are small enough, the iteration in Equation (20) is stable and the errors in results are limited.