## 3 Design of Distributed Controller and Filters
### Filters Design  
Denote  
\[z_{i,j}=\sum_{k=1}^{N}a_{i,k}(q_{i,j}-q_{k,j})+\mu_{i}(q_{i,j}-y_{d}^{(j-1)}),j =1,2 \tag{5}\]  
Then, design the filters as  
\[\left\{ \begin{array}{l}\dot{q}_{i,1}=q_{i,2}\\ \dot{q}_{i,2}=v_{i}\end{array} \right. \tag{6}\]  
with  
\[v_{i}=-c_{1}\underline{z}_{i}-c_{0}q_{i,2}-c_{0}\text{sgn}(\underline{z}_{i}) \sum_{j=1}^{2}\hat{F}_{i,j} \tag{7}\]  
\[\dot{\hat{F}}_{i,j}=\sum_{k=1}^{N}a_{i,k}(\hat{F}_{k,j}-\hat{F}_{i,j})+\mu_{i} (F_{j}-y_{d}^{(j-1)}),\ j=1,2 \tag{8}\]  
where \(\underline{z}_{i}=c_{0}\underline{z}_{i,1}+z_{i,2}\), \(y_{d}^{(0)}=y_{d}\) and \(y_{d}^{(1)}=\dot{y}_{d,i}\) and \(c_{0},c_{1}\) are design parameters chosen as \(c_{0}\geq 1\) and \(c_{1}>c_{0}+1\). We then have the following lemma. **Lemma 2**. : _Consider a closed-loop system consisting of Nfilters (6) satisfying Assumption \(1\) with local controller (7). The asymptotic consensus tracking of all the filter's outputs to \(y_{d}(t)\) is achieved, i.e., \(\underset{t\rightarrow+\infty}{\lim}|q_{i,1}-y_{d}(t)|=0\). Moreover, \(q_{i,1}\) and \(q_{i,2}\) are bounded._  
Proof (of Lemma 2). : Consider the following Lyapunov function  
\[V_{z}=\frac{1}{2}z^{T}Pz+\frac{1}{2\gamma}\sum_{j=1}^{2}\tilde{F}_{j}^{T}P \tilde{F}_{j} \tag{9}\]  
where \(z=[\underline{z}_{1},\underline{z}_{2}\ldots,\underline{z}_{N}]^{T}\), \(\tilde{F}_{j}=\hat{F}_{j}-F_{j}\), \(\hat{F}_{j}=[\hat{F}_{1,j},\hat{F}_{2,j}\ldots,\hat{F}_{N,j}]^{T}\), \(F_{j}=[F_{1,j},F_{2,j}\ldots,F_{N,j}]^{T}\), and \(\gamma>0\) is a constant satisfying \(\gamma<\frac{2\gamma_{2111}^{2}(Q)}{\varphi^{2}}\) with \(\varphi=\|P(L+B)\|\). Denote \(z_{j}=[z_{1,j},z_{2,j}\ldots,z_{N,j}]^{T}\) and \(q_{j}=[q_{1,j},q_{2,j}\ldots,q_{N,j}]^{T}\). Then, we have  
\[\dot{z}=(L+B)(c_{0}q_{2}-c_{0}y_{d}^{(1)}+v-y_{d}^{(2)}) \tag{10}\]  
Using (9) and (10), the time derivative of \(V_{z}\) is\[\dot{V}_{z}=z^{T}P(L+B)(-c_{1}z-c_{0}\sum\limits_{j=1}^{2}\text{sgn}(z) \cdot*\underline{E}_{j} \tag{11}\] \[+c_{0}\sum\limits_{j=1}^{2}\varepsilon(\underline{z})\cdot* \underline{E}_{j}-c_{0}y_{d}^{(1)}-y_{d}^{(2)})\] \[-\frac{1}{\tau}\sum\limits_{j=1}^{2}\widetilde{F}_{j}^{T}P(L+B) \widetilde{F}_{j}\] \[\leq-c_{1}z^{T}Qz-\frac{1}{\tau}\sum\limits_{j=1}^{2}\widetilde{F }_{j}^{T}Q\widetilde{F}_{j}-c_{0}\sum\limits_{j=1}^{2}z^{T}P\text{Asgn}( \underline{z})\cdot*\underline{E}_{j}\] \[+c_{0}\sum\limits_{j=1}^{2}z^{T}P\text{Asgn}(\underline{z})\cdot *\underline{E}_{j}-c_{0}\sum\limits_{j=1}^{2}z^{T}P\text{Bsgn}(\underline{z}) \cdot*\underline{E}_{j}\] \[-\sum\limits_{j=1}^{2}z^{T}P(L+B)(c_{0}y_{d}^{(1)}+y_{d}^{(2)})+ c_{0}\sum\limits_{j=1}^{2}\|z\|\|P(L+B)\|\|\widetilde{\underline{E}}_{j}\|\]  
where \(\text{sgn}(\underline{z})=[\text{sgn}(\underline{z}_{1})\,,\ldots,\text{ sgn}(\underline{z}_{N})]^{T}\). By noting  
\[\dot{\hat{F}}_{i,j}=\sum_{k=1}^{N}a_{i,k}(\hat{F}_{k,j}-\hat{F}_{i,j})+\mu_{i} (F_{j}-y_{d}^{(j-1)}),\ j=1,2 \tag{8}\]0  
\[\dot{\hat{F}}_{i,j}=\sum_{k=1}^{N}a_{i,k}(\hat{F}_{k,j}-\hat{F}_{i,j})+\mu_{i} (F_{j}-y_{d}^{(j-1)}),\ j=1,2 \tag{8}\]1  
\[\dot{\hat{F}}_{i,j}=\sum_{k=1}^{N}a_{i,k}(\hat{F}_{k,j}-\hat{F}_{i,j})+\mu_{i} (F_{j}-y_{d}^{(j-1)}),\ j=1,2 \tag{8}\]2  
\[\dot{\hat{F}}_{i,j}=\sum_{k=1}^{N}a_{i,k}(\hat{F}_{k,j}-\hat{F}_{i,j})+\mu_{i} (F_{j}-y_{d}^{(j-1)}),\ j=1,2 \tag{8}\]3  
\[\dot{\hat{F}}_{i,j}=\sum_{k=1}^{N}a_{i,k}(\hat{F}_{k,j}-\hat{F}_{i,j})+\mu_{i} (F_{j}-y_{d}^{(j-1)}),\ j=1,2 \tag{8}\]4  
we have  
\[\dot{\hat{F}}_{i,j}=\sum_{k=1}^{N}a_{i,k}(\hat{F}_{k,j}-\hat{F}_{i,j})+\mu_{i} (F_{j}-y_{d}^{(j-1)}),\ j=1,2 \tag{8}\]5  
where \(c_{2}=\lambda_{\min}(Q)(c_{1}-c_{0})\), \(\gamma^{*}=\lambda_{\min}(Q)\left(\frac{1}{\tau}-\frac{\varphi^{2}}{2\lambda_{ \min}^{2}(Q)}\right)\). It is easily verified that \(c_{2}>0\) and \(\gamma^{*}>0\), therefore, it follows from (17) that \(\lim\limits_{t\rightarrow+\infty}\|z\|=0\) and hence \(\lim\limits_{t\rightarrow+\infty}|q_{i,1}-y_{d}(t)|=0\). From the boundedness of \(V_{z}\) and \(\|z\|\), the boundedness of \(q_{i,1}\) and \(q_{i,2}\) are easily obtained. This completes the proof. **Remark 3**.