## 4 Simulation Examples  
This Section contains some numerical simulation examples to illustrate the theoretical results introduced in Section 3. **Example 1**. : _Consider the interconnected linear system with less inputs than outputs given by, [19]:_  
\[ \begin{array}{c}\dot{x}_{1}(t)=A_{11}x_{1}(t)+A_{12}x_{2}(t)+B_{1}u_{1}(t)\\ \dot{x}_{2}(t)=A_{21}x_{1}(t)+A_{22}x_{2}(t)+B_{2}u_{2}(t)\\ y_{1}(t)=C_{1}x_{1}(t)\\ y_{2}(t)=C_{2}x_{2}(t)\end{array} \]  
_with \(x_{1}(t)^{T}=\left[ \begin{array}{cc}x_{11}(t)&x_{12}(t)\\ \end{array} \right]x_{2}(t)^{T}=\left[ \begin{array}{cc}x_{21}(t)&x_{22}(t)\\ \end{array} \right]\) and matrices defined by:_  
\[ \begin{array}{c}A_{11}=\left[\begin{array}{cc}1&0\\ 0&2\\ \end{array}\right]\text{, }A_{12}=A_{21}=\left[\begin{array}{cc}-1&0\\ 0&-1\\ \end{array}\right]\text{, }A_{22}=\left[\begin{array}{cc}-1&0\\ 0&-2\\ \end{array}\right]\text{, }B_{1}=\left[\begin{array}{c}1\\ 2\\ \end{array}\right]\text{, }B_{2}=\left[\begin{array}{c}2\\ 1\\ \end{array}\right]\end{array} \]  
\[C_{1}=C_{2}=I_{2}\]  
This system can be cast into the form of Equations (1) and (2) by composing the matrices:  
\[A_{c}=\left[ \begin{array}{cccc}1&0&-1&0\\ 0&2&0&-1\\ -1&0&-1&0\\ 0&-1&0&-2\\ \end{array} \right]\text{, }B_{c}=\left[ \begin{array}{cc}1&0\\ 2&0\\ 0&2\\ 0&1\\ \end{array} \right]\text{, }C_{c}=I_{4}\text{, }D_{c}=0\]  
Note that matrix \(A_{c}\) is unstable with eigenvalues given by \(\{2.36,.141,-1.41,-2.23\}\). A static feedback output controller of the form of Equation (3) can be designed for this system, which leads to the following gain:  
\[K_{c}=\left[ \begin{array}{cccc}-9.3&8.2&-0.015&0.02\\ 0.01&-0.01&0.375&0.25\\ \end{array} \right]\]  
that places the closed-loop poles at \(\{-0.84,-1.1,-2.8,-3.3\}\) and thus stabilizes the closed-loop system. The static feedback gain \(K_{c}\) corresponds to a centralized controller as it can be readily observed. The question that arises now is whether a decentralized controller defined by:  
\[K_{d}=\left[ \begin{array}{cccc}-9.3&8.2&0&0\\ 0&0&0.375&0.25\\ \end{array} \right]\]  
is enough to stabilize the system or not. Note that \(K_{d}\) is a block-diagonal matrix with zero off-block-diagonal entries. Theorem 1 enables us to guarantee the asymptotic stability of the above system when the decentralized controller \(K_{d}\) is used. Therefore, \(A_{d}=A_{c}\), \(B_{d}=B_{c}\) and \(C_{d}=C_{c}\) while the feedback gain \(K_{d}\) is restricted to the proposed particular structure. In this way, consider now \(A_{cc}=A_{c}+B_{c}K_{c}C_{c}\), \(A_{dc}=A_{c}+B_{c}K_{d}C_{c}\) and \(\widetilde{A}_{dc}=B_{c}(K_{c}-K_{d})C_{c}\). Condition 2 of Theorem 1 _(i)_ yields:  
\[0.055=\|\widetilde{A}_{dc}\|_{2}<\underbrace{\frac{1}{sup\limits_{\omega\in R _{0+}}\left\|(ioI_{4}-A_{cc})^{-1}\right\|_{2}}}_{\omega\in R_{0+}}=0.07\]Consequently, we can conclude from Theorem 1 that the closed-loop system controlled by the decentralized static output gain \(K_{d}\) is asymptotically stable. Thus, we have been able to easily analyze the stability of the decentralized case from the stability property of the centralized one. Figure 1 shows the trajectories of the closed-loop system when the gain \(K_{d}\) is deployed with initial conditions given by \(x_{1}(t)^{T}=\left[ \begin{array}{cc}-3&-4\end{array} \right]\), \(x_{2}(t)^{T}=\left[ \begin{array}{cc}5&6\end{array} \right]\). It can be observed in Figure 1 that all the states converge to zero as predicted by Theorem 1. **Example 2**. : _Consider the linear system with the same number of inputs and outputs composed of two identical pendulums THAT are coupled by a spring and subject to two distinct inputs, as displayed in Figure 2, [19]._  
The mathematical model of such interconnected system is given by:  
\[ \begin{array}{l}\dot{x}_{1}(t)=A_{11}x_{1}(t)+A_{12}x_{2}(t)+B_{1}u_{1}(t)\\ \dot{x}_{2}(t)=A_{21}x_{1}(t)+A_{22}x_{2}(t)+B_{2}u_{2}(t)\\ y_{1}(t)=C_{1}x_{1}(t)\\ y_{2}(t)=C_{2}x_{2}(t)\end{array} \]  
with \(x_{1}(t)^{T}=\left[ \begin{array}{cc}\theta_{1}&\dot{\theta}_{1}\end{array}  \right]x_{2}(t)^{T}=\left[