## 4 Simulation Examples
\begin{array}{cc}\theta_{2}&\dot{\theta}_{2}\end{array} \right]\) and matrices defined by:  
\[A_{11}=A_{22}=\left[ \begin{array}{cc}0&1\\ \frac{g}{1}-\frac{k\dot{\theta}^{2}}{m\dot{\theta}^{2}}&-\mu\end{array} \right] A_{12}=A_{21}=\left[ \begin{array}{cc}0&0\\ \frac{k\dot{\theta}^{2}}{m\dot{\theta}^{2}}&0\end{array} \right],\ B_{1}=B_{2}= \left[ \begin{array}{cc}0\\ \frac{1}{m\dot{\theta}^{2}}\end{array} \right]\]  
Figure 1: States evolution when the decentralized controller \(K_{d}\) is employed. Figure 2: Two inverted pendulums coupled by a spring. \[C_{1}=C_{2}=\left[ \begin{array}{cc}1&0\end{array} \right]\]  
where \(g\) represents the gravity, \(\mu\) accounts for the friction, \(m=m_{1}=m_{2}\) are the masses of both pendulums, \(k\) is the spring constant and the meanings of the geometrical parameters are shown in Figure 2. This linear model corresponds to the linearization of the pendulum nonlinear equations around the up-right position equilibrium point. The following values were used in simulation, [19]:  
\[\frac{g}{l}=1,\;\frac{1}{ml^{2}}=1,\;\mu=1,\;\frac{k}{m}=2,\;\frac{a}{l}=0.5\]  
This system can be cast into the form of Equations (1) and (2) as:  
\[A_{c}=\left[ \begin{array}{cccc}0&1&0&0\\ 0.5&-1&0.5&0\\ 0&0&0&1\\ 0.5&0&0.5&-1\end{array} \right],\;B_{c}=\left[ \begin{array}{cccc}0&0\\ 1&0\\ 0&0\\ 0&1\end{array} \right],\;C_{c}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&0&1&0\end{array} \right],\;D_{c}=0\]  
A static output feedback controller can be designed for this system to achieve its asymptotic stability. In this way, the feedback gain  
\[K_{c}=\left[ \begin{array}{cc}2.92&0.65\\ 0.64&2.80\end{array} \right]\]  
places the closed-loop poles at \(\{-0.5\pm 1.5i,-0.5\pm 1.4i\}\) with negative real parts. Now, we implement a decentralized controller with feedback gain given by:  
\[K_{d}=\left[ \begin{array}{cc}2.92&0\\ 0&2.80\end{array} \right]\]  
Theorem 2 is now used to analyze the stability of the closed-loop system when this controller is employed. This case is of practical importance and corresponds to the situation when the local controller has only available for control purposes the information regarding the local output, and not the output of the complete system. Thus, the centralized and decentralized systems are the same and only the static feedback gain changes. Theorem 2 conditions are applied with \(\widetilde{\sigma}_{A}=\widetilde{\sigma}_{B}=\widetilde{\sigma}_{C}= \widetilde{\sigma}_{D}=0\), \(\widetilde{\sigma}_{K}=\|K_{c}-K_{d}\|_{2}=0.65\) while the stability condition for this special case (see Appendix B) reads:  
\[0.65=0.65\times 1\times 1=\|K_{c}-K_{d}\|_{2}\|B_{c}\|_{2}\|C_{c}\|_{2}<1\]  
Accordingly, the closed-loop system attained with the decentralized controller is asymptotically stable and all the outputs will converge to zero asymptotically. Figure 3 displays the evolution of both angles from initial conditions \(x_{1}(t)^{T}=\left[ \begin{array}{cc}0.5&-0.5\end{array} \right],x_{2}(t)^{T} =\left[ \begin{array}{cc}0.15&0.5\end{array} \right]\), where it can be observed that both pendulums are stabilized in the up-right position with the decentralized controller. **Example 3**. : _Consider the linear interconnected system given by:_  
\[ \begin{array}{l}\dot{x}_{c}(t)=A_{c}x_{c}(t)+B_{c}u_{c}(t)\\ y_{c}(t)=C_{c}x_{c}(t)+D_{c}u_{c}(t)\end{array} \]  
_with matrices defined by:_  
\[A_{c}=\left[ \begin{array}{ccc}-1&0.4&0.3\\ 0.2&-2&0.1\\ -0.1&0.2&-3\end{array} \right],B_{c}=\left[ \begin{array}{ccc}1&0.1&0\\ 0&2&0\\ 0&0&2\end{array} \right],C_{c}=\left[ \begin{array}{ccc}1&0.1&0\\ 0&1&0.1\\ 0&0&1\end{array} \right],D_{c}=0.1I_{3}\]  
This system is controlled by the static output feedback gain given by:  
\[K_{c}=\left[ \begin{array}{ccc}0.19&0.05&0.04\\ 0.05&-0.01&0.03\\ -0.02&0.02&-0.46\end{array} \right]\]  
which places the closed-loop poles at \(\{-1.17,-2,-2.1\}\). The decentralized system is now given by:  
\[A_{d}=A_{c},B_{d}=\left[ \begin{array}{ccc}1&0&0\\ 0&2&0\\ 0&0&2\end{array}