## 5 Robust Control
### Robust PI Control  
A PI controller based on robust stability analysis is designed to investigate the effectiveness of the proposed design method. The (nominal) open loop transfer function of system in Equation (4) with the controller Equation (6) is expressed as  
\[L(s)=\left(K_{p}+\frac{K_{i}}{s}\right)\left(\frac{\bar{K}}{1+s\widetilde{T}} \right). \tag{24}\]  
The (nominal) associated sensitivity and complementary sensitivity function are given by  
\[\widetilde{S}(s)=\frac{s(1+s\widetilde{T})}{s^{2}\widetilde{T}+s(\bar{K}K_{p}+ 1)+\bar{K}K_{i}}, \tag{25}\]  
\[\widetilde{T}(s)=\frac{\widetilde{K}(sK_{p}+K_{i})}{s^{2}\widetilde{T}+s(\bar{ K}K_{p}+1)+\bar{K}K_{i}}. \tag{26}\]  
By substituting the above equations in the robust stability condition of each uncertainty model, the range of the controller parameters for which the robust stability of the system is guaranteed can be obtained. To do so, the norms of auxiliary transfer functions occurring in the stability conditions of Table 1 are calculated and then plotted for different values of \(K_{p}\) and \(K_{i}\) as it is shown in Figures 7 and 8. Note that for our model the auxiliary transfer functions resulting from the additive and multiplicative model are the same. The level sets of function values of the corresponding Hardy space norm less than one fulfill the conditions and ensure robust stability.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Uncertainty Type** & **Robust Stability Condition** \\ \hline \(\widetilde{G}+W\Delta\) & \(\|W\widetilde{C}\bar{S}\|_{\infty}<1\) \\ \(\widetilde{G}\left(1+W\Delta\right)\) & \(\|W\widetilde{T}\|_{\infty}<1\) \\ \(\widetilde{G}\) & \(\|W\widetilde{G}\bar{S}\|_{\infty}<1\) \\ \(\widetilde{G}\) & \(\|W\widetilde{S}\|_{\infty}<1\) \\ \(\widetilde{1+W\Delta}\) & \(\|W\widetilde{S}\|_{\infty}<1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Robust stability conditions for different uncertainties [32; 34].  
Figure 7: Level sets of \(\|W\widetilde{C}\bar{S}\|_{\infty}\) and \(\|W\widetilde{T}\|_{\infty}\) for different values of \(K_{p}\) and \(K_{i}\) for the system with additive and multiplicative uncertainty.  
To verify the performance of the robust PI controller, output position control of the textile reinforced composite for a desired deflection was tested. The selected gain values considering the additive/multiplicative uncertainty are \(K_{p}=2\) and \(K_{i}=0.004\). The Bode plot of the auxiliary transfer functions, \(WC\bar{S}\) and \(W\bar{T}\), is shown in Figure 9. Clearly, the magnitude plot stays below \(1=0_{\text{dB}}\) such that robust stability condition is fulfilled. The result of the experiment for a reference deflection of 22 mm is shown in Figure 10. It can be seen that the control action is really slow which is due to small value of the integral gain.  
Figure 8: Level sets of \(\|W\bar{S}\|_{\infty}\) for different values of \(K_{p}\) and \(K_{i}\) for the system with multiplicative feedback uncertainty.  
Figure 10: Deflection of the controlled system with multiplicative uncertainty for 22 mm reference using a robust PI controller.  
Figure 9: Bode diagram of \(W\overline{S}\) for \(K_{p}=2\) and \(K_{i}=0.004\) considering multiplicative uncertainty.  
The same experiment was repeated for the feedback multiplicative uncertainty case using the gain values \(K_{p}=3.2\) and \(K_{i}=0.16\). The Bode plot of the auxillary transfer function \(W\widetilde{S}\) occurring in the stability condition using these gain values is shown in Figure 11. Experimental results in Figure 12 show that this controller has a better performance in comparison to other uncertainty models and classical controllers. It takes less time to reach the reference value. Furthermore, it provides stability for the whole set of plants described by Equation (11).