Robustness analysis with sector-bounded and slope-restricted nonlinearities

The file Demo_006.m is found in IQClab’s folder demos. This demo performs an IQC robustness analysis for an uncertain plant that is affected by sector-bounded and slope-restricted scalar nonlinearities. Here it is possible to the following input:

The uncertainty block:

3 different sector-bounded scalar nonlinearities
3x repeated sector-bounded scalar nonlinearity
3 different sector-bounded and slope-restricted scalar nonlinearities
3x repeated sector-bounded and slope-restricted scalar nonlinearity

The uncertain system is given by $\Delta\star M$ with the open-loop LTI plant $M=ss(A,B,C,D)$ , where

$A=\left(\begin{array}{cc}-0.4&-1\\1&0\end{array}\right)$ , $B=\left(\begin{array}{cccc}-0.2&-1&-0.25&1\\0&0&0&0\end{array}\right)$ , $C=\left(\begin{array}{cc}1&0\\0&1\\0&0\\0&-0.2\end{array}\right)$ , $D=\left(\begin{array}{cccc}0&0&0&0 \\ 0&0&0&0\\0&1&0&0\\-0.1&0&0&1\end{array}\right)$ .

On the other hand, the uncertainty block is defined by:

$\Delta=\Delta_1=\mathrm{diag}(\delta_1,\delta_2,\delta_3)$ with $\delta_i\in\mathrm{sector}(0,\alpha)$ , $i\in\{1,2,3\}$
$\Delta=\Delta_2=\delta I_3$ with $\delta\in\mathrm{sector}(0,\alpha)$
$\Delta=\Delta_3=\mathrm{diag}(\delta_1,\delta_2,\delta_3)$ with $\delta_i\in\mathrm{sector}(0,\alpha)\cap\mathrm{slope}(0,\alpha)$ , $i\in\{1,2,3\}$
$\Delta=\Delta_4=\delta I_3$ with $\delta\in\mathrm{sector}(0,\alpha)\cap\mathrm{slope}(0,\alpha)$

If running scenarios 1. and 2., we obtain the results shown in the following two figures. As can be seen, considering three independent blocks leads to lower bounds on the worst-case $L_2$ -gains if compared to one block that is repeated trice. In addition, the third and fourth figure demonstrate the advantage of considering both a sector and a slope constraint (instead of just a sector constraint) in the analysis by the favour of the Zames-Falb multiplier.