Robustness analysis with rate-bounded time-varying parametric uncertainties

The file Demo_005.m is found in IQClab’s folder demos. This demo performs an IQC robustness analysis for an uncertain plant that is affected by rate-bounded time-varying parametric uncertainties. Here it is possible to vary several inputs:

The uncertainty block:
1. One rate-bounded parametric uncertainty that is repeated twice, or
2. Two rate-bounded parametric uncertainties that are repeated once.
Relaxation type:
1. DG-scalings
2. Convex hull relaxation
3. Partial convexity
4. Zeroth order Polya relaxation

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

$A=\left(\begin{array}{cc}-2&-3\\1&1\end{array}\right)$ , $B=\left(\begin{array}{ccc}1&0&1\\0&0&0\end{array}\right)$ , $C=\left(\begin{array}{cc}1&0\\0&0\\1&0\end{array}\right)$ , $D=\left(\begin{array}{ccc}1&-2&0\\1&-1&1\\0&1&0\end{array}\right)$ ,

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

$\Delta=\Delta_1=\delta_1I_2$ with $\delta_1\in[-0.3,0.3]$ , $\dot{\delta}_1\in[-\alpha,\alpha]$ for Option 1.1, or
$\Delta=\Delta_2=\mathrm{diag}(\delta_1,\delta_2)$ with $\delta_i\in[-0.,0.3]$ , $\dot{\delta}_i\in[-\alpha,\alpha]$ , $i\in\{1,2\}$ for Option 1.2.

The demo file Demo_005.m runs an IQC-analysis for various values of $\alpha\in[0,0.2]$ and within the file you can change the inputs mentioned above. If running the analysis for $\Delta=\Delta_1$ for all available relaxation schemes, we obtain the following results shown in the following figures. As can be seen, the $\mu$ -analysis provides a lower-bound, while the IQC analysis with static IQC-multipliers provides an upper-bound on the worst-case induced $L_2$ -gain. In addition, if increasing the length of the basis function, and thereby introducing dynamics in the IQC multipliers, all the results show an increasing worst-case induced $L_2$ -gain for increasing rate-bounds on the uncertainties. Moreover, if changing the relaxation scheme, one can clearly see the benefit of the Partial concexity and zeroth order Polya relaxation over the convex hull relaxation and the DG-scaling; this at the cost of a higher computational load.