Robustness analysis with arbitrarily fast time-varying parametric uncertainties

The file Demo_004.m is found in IQClab’s folder demos. This demo performs an IQC-robustness analysis for an uncertain plant that is affected by arbitrarily fast time-varying parametric uncertainties. Here it is possible to vary several inputs:

The uncertainty block:
1. Two LTV parametric uncertainties that are diagonally repeated once, or
2. Four LTV parametric uncertainties packed in a full $2\times 2$ delta-block
Performance metric:
1. Induced $L_2$ -gain
2. $H_2$ -norm
3. Robust stability test

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

$A=\left(\begin{array}{cc}-2&-3\\1&1\end{array}\right)$ , $B=\left(\begin{array}{ccc}1&0&1\\0&0&0\end{array}\right)$ , $C=\left(\begin{array}{cc}1&0\\0&0\\1&0\end{array}\right)$ , $D=\left(\begin{array}{ccc}1&-2&0\\1&-1&d_{23}\\0&1&0\end{array}\right)$ ,

while:

$d_{23}=1$ for Option 2.1 and Option 2.3,
$d_{23}=0$ for Option 2.2.

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

$\Delta=\Delta_1=\mathrm{diag}(\delta_1,\delta_2)$ with $\delta_i\in[-\alpha,\alpha]$ , $i\in\{1,2\}$ , for Option 1.1, or
$\Delta=\Delta_2=H_1\delta_1+H_2\delta_2+H_3\delta_3+H_4\delta_4$ for Option 1.2, where $H_1=\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$ , $H_3=\left(\begin{array}{cc}1&1\\0&0\end{array}\right)$ , $H_3=\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ , $H_4=\left(\begin{array}{cc}1&0\\1&0\end{array}\right)$ , and $\delta_i\in[-\alpha,\alpha]$ , $i\in\{1,2,3,4\}$ .

The demo file Demo_004.m allows to run an IQC-analysis for various values of $\alpha\in[0,0.4]$ and different relaxation schemes. Within the file one can change the inputs mentioned above. For illustration purposes, the following code specifies an IQC-analysis for the uncertain plant $\Delta_2(\delta_1,\delta_2,\delta_3,\delta_4 )\star M$ , $\delta_i\in[-\alpha,\alpha]$ , $\alpha=0.1$ , and the induced $L_2$ -gain as performance metric. In addition, the following parameters are considered:

Relaxation type: ‘CH’
Solution check: ‘on’
Enforce strictness of the LMIs: $\epsilon=1e-8$

% Define uncertain plant
M = ss([-2,-3;1,1],[1,0,1;0,0,0],[1,0;0,0;1,0],[1,-2,0;1,-1,1;0,1,0]);

% Define uncertainty block
H{1} = [1,0;0,1]; H{2} = [1,1;0,0]; H{3} = [0,1;1,0]; H{4} = [1,0;1,0];

La = polydec(pvec('box',0.1*[-1,1;-1,1;-1,1;-1,1]))';

de = iqcdelta('de','InputChannel',1:2,'OutputChannel' ,1:2,'Structure','FB','UncertaintyMap',H,'Polytope',La,'TimeInvTimeVar','TV');

% Assign IQC-multiplier to uncertainty block
de = iqcassign(de,'ultv','RelaxationType','CH');

% Define performance block
pe = iqcdelta('pe','ChannelClass','P','InputChannel', 3,'OutputChannel',3,'PerfMetric','L2');

% Perform IQC-analysis
prob = iqcanalysis(M,{de,pe},'SolChk','on','eps',1e-8);

To continue, if running the IQC-analysis in Demo_004.m for

$\Delta_2$ (Option 1.2)
$H_2$ -gain performance (Option 3.2)

you obtain the worst-case $H_2$ -norm for increasing values of $\alpha\in[0,0.2]$ computed by the IQC-tools. This yields the results shown in the following figure. As can be seen, the IQC-analysis produces roughly the same worst-case $H_2$ -norms for the two considered relaxation schemes.