Robustness analysis with LTI dynamic uncertainties

The file Demo_002.m is found in IQClab’s folder demos. This demo performs a $\mu$ – and IQC-robustness analysis for an uncertain plant that is affected by LTI dynamic uncertainties. Here it is possible to vary several inputs:

The uncertainty block:
1. One LTI dynamic $2\times 2$ uncertainty block, or
2. An LTI dynamic scalar uncertainty that is repeated twice
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 3.1 and Option 3.3,
$d_{23}=0$ for Option 3.2.

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

$\Delta=\Delta_1$ with $\|\Delta_1\|_\infty<\alpha$ , for Option 1.1, or
$\Delta=\Delta_2=\delta_2I_2)$ with $\|\delta_2\|_\infty<\alpha$ for Option 1.2.

The demo file Demo_002.m allows to run an IQC-analysis for various values of $\alpha\in[0,0.5]$ and within the file one can change the inputs mentioned above. For illustration purposes, the following 5 lines of code specify an IQC-analysis for the uncertain plant $\Delta_1\star M$ , $\|\Delta_1\|_\infty<\alpha$ for $\alpha=0.2$ and the $H_2$ -norm as performance metric. In addition, the following parameters are considered:

Length of the basis function: 3
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,-1,0;0,1,0]);

% Define uncertainty block
de = iqcdelta('de','InputChannel',[1;2], 'OutputChannel'[1;2],'StaticDynamic','D', 
'NormBounds',0.2);

% Assign IQC-multiplier to uncertainty block
de = iqcassign(de,'ultid','Length',3);

% Define performance block
pe = iqcdelta('pe','ChannelClass','P','InputChannel',3, 'OutputChannel',3,'PerfMetric','H2');

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

To continue, if running the IQC-analysis in Demo_002.m for

$\Delta=\Delta_1$ (Option 1.1)
Induced $L_2$ -gain performance (Option 3.1)

you obtain as output the worst-case induced $L_2$ -gain for increasing values of $\alpha\in[0,0.5]$ computed by the $\mu$ -tools (command: wcgain) and the IQC-tools for different lengths of the basis function. This yields the results shown in the following figure. As can be seen, the IQC-analysis produces worst-case induced $L_2$ -gains (i.e. $H_\infty$ -norms in this example), which are identical to the $\mu$ -analysis. In addition, note that for any length of the basis function, the same results are obtained. This means that static (i.e. non-dynamic) multipliers are sufficient in this analysis.