# Robustness analysis with arbitrarily fast time-varying parametric uncertainties

The file Demo_004.m is found in IQClabâ€™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:

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

The uncertain system is given by with the open-loop LTI plant , where

, , , ,

while:

• for Option 2.1 and Option 2.3,
• for Option 2.2.

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

• with , , for Option 1.1, or
• for Option 1.2, where , , , , and , .

The demo file Demo_004.m allows to run an IQC-analysis for various values of 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 , , , and the induced -gain as performance metric. In addition, the following parameters are considered:

• Relaxation type: ‘CH’
• Solution check: ‘on’
• Enforce strictness of the LMIs:
% 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

1. (Option 1.2)
2. -gain performance (Option 3.2)

you obtain the worst-case -norm for increasing values of 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 -norms for the two considered relaxation schemes.

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