# Robustness analysis with LTI parametric uncertainties

The file Demo_001.m is found in IQClabâ€™s folder demos. This demo performs a – and IQC-robustness analysis for an uncertain plant that is affected by LTI parametric uncertainties. Here it is possible to vary several inputs:

1. The uncertainty block:
1. One parametric uncertainty that is repeated twice, or
2. Two different parametric uncertainties that are repeated once
2. Relaxation type:
1. DG-scalings
2. Convex hull relaxation
3. Partial convexity
4. Zeroth order Poyla relaxation
3. 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 3.1 and Option 3.3,
• for Option 3.2.

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

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

The demo file Demo_001.m allows to run an IQC-analysis for various values of 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 , for and the induced -gain as performance metric. In addition, the following parameters are considered:

• Relaxation type: ‘CH’ (Option 2.2)
• Length of the basis function: 2
• 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
de = iqcdelta('de','InputChannel',1:2,'OutputChannel',1:2,'Bounds',[-0.5,0.5]);

% Assign IQC-multiplier to uncertainty block
de = iqcassign(de,'ultis','Length',3,'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_001.m for

1. (Option 1.1)
2. Convex hull relaxation (Option 2.2)
3. Induced -gain performance (Option 3.1)

you obtain as output the worst-case induced -gain for increasing values of computed by the -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 -gains (i.e. -norms in this example), which are consisted with the -analysis. For static multipliers (case: , red dashed-line) the -level increases faster than the others, at the benefit of a lower computational load. On the other hand, for basis-lengths larger than , the results coincide the -analysis up to values of . Choosing higher basis-lengths would allow to converge even further.

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