# Robust estimator design for a mass-spring-damper system

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The file Demo_014.m is found in IQClab’s folder demos; it performs a robust estimator synthesis with the function fRobest for a mass-spring-damper problem derived from the example presented in . The setup is shown in the following figure.

The numerical values are , , , and with . The first mass is disturbed by an external force . The objective is to estimate using a measument of corrupted by measurement noise . The linear fractional representation can then be written as with , .

For comparison reasons, we design a nominal estimator, , using the function ‘h2syn’, which is part of MATLAB’s robust control toolbox, as well as a robust estimator, , using the function ‘fRobest’, which is part of IQClab.

The code for designing the nominal estimator is found in the file Demo_014.m. Designing the robust estimator proceeds as follows:

% Define plant (see script for system matrices)
H = ss(A,[Bp,Bw],[Cq;Cz;Cy],[Dqp,Dqw;Dzp,Dzw;Dyp,Dyw]);

% Define uncertainties
de1 = iqcdelta('de1','InputChannel',1,'OutputChannel',1, 'Bounds',[-1,1]);
de1 = iqcassign(de1,'ultis','Length',3);

de2 = iqcdelta('de2','InputChannel',2,'OutputChannel',2, 'Bounds',[-1,1]);
de2 = iqcassign(de2,'ultis','Length',3);

% set options
options.perf      = 'H2';
options.StrProp   = 'yes';
options.subopt    = 1.03;
options.constants = 1e-8*ones(1,4);
options.Pi11pos   = 1e-8;

[Erob,gamrob]     = fRobest(H,{de1,de2},[2,1,1],[2,2],options);

We obtain a nominal and robust estimator, and respectively, which guarantee a nominal and robust performance level of and respectively. The corresponding Bode-magnitude plots are depicted in the following figure.

Given and we can also perform a robustness analysis using the function ‘iqcanalysis’. This yields the following computed worst-case perfomormance levels for increasing levels of with , .

As can be seen, the nominal estimator, , shows better performance levels for smaller values of , while for the robust estimator, , outperforms .

To continue, let us also show some time-domain simulations, where we perturb the system with a random force disturbance and for different values of . The results are shown in the following two figures for and respectively.

As can be seen, the robust estimator does a better job, if compared to . This is confirmed by the computed RMS values which are given in the following table.