SYSM 6325
1 0 ) UT % vector fresp contains ( complex-valued) frequency responses % vector w, contains frequencies at which measurements were taken load...
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# There is one more matlab file to solve this question. Let me know if you can solve this problem as soon as

possible.    1 0 ) UT
% vector fresp contains ( complex-valued) frequency responses
% vector w, contains frequencies at which measurements were taken
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fresp=02i1(10:10: 650) ;
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frequency vector (in Hz)
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freq=02i1x (10 : 10 :650) ;
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w = 2*pi*freq;
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%non-parametric model
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G_h = frd( fresp, w) ;
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bode (G_h)
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fresp=02i1 (20:1:1200) ; %frequency response vector
% frequency vector (in Hz)
freq=02i1x(20:1:1200) ;
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w = 2*pi*freq;
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%non-parametric model
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G_h = frd( fresp, w);
% SYS = frd (RESPONSE, FREQS)
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bode (G_h)
% SYS = frd (RESPONSE, FREQS, TS) creates a discrete-time frd model with
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sample time TS (a positive value)
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% Frequency Response Data ( frd) models store the frequency response of linear
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% systems, for example, experimental data collected with a frequency analyzer Consider the problem of identifying transfer function of a linear time—invariant single—
input single—output system from frequency domain data. Let us assume that the transfer function of the system can be described by transfer function B( )
s
G“) ‘ A(s)
where
B(s) = be + bls + b282 + . .. + bn_1s&quot;_1
A(s) = l+a13+a232+...+ansn We perform experiments that involve applying sine waves to the system at frequencies
£01,002, . . . ,wN and determining the input—output frequency response at those frequencies, A C(jwi). Note that we use G instead of G, as there will be unavoidable errors in the measurements. (i) Show that the problem of ﬁnding A(s) and 3(3) can be cast as a least squares
optimization problem. Deﬁne a least squares cost function and ﬁnd the optimal solution. Hint: Note that
AUCUWUW) = BUM), and that the error corresponding to each frequency wi may be deﬁned as
6w: = AUWQGUM) — BUM)
Also note that 6,; is complex-valued. (ii) Download the data ﬁle G11QA3.mat and the accompanying Matlab ﬁle ex—l—data.m.
Running ex— 1—data.m produces the frequency response function of the ﬁrst mode of
a cantilever beam. Use the method above to identify a model for the system with
this frequency response data. First, you will need to make an estimate of the order of this system. Explain your rationale. (iii) Evaluate and plot the least squares cost associated with increasing order of the system and use that to determine if your initial estimate was correct. (iv) Can you obtain a better ﬁt to the frequency response data by incorporating a weight— ing function into the least squares cost? Explain how. (v) Download and run ex—l—threemodes.m. This data set contains the frequency response
of the ﬁrst three modes of the same structure. Repeat steps (ii)—(iv) with this data.
Can you achieve an acceptable outcome by selecting a subset of frequency data points? Explain your observations.

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