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"_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.