Frequency domain parametric system identification

Question 1

steps of the identification procedure

Answer 1

Classical steps of parametric techniques:

1) experiment design and data preprocessing
2) selection of a parametric model class
3) definition of a performance index
4) optimization, to find the optimal parameters

Question 2

general idea of the method

Answer 2

• make a set of single-sinusoid excitation experiments
• from each experiment estimate a single point of the frequency response function of the system
• using the estimated points, find a parametric transfer function that fits the estimated points

Question 3

Experiment design

Answer 3

H independent experiments
set of frequencies {ω1, ω2, …, ωH}
Input of the i-th experiment
ui(t) = Ai*sin(ωi*t)
Output of the i-th experiment yi(t)

Question 4

On the amplitude of the sinuisoids

Answer 4

Amplitudes Ai usually decrease with increasing frequency, to have constant power required

Question 5

Output of the i-th experiment

Answer 5

yi(t) in real application is not a perfect sinusoid, because of

• noise on measurement
• noise on the system
• small non-linear effects

Assuming the system to be linear time invariant, it’s possible to extract the real sinusoid at frequency ωi by a parametric identification
y^i(t) = Bisin(ωit+φi) or
y^i(t) = aisin(ωit) + bicos(ωit)

Question 6

output of the H experiments

Answer 6

The data set:
H points of the frequency response from the input u(t) to the output y(t)
{B1^/Ai * e^jφ1^,…,BH^/AH * e^jφH^}

Question 7

Model class selection

Answer 7

m(θ) : W(z; θ) = (b0+b1z^-1+…+bpz^-p)/(1+a1z^-1+…+amz^-m) * z^-1

θ = [a1, a2, … , am, b0, b1, …, bp]’

the order (m,p) of the model class can be decided using cross validation, or visual inspection

Question 8

Performance index

Answer 8

JH(θ) = 1/H * sum(i=1,H) ( W(e^jωi; θ) - Bi^/Ai*e^jφ1^ )^2

Question 9

Optimization

Answer 9

θ^ = argminθ {JH(θ)}

Usuallu JH(θ) is a type 3 performance index (not quadratic, not convex), so iterative optimization methods are required

Question 10

On the frequencies selection

Answer 10

Theoretically, H points evenly spaced between 0 and Nyquist frequency ωN = 1/2 * ωs

In practice, it’s better to focus on a smaller bandwidth:
ωH 2 or 3 times larger than the expected cutoff frequency of the control system

Question 11

Use of weighted frequencies

Answer 11

To have more accuracy in some regions of frequency, for example around cutoff freq. or around resonances of the system, different weights γi can be used:
the performance index becomes
JH(θ) = 1/H * sum(i=1,H) γi * ( W(e^jωi; θ) - Bi^/Ai*e^jφ1^ )^2
In this way errors in a given frequency range can be considered more.

Question 12

single experiment

Answer 12

It’s a different way to obtain the data set (step 1).
Instead of H independet experiments, one single very long sweep experiment can be used, where the input varies its frequency from ω1 to ωH (sinusoidal sweep).
Then the signal can be cut in H pieces, or the ratio of complex spectra of output over input can be directly computed, obtaining an estimated W^(e^jω) (data set).

Question 13

Comparison between time domain and frequency domain identification methods

Answer 13

Pros of f.d. methods:

• robust and reliable
• intuitive and easy to understand
• consistent with classical loop-shaping control design techniques

Cons of f.d. methods:

• experiment step is more demanding
• no model of the noise is obtained

The two methods, if done properly, should give very similar results