Andrea Zanchettin – Automatic Control
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AUTOMATIC CONTROL
Andrea M. Zanchettin, PhD
Spring Semester, 2016
Identification of LTI systems from time response
Andrea Zanchettin – Automatic Control
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Motivations
During classes we have learned how to analyse the
behaviour of a linear system based on a mathematical
description in terms of
1.  differential equations
2.  frequency response (transfer function)
However, a common problem in engineering is to infer the
mathematical model once the behaviour of the system is
experimentally observed.
In this series of (two) lectures, we will see how to handle
this problem.
Andrea Zanchettin – Automatic Control
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Step response - definitions
We are interesting in understanding the time behaviour of a
given LTI system by looking at its step response.
time
4
Andrea Zanchettin – Automatic Control
First order step response
We consider a (a.s., T > 0) first order system
or
As we are interested in its step response, we know
and
Heaviside
Andrea Zanchettin – Automatic Control
First order step response – cont’d
Let’s compute the initial and final values
5
6
Andrea Zanchettin – Automatic Control
First order step response – cont’d
Can we say something about its derivative?
… and its initial value?
For the case
we have
Andrea Zanchettin – Automatic Control
First order step response – cont’d
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8
Andrea Zanchettin – Automatic Control
First order step response – cont’d
Assume we are observing the step response of a system
which looks as follows
Step Response
3
2.5
Amplitude
2
1.5
1
0.5
0
0
1
2
3
4
Time (seconds)
5
6
7
9
Andrea Zanchettin – Automatic Control
First order step response – cont’d
We can assume that is the step response of
Step Response
3
2.5
Amplitude
2
1.5
1
0.5
0
0
1
2
3
4
Time (seconds)
5
6
7
10
Andrea Zanchettin – Automatic Control
First order step response – cont’d
Unfortunately, realistic responses are rather like this one
Step Response
3.5
3
2.5
Amplitude
2
1.5
1
0.5
0
-0.5
0
1
2
3
4
Time (seconds)
5
6
7
Andrea Zanchettin – Automatic Control
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First order step response – cont’d
Let’s stick to the same hypothesis and consider the
corresponding time behaviour (analytical)
We need to estimate three parameters:
•  the gain
•  the time constant
•  the delay
We are going to see how to do that robustly wrt noise!
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Andrea Zanchettin – Automatic Control
First order step response – cont’d
As for the gain we know that
In practice, we can compute the average of the last values.
Step Response
3.5
3
2.5
Amplitude
2
1.5
1
0.5
0
-0.5
0
5
10
Time (seconds)
15
13
Andrea Zanchettin – Automatic Control
First order step response – cont’d
As we now know the value of we can focus on and .
Consider again the analytical form of the response.
Step Response
3
2.5
Amplitude
2
1.5
Are we able to analytically
determine this area?
1
0.5
0
0
1
2
3
4
Time (seconds)
5
6
7
Andrea Zanchettin – Automatic Control
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First order response – cont’d
Indeed, we can compute such an area as
With this method we are able to evaluate
our previous knowledge of .
based on
15
Andrea Zanchettin – Automatic Control
First order response – cont’d
As we know know
area
we can try to evaluate another
Step Response
3
2.5
Amplitude
2
1.5
Are we able to analytically
determine this area?
1
0.5
0
0
1
2
3
4
Time (seconds)
5
6
7
16
Andrea Zanchettin – Automatic Control
First order step response – cont’d
Indeed, we can compute such an area as
from which we can evaluate
.
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Andrea Zanchettin – Automatic Control
First order step response – cont’d
Step Response
3.5
3
2.5
Amplitude
2
1.5
1
0.5
0
-0.5
0
5
10
Time (seconds)
15
18
Andrea Zanchettin – Automatic Control
First order approximation
This method can be also used to approximate higher order
responses with a first order model (with delay).
Step Response
1
0.9
0.8
0.7
Amplitude
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
Time (seconds)
15
19
Andrea Zanchettin – Automatic Control
First order approximation – cont’d
This method can be also used to approximate higher order
responses with a first order model (with delay).
Step Response
1
0.8
Amplitude
0.6
0.4
0.2
0
-0.2
0
5
10
Time (seconds)
15
Andrea Zanchettin – Automatic Control
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Second order step response
We consider an asymptotically stable second order system
of the following kind (complex and conjugate poles)
The corresponding step response will be as follows
Andrea Zanchettin – Automatic Control
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Second order step response – cont’d
time
Andrea Zanchettin – Automatic Control
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Second order step response – cont’d
From the explicit form of the step response we can
evaluate
whilst from the two exponential envelopes we have the
settling time
Andrea Zanchettin – Automatic Control
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Second order step response – cont’d
If, instead, we are interesting in estimating parameters
we can simply invert the previous formulas, i.e.
From the maximum and the asymptotic values
From the time instant of the first peak