27
Frequency Response Methods
contains a complete description of
the linear dynamics of the system.
The impulse response is
The frequency response characteristics
maybe determined from the Fourier
Transform of
Then
28
Given the Fourier Transform of the input
and output,
may be determined by
Consider this ratio at different
frequencies:
Assume
then
after the transients have decayed.
Then,
for frequencies over the full bandwidth
of the system.
This information maybe obtained
experimentally, resulting in a measured
A model transfer function
is then
constructed to match
.
29
Bode Plot Construction:
Consider the frequency response of the
transfer function in factored form.
If the magnitude in db(
) is
plotted against frequency
on a log scale,
each factor may be plotted separately and
then summed together to generate the
overall magnitude plot. Also, if the angle
is plotted against frequency on a log scale,
each factor may be plotted separately and
then summed together to generate the
overall phase plot.
These magnitude and phase plots are
referred to as the Bode plot of the
system.
Consider the types of individual
components.
30
60
Poles and Zeroes at the Origin
Magnitude (dB)
40
20
0
-20
-40
-60
180
Phase (deg)
90
0
-90
-180
-270
-1
10
0
1
10
10
Frequency (rad/sec)
2
10
31
First Order Pole and Zero
30
Magnitude (dB)
20
10
0
-10
-20
-30
90
Phase (deg)
45
0
-45
-90
-1
10
0
10
Normalized Frequency (rad/sec)
1
10
32
Second Order Poles
40
Magnitude (dB)
20
0
-20
-40
0
Phase (deg)
-45
-90
-135
-180
-1
10
0
10
Frequency (rad/sec)
1
10
33
Reconstruction of G(s) From G(j ):
Consider the general factored form of G(s);
s=j
This may he written in the Bode form
Then
and
The total characteristic is the sum of the
individual factored characteristics.
34
Guidelines for the Synthesis of G(s):
- Initial slope
# of free poles (zeroes)
- Low frequency gain is
- Final slope is (n-m)(-20 db/dec)
- For well separated simple poles and zeroes,
asymptotic slope changes locate the corner
frequencies
- For complex pales and zeroes, the peak
magnitude
damping
- For close poles and zeroes, some
experimentation using slopes and -3db (-6db)
information must be used
- Use the angle plot to detect e
terms
and (1-j
)
- Construct G (j ) in the Bode form and then
transform it to other standard forms
- Plot the final resulting G(j ) and compare with
the measured data
35
Frequency Response Identification Example 1
Magnitude Plot for Example 1.
60
50
40
Magnitude, db
30
20
10
0
-10
-20
-2
10
-1
10
0
1
10
10
Frequency, w
2
10
3
10
36
Phase Plot for Example 1.
20
0
-20
Phase, degrees
-40
-60
-80
-100
-120
-140
-2
10
-1
10
0
1
10
10
Frequency, w
2
10
3
10
37
Frequency Response Identification Example 2
Magnitude Plot for Eample 2
60
40
Magnitude, db
20
0
-20
-40
-60
-2
10
-1
10
0
10
Frequency, w
1
10
2
10
38
Phase Plot for System Example 2
0
-50
Phase, degrees
-100
-150
-200
-250
-300
-2
10
-1
10
0
10
Frequency, w
1
10
2
10
39
Correlation Methods
These methods have the following
characteristics:
- use noise inputs
- test inputs may be superimposed on
normal operating signals
- works with small signal to noise ratios
- uses an averaging process which reduces
noise effects
- the system model is in the form of an
impulse response
Convolution:
40
Note: the independent variable may be
changed as follows:
Define
then
This is the form we will use.
41
Correlation:
Consider two signals
If there is no consistent relationship
between the signals (i.e. no correlation),
then the average of the product
as
should be zero. A measure of
the correlation between two signals is the
cross correlation function
The auto correlation function is defined in
the same manner where
42
Consider the correlation between a
system input and output.
Reverse the order of integration
define
Therefore
43
White Noise Inputs:
If there is a white noise input, then the
correlation function is
Then, the input/output cross correlation
function becomes
or
44
Other Inputs and Measurement Noise:
Consider the cross correlation between
u(t) and z(t) .
Again, by changing the order of integration
and change of dummy variables
Now:
45
Implementation:
46
Pseudo Random Binary Noise Sequences:
White noise signals:
- ideal impulse autocorrelation
- requires long averaging times
- difficult to generate and transmit
without distortion
Binary test signals:
- switches between two levels
- switching times are random
o pseudo random telegraph signals
o pseudo telegraph signal
ƒ random switching decision at
evenly space times ( sec.)
47
These signals still require long
averaging times.
o pseudo random binary noise
sequences
ƒ requires averaging over a specific
number of samples.
Generation of pseudo random binary noise
sequences (PRBNS):
A m-length sequence is generated using
a m stage shift register with “modulo
two” feedback of several stages.
D
D
D
D
D
D
D
48
The maximum length of the sequence
before it repeats is M = 2 - 1 (the period
T = M ; = sample period)
Specific modulo two feedback is required
for different number of stages, m.
m
Feedback
2
(D
O
D) x = x
3
(D
O
D) x = x
5
(D
O
D ) x = x
7
(D
O
D ) x = x
8
(D
O
D
9
(D
O
D ) x = x
10
(D
O
O
D
O
D ) x = x
D ) x = x
These sequences are sometimes called
maximum length null sequences (MLNS)
49
Autocorrelation of PRBNS:
For a sequence of length M = 2 - 1, a
clock period of , and an amplitude of a,
the autocorrelation is
This autocorrelation is obtained by
averaging over a multiple of the sequence
length, M.
Discrete Correlation:
The discrete correlation function is defined as
50
Impulse Response Identification Using
Discrete Correlation and PRBNS:
Consider the discrete approximation of the
convolution integral
Ts = the settling time or greater
The discrete approximation is
where
51
Discrete correlation of input and output:
Again, consider the precious system
Interchanging the order of summation
52
Now if
is a PRBNS then
If N = k ( 2 - 1 ) ; k = 1, 2, . . .
then
If
and
are not correlated,
and
and
are not correlated,
(theoretically
), then
and
53
Some practical notes:
If we use
the calculated pulse response will be
backwards.
1. Apply one full sequence before starting
the correlation (initial transients decay)
2. Sampling considerations; choose small
enough to capture the fast dynamics
and M larger than the settling time.
54
3. Use the non zero steady state or area
under the impulse response to estimate
the bias.
4. The averaging time
should be large
and a multiple of the sequence length
5. The PRBNS amplitude should be chosen
using noise level and nonlinearity
considerations.