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شناسایی سیستم ها
Lecture 5
باسمه تعالی
Input Design
Systems Identification
by: Dr B. Moaveni
2
Input Signals
• Commonly used signals
– Step function
– Autoregressive, moving average process
– Periodic signals: sum of sinusoids
– Pseudo-Random Binary Sequence (PRBS)
• Notion of “sufficient excitation”. Conditions !
• Degeneracy of input design.
• Relations between PBRS & white noise.
• Frequency domain properties of input signals.
2
Systems Identification
by: Dr B. Moaveni
3
Commonly used signals
Step function
• where the amplitude is arbitrarily chosen
• The step response can be related to rise time, overshoots, static
gain, etc.
• It is useful for systems with a large signal-to-noise ratio
0
0 0(t)
0
tu
u t
0u
Systems Identification
by: Dr B. Moaveni
4
Commonly used signals
Autoregressive, moving average process
1
1
( )
( )k k
D zu e
C z
3
Systems Identification
by: Dr B. Moaveni
5
Commonly used signals
Autoregressive, moving average process
1
1
( )
( )k k
D zu e
C z
Systems Identification
by: Dr B. Moaveni
6
Characteristics of Input Signals
Characterization of ARMA process
where,
• The distribution of e(t) is often chosen to be Gaussian
• are chosen such that have zeros outside the unit
circle.
• Different choices of lead to inputs with various spectral
characteristics.
1
1
( )
( )k k
D zu e
C z
1 1 2
1 2
1 1 2
1 2
( ) 1
( ) 1
p
p
q
q
D z d z d z d z
C z c z c z c z
,i ic d 1 1( ), ( )C z D z
,i ic d
4
Systems Identification
by: Dr B. Moaveni
7
Characteristics of Input Signals
Spectrum of ARMA process
Let e(t) be a white noise with variance .
The spectral density of ARMA process is
2
Systems Identification
by: Dr B. Moaveni
8
Commonly used signals
Periodic signals: sum of sinusoids
• where the angular frequencies are distinct,
• The amplitudes and phases , should be chosen by the user.
1
( ) sinm
k k k
k
u t a t
k
1 20 m
kak
5
Systems Identification
by: Dr B. Moaveni
9
Commonly used signals
Periodic signals: sum of sinusoids
0 100 200 300 400 500 600 700 800 900 1000-1.5
-1
-0.5
0
0.5
1
1.5
* Matlab function:
“idinput”
Systems Identification
by: Dr B. Moaveni
10
Characteristics of Input Signals
Characterization of sinusoids
6
Systems Identification
by: Dr B. Moaveni
11
Characteristics of Input Signals
Spectrum of sinusoidal inputs
Systems Identification
by: Dr B. Moaveni
12
Commonly used signals
Pseudo-Random Binary Sequence (PRBS)
• periodic signal
• switches between two levels in a certain fashion
• levels = ± a
* Matlab function:
“idinput”
Special Case: Square Wave
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
7
Systems Identification
by: Dr B. Moaveni
13
Commonly used signals
Pseudo-Random Binary Sequence (PRBS)
Systems Identification
by: Dr B. Moaveni
14
Commonly used signals
Pseudo-Random Binary Sequence (PRBS)
• Every initial state is allowed except the all-zero state ( )
• The feedback coefficients a1, a2, . . . , an are either 0 or 1.
• All additions is mod 2 operation
• The sequences are two-state signals (binary)
• There are possible 2n − 1 different state vectors x (all-zero state is
• excluded)
• A PRBS of period equal to M = 2n − 1 is called a maximum length
• PRBS (ML PRBS)
• For maximum length PRBS, its characteristic resembles white
random noise (pseudorandom)
(0) 0x
8
Systems Identification
by: Dr B. Moaveni
15
Commonly used signals
ML-PRBS
Systems Identification
by: Dr B. Moaveni
16
Commonly used signals
ML-PRBS
معادله PRBSنشان دهنده یک مرحله تاخیر زمانی است و پر واضح است که ورودی :زیر را برآورده می کند
Theorem, This equation has only solutions of period M = 2n − 1 if and only if
1. The binary polynomial is irreducible, i.e., there do not exist any two
polynomial and such that
2. is a factor of but is not a factor of for any p < M.
1z
1
1 1 2
1 2
( ) ( ) 0
,
( ) 1 n
n
A z y k
where
A z a z a z a z
1( )A z
1( )A z
1
1( )A z 1
2 ( )A z
1 1 1
1 2( ) ( ) ( )A z A z A z
1 Mz 1 Pz
9
Systems Identification
by: Dr B. Moaveni
17
Commonly used signals
Generating ML-PRBS
Systems Identification
by: Dr B. Moaveni
18
Commonly used signals
Properties of ML-PRBS
10
Systems Identification
by: Dr B. Moaveni
19
Commonly used signals
Mean and Variance of ML-PRBS
The mean of y(t) as ML-PRBS is:
Using , we have the variance as
1
1 1 10.
1 1( )
2 2 25
M
i
Mm y t
M M M
2 ( ) ( )y t y t
22 2 2
2 21
1 1 1 1(0) var( ( )) ( ) m
4 4 40.25
M
i
MC y t y t m m
M M M
Systems Identification
by: Dr B. Moaveni
20
Commonly used signals
Covariance of ML-PRBS
Covariance of y(t) as ML-PRBS is:
2 2
1 1
2
2
1 1( ) ( ) ( ) ( ) ( ) ( ) ( )
1
2 4
, 1,2, , 1
M M
t t
C y t y t m y t y t y t y t mM M
m Mm m
M
where M
11
Systems Identification
by: Dr B. Moaveni
21
Commonly used signals
ML-PRBS vs. White Noise
Define so that its outcome is either −1 or 1
When M is large, the covariance function of PRBS has similar properties to a
white noise.
( ) 1 2 ( )y t y t
2
2
11 2
1(0) 4 (0) 1
1 1( ) 4 ( ) ,
0
1
0 1, 2, , 1
M
M
M
m mM
C CM
C C MM M
Systems Identification
by: Dr B. Moaveni
22
Commonly used signals
ML-PRBS vs. White Noise
Define so that its outcome is either −1 or 1
When M is large, the covariance function of PRBS has similar properties to a
white noise
However, their spectral density matrices can be drastically different.
( ) 1 2 ( )y t y t
2
2
11 2
1(0) 4 (0) 1
1 1( ) 4 ( ) ,
0
1
0 1, 2, , 1
M
M
M
m mM
C CM
C C MM M
12
Systems Identification
by: Dr B. Moaveni
23
Commonly used signals
Spectral Density of ML-PRBS
The output of PRBS sequence is shifted to values −a and a with period M, the
autocorrelation function is also periodic and given by
Since is periodic with period M, it has a Fourier representation:
2
2
2
0, , 2 ,
( )
a M M
R aotherwise
M
( )R
Systems Identification
by: Dr B. Moaveni
24
Commonly used signals
Spectral Density of ML-PRBS
Therefore, the spectrum of PRBS is an impulse train:
Using the expression of , we have
Therefore,
It does not resemble spectral characteristic of a white noise (flat spectrum).
1
0
2( ) ( )
M
k
k
kS C
M
( )R
2 2
0 2 2, ( 1), 1,2,k
a aC C M k
M M
2 1
21
2( ) ( ) ( 1) ( )
M
k
k
a kS M C
M M
13
Systems Identification
by: Dr B. Moaveni
25
Commonly used signals
ML-PRBS vs. Colored Noise
Systems Identification
by: Dr B. Moaveni
26
Commonly used signals
ML-PRBS vs. Colored Noise
14
Systems Identification
by: Dr B. Moaveni
27
Commonly used signals
Design a PRBS (Rivera and Jun 2000):
1. Number of shift registers: ?
2. Switching Time (minimum time between changes in the level of the
signal): ?
Consider the desired frequency range for PRBS signal: min max,
( )
min
( )
max
1Hz
H
s dom
Hz s
L
dom
Typically, αs = 2 and βs = 3 (Deshpande, 2013)
Systems Identification
by: Dr B. Moaveni
28
Commonly used signals
Design a PRBS:
1. Number of shift registers: ?
2. Switching Time (minimum time between changes in the level of the
signal): ?
Rivera and Jun (2000) showed that the switching time and the maximum
length of signal (M) can be calculated as:
max
2.8 2.8L
domsw
s
T
min
2 . . 22 1
Hn s dom
sw sw
MT T
15
Systems Identification
by: Dr B. Moaveni
29
Persistent excitation
As seen in the Lecture 3, in order for the LS estimate to give unique solutions, we
need to have that the associated sample covariance matrix is full rank. The
notion of Persistency of Excitation (PE) is an interpretation of this condition
when the LS estimate is applied for estimating the parameters of a dynamical
model based on input- and output behavior.
The following example help us to clear the subject:
Systems Identification
by: Dr B. Moaveni
30
Persistent excitation
16
Systems Identification
by: Dr B. Moaveni
31
Persistent excitation
Systems Identification
by: Dr B. Moaveni
32
Persistent excitationDefinition. A signal u(t) is persistently exciting of order n if
1. The following limit exists:
2. The following matrix is positive definite
1
1( ) lim ( ) ( )
N
uN
t
R u t u tN
(0) (1) ( 1)
(1) (0) ( 2)( )
( 1) ( 2) (0)
u u u
u u u
u u u
R R R n
R R R nn
R n R n R
R
17
Systems Identification
by: Dr B. Moaveni
33
Persistent excitation
Systems Identification
by: Dr B. Moaveni
34
Persistent excitation
18
Systems Identification
by: Dr B. Moaveni
35
Persistent excitation
Theorem. Let u be a quasi-stationary input of dimension nu, with
spectrum . Assume that for at least n distinct
frequencies. Then u is PE of order n.
Theorem (Scalar):
If is PE of order n for at least n-points.
Systems Identification
by: Dr B. Moaveni
36
Persistent excitation
( )uS ( ) 0uS
( )u t ( ) 0uS
19
Systems Identification
by: Dr B. Moaveni
37
Persistent excitation
Example
Systems Identification
by: Dr B. Moaveni
38
Persistent excitation
20
Systems Identification
by: Dr B. Moaveni
39
Persistent excitation
Systems Identification
by: Dr B. Moaveni
40
Persistent excitation
21
Systems Identification
by: Dr B. Moaveni
41
Ergodicity
Stationary Process
Wide Sense Stationary Process
1 , pdf is not a function of time
2 ( , ) ( ) Autocorrelation is not a function of time
x x
x x
pdf t x pdf xconditions
R t R
2 21 ( ) & ( )
2 ( , ) ( ) Autocorrelation is not a function of timex x
t tconditions
R t R
Systems Identification
by: Dr B. Moaveni
42
Ergodicity
Ensemble Averaging and Time Averaging
1 1 1 1
Stationary
x p x dx
22
Systems Identification
by: Dr B. Moaveni
43
Ergodicity
Ergodicity Condition
( )
1
:
( ) ( )i
Ergodicity Conditi
x t x
on
t
Systems Identification
by: Dr B. Moaveni
44
Ergodicity
Non Ergodic Process
1(t)v
2(t)v
(t)Nv
t
t
t
* 1( ) 0E v t
* 1( ) ( )iE v t E v t
Ergodic Process
1R
1 v
2R
2 v
NR
Nv
1(t)v
2(t)v
* 1( ) ( )iE v t E v t
23
Systems Identification
by: Dr B. Moaveni
45
Ergodicity
Stationary, Wide Sense Stationary and Ergodic process
*** Why Ergodicity is important issue?
Random Process
Wide S. S.
Stationary Process
Ergodic Process
References
1- Prof. Munther A. Dahleh, System Identification, MIT, (Lectrue Notes).
2- T. Soderstrom and P. Stoica, System Identification, Prentice Hall, 1989.
3- Kristiaan Pelckmans, Lecture Notes for a course on System Identification,
v2012, Uppsala University, Sweden.
4- http://www.youtube.com/watch?v=k6y2kzayV6A, Published on Mar 19, 2013,
By Dr Ivica Kostanic from communications theory course ,Florida tech.
5- Rivera, D. E. and K. S. Jun (2000). "An integrated identification and control
design methodology for multivariable process system applications." Control
Systems, IEEE 20(3): 25-37.
6- Deshpande, S. and D. E. Rivera (2013). Optimal input signal design for data-
centric estimation methods. American Control Conference (ACC), 2013, IEEE.
Systems Identification
by: Dr B. Moaveni
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