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شناسایی سیستم ها

Lecture 5

باسمه تعالی

Input Design

Systems Identification

by: Dr B. Moaveni

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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.

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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


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Autoregressive, moving average process

1

1

( )

( )k k

D zu e

C z

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Autoregressive, moving average process

1

1

( )

( )k k

D zu e

C z


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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

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Spectrum of ARMA process

Let e(t) be a white noise with variance .

The spectral density of ARMA process is

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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

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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”


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Characterization of sinusoids

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Spectrum of sinusoidal inputs


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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

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• 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

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ML-PRBS


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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

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Generating ML-PRBS


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Properties of ML-PRBS

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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


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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

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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


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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

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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


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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

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ML-PRBS vs. Colored Noise


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ML-PRBS vs. Colored Noise

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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)


by: Dr B. Moaveni

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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

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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:


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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

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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.


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( )uS ( ) 0uS

( )u t ( ) 0uS

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Example


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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


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Ergodicity

Ensemble Averaging and Time Averaging

1 1 1 1

Stationary

x p x dx

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Ergodicity

Ergodicity Condition

( )

1

:

( ) ( )i

Ergodicity Conditi

x t x

on

t


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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

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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.


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http://www.youtube.com/watch?v=k6y2kzayV6A

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