system identificationkarimpor.profcms.um.ac.ir/imagesm/354/stories/iden/system_iden4.pdf ·...

SYSTEMSYSTEMIDENTIFICATIONIDENTIFICATION

Ali KarimpourAssociate Professor

Ferdowsi University of Mashhad

کراريشناسائی سیستم تالیف دکتر مهدي : مراجعAdvanced PID Control Technologies by Karl J. Astrom and Tore Hagglund

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Ali Karimpour Jan 2014

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Lecture 4Classical System Identification Classical System Identification

MethodsMethodsTopics to be covered include:

Introduction.

Classical Identification Methods for Discrete Systems.

Classical Identification Methods for Continuous Systems.

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Classical System Identification MethodsClassical System Identification Methods

Topics to be covered include:

Introduction.



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Introduction

. است LTIمنظور از روشهاي کالسیک روشهاي مبتنی بر نظریه سیستمهاي

مبتنی بر پاسخ پله •مبتنی بر پاسخ ضربه •

مبتنی بر پاسخ فرکانسی •مبتنی بر پاسخ ورودي هاي ساده •

:موارد کاربرد این روشها

. باشد LTIسیستم واقعی که در حال شناسائی آن هستیم، در واقع •

.نویز در سیستم تقریبا قابل اغماض باشد •

.چنانچه سیستم با این روشها شناسائی نشد روشهاي پیچیده تر خطی و غیر خطی استفاده می شود

.مناسب جهت طراحی کنترلر •

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Introduction.



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Classical Identification Methods for Discrete Systems

Classical Identification Methods for discrete systems are:

1. Time domain identification.

2. Frequency domain identification.(It is a parametric procedure such as linear parametric models)

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In causal discrete LTI systems relation between input and output is:

+u(t) y(t)

v(t))()()()(

0tvktukgty

k

g(k): impulse response

In stable causal discrete LTI systems relation between input and output can be approximated by:

)()()()(0

tvktukgtym

k

System Identification: Determination of impulse response (g(k)) of system.

Classical Identification Methods (Time domain identification)

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1. Impulse response test.

2. Step response test.

3. Arbitrary input test.

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I) Impulse response test:

)()()(0

ktukgtym

k

0....)2()1(,1)0( uuu


)0()0( gy )1()1( gy

.........)()( mgmy

II) Step response test:

)()()(0

ktukgtym

k

1....)2()1()0( uuu

)0()0( gy )1()0()1( ggy

.........)(....)1()0()( mgggmy

)1()()( kykykg

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III) Arbitrary applied signal test:

)()()(0

ktukgtym

k

muuuu ,...,,, 210


0)0()0( ugy

.........

01 )(....)1()0()( umgugugmy mm

0/)0()0( uyg

01 )1()0()1( ugugy

01 /))0()1(()1( uugyg .........

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Example 5-1: Let y(t)=12u(0)+8u(-1)+4u(-2)+2u(-3)+e(t) e(t) is a zero mean white noise with σ2=0.25.


I) Impulse response test: y=[11.5283 8.4696 3.8795 1.5283]

II) Step response test: y=[11.5283 20.4696 23.8795 25.3708 ]

III) Arbitrary applied signal test: u= [1;0.5 ;-0.5;-1] , y=[11.5283 14.4696 1.8795 -12.4717]

,5283.1)3(ˆ,8795.3)2(ˆ,4696.8)1(ˆ,5283.11)0(ˆ gggg 8247.0ˆ2 gg

,4914.1)3(ˆ,4098.3)2(ˆ,8.9413)1(ˆ,5283.11)0(ˆ gggg 3098.1ˆ 2 gg

,2503.1)3(ˆ,3491.4)2(ˆ,.99807)1(ˆ,5283.11)0(ˆ gggg 9521.0ˆ 2 gg

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Introduction.



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Classical Identification Methods for Continuous Systems

Classical Identification Methods for continuous systems are:

1. Time domain identification.

2. Frequency domain identification.

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Classical Identification Methods for Continuous SystemsTime domain identification.

In causal discrete LTI systems relation between input and output is:

+u(t) y(t)

v(t))()()()( 0 tvdtuhtyt

h(t): impulse response

System Identification: Determination of impulse response (h(t)) of system.


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1. Impulse response test.

2. Step response test.

3. Doublet-Pulse Method

• First order systems.

• Second order systems.

• n-order systems.

• Prony analysis


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I) Impulse response test:


What is the draw backs of impulse response test?

dtuhty t )()()( 0

dthty t )()()( 0 )(th

II) Step response test:

dtuhty t )()()( 0 dhtyt

0 )()( dtdyth )(

What is the draw backs of step response test?


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Dynamic models should give the relation between the input and the output signal during transients.

Step response of the system is used to describe the dynamical behavior.

Step response of process control.

Temperature control, level control, ….

Step response that is uncommon in process control.

Oscillation process, spring constructions, ….


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Step response of the integration process.

Temperature control in a well isolated chamber, level control, ….

Step response of a system with long time delay.In transportation systems, ….

Step response of a non-minimum phase system.Water level in boilers, generation of hydro generators, ….


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dsTp eTsK

susysg

1)()()(

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1- FO systems: First order systems.

2- FOTD systems: First order system with time delay.

3- SOTD systems: Second order system with time delay.

dsTp esTsT

Ksusysg

)1)(1()()()(

21

1)()()(

TsK

susysg p

5- FOTDI: First order system with time delay with integration. sLv e

sTsK

susysg

)1()()()(

4- Integral and time delay processes:sLv e

sK

susysg )()()(


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6- Non-interacting tanks or multiple lags

n

p

TsK

susysg

)1()()()(

7- Oscillatory Systems

12)()()()(

2

sTsTK

susysg p

8- System with inverse responses.

14.11

)()()(

2

sssT

susysg


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Step response test (The Bump Test)

The bump test is a simple procedure that is commonly used in process control.

FOTD systems can be find by bump test. sLp eTsK

susysg

1)()()(


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Integral and time delay process sLv esK

susysg )()()(

LaK v




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Oscillatory Systems

21/2 ed21/2 npT

2)ln/2(11

d

p

n T212



22

2

2)()()(

nn

np

ssK

susysg

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Double step response test (The Doublet-Pulse Method)

Tp is chosen a little longer than the time delay of the process.

The pulse amplitude a is chosen so that the response is well above the noise level.

FOTD systems can be find by bump test. sLp eTsK

susysg

1)()()(

pp

p

p

TtLorTtLyyT

T

ayyK

2

)1ln(

minmax

max

min

min

2

max


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Prony analysis


systemorder High )( sg

n

iii

t

i tfeAty i1

)2cos()(

n

i

jtfjjtfjt

i

iiii

ieeeeeAty

1

22

22)(

n

i

kTfjji iiieeAkTy1

)2(

2)(

....................

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Bode Diagram Method

Classical Identification Methods for Continuous SystemsFrequency domain identification.

1)()()(

sk

susysh

Ak log20

?1

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Bode Diagram Method


22

2

2)()()(

nn

n

ssk

susysg

220 I

2121log20

rM

221 nr

122

II

2log20rd

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Bode Diagram Method


Minimum phase and non-minimum phase systems?

Systems with time delay?

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Bode Diagram Method


Noise reduction in frequency domain.

)())(cos()( tejHtjHA

))(sin()(2

jHjHAys

))(cos()(2

jHjHAyc

222)( cs yyAjH

c

s

yyjH 1tan)(

Integration reduce the noise effect.

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Relay Feedback

frequency cross Phase 180 daK

4180

d-dRelay method is used to derive phase

crossover frequency and gain of systemat that frequency.


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Relay Feedback

frequency cross Phase 180 daK

4180


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