Lecture8 ID

8/12/2019 Lecture8 ID

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EE392m - Winter 2003 Control Engineering 8-1

Lecture 8 - Model Identification

What is system identification?

Direct pulse response identification Linear regression

Regularization

Parametric model ID, nonlinear LS


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What is System Identification?

White-box identification

estimate parameters of a physical model from data

Example: aircraft flight model

Gray-box identification

given generic model structure estimate parameters from data

Example: neural network model of an engine

Black-box identification determine model structure and estimate parameters from data

Example: security pricing models for stock market

Data Identification ModelExperiment

Plant

Rarelyusedin

real-lifecontr

ol


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Industrial Use of System ID Process control - most developed ID approaches

all plants and processes are different

need to do identification, cannot spend too much time on each

industrial identification tools

Aerospace

white-box identification, specially designed programs of tests Automotive

white-box, significant effort on model development and calibration

Disk drives

used to do thorough identification, shorter cycle time

Embedded systems

simplified models, short cycle time


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Impulse response identification Simplest approach: apply control impulse and collect the

data

Difficult to apply a short impulse big enough such that the

response is much larger than the noise

0 1 2 3 4 5 6 7

0

0.2

0.4

0.6

0.8

1IMPULSE RESPONSE

TIME

0 1 2 3 4 5 6 70. 5

0

0. 5

1NOISY IMPULSE RESPONSE

TIME Can be used for building simplified

control design models from complex sims


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Step response identification Step (bump) control input and collect the data

used in process control

Impulse estimate still noisy: impulse(t) = step(t)-step(t-1)

0 200 400 600 800 1000

0

0.5

1

1.5STEP RESPONSE OF P APER WEIGHT

TIME (SEC)

0 100 200 300 400 500 600

0

0.1

0.2

0.3

IMPULSE RESPONSE OF PAPER WEIGHT

TIME (SEC)

Actuator bumped


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Noise reductionNoise can be reduced by statistical averaging:

Collect data for mutiple steps and do more averaging toestimate the step/pulse response

Use a parametric model of the system and estimate a few

model parameters describing the response: dead time, rise

time, gain

Do both in a sequence

done in real process control ID packages

Pre-filter data


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Linear regression Mathematical aside

linear regression is one of the main System ID tools

)()()(1

tetty j

N

j

j +=!=

Data Regression weights Regressor Error

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!!

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Linear regression Makes sense only when matrix is

tall, N >K, more data available thanthe number of unknown parameters.

Statistical averaging

Least square solution: ||e||2min

Matlab pinv or left matrix division \

Correlation interpretation:

( ) yTT = 1

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cR 1 =


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Example: linear first-order model

Linear regression representation

)()1()1()( tetgutayty ++=

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$%&

'=

=

=

g

a

tut

tyt

)1()(

)1()(

2

1

This approach is considered in most of the technical

literature on identification

Matlab Identification Toolbox

Industrial use in aerospace mostly Not really used much in industrial process control

Main issue:

small error in a might mean large change in response

Lennart Ljung, System Identification: Theory for the User, 2nd Ed, 1999

( ) yTT = 1


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Regularization Linear regression, where is ill-conditioned

Instead of ||e||2

min solve a regularized problem

r is a small regularization parameter

Regularized solution

Cut off the singular values of that are smaller than r

ey +=

T

min22+ re

( ) yrI TT += 1


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Regularization Analysis through SVD (singular value decomposition)

Regularized solution

Cut off the singular values of that are smaller than r

n

jj

mmnnTsSRURVUSV 1

,, }diag{;;;=

==

( ) yUrs

sVyrI T

n

jj

jTT

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()

(*+

(,

(-.

+=+=

=

1

2

1diag

0 1 2 3 4 50

0.5

1

1.5

2

s

REGULARIZED INVERSE

1.02 +ss


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Linear regression for FIR model

Linear regression representation

)()()()( 1 tektukhty

K

k

+=

!=

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=

=

)(

)1(

)()(

)1()(1

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h

Ktut

tut

K

!!

( ) yrI TT += 1

Identifying impulse response by

applying multiple steps

PRBS excitation signal

FIR (impulse response) model 0 10 20 30 40 50-1

-0.5

0

0.5

1

PRBS EXCITATION SIGNAL

PRBS =

Pseudo-Random Binary Sequence,see IDINPUTin Matlab


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Example: FIR model ID PRBS excitation

input

Simulated system

output: 4000

samples, random

noise of theamplitude 0.5

0 200 400 600 800 1000

-1

-0.5

0

0.5

1

PRBS excitation

0 200 400 600 800 1000

-1

-0.5

0

0.5

1

SYSTEM RESPONSE

TIME


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Example: FIR model ID

Linear regression

estimate of the FIR

model

Impulse response

for the simulated

system:T=tf([1 .5],[1 1.1 1]);

P=c2d(T,0.25);

0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15

0.2

FIR es timate

Impulse Res pons e

Time (sec)

0 1 2 3 4 5 6 7-0.05

0

0.05

0.1

0.15

0.2


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Nonlinear parametric model ID Prediction model depending on

the unknown parameter vector

Loss index

Iterative numerical optimization.

Computation of Vas a subroutine sim

Model including the

parameters

Optimizer

Loss Index V

)(tu

! =

2)|()( tytyV

! =2

)|()( tytyJ

)|()MODEL()( tytu

)(ty

Lennart Ljung, Identification for Control: Simple Process Models,

IEEE Conf. on Decision and Control, Las Vegas, NV, 2002


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Parametric ID of step response First order process with deadtime

Most common industrial process model

Response to a control step applied at tB

Example:

( )

,

-.

>+=

DB

DB

Ttt

Ttt

Tttegty

DB

for,0

for,1)|(

/)(

Papermachine

process

DT

g

"""

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#

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&

'

=

DT

g


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Gain estimation For given , the modeled step response can be

presented in the form

This is a linear regression

Parameter estimate and prediction for given

),|()|( 1 DTtygty +=

!==

2

1

)()|(k

kk twty

DT,

( ) yTw TTD = 1

),( ),|(),|( 1 DD TtygTty +=

1)(),|()(

2

11

2

1

=

=

=

=

tTtyt

wgw D

DT,


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Examples: Step response ID Identification results for real industrial process data

This algorithm works in an industrial tool used in 500+

industrial plants, many processes each

0 10 20 30 40 50 60 70 80-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Proce ss parameters: Gain = 0.134; Tdel = 0.00; Trise = 119.8969

time in se c.; MD res ponse - solid; es timated res ponse - das hed

Linear

Regression ID

of the first-ordermodel

Nonlinear

Regression ID

Nonlinear

Regression ID


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Linear filtering

Lis a linear filtering operator, usually LPF

A trick that helps: pre-filter data

Consider data model

euhy += *

$$

)(**)()*(

)*(

LuhuLhuhL

LeuhLLyf

fey

==

+=

Can estimate hfrom filteredyand filtered u

Or can estimate filtered h from filteredyand raw u

Pre-filter bandwidth will limit the estimation bandwidth


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Multivariable ID Apply SISO ID to various input/output pairs

Need ntests - excite each input in turn

Step/pulse response identification is a key part of the

industrial Multivariable Predictive Control packages.

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Lecture8 ID