Lecture 8 - Model Identification - Stanford University · PDF fileLecture 8 - Model Identification ... System Identification: Theory for the User, ... • Identification results for

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


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

Rar

ely

used

inre

al-li

fe c

ontro

l


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


Impulse response identification

• Simplest approach: apply control impulse and collect thedata

• Difficult to apply a short impulse big enough such that theresponse is much larger than the noise

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1IMPULSE RESPONSE

TIME

0 1 2 3 4 5 6 7-0.5

0

0.5

1NOISY IMP ULSE RESPONSE

TIME• Can be used for building simplifiedcontrol design models from complex sims


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 PAPER 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


Noise reduction

Noise can be reduced by statistical averaging:• Collect data for mutiple steps and do more averaging to

estimate the step/pulse response• Use a parametric model of the system and estimate a few

model parameters describing the response: dead time, risetime, gain

• Do both in a sequence– done in real process control ID packages

• Pre-filter data


Linear regression

• Mathematical aside– linear regression is one of the main System ID tools

)()()(1

tetty j

N

jj +=∑

=

ϕθ

Data Regression weights Regressor Error

ey +Φ= θ

=

=

=Φ

=

)(

)1(,,

)()(

)1()1(,

)(

)1( 1

1

1

Ne

ee

NNNy

yy

KK

K

MM

K

MOM

K

M

θ

θθ

ϕϕ

ϕϕ


Linear regression

• Makes sense only when matrix Φ istall, N > K, more data available thanthe number of unknown parameters.– Statistical averaging

• Least square solution: ||e||2 → min– Matlab pinv or left matrix division \

• Correlation interpretation:

( ) yTT ΦΦΦ= −1θ̂

ey +Φ= θ

=

=

∑

∑

∑∑

∑∑

=

=

==

==

N

tK

N

t

N

tK

N

tK

N

tK

N

t

tyt

tyt

Nc

ttt

ttt

NR

1

11

1

2

11

11

1

2

)()(

)()(1,

)()()(

)()()(1

1

ϕ

ϕ

ϕϕϕ

ϕϕϕ

M

K

MOM

K

cR 1ˆ −=θ


Example: linear first-order model

• Linear regression representation

)()1()1()( tetgutayty +−+−=

=−=−=

ga

tuttyt

θϕϕ

)1()()1()(

2

1

• This approach is considered in most of the technicalliterature 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 ΦΦΦ= −1θ̂


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 Φ that are smaller than r

ey +Φ= θ

ΦΦT

min22 →+ θre

( ) yrI TT Φ+ΦΦ= −1θ̂


Regularization• Analysis through SVD (singular value decomposition)

• Regularized solution

• Cut off the singular values of Φ that are smaller than r

njj

mmnnT sSRURVUSV 1,, }diag{;;; ==∈∈=Φ

( ) yUrs

sVyrI T

n

jj

jTT

+=Φ+ΦΦ=

=

−

12

1 diagθ̂

0 1 2 3 4 50

0.5

1

1.5

2

s

REGULARIZED INVERSE

1.02 +ss


Linear regression for FIR model

• Linear regression representation

)()()()(1

tektukhtyK

k+−=∑

=

=

−=

−=

)(

)1(

)()(

)1()(1

Kh

h

Ktut

tut

K

MM θϕ

ϕ( ) yrI TT Φ+ΦΦ= −1θ̂

• Identifying impulse response byapplying multiple steps

• PRBS excitation signal• FIR (impulse response) model 0 10 20 30 40 50

-1

-0.5

0

0.5

1PRBS EXCITATION SIGNAL

PRBS =Pseudo-Random Binary Sequence,see IDINPUT in Matlab


Example: FIR model ID

• PRBS excitationinput

• Simulated systemoutput: 4000samples, randomnoise of theamplitude 0.5

0 200 400 600 800 1000-1

-0.5

0

0.5

1PRBS excita tion

0 200 400 600 800 1000

-1

-0.5

0

0.5

1

SYSTEM RESPONSE

TIME


Example: FIR model ID

• Linear regressionestimate of the FIRmodel

• Impulse responsefor the simulatedsystem:

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 tima te

Impuls e Res pons e

Time (s ec)0 1 2 3 4 5 6 7

-0.05

0

0.05

0.1

0.15

0.2


Nonlinear parametric model ID

• Prediction model depending onthe unknown parameter vector

• Loss index

• Iterative numerical optimization.Computation of V as a subroutine sim

Model including theparameters

Optimizerθ Loss Index V

)(tu

∑ −= 2)|(ˆ)( θtytyV

θ

∑ −= 2)|(ˆ)( θtytyJ

)|(ˆ)MODEL()( θθ tytu →→

)(tyθ

Lennart Ljung, “Identification for Control: Simple Process Models,”IEEE Conf. on Decision and Control, Las Vegas, NV, 2002


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

DBTtt

TttTtteg

tyDB

for ,0for ,1

)|(/)( τ

γθ

Papermachineprocess

DT

τ

g

=

DT

gτ

γ

θ


Gain estimation

• For given , the modeled step response can bepresented in the form

• This is a linear regression

• Parameter estimate and prediction for given

),|()|( 1 DTtygty τγθ ⋅+=

∑=

=2

1

)()|(k

kk twty ϕθ

DT,τ

( ) yTw TTD ΦΦΦ= −1),(ˆ τ ),|(ˆˆ),|(ˆ 1 DD TtygTty τγτ ⋅+=

1)(),|()(

2

11

2

1

==

==

tTtyt

wgw D

ϕτϕ

γ

DT,τ


Rise time/dead time estimation

• For given , the loss index is

• Grid and find the minimum of

DT,τ

∑=

−=N

tDTtytyV

1

2),|(ˆ)( τ

),( DTVV τ=DT,τ


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 s s pa ra me te rs : Ga in = 0.134; Tde l = 0.00; Tris e = 119.8969

time in s ec.; MD res pons e - solid; e s tima te d re spons e - da s hed

Linear Regression ID

of the first-ordermodel

Nonlinear Regression ID

Nonlinear Regression ID


Linear filtering

• L is a linear filtering operator, usually LPF

• A trick that helps: pre-filter data• Consider data model

euhy += *

{ {

)(**)()*(

)*(

LuhuLhuhL

LeuhLLyff

ey

==

+=

• Can estimate h from filtered y and filtered u• Or can estimate filtered h from filtered y and ‘raw’ u• Pre-filter bandwidth will limit the estimation bandwidth


Multivariable ID• Apply SISO ID to various input/output pairs• Need n tests - excite each input in turn

• Step/pulse response identification is a key part of theindustrial Multivariable Predictive Control packages.

Lecture 8 - Model Identification - Stanford University · PDF fileLecture 8 - Model Identification ... System Identification: Theory for the User, ... • Identification results for

Documents