06 model robustness cts

7/21/2019 06 Model Robustness CTS

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February 21, 2016 Reliance Technology Group – APC / RTO For Internal Circulation1

Process Engineering GroupProcess Engineering Group

Model RobustnessModel Robustness

Rajalingam RRajalingam R




1) Importance of Model Robustness

2) Tools for making models robust

) Model Identification ! "M#

FIR Identification

$ubspace identification

%) Model Robustness

Scope



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&'ample

Condition No: 23957 - ILL Conditioned

Condition No: 35 - !ell Conditioned

Solution"#uation

Impo$tance o% Model Robustness



Model Robustness

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Model Robustness is important from mat(ematical stability

* Model Identification+ Finite Impulse Response FIR) and $ubspace

* FIR is Multiple input and $ingle -utput Model MI$-)

* $ubspace is Multiple input Multiple -utput Model MIM-)

* Model .ncertainty+* Time "omain and Fre/uency "omain nalysis

* #orrelations+

* uto correlation #ross correlation analysis

* #ondition number and R+

* #ondition 3umber * Relati4e ain analysis




Model IdentifcationModel Identifcation



Linea$ Models


$tructure &mpirical9data!based First principles

Mat(ematicalform

:inear9non!linear :inear9non!linear

"'amples o% Linea$ models+

$tatic pressure ;it( respect to dept( of li/uid

<olume of a 4ertical cylinder ;it( respect to le4el

"'amples o% Non-linea$ models:

In distillation column, t(e c(ange in purity ;it( Reboiler "uty ;ill be more ;(en

Reflu' Ratio is small= and 4ice 4ersa

Linea$i(ation

If non!linear, t(en use a suitable transform like :- transform) to make t(erelation linear


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)ene%its o% Linea$ models


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@I1 *+ I+,+

I2,+*+

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C. C+ C2 C3 C/ C5 C0 C7

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#<

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Identi%ication

δ #< A B ∆I, IAM<

Cno;n

Calculate

∆∆

∆∆

=

∂∂∂∂∂∂∂

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2

1

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2%%

12%

12

12

1

8

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Cno;n



Response to a c(ange in t(e independent+

#<1 ! #<0 A a1 B DI)

#<2 ! #<0 A a2 B DI)

#< ! #<0 A a B DI)#<% ! #<0 A a% B DI)

#<5 ! #<0 A a5 B DI)

#<6 ! #<0 A a5 B DI)

DIInd

a1a2

aa% a5

Ereser4ation of $cale

#<1

#<2

#<

#<% #<5

#<0

Time 0 1 2 % 5

"ep

Identi%ication


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0 2 % 6 >

Ind 1!2

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!1

!5

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!1%

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#<

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!8,!2)B8,0)B8,1)B8#<#<

!5,!2)B6,0)B8,1)B8#<#<

!1,!2)B%,0)B6,1)B8#<#<

%,!2)1B,0)B%,1)B6#<#<

%,0)1B,1)B%#<#<

1,1)1B#<#<

08

06

05

0%

0

02

01

=++=−

=++=−

=++=−

=++=−

=++=−=+=−

==−

Identi%ication

Time 0 1 2 % 5 6 8



Finite Impulse Response FIR) Identification

ny impulse can be represented like t(is+ G1,0,0,H0



Finite Impulse Response FIR) Identification

-utput 4ector can be ;ritten as a combination of inputs 4ector+

yJtK A (0 'JtK (1 'Jt!1K (2 'Jt!K HHHH (k!1 'Jt!k!1)K (k 'Jt!kK

* 'JtK L c(ange in independent 4ariable* yJtK L c(ange in dependent 4ariable* k A Time to $teady $tate

* t A time

We usually assume that h0= 0, i.e., the system

does not react immediately to the input.

3o;, by least s/uare obecti4e function, ;e can find

out t(e step response coefficients G(0,(1,(2,H,(k

where N = total number of samples in a dataset




FIR I"H$tep Response #oefficients



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$tep!Response coefficients

FIR I"Hstep response coefficients

Impulse $esponse coe%%icients :

Step $esponse coe%%icients +

ny step response can be represented as superposition of 4arious impulses+

G1,1,1H,1 A G1,0,0,H,0 G0,1,0,H,0 G0,0,1H,0 H G 0,0,0H,1

G0,(1,(2,(,H,0 G0,0,(1,(2,(H,0 G0,0,0,(1,(2,(H,0 H G 0,0,0H,(n



FIR Identification+ T(e Matri'

Steps to dete$mine t1e unit step $esponses

4u$ing a test pe$iod, collect time-stamped p$ocess data

%$om t1e sstem %o$ all independent and dependent

6a$iables

&$om t1e collected data, calculate C and I

Sol6e t1e cont$ol e#uation, C 8 I, %o$ t1e unit step

$esponse coe%%icients, t1e mat$i'



Erocess reac(es steady!state

after % inter4als $o, t(ere are

only % NaO coefficients

FIR Identification

"#uations %o$ p$edicting t1e linea$ sstem;

0B)IIaB)IIaB)IIaB)IIaB)II#<#<

aB)IIaB)IIaB)IIaB)II#<#<

aB)IIaB)IIaB)II#<#<

aB)IIaB)II#<#<

aB)II#<#<

01%1222%1%505

%0112221%0%

01212120

20111202

10101

−+−+−+−+−=−−+−+−+−=−

−+−+−=−−+−=−

−=−



FIR Identification+ N$tep!ResponseO Form

Step-Response &o$m o% t1e Identi%ication <$oblem

%1222%1%505

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aB)IIaB)IIaB)II#<#<

aB)IIaB)II#<#<aB)II#<#<

−+−+−+−=−−+−+−+−=−

−+−+−=−−+−=−

−=−

∆

∆

∆∆

=

∂

∂

∂

∂∂

∂

∂

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2

1

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2%%

12%

12

12

1

8

6

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I

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a

#<

#<

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#<

#<

#<

#<

*

∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆

∆∆∆

∆∆∆

=

∂∂∂∂∂

∂∂

%

2

1

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%56

2%5

12%

12

12

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8

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II

I

#<

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a

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#omplete e/uations

useful)



FIR Identification L Pad "ata

period of bad data s(ould not in4alidate t(e entire data set

In t(e step!form of t(e algorit(m

#<0 appears in eac( e/uation

-nly one continuous section of data can be analyQed using t(is algorit(m L step

response algorit(m

Pad data ;it(in a data set in4alidates t(e data set

L Erocess problem9upset

L #omputer problem

L ard;are9instrumentation failure

L <al4e saturation

L typical disturbance

-t(er forms of t(e algorit(m L eliminate #<07



FIR Identification L Impulse Form 1)

0)))

)))

))

)

+−−+−−+−−+−=−

−−−−−−−−=+

−+−+−+−+−=−

−−+−−+−−+−=−

−−−−−−=+

−+−+−+−=−

−−+−−+−=−−−−−=+

−+−+−=−

−+−=−

−−=−

−+−=−

−=−

%122212%1%5%5

%0112221%0%

%01%1222%1%505

%012121221%%

01212120

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2011212122

20111202

01212120

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10101

20111202

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a,aB)I,Ia,aB)I,Ia,aB)I,IaB)I,I#<#<

aB)I,IaB)I,IaB)I,IaB)I,I#<#<!

aB)I,IaB)I,IaB)I,IaB)I,IaB)I,I#<#<

a,aB)I,Ia,aB)I,Ia,aB)I,IaB)I,I#<#<

aB)I,IaB)I,IaB)I,I#<#<!

aB)I,IaB)I,IaB)I,IaB)I,I#<#<

a,aB)I,Ia,aB)I,IaB)I,I#<#<aB)I,IaB)I,I#<#<!

aB)I,IaB)I,IaB)I,I#<#<

a!,aB)I,IaB)I,I#<#<

aB)I,I#<#<!

aB)I,IaB)I,I#<#<

aB)I,I#<#<

$ubtract pairs of

e/uations to

remo4e #<0



FIR Identification L Impulse Form 2)

"efine NbiO as t(e impulse coefficient+ ai ! ai!1

3o;, t(e Nimpulse!formO of t(e "M#plus Model e/uations

%1222%1%5%5

%0112221%%

01212122

20111212

10101

bB)IIbB)IIbB)IIbB)II#<#<

bB)IIbB)IIbB)IIbB)II#<#<

bB)IIbB)IIbB)II#<#<

bB)IIbB)II#<#<

bB)II#<#<

−+−+−+−=−−+−+−+−=−

−+−+−=−

−+−=−−=−

%%

2

122

11

aab

aab

aab

ab

−=

−=−=

=



FIR Identification+ Impulse Response Model

#(aracteristics

Penefit+

"ata $licing is llo;ed+ #(ange in t(e dependent 4ariable is only a

function of t(e past c(anges in independent 4ariables for a time e/ual

to t(e time to steady!state

Eenalty+ dditional noise results from taking t(e deri4ati4e of t(e dependent

4ariable



FIR Identification+

$imultaneous Independent <ariable Mo4es

In a typical process, it is generally impossible to N(oldO an independent step

for a full time to steady!state ;it(out ot(er mo4ement in ot(er independent

4ariables

-perations needs to make a mo4e

Feedfor;ard, disturbance, 4ariable mo4es

simple, straig(t!for;ard solution to t(e control problem ;ill not (andle

t(ese additional mo4esH

e need a solution met(od t(at is tolerant of multiple mo4es ;it(in a single

Tss



FIR Identification+

$imultaneous Identification of Models

Multiple Input, $ingle -utput MI$-) $imultaneousidentification

of model

coefficients

∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆

∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆

∆∆∆∆∆∆∆∆∆∆ ∆∆

=

∆∆∆∆

∆∆∆∆∆

2,%

2,

2,2

2,1

1,%

1,

1,2

1,1

2,62,82,>2,?1,61,81,>1,?

2,52,62,82,>1,51,61,81,>

2,%2,52,62,81,%1,51,61,8

2,2,%2,52,61,1,%1,51,6

2,22,2,%2,51,21,1,%1,5

2,12,22,2,%1,11,21,1,%

2,12,22,1,11,21,

2,12,21,11,2

2,11,1

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>

8

6

5

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2

1

IIIIIIII

IIIIIIII

IIIIIIIIIIIIIIII

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#<

#<

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#<

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b

b

b

b

b

b

b

b

*



FIR Identification+

$imultaneous Identification of Models



First, letSs con4ert t(e control e/uation to a

residual form+ C - I 8 R

FIR Identification+ $imultaneous Identification

=

∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆

∆∆∆∆∆∆∆∆∆∆

∆∆

−

∆∆∆∆∆∆∆∆

∆

!

"

#

$

%

&

'

(

*

r

r

r

r

r

r

r

r

r

b

b

b

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b

bb

2,%

2,

2,2

2,1

1,%

1,

1,2

1,1

2,62,82,>2,?1,61,81,>1,?

2,52,62,82,>1,51,61,81,>

2,%2,52,62,81,%1,51,61,8

2,2,%2,52,61,1,%1,51,6

2,22,2,%2,51,21,1,%1,5

2,12,22,2,%1,11,21,1,%

2,12,22,1,11,21,

2,12,21,11,2

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FIR Identification+ $imultaneous Identification L $olution

3o;, ;e can use a N:east $/uaresO regression met(od to sol4e fort(e response coefficients ;(ile minimiQing NR2O

#alculate t(e sum of t(e s/uared residual terms+

Residuals$/uaredof $um=

=

++++++++=

∑ '

'

'

!

'

"

'

#

'

$

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%

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i

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r

r r r r r r r r r r r



FIR Identification+ $imultaneous Identification L

$ummary

Penefit

$imultaneous solution allo;s for c(anges in more t(an one

independent at a time during t(e test

Implication9Eenalty

Identifying t(e indi4idual responses re/uires uncorrelated mo4ement in

t(e independent 4ariables during t(e plant test

Models are identified simultaneously so c(anging t(e list of

independents may c(ange t(e model coefficients



FIR Identification "esign #riteria

period of bad data s(ould not in4alidate t(e entire

data set

$ol4e t(e impulse!form rat(er t(an t(e step!form of t(e

algorit(m

More t(an one independent 4ariable s(ould be allo;ed

to c(ange at t(e same time

$ol4e for all independent9#< response coefficients

simultaneously as a least!s/uares MI$- problem



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FIR parameters

+ =ime to stead state ==SS

2 Numbe$ o% coe%%icients

3 Smoot1ing %acto$



FIR Identification Earameters

Erocess settling time steady!state time)

:engt( of t(e step response models in time

3umber of coefficients

More coefficients re/uired for accurate modeling of fast responses less time

bet;een coefficients)

$moot(ing

Eenalty for c(ange from one coefficient to ne't

Model fit is not significantly degraded



Puild t(e #ontroller Model+ 3umber of Model #oefficients

T(e number of model coefficients, or t(e number of points used to Ndra;O t(e

line is related to t(e controller e'ecution fre/uency and t(e Tss+

#ontrol inter4al needs to be fast enoug( so t(at t(e controller can respond to

t(e fastest measured9 unmeasured independent disturbances in t(e system

Ideally, t(e controller is running fast enoug( t(at it (as 5 to 10 e'ecutions to (andle

a disturbance before it becomes a problem

−

= Interval

Collection

Data

tsCoefficien Model of Number

StateSteadytoTime

Interval

ExecutionController *



Puild t(e #ontroller Model+ 3umber of Model #oefficients

f # ff




FIR parametersH3umber of #oefficients

It decides o4ersampling ratio in model identifications

$ampling time A $ettling time 9 $tep!response coefficients

FIR 3 b f # ffi i



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FIR parametersH3umber of #oefficients

-4ersampling Ratio A 1 -4ersampling Ratio A 2

3otice gain mismatc( in FIR $$ FIR and $$ results no; matc( better

FIR t $ t(i F t



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FIR parametersH$moot(ing Factor

FIR smoot(ing algorit(m ;ill ad4ersely affect model responses ;it( faster initial

dynamics, like 4al4e positions e can use un!smoot(ed response models

instead, and use cur4e operations to remo4e noise and s/uiggles from t(e model

FIR t $ t(i F t



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FIR parametersH$moot(ing Factor

$FA 5 $FA 001

Ti " i (




Time "omain pproac(

FI I is time domain approach. Consider the followin+ euations-

i/en the compleity of differential euations, why would we e/er want to wor1 in the time

domain2

3he 4aplace transform mo/es us out of the time5domain into the comple freuency domain, so

that

we can study and manipulate our systems as al+ebraic polynomials instead of linear 67s.8( 9 's 9 $s'):8s) = 8( 9 s);8s)

3hat<s ri+ht, the 4aplace transform is hidin+ the fact that we are actually dealin+ with second5order

differential euations. In the 4aplace or time domain, if we want to account for systems with

multiple inputs and multiple outputs, we are +oin+ to need to rely on the principle of

superposition to create a system of simultaneous 4aplace euations 8or time5domain based

euations) for each output and each input. For such systems, the classical approach doesn<tsimplify the situation in MIMO case.

FIR models are $ingle input single output models $I$-) and can be e'tended for

Multiple input single output MI$-) $o, a case containing multiple #<s is run as

combination of MI$- models for eac( #<

$t t (




$tate!space pproac(

It turns out t(at if ;e decompose our (ig(er!order differential e/uations into multiple first!order

e/uations, ;e can find a ne; met(od for easily manipulating t(e system in MIM> models T(e

solution to t(is problem is state space app$oac1

State

#entral to t(e state!space notation is t(e idea of a state state of a system is t(e current 4alue

of internal elements of t(e system, t(at c(ange separately to but not completely unrelated to)

t(e output of t(e system ere are some e'amples+

*Consider a chemical reaction where certain reagents are poured into a mixing container, and the output is the amount of the

chemical product produced over time. The state variables may represent the amounts of un-reacted chemicals in the container,

or other properties such as the quantity of thermal energy in the container (that can serve to facilitate the reaction).

e denote t(e input 4ariables ;it( u, t(e output 4ariables ;it( y, and t(e state 4ariables ;it( ' In essence,

;e (a4e t(e follo;ing relations(ip+

y A f', u)

(ere f', u) is our system lso, t(e state 4ariables can c(ange ;it( respect to t(e current state and t(e

system input+

' A g', u)

(ere ' is t(e rate of c(ange of t(e state 4ariables

$tate "efinition



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$tate "efinition

$tate 4ariables A memory elements A contains all t(e information from t(e

past t(at is rele4ant for predicting t(e future

If ;e describe system as an operating mapping from t(e space of input to

t(e space of output, t(en ;e may need t(e entire input!output (istory of

t(e systems toget(er ;it( t(e planned input in order to compute future

output 4alues lternati4ely, ;e may use state 4ariables ;(ic( (as all

(istory of input and -utput 4ariables

-rder of model A number of states n)

#< t i i $ b id tifi ti



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#<s uto!grouping in $ubspace identification

$ubspace is MIM- model identification= but ;(en number of unrelated #<s are large in

case, t(en it increases computation time, and benefits ;ere limited compared to running

multiple smaller cases

T(e subspace identification algorit(m can optimiQe case runs by using #< grouping T(e

#< cross correlation is used to determine groups of #< t(at s(are common dynamics,

and create internal sub!case for t(ose related #<s MIM- for eac( internal sub case)

Py keeping related #<s toget(er t(e underlying states ;ill be identified ;it( more

certainty Py remo4ing unrelated #<s t(e computation time per subgroup is decreased

#< grouping alternati4es

1)roup Related #<s uto!grouping, "efault) L ma'imum of 10 #<s for one sub!case

2)-ne #< per roup Forced MI$- I") L ;it( t(is option, t(e users are able to compare

t(e MI$- subspace I" ;it( t(e FIR I" in a similar internal setup

)ll #<s in -ne roup Forced :arge MIM- I") L set as one large MIM-

%)$et Ma'imum #< roup $iQe MIM- I" ;it( certain number #<s per group) L gi4es

fle'ibility to modify t(e default ma'imum #<s of 10 per group

llows users to gain!validate their "nowledge about the C#s correlation in their

process, compare models with different setups, and retune the default parameters when

necessary.

#<s to gro ping in $ bspace identification




#<s uto!grouping in $ubspace identification

Time to steady state in $ubspace identification





T(eoretically, TT$$ 4alue (a4e no direct impacts on t(e subspace I", because TT$$ is

not a dependent parameter in subspace I" and it only determines (o; many model cur4e

coefficients ;ill be generated from a subspace model identified

o;e4er, "etrending filters t(at ;ill be calculated upon TT$$ 4alues affect $ubspace I"

for different TT$$






ll parameters are t(e same

e'cept TT$$




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Ma'imum states per #< group+ $ubspace identification



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$tates do not necessarily (a4e a direct p(ysical interpretations, but t(ey (a4e a

conceptual rele4ance

Ma'imum states per #< group, ;(ic( is an upper bound of t(e model order ie

ma'imum number of states allo;ed) for eac( #< group sub!case

T(is parameters allo;s user to set a model order searc( range, or force t(e subspace

algorit(m to fit a lo;!order model ;it( truncation error

(en to reduce t(e ma'imum states per #< group7






ig(!fre/uency dynamic be(a4ior of model is caused by a (ig( model order, and a model

reduction can (elp to damp t(e (ig( fre/uency noise






Ma'imum order per I9- pair+ $ubspace identification





T(is parameter is t(e identification (oriQon used in t(e identification It is e'actly

t(e number of data points in t(e future and past data (oriQon T(e larger t(e

Ma'imum order per I9- pair, t(e (ig(er order model ;ill be identified in order toco4er longer comple') dynamics T(e computation ;ill also be (ea4ier




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If 4ery different ma'imum order per I9- are used in subspace case, t(en e4en if t(e same

final model order is selected, t(e calculated models are bound to be different

T(e amount of data used for initialiQing t(e identification run depends on ma'imum order

not t(e final order it e4entually selects), so if you select a (ig( ma' order, more data is

lost If useful data is lost, eg t(e only big step you (a4e) t(en t(e model can degrade If

bad data is lost, t(en t(e model can actually impro4e $o as before ;it( "MI, be careful

;it( slicing t least you lose muc( less data t(an before

:ets assume ma' order A n T(en n pre4ious n data points are t(en used to predict t(ene't n steps into t(e future T(e cost function is determined from t(e difference bet;een

predicted #<s and obser4ed #<s o4er t(ese n steps T(is means t(at a (ig( ma' order

fa4ors t(e long term prediction accuracy, ;(ile lo;er ma' orders ;ill fa4or t(e s(ort term

prediction accuracy of t(e models






"rift Remo4al ! $ubspace




"rift Remo4al $ubspace

ll process units are disturbed by

se/eral drift disturbances.

rift disturbances are 1ind of

disturbances for which you don’tha/e measure, and it is not possible to

include them as FF si+nal. >ecause of

these drift disturbances, your process

+ain chan+es o/er time.

7ample- in heat echan+er, process

stream is bein+ heated by ?@ steam.

Now, heat echan+er will be

pro+ressi/ely foulin+. Ao o/er a

period time , heat transfer coefficient

will come down, and for the same

amount of steam chan+e 8say (tph)Btemperature chan+e of the outlet

process steam will not be the same

8will be the less as more and more

foulin+ happens).

"ata Ere!processing+ $ubspace identification



February 21, 2016 Reliance Technology Group – APC / RTO For Internal Circulation5?

"ata Ere processing+ $ubspace identification

4ailable option on data preprocessing are listed belo;+

1)"etrending

2)"ifferencing

)Uero!mean

%)"ouble "iff

Detrending- @rocess test data often contains low freuency drift that is introduced by un1nown disturbances. 3hese low

freuency disturbances will create ne+ati/e impacts on identification and has to be eliminated. 3he option “etrendin+” is

desi+ned to do that ob.

ssume a measured si+nal can be di/ided into a process si+nal and a trend si+nal.

3he pre5process of data with etrendin+ will estimate the trend si+nal and remo/e it from the system . >y default, the time5

constant of etrendin+ filter is calculated based on the caseDs 33AA-

>asis for &*33AA and #*33AA 5 how much etrendin+ filter should be used2

If central average is calculated over a period of 3*TTSS t!en filter operation tends to remove only slo" drift

disturbances and leave dynamic information mostly intact# $ too small TTSS value for a case may lead to an “ over%

Detrending ” t!at may remove useful dynamic information from t!e data and cause inaccurate models#






"ifferencing+ yA't1) L 't)Aimilar to etrendin+, differencin+ is also intended to remo/e slow drifts from the process data. 3he

FI model I has been usin+ differencin+ as a data pre5processin+ strate+y since the ?C was first

built. ifferencin+ can remo/e the slow disturbance, but it reduces Ai+nal to Noise 8AEN) ratio.

3herefore, for better model +ains and less strin+ent reuirements on AEN, the etrendin+ is

recommended in place of differencin+.

Uero mean+ It simply remo/es only an offset from raw data. It should be aware that there will be

multiple offsets if data consists of se/eral slices. 3he ero5mean operation will be performed on each

data slices.





p g p

"ouble "ifferencing+ Q A y t1) L y t) A 't1)! 't) L 't) ' t!1)

A 't1) L 2 't) ' t!1)

"ouble differencing ;as originally designed for Ramp and Eseudo!Ramp #< in FIR

I" operation For subspace I", it is no longer t(e only ;ay to preprocess Ramp

#<s &it(er "ifferencing or "ouble "ifferencing as an option ;as selected, a single

"ifferencing ;ill be first applied to a Ramp or Eseudo!Ramp #<

In addition, Ramp #< or Eseudo!Ramp #< are al;ays put into one or more Ramp

#< group by t(e uto!grouping algorit(m in order to separate Ramp #<s from non!

Ramp #<s

-4er $ampling Ratio+ $ubspace identification




p g p

In t(e case ;(ere t(e process data are collected in a

(ig( fre/uency sampling and t(e final model for on!linecontrol ;ill be run in slo;er control sampling fre/uency, a

4alue of greater t(an one can be used to matc( t(e

modelSs sampling rate ;it( t(e controllersS

measurements &g L # analyQers

-4er $ampling Ratio+ $ubspace identification




p g p

T(e model you get by setting -4er!$ampling Ratio A 1 ;ill

(a4e less precise gain but less delay ;it( t(e dynamics

"ynamic delay means t(e model calculated by $$ isslo;er t(an real system

:ast cur4e is ;it(

$RA10, and it is laggingbe(ind $RA1 cur4e

"irect Term "!matri')+ $ubspace identification




) p

#ase ;it( non!e'istent in4erse response




#ase ;it( non e'istent in4erse response

#ase ;it( non!e'istent initial kick




In <al4e -9E, " term must

"ata $licing L FIR 4s $ubspace




g p

In subspace algorit(m, (o; muc( data loss ;ill (appen on data

slicing depends on t(e ma'imum order n) per I9- pair NnO data pointsare lost after eac( bad slice

In FIR algorit(m, (o; muc( data loss ;ill (appen on data slicing

depends on t(e time to steady state Tss) "ata of time period e/ual

to one Time to $teady state is lost after eac( bad slice

Identification speed L FIR 9$$ I"




p

* $ubspace identification takes more computation time t(an t(e FIR

identification, in particular for models ;it( a large number of independent

and dependent 4ariables

*T(is occurs because t(e subspace identification does a true multi!input

multi!output MIM-) model identification t(at needs more intensi4e

computation t(an t(e multi!input single!output MI$-) FIR identification

*T(e benefit is t(at a true MIM- model (as less uncertainty, and is

statistically more accurate

*T(e computation time (as a cubical relation ;it( t(e NMa'imum -rderO

FIR I" 9 $$ I" $ummary




* FIR algorit(m is like cur4e!fitting non!parametric) , $ubspace

algorit(m is finding parameter for pre!defined model structure

* utomatic model order determination! $$ I" auto!determines t(eoptimal model order, to capture balanced (ig( and lo; fre/uency

dynamics , ;it(out o4er!fitting= ;(ereas in FIR I" model order is pre!

specified based on TT$$ and number of coefficients T(e goodness of

data fitting is t(e only obecti4e of FIR I"

*FIR is MI$-, $$ is truly MIM-*FIR fails ;(en $93 is lo;, $$ performs better *&fficient $licing L $$ I" is more efficient in term of slicing

o;e4er, because of (ig( degree of freedom t(at a FIR model can

offer ;(en fitting a model to a dataset, t(e FIR models can easily

capture (ig( order effects




Model RobustnessModel Robustness

Model .ncertainty




When we measure some physical quantity with an instrument and obtain anumerical value, we want to know how close this value is to the true value.

The difference between the true value and the measured value is the error.Unfortunately, the true value is in general unknown and unknowable. Sincethis is the case, the exact error is never known. We can only estimate error.

=1e$e?s no suc1 t1ing as a pe$%ect measu$ement@@=1e$e?s no suc1 t1ing as a pe$%ect measu$ement@@

Pode Elot L Fre/uency "omain .ncertainty




Pode plot or fre/uency response) of any dynamic system, is defined as t(e

magnitude same as amplitude) of t(e sine ;a4e obser4ed in a specific #<, for a

constant fre/uency sine ;a4e ;it( a peak amplitude of 10 in t(e particular M< st(e fre/uency of sine ;a4e 4aries from t(e real slo; smaller TT$$) to 4ery fast

larger TT$$), t(e amplitude of t(e sine ;a4e 4aries in t(e #< significantly

* Input+ ct) A sinVt

* For simple first order systems) A Cp9Ws1)

* -utput ;ill be + yt) A S sinVt X)

* (ere S A f , V,W, Cp)

Pode Elot L Fre/uency "omain .ncertainty




Time "omain .ncertainty




.ncertainty Results




T(e first in4erse response is in red Qone, and it is good to appro'imate

as dead time




Importance of Model .ncertainty analysis during $tep Test+1) .sed to monitor (o; accurate step test is going on

2) .sed to find out ;(en to slice out data

) Retuning step siQe

%) .sed to find out missing Feed for;ard 4ariables

5) .sed to take decision to ;indup step test



February 21, 2016 Reliance Technology Group – APC / RTO For Internal Circulation>0

$tep Test Monitoring




Retuning $tep $iQe




$tep test completion



February 21, 2016 Reliance Technology Group – APC / RTO For Internal Circulation>

ddition 9 "eletion of Feed for;ard 4ariables

06 model robustness cts

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