Dynamic and Time Series Modeling for Process...

1

Dynamic and Time Series Modeling

for Process Control

Sachin C. PatwardhanDept. of Chemical Engineering

IIT Bombay

1/18/2006 System Identification 2

Automation LabIIT Bombay

Why Mathematical Modeling?

• Process Synthesis and Design (offline)• Operation scheduling and planning • Process Control

- Soft sensing / Inferential measurement - Optimal control (batch operation)- On-line optimization (continuous operation)- On-line control (Single loop / multivariable)

• Online performance monitoring Fault diagnosis / fault prognosis

Key Component of All Advanced Monitoring,Control and Optimization Schemes

2



Long Term Scheduling and Planning

On-line Optimizing Control

Model Predictive Control

PID & Operating constraint Control

Plant

Plant-ModelMismatch

MarketDemands /Raw materialavailability

Setpoints

Setpoints

MV Load Disturbances

PV

PV

PV

Plant Wide Control Framework

Layer 4

Layer 3

Layer 2

Layer 1



Models for Plant-wide Control

Aggregate Production Rate Models

Steady State / Dynamic First Principles Models

Dynamic Multivariable Time Series Models

SISO Time Series Models, ANN/PLS/Kalman Filters

(Soft Sensing)

Layer 4

Layer 3

Layer 2

Layer 1

3



Mathematical ModelsQualitative

Qualitative Differential Equation Qualitative signed and directed graphs Expert Systems

QuantitativeDifferential Algebraic systems Mixed Logical and Dynamical Systems Linear and Nonlinear time series modelsStatistical correlation based (PCA/PLS)

MixedFuzzy Logic based models



White Box Models

First Principles / Phenomenological/ Mechanistic

Based on energy and material balancesphysical laws, constitutive relationshipsKinetic and thermodynamic models heat and mass transfer models

Valid over wide operating range Provide insight in the internal working of systemsDevelopment and validation process:

difficult and time consuming

4



Example: Quadruple Tank System

21

21

24

224

4

44

13

113

3

33

22

224

2

42

2

22

11

113

1

31

1

11

h and h :Outputs Measuredv and v :Inputs dManipulate

)1(2

)1(2

22

22

vA

kghAa

dtdh

vA

kghAa

dtdh

vAkgh

Aagh

Aa

dtdh

vAkgh

Aagh

Aa

dtdh

γ

γ

γ

γ

−+−=

−+−=

++−=

++−=

Pump 2V2

Pump1V1

Tank3

Tank 2

Tank 1

Tank 4


Automation LabIIT BombayExample: Non-isothermal CSTR

)()/(

Jacket Cooling to Tranfer Heat)/exp()(

)(

Balance Enery

)/exp()(

Balance Material:Reaction

cinpcc

bc

bc

Arxn

pp

AAAA

TTCaFF

aFQ

CRTEVkH

QTTFCdtdTcV

CRTEVkCCFdtdCV

BA

−+

=

−∆−+

−−=

−−−=

⎯→⎯

+

ρ

ρρ

2

1

0

0

00

5



Example: Fed-Batch Fermenter

),(7.117),(;05.1/),(),(0303.00.2

086.0),(

Volume) Reactor :(

Conc.) Product :(),()(

Conc.) 2-Substrate:(),()(

Conc.) Biomass:(),()(

21311.0

2121212

211

2121

2

21

2212222

21

2 SSeSSSSSSSS

SSSS

VFdtdV

PkPVXVSSdtPVd

SXVSSSFdt

VSd

XXVSSdtXVd

S

F

µπµσ

µ

π

σ

µ

−==

++=

=

−=

−=

=



Material Balances (Distributed Parameter System)

Energy Balances

1 rE / RTA Al 10 A

C Cv k e Ct z

−∂ ∂= − −∂ ∂

1 r 2 rE / RT E / RTB Bl 10 A 20 B

C Cv k e C k e Ct z

− −∂ ∂= − + −∂ ∂

( )

( ) ( )

1 r

2 r

r1 E / RTr rl 10 A

m pm

r2 E / RT w20 B j r

m pm m pm r

HT Tv k e Ct z C

H U k e C T TC C V

−

−

−∆∂ ∂= − +∂ ∂ ρ

−∆+ + −

ρ ρ

( )j j wjr j

mj pmj j

T T Uu T T

t z C V∂ ∂

= + −∂ ∂ ρ

……..Reactant A

……..Product B

……..Reactor Temp.

……..Jacket Temp.

Fixed Bed Reactor

6



Grey Box Models

Semi-Phenomenological

Part of model developed from the first principles and part developed from data

Example: dynamic model for reactor model using energy and material balance and Reaction kinetics modeled using neural network

Better choice than complete black box models



Heater Coil

LT

TT-1

Cold Water Flow Cold Water

Flow CV-2

ThyristerControl Unit

TT-3

CV-1

4-20 mAInput Signal

4-20 mAInput

Experimental Setup: Schematic Diagram

TT-2

Example: Stirred Tank Heater-Mixer

7




[ ]

)(;/5.139

0093.071.0279.3)(0073.0989.0979.7)(

)()()(1

)(1

)()(

202

32

22222

31

2111

2222211

22

2

2212

2

1

111

1

11

hhkhFKsmJU

IIIIFIIIIQ

CTTUATTFTTF

AhdtdT

FIFFAdt

dhCVIQTT

VF

dtdT

p

atmi

pi

−==

+−+=

−+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−−+−=

−+=

+−=

ρ

ρ

valve control to input current%:Icontrollerpowerthyristertoinputcurrent % :

2

1I



0 10 20 30 40 500

500

1000

1500

2000

2500

3000

3500

4000

4500

5000Heat Input to Tank 1

% Current input to the power module

Hea

t sup

plie

d to

wat

er in

tank

one process

predicted


0 20 40 60 80 1000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

% valve opening

flow

rat

e in

ml/m

in

Charecteraisation of Control Valve CV-2

data 1 cubic

Control Valve Characterization

Thyrister Power Controller Characterization

8



Model Validation: Input Excitations

0 200 400 600 800 1000

10

12

14

16

18

Sampling instant

Hea

ter i

nput

(mA

)

Input Perturbations

0 200 400 600 800 100012

14

16

18

Sampling instant

Val

ve tw

o in

put(m

A)



Model Validation: Level Variations

0 200 400 600 800 100020

25

30

35

40

45

50

55

60

65

Sampling instant

Hei

ght(

cm)

Simulation Validation data

Model PredictionsMeasured Output

9


Automation LabIIT BombayModel Validation:

Temperature Profiles

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 03 1 5

3 2 0

3 2 5

3 3 0

3 3 5

3 4 0

3 4 5

S a m p l i n g I n s t a n t

Tank

1 T

emp(

K)

S im u l a t i o n V a l i d a t i o n d a t a

M o d e l P r e d i c t i o n sM e a s u r e d O u t p u t

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 03 0 8

3 1 0

3 1 2

3 1 4

3 1 6

3 1 8

S a m p l i n g i n s t a t

Tank

2 te

mp(

K)

M o d e l P r e d i c t i o n sM e a s u r e d O u t p u t



Dynamic Models for Control

Linear perturbation models: Regulatory operation around fixed operating point of mildly nonlinear processes operated continuously. Developed using

Local linearization of white/gray box modelsIdentification from input output data

Why use approximate Linear Models?Linear control theory for controller synthesis and closed loop analysis is very well DevelopedFor small perturbations near operating point, processes exhibit linear dynamics

Nonlinear dynamic models: strongly nonlinear systems, operation over wide operating range, batch / semi-batch processes

10


Automation LabIIT BombayLocal Linearization

;D-D(t)d(t) ;U-U(t)u(t);Y-Y(t)y(t);X-X(t)x(t)

variables onPerturbatix

model onperturbati linear develop to ),,( around expansion series Taylor apply we

),,,( point operating state steady and)(;),,(dX/dt

modelparameterlumpedaGiven

==

==

=++=

==

Cy;HdBuAxdx/dt

DUXDUX

XGYDUXF


Automation LabIIT BombayLocal Linearization

[ ] [ ][ ] [ ]

),,( at computed/;/

;/;/where

DUXXGCDFH

UFBXFA∂∂=∂∂=

∂∂=∂∂=

Transfer Function Matrix:Can be obtained by taking Laplace transform together with assumption (i.e. initial state of the process corresponds to operating steady state)

[ ] [ ] HAsICsGBAsICsG

sdsGsusGsy

du

dp11 )(;)(

)()()()()(−− −=−=

+=

00 =)(x

11



),,,,,(

),,,,,(

02

01

cinAcA

cinAcAA

TCFFTCfdtdT

TCFFTCfdt

dC

=

=

[ ] [ ]

[ ][ ]cinm

A

Tc

TA

TDC

FFUTYTCX

≡≡

≡

≡≡

)( esDisturbancMeasured)(D esDisturbancUnmeasured

][)(Inputs dManipulate)(OutputMeasured)( States

u 0

Consider non-isothermal CSTR dynamicsfeed flow rate

coolant flow rate

Feed conc.

Cooling waterTemp.

Perturbation Model for CSTR


Automation LabIIT BombayCSTR: Model Parameters and

Steady state Operating Point

K 8330.1 = R E/ ;.; min/10 x 1.678 = a

; cal/kmol 10 x 130 = reaction) of (Heat H-

;g/m 10 = ) density Coolent (;g/m 10 = density) liquid (Reacting

; K) /(g cal 1 = ) coolent of heat (spacific C ; 92 = ) eTemperatur Inlet (Coolent Tcin

; K) cal/(g 1= mixture) reacting of heat Specific(C; /minm 15 = flow) (Coolant F;C 50 = e)temperatur (Inlet T

;kmol/m 2.0 = A) of ionconcentrat Inlet (C

; /minm 1 = flow) (Inlet F ; m 1 = ) volume Reactor (V

6

6rnx

36c

36pc

0p

300

3A0

33

50=

∆

bcal

C

ρρ

CT 0121=e)Temperatur Reactor(

kmol/m0.265 = A) of tion(ConcentraC 3A Operating Steady

State

12


Automation LabIIT BombayDiscrete Dynamic Models

Computer control relevant discrete models

........!2

)exp(

Definition

)exp(;)exp(

)()()()()1(

22

0

+Φ+Φ+==Φ

=Γ=Φ

=Γ+Φ=+

∫

TTIAT

dBAAT

kCxkykukxkx

Tττ

Note: Assumption of piece-wise constant inputs holds only for manipulated inputs and NOTfor the disturbances or any other input


Automation LabIIT BombayTransfer Function Matrix

[ ][ ]

[ ] Function Transfer Pulse :)(

)(-zI)()(

)(-zIx(z)

0x(0) When)()()0()(

equationdifferenceofsidesbotheontransform-ztaking ely,Alternativ

1

1

1

ΓΦ−=

ΓΦ==

ΓΦ=

=

Γ+Φ=−

−

−

−

zICzGzuzCxzy

zu

zuzxxzzx

p

[ ]

1)-f(kf(k)q;1)f(kqf(k)

Operator Shift :)(

)()()(

1-

1

=+=

ΓΦ−=

=−

qqICqG

kuqGky

p

p

q-Transfer Function Matrix: Can be obtained by taking q-transform together with assumption 00 =)(x

13



Computation of System Matrices

B1-T

0

1-

1-

)dexp(

T)exp(

columns as A of rseigenvecto with matrix :diagonal maion on appearing seigenvalue with

matrix diagonal is where A Let :1MethodnCompouatio

Ψ⎥⎥⎦

⎤

⎢⎢⎣

⎡ΛΨ=Γ

ΨΛΨ=Φ

Ψ

ΛΨΛΨ=

∫ ττ

[ ] ( )[ ]

[ ] [ ] BAIBAIAT

BA

sILTsI

ItAdtd

Att

Tt

11

T

0

111

)exp( matrix invertible is When

)dexp(

)((s)(s)A(0)-(s)s

Transform Laplace Taking

)0(;)(

IVP-ODE of solution is )exp()(:2MethodnCompouatio

−−

=−−−

−Φ=−=ΓΑ

⎥⎥⎦

⎤

⎢⎢⎣

⎡=Γ

Φ−=Φ⇒Φ−=Φ⇒Φ=ΦΦ

=ΦΦ=Φ=Φ

∫ ττ



CSTR: Continuous Perturbation Model

[ ] )()(

)(70.95-6.07- 1.735

)(5.77852.720.09-7.56-

/

;)()(

)(;)()(

)(

txty

tutxdtdx

FtFFtFtu

TtTCtCtx

cc

AA

10

0

=

⎥⎦

⎤⎢⎣

⎡+⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡−−

=

Continuous linear state space model

Laplace Transfer Function

35.83 + s 1.79 + s943.5+s70.95-

35.8 + s 1.79 + s45.9-s6.07 - )( 22 ⎥⎦

⎤⎢⎣⎡=sGp

14



Models for Computer Control

ProcessAnalog To Digital Converter

Digital To Analog Converter

Manipulated Inputs From Control Computer

Measured Outputs To Control Computer

Control Computer /DCS

Computer controlled system / Distributed Digital Control system


Automation LabIIT BombayDigital Control: Measured Outputs

0 5 10 15 201.8

2

2.2

2.4

2.6

2.8

3

3.2

Sampling Instant

Mea

sure

d O

utpu

t

0 5 10 15 201.8

2

2.2

2.4

2.6

2.8

3

3.2

Sampling Instant

Mea

sure

d O

utpu

t

ADC

Continuous Measurement from process

Sampled measurement sequence to computer

Output measurements are available onlyat discrete sampling instant Where T represents sampling interval

,....,,: 210== kkTtk

15



Digital Control: Manipulated Inputs

In computer controlled (digital) systems Manipulated inputs implemented through DACare piecewise constant

0 2 4 6 8 10 12 14 16 18 202

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

Sampling Instant

Man

ipul

ated

Inpu

t

0 2 4 6 8 10 12 14 16 18 202

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

Sampling Instant

Man

ipul

ated

Inpu

t Seq

uenc

e

DAC

Input Sequence Generated by computer

Continuous input profile generated by DAC

1+≤≤≡= kkk tttforkututu )()()(



CSTR: Discrete Perturbation Model

[ ] )()(

)(1.797-0.7335-0.1340.0026

)(1.33373.4920.008- 0.185

)(

kxky

kukxkx

10

1

=

⎥⎦

⎤⎢⎣

⎡+⎥⎦

⎤⎢⎣

⎡=+

Discrete linear state space modelSampling Time (T) = 0.1 min

Discrete q-transfer function model

⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡=

−2-1-

2-1-

2-1-

2-1-

22

35.83q + q 1.79 + 1943.5q + q 70.95-

35.83q + q 1.79 + 145.9q - q6.07 -)(

35.83 + q 1.79 + q943.5+q70.95-

35.83 + q 1.79 + q45.9-q6.07 -)(

1qG

qG

p

p

16



Black Box ModelsData Driven / Black Box Models

Static maps (correlations)/ dynamic models (difference equations) developed directly fromhistorical input-output data

Valid over limited operating range

Provide no insight into internal working of systems

Development process: much less time consuming and comparatively easy



Black Box Models Dynamic Models: Given observed data

[ ][ ])(...)()( :Outputs Measured

)(...)()( :Inputs past of Set)(

)(

kyyyYkuuuU

k

k

21

21

=

=

we are looking for relationship

( ) )(,,)( )1()1( keYUk kk +Ω= −− θy

such that noise (residuals) e(k) are as small as possible

vectorparameterrepresents dR∈θ

17



Tools for Black Box Modeling

Linear Difference equation (time series) models

Principle component analysis (PCA) / Projection Of latent structures (PLS) /Statistical models based on linear correlation analysis of historical data

Artificial Neural Networks/Wavelet NetworksExcellent for capturing arbitrary nonlinear maps

Fuzzy Rule Based Models Quantification of qualitative process knowledge


Automation LabIIT BombaySteps in Model Development

Selection of model structure Planning of experiments for estimation of unknown model parameters

Design of input perturbation sequencesOpen loop / closed loop experimentation

Estimation of model parameters from experimental data using optimization techniques Model validation

Prediction capabilities Steady state behavior

18



Model Structure Selection Issues in Model Selection

• Process application (batch / continuous)• Time scale of operation • Type of application

(scheduling/optimization/MPC/Fault Diagnosis)• Availability of physical knowledge /

historical data • Development time and efforts

Model granularity decides how well we can make control / planning moves or diagnose / analyze process behavior



Data Driven ModelsDevelopment of linear state space/transfermodels starting from first principles/gray box models is impractical proposition. Practical Approach• Conduct experiments by perturbing process

around operating point • Collect input-output data • Fit a differential equation or difference

equation model Difficulties • Measurements are inaccurate • Process is influenced by unknown disturbances• Models are approximate

19


Automation LabIIT BombayDiscrete Model Development

0 2 4 6 8 10 12 14 16 18 202

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

Sampling Instant

Man

ipul

ated

Inpu

t

Excite plant around the desired operating point by injecting input perturbations

Process

0 5 10 15 201.8

2

2.2

2.4

2.6

2.8

3

3.2

Sampling Instant

Mea

sure

d O

utpu

t

Input excitation for model identification

Unmeasured Disturbances Measured output

response

Measurement Noise


Automation LabIIT BombayCSTR: Input Excitation

0 5 10 15 2013

14

15

16

17

Sampling Instant

Coo

lent

Flow

Manipulated Input Excitation

0 5 10 15 200.85

0.9

0.95

1

1.05

1.1

Sampling Instant

Rea

ctan

t Inf

low

PRBS: Pseudo Random Binary Signal

20



CSTR: Identification Experiments

Dotted Line: Data without noise

Continuous Line: Data with noise

0 5 10 15 200.8

1

Sampling Instant

Inle

t Con

c.(m

od/m

3)Effect of Inlet Concentration Fluctuations On Reactor Temperature

0 5 10 15 20

390

400

Sampling Instant

Mea

sure

d Te

mpe

ratu

re (K

)


Automation LabIIT BombayCSTR: Noise Component

0 50 100 150 200-4

-3

-2

-1

0

1

2

3Unmeasured Disturbance Component in Measured Outputs

Sampling Instant

v(k)

= y(

k) -

G(q

) u(k

), (D

eg K

)

21



Two Non-Interacting Tanks Setup

Tank2

Tank1

Non Interacting Tank Level Control setup

SumpPump1 Pump2

CV1 CV2

LT

SISO System Output: Level in tank 2Manipulated Input :

Valve Position CV-2Disturbance:

Valve Position CV-1


Automation LabIIT BombayInput Output Data

0 50 100 150 200 2504.5

5

5.5

6

6.5

Out

put (

mA

)

Raw Input and output signals

0 50 100 150 200 250

10

12

14

Time

Inpu

t (m

A)

22



Perturbation Data for Identification

0 50 100 150 200 250-1

0

1

Out

put

Input and output signals

0 50 100 150 200 250

-2

0

2

Time

Inpu

t

Mean values removed from Input and Output data


Automation LabIIT BombayImpulse Response Model

[ ]

[ ] [ ]

⎥⎦⎤

⎢⎣⎡=

−=−⎥⎦⎤

⎢⎣⎡=

+−⎥⎦⎤

⎢⎣⎡+−⎥⎦

⎤⎢⎣⎡=

−=

==

=

∫

∑∑ ∫

∫∫

∫

∞

=

∞

=

∞

T

j

jT

j

T

T

T

T

dgg

jkujgjkudgky

TkudgTkudgkTy

duugty

sg

sgsusy

0

11 0

2

0

0

1-impulse

)( :tsCoefficien Response Impulse

)()()()()(

...)2()()1()()(

inputs constant wise-piece For

)()()(:IntegralnConvolutio

)(Lg(t)(t)y

input impulse with )()()( T.F. Consider

ττ

ττ

ττττ

τττ

23


Automation LabIIT BombayImpulse Response Model

)()()(

operator transfer Defining

)()()(

ModelResponseImpulse

1

11

kuqGky

qgG(q)

kuqgjkugky

j

jj

j

jj

jj

=

=

=−=

∑

∑∑∞

=

−

∞

=

−∞

=

Current output y(k) is viewed as weighted sum of all past inputs moves.

Impulse response coefficients determine weighting of each past move

G(q) is open loop BIBO stable if

∞<∑∞

=1jjg


Automation LabIIT BombayDiscrete Model Forms

∑=

−≅

∞→→N

jj jkugky

1

k

)()(

k as0gsystemsstableloopopen For

Finite Impulse Response (FIR) Model

Discrete Transfer Function Model

1)-u(kb1)-u(k2)-y(k1)-y(k-a(k)to equivalent is whichqa + q a + 1

qb q b)()(

Exampleq..a+ q a + 1

qb...qbqb)()(

2121

22

1-1

2-2

1-1

1

1

n1-

1

2-n

2-2

1-1

1

1

++−=

+=

++=

−−

−

−−

−

bay

quqy

quqy

n

24


Automation LabIIT BombayOutput Error (OE) Model

Data collected through experiments

)(),.......,(),(),(:)( Sequence Input of Set

)(),.......,(),(),(:)(tsMeasuremenoutputN of Set

NuuuukuU

NyyyykyY

N

N

210

210

≡

≡

Output / Measurement Error Model Measured Value ofOutput)()()()( kvkuqGky +=

Deterministic component

Residue: unmeasured disturbances + measurement noise


Automation LabIIT BombayEstimation of FIR Model

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡+

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡+

+++=

++++=++++=

+++=

v(N)...

1)v(nv(n)

g...gg

)(....)1(........

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

y(N)...

1)y(ny(n)

form matrix in Arrangingv(N)n)-u(Ng......1)-u(Ngy(N)

.....................1)v(nu(1)g....u(n)g1)y(n

v(n)u(0)g....1)-u(ngy(n)write can we data alexperiment Usingv(k)n)-u(kg......1)-u(kgy(k)

scoeffieintnwithmodelFIRConsider

n

2

1

n1

n1

n1

n1

nNuNu

ununuununu

25


Automation LabIIT BombayLeast square estimation

[ ] [ ]

[ ]

[ ] [ ] [ ][ ]

[ ] [ ] [ ] TTT

T

TT

TTT

T

T

EE θVAAAθθ

VAAAθ

VAθAAAAYAAθ

VAθY

θAYAAθ

AθYAθYθ

VVθ

θ

VAθY

T

T

T

T

=+=

+=

+==

+=

=

−−==

+=

−

−

−−

−

1

1

11

T

1

ˆ

ˆ

i.e. vector, parameter the of value truerepresent let and mean zero have v(k) sequence noise the Let

ˆ

minminˆ

estimation parameter square Least

parametersinlinearismodelResulting


Automation LabIIT BombayEstimated FIR Model

0 10 20 30 40 50 60 70 80-0.01

0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

FIR (69 coeff)ARMAX(2,2,2,1) ARX(6,6,1)OE(2,2,1)

Comparison of Impulse Responses

0 10 20 30 40 50 60 70-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Impu

lse

Res

pons

e C

oeffi

cien

ts

26


Automation LabIIT BombayFIR Model Fit

0 50 100 150

-0.5

0

0.5y(

k)

0 50 100 150-0.1

0

0.1

0.2

Sampling Time

v(k)

PlantModel


Automation LabIIT BombayEstimated step Response

∑=

=i

jji ga

1 :tCoefficien Response Step Unit

0 10 20 30 40 50 60 70-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Unit Step Response

Time

y(k)

2 4 6 8 10

-0.1

-0.05

0

0.05

0.1

0.15

Unit Step Response

Time

y(k)

Step response can be estimated from impulse response coefficients

Unit Delay

27


Automation LabIIT BombayFeatures of estimation

[ ]

[ ] [ ] ( )( )

[ ] [ ] [ ] [ ] 1211

2

2

ˆˆˆCov

Cov then , variance with sequence noise white is v(k) If

0 when parameters model of estimate unbiasedangeneratesestimationsquareleast the Thus,

−−−==

⎥⎦⎤

⎢⎣⎡ −−=

==

=

AAAAAVVAAA

θθθθθ

VVV

V

TTTTT

T

TT

T

E

E

IE

E

σ

σσ

( )[ ] 1

2

2

ˆˆ1ˆ,ovC

is matrix covariance parameter estimated Thus,

)ˆ()ˆ(1ˆˆ1ˆ

asestimatedbecan

−

−=

−−==

AAVVθθ

θAYθAYVV

TT

T

nN

NNTσ

σ


Automation LabIIT BombayDifficulties with FIR Model

Variances of parameter estimates can be reduced by increasing data length (N)

Disadvantages:Large number of parameters for MIMO caseLarge data set required to get good parameter

estimates, which implies long time for experimentation. Alternate Model Form

[ ]n)-1/(N )var(ParametersModelFIRinErrors Variance

αig

Advantages: Method can be easily extended to multiple input case

Difficulty:

)()(qa... + q a + 1

qb...qb)( 2n

1-1

-2n

1-1 kvkuqky d +

+++

= −−

Output Error

28


Automation LabIIT BombayParameterized OE model

v(k)x(k)y(k) 3)-u(kb2)-u(k2)-x(k1)-x(k-a(k)

to equivalent is which

)(qa + q a + 1

qb q b)(

1dbetofoundwastime)(deaddelay time Since

2121

12

21-

1

-22

1-1

+=++−=

+=

=

−−

bax

kuqkx

Two tank system under consideration is expected to have second order dynamics

)(1

)(

model time discrete order nd2' to equivalent is which

)()1)(1(

)(

22

11

22

11

21

kuqaqa

qbqbkx

suss

ksx p

−−

−−

+++

=

++=

ττ


Automation LabIIT BombayParameter Estimation

x(k) : true value Y(k) : measured valuev(k) : measurement noise / disturbance

Difficulty: Only y(k) sequence is known. Sequence x(k) is unknown

Consequence: Linear least square method can’t be used for parameter estimation

NkforkxkykvNubNubNxaNxaNx

ububxaxaxububxaxax

xxxbbaa

,....4,3)()()()3()2()2()1()(

..........)1()2()2()3()4()0()1()1()2()3(

as x(k) estimate yrecursivel can we1dand))2(),1(),0(,,,,( Given

2121

2121

2121

2121

=−=−+−+−−−−=

++−−=++−−=

=

29


Automation LabIIT BombayParameter Estimation

[ ] [ ]

)3()2()2()1()()()()(

))1(),0(,,,,( to respect with minimized is

v(k))(),...0(

thatsuch))1(),0(,,,,(Estimate

2121

2121

2

2

2121

−+−+−−−−=−=

=Ψ ∑=

kubkubkxakxakxkxkykv

xxbbaa

kee

xxbbaaN

k

Simplification : Choose x(0) = x(1) = 0

Nonlinear Optimization Problem

Identified Model Parameters

y(k) = [B(q)/A(q)]u(k) + v(k)

B(q) = 4.567e-006 q^-2 + 0.01269 q^-3

A(q) = 1 - 1.653 q^-1 + 0.6841 q^-2


Automation LabIIT BombayOE Model

50 100 150 200

-0.5

0

0.5

y(k)

OE(2,2,2): Measured and Simulated Outputs

50 100 150 200

-0.050

0.050.1

0.15

v(k)

30


Automation LabIIT BombayOE Model : Autocorrelation

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Auto Cotrelation for OE(2,2,2)

95% Confidence

Interval



Unmeasured Disturbance Modeling

The measured output y(k) contains contributions due to

Measurement errors (noise) Unmeasured disturbances

In additions modeling (equation) errors arise while developing approximate linear perturbation models Thus, in order to extract true model parameters from the data, we need to carry out modeling of

unmeasured disturbances (or noise)

Noise is modeled as a stochastic process (sequence of random variables, which are

correlated in time)

31


Automation LabIIT BombayNoise Modeling

Note: Information about unmeasured disturbances in the past is contained in the output measurement record. Thus, an obvious choice of model structure is

)()()()( kvkuqGky +=

Deterministic component

Residue: unmeasured disturbances + measurement noise

[ ] )(p)-y(k.....,1),-y(km),-u(k1),...,-u(k)( kefky +=


Automation LabIIT BombayEquation Error Model

A discrete linear model, which captures the effect of past unmeasured disturbances,can be proposed as

time dead / delay Time : d)()(...)1(

)(...)1()(

1

1

kenkyakyamdkubdkubky

n

m

+−−−−−−−++−−=

How many past outputs do we include in the model? We can choose n such that

error e(k) becomes uncorrelated with y(k) andcontains no information about past disturbancesError e(k) is like a random variable uncorrelated with e(k-1), e(k-2),…

How do we mathematically state above requirement?

32



White NoiseLet us define auto-correlation in a randomprocess e(k): k= 1, 2, … as

[ ]

∑=

−−∞→

=

−=N

k

v

kekeNN

kekeR

ττ

τ

ττ

)()()(

1lim)(),(cov)(

Equation error sequence e(k) in ARX model should be independent and equally distributed random variable sequence, i.e.

⎩⎨⎧

±±==

=,....,

)(21002

ττσ

τfor

forree

Such sequence is called discrete time white noise


Automation LabIIT BombayExample: White Noise

0 50 100 150 200-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Sampling Instant

e(k)

Mean = 0 Variance = 1

33


Automation LabIIT BombayWhite Noise: Autocorrelation

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF) of White Noise


Automation LabIIT BombayARX Model Development

NkforkykykeNubNubNyaNyaNy

ububyayayububyayay

,......4,3)(ˆ)()()3()2()2()1()(ˆ

..........)1()2()2()3()4(ˆ)0()1()1()2()3(ˆ

as(k)yestimateyrecursivelcan We

2121

2121

2121

=−=

−+−+−−−−=

++−−=

++−−=

)()3()2()2()1()(1dwithmodelARXordernd2' Consider

2121 kekubkubkyakyaky +−+−+−−−−==

Advantages: Sequences y(k) and (u(k) are known Linear in parameter model – optimum can be computed analytically

34


Automation LabIIT BombayARX : Parameter Identification

[ ] [ ]

[ ] AYAAθ

AθYAθYθ

eeθ

θ

eAθY

T

1

2

1

2

1

ˆ

minminˆ

estimation parameter square Least

parameters in linear is model Resultinge(N)

...e(3)e(2)

bbaa

)2()1()2()1(........

)1()2()1()2()0()1()0()1(

y(N)...

1)y(ny(n)

formmatrixinArranging

−=

−−==

+=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

−−−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡+

T

T

NuNuNyNy

uuyyuuyy

Choose model order n such that sequence e(k) becomes white noise


Automation LabIIT BombayARX: Order Selection

2 3 4 5 62.5

3

3.5

4

4.5

5

5.5x 10-4 ARX Order Selection

Model Order

Obj

ectiv

e F

unct

ion

Val

ue

Time Delay (d) = 1

35



0 5 10 15 20-0.5

0

0.5

1S

ampl

e A

utoc

orre

latio

n Auto Cotrelations for ARX(2,2,2)

0 5 10 15 20-0.5

0

0.5

1

Lag

Sam

ple

Aut

ocor

rela

tion Auto Cotrelations for ARX (4,4,2)

Model Residuals are

not white



0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Auto Cotrelation for ARX(6,6,2)

Model Residuals

white

36


Automation LabIIT BombayARX: Identification Results

0 50 100 150 200 250-1

0

1

y(k)

ARX(6,6,2): Measured and Simulated Output

50 100 150 200 250

-0.04

-0.02

0

0.02

0.04

0.06

e(k)

Time


Automation LabIIT Bombay6’th Order ARX Model

Identified ARX Model Parameters

A(q)y(k) = B(q)u(k) + e(t)

A(q) = 1 - 0.8135 q-1 - 0.1949 q-2 - 0.07831 q-3 + 0.1107 q-4

+ 0.03542 q-5 + 0.01755 q-6

B(q) = 0.00104 q-2 + 0.013 q-3 + 0.01176 q-4 + 0.004681 q-5

+ 0.002472 q-6 + 0.002197 q-7

Error statistics

e(k) is practically a zero mean white noise sequence

4-2

-3

102.5496ˆ:VarianceEstimated

104.8813Ee(k):MeanEstimated

×=

×=

λ

37



ARX: Estimated Parameter Variances

Value Value

-0.8135 0.0674 0.001 0.0009

-0.1949 0.0868 0.013 0.0011

-0.0783 0.0863 0.0118 0.0014

0.1107 0.0863 0.0047 0.0015

0.0354 0.0871 0.0025 0.0015

0.0175 0.0484 0.0022 0.0013

1a

2a

3a

4a

6a

5a

1b

2b

3b

4b

5b

6b

σ σ


Automation LabIIT BombayARX Model

Auto Regressive with Exogenous input (ARX)

)()(...)1()(...)1()(

1

1

kenkyakyamkubkubky

n

m

+−−−−−−++−=

Using shift operator (q), ARX model can be expressed as

mm

nn

d

qbqbqBqaqaqA

keqA

kuqqAqBky

−−−

−−−

−−

−

−

++=

+++=

+=

....)(....1)(

)()(

1)()()()(

11

1

11

1

11

1

where e(k) is white noise sequence Disadvantage

Large model order required to get white residuals

Noise Model

38


Automation LabIIT BombayNoise Models

)(....)1()(v(k)Process (MA) Average Moving

)1()()()(

)(.......1)(

1division long by then,

circle, unit inside are A(q) of poles if ely,Alternativ)()(...)1()(

Model (AR) Regressive Auto variance with process noise white mean Zero :)(

)()(

1)(

1

0

122

111

1

2

1

nkehkehke

kehkeqHkv

qHqhqhqA

kenkvakvakv

ke

keqA

kv

n

ii

n

−+−+=

−==

=+++=

+−−−−−=

=

∑∞

=

−−−−

−

λ


Automation LabIIT BombayARMA Model

)(.......1)()(

division long by then, circle, unit inside are A(q) of poles If variance with process noise white mean Zero :)(

)()()()( or

)(...)1()()(...)1()( model ARMA general more aformulatetocombinedbecanmodels MA and AR

122

111

1

2

1

111

−−−−

−

−

−

=+++=

=

−++−++−−−−−=

qHqhqhqAqC

ke

keqAqCkv

mkeckeckemkvakvakv mn

λ

Advantage: Parsimonious in parameters (significantly less number of model parameters required

than AR or MA models for capturing noise characteristics)

39


Automation LabIIT BombayExample: Colored Noise

0 50 100 150 200-4

-3

-2

-1

0

1

2

3

4White Noise Passing Through Filter

Sampling Instant

Filte

r O

utpu

t

)(..

..)( keqq

qqkv 21

21

805114080

−−

−−

+−−=

Properties Of e(k) Mean = 0 Variance = 1


Automation LabIIT BombayColored Noise: Autocorrelation

0 5 10 15 20 25 30-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Lag

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF)

40


Automation LabIIT BombayParameterized Models

ARMAX: Auto Regressive Moving Average with exogenous input (ARMAX)

)()()()(

)()()( Or

)(...)1()()(...)1(

)(...)1()(

1

1

1

11

1

1

keqAqCkuq

qAqBky

rkeckeckenkyakya

mdkubdkubky

d

r

n

m

−

−−

−

−

+=

−++−++−−−−−

−−++−−=

Box-Jenkins (BJ) model: most general representation of time series models

)()()()(

)()()( 1

1

1

1

keqDqCkuq

qAqBky d

−

−−

−

−

+=

e(k) is white noise sequence in both the cases



Parameter Identification Problem

)(),.......,2(),1(),0(:)(

)(),.......,2(),1(),0(:)(NuuuukuUNyyyykyY

N

N

≡

≡

Given input output data collected from plant

Choose a suitable model structure for the time series model and estimate the parametersof the model (coefficients of A(q), B(q), C(q)

polynomials) such that some objective function of the residual sequence e(k)

is minimized.The resulting residual sequence e(k) should be a

white noise sequence

[ ])(),.......1(),0( NeeeΨ

41


Automation LabIIT BombayARMAX: One Step Prediction

)(11)(

1)(

)2()1()()3()2()2()1()(

1dwithmodelARMAXordernd2' Consider

22

11

22

11

22

11

32

21

21

2121

keqaqaqcqcku

qaqaqbqbky

keckeckekubkubkyakyaky

−−

−−

−−

−−

++++

+++

+=

−+−++−+−+−−−−=

=

Difficulties: Sequences y(k) and u(k) are known but e(k) is unknownNon-Linear in parameter model – optimum can’t be computed analytically

Solution Strategy Problem solved numerically using nonlinear optimization procedures


Automation LabIIT BombayInevitability of Noise Model

∞<

−==

∞<

−==

∑

∑

∑

∑

∞

=

∞

=

−

∞

=

∞

=

0

1-

0

1

0

0

~ i.e. stable is (q)H

)(~)()()(

i.e. stable is H(q)

)()()()(

ii

ii

ii

ii

h

ikehkvqHke

h

ikehkeqHkv

Crucial Property of Noise Model : Noise model and its inverse are stable and all its poles and zeros are inside unit circle

Key problem in identification is to find such H(q) and a white noise sequence e(k)

1hi.e.polynomialmonic''alwaysis H(q) :Note 0 =

42



Example: A Moving Average Process

∑

∑

∞

=

∞

=

−−

−−=

<−=+

=

==+=+=

+=

0

01

1-

1-

)()(e(k)

v(k) of tsmeasuremen from recovered be can e(k) and

1c if)(1

1)(H Then,

c- q at zero and 0q at pole a hascq1H(q) i.e.

sequence noise white a is e(k) where 1)-ce(ke(k)v(k)

processMAorderfirstaConsider

i

i

i

ii

ikvc

qccq

q

qcq

Inversion of Noise Model plays a crucial role in the procedure for model identification


Automation LabIIT BombayOne Step Prediction

[ ]

[ ] ∑

∑

∑∑

∞

=

−

∞

=

∞

=

∞

=

−−=−=−

−=

−=−=−

−

−=−

=−+=−=

≤

1

1

1

10

)(~)(1)()1|(ˆ

)()(

1)()(1)()()()1|(ˆ

1)-(k upto ninformatio on based v(k) of nexpectatio lConditiona :)1|(ˆ

mean zero has e(k) as)()1|(ˆ

)()()()()(

1)-(k time upto tsmeasuremen on based v(k) predict to want weand1)-(ktuptov(t)observedhavewe Suppose

ii

ii

ii

ii

ikvhkvqHkkv

kvqH

qHkeqHkekvkkv

kkv

ikehkkv

keikehkeikehkv

43


Automation LabIIT BombayOne Step Output Prediction

[ ]

[ ][ ]

[ ]

[ ]y(k)1)()()()1|(ˆ)( or

y(k))(1)()()()1|(ˆ have we gRearrangin

G(q)u(k)-y(k))(1)()()1|(ˆG(q)u(k)-y(k)v(k) However,

)()(1)()()1|(ˆ)()()1|(ˆ

1)-(k time upto ninformatio on based y(k)predict to want we and v(k)G(q)u(k) y(k)have We

and1)-(ktuptou(t)andy(t)observed have we Suppose

11

1

1

−+=−

−+=−

−+=−

=−+=

−+=−

+=≤

−−

−

−

qHkuqGkkyqH

qHkuqGqHkky

qHkuqGkky

kvqHkuqGkkvkuqGkky


Automation LabIIT BombayARX: One Step Predictor

)1|(ˆ)()( as estimated be can instant thk' at Residual

known are RHS in terms All :Advantage)3()2(

)2()1()1|(ˆ equation difference to equivalent is which

)(1

)(1

)1|(ˆ

is model this for predictor ahead step One

)(1

1)(1

)(

1dwithmodelARXordernd2' Consider

21

21

22

11

32

21

22

11

22

11

32

21

−−=

−+−+−−−−=−

⎥⎦

⎤⎢⎣

⎡ −−+⎥

⎦

⎤⎢⎣

⎡ +=−

⎥⎦

⎤⎢⎣

⎡++

+⎥⎦

⎤⎢⎣

⎡++

+=

=

−−−−

−−−−

−−

kkykyke

kubkubkyakyakky

kyqaqakuqbqbkky

keqaqa

kuqaqa

qbqbky

44


Automation LabIIT BombayARMAX: One Step Predictor

( ) ( )

)1|(ˆ)()( as estimated be can instant thk' at Residual)2()1(

)3()2()3|2(ˆ)2|1(ˆ)1|(ˆ

equation difference to equivalent is which

)(1

)()()(1

)1|(ˆ

is model this for predictor ahead step One

)(11)(

1)(

1dwithmodelARMAXordernd2' Consider

2211

21

21

22

11

222

111

22

11

32

21

22

11

22

11

22

11

32

21

−−=

−−+−−+−+−+

−−−−−−=−

⎥⎦

⎤⎢⎣

⎡++

−+−+⎥

⎦

⎤⎢⎣

⎡++

+=−

⎥⎦

⎤⎢⎣

⎡++++

+⎥⎦

⎤⎢⎣

⎡++

+=

=

−−

−−

−−

−−

−−

−−

−−

−−

kkykyk

kyackyackubkub

kkyckkyckky

kyqcqc

qacqackuqcqc

qbqbkky

keqaqaqcqcku

qaqaqbqbky

ε


Automation LabIIT BombayARMAX: One Step Predictor

[ ][ ]

u(k). and y(k) sequences using (k) sequence generate can we

),,,,,,(a parameters model given and,0(1)(0)

guesses initial with prediction start can We)1|(ˆ)()(

)2()1()3()2()2()1()1|(ˆ

as predictor step one rearrange can we )3|2(ˆ)2()2()2|1(ˆ)1()1(instancespreviousatresidualsusingely,Alternativ

212121

21

2121

ε

εε

εεε

εε

ccbba

kkykykkckc

kubkubkyakyakky

kkykykkkykyk

==

−−=

−+−+−+−+−−−−=−

−−−−=−

−−−−=−

45


Automation LabIIT Bombay2’nd Order ARMAX Model

Identified Model Parameters

A(q) = 1 - 1.651 q^-1 + 0.68 q^-2

B(q) = 0.001748 q^-2 + 0.01154 q^-3

C(q) = 1 - 0.8367 q^-1 + 0.2501 q^-2

Residual e(k) Statistics

004-2.6813eˆ:Variance Estimated

003-4.3601eEe(k):Mean Estimated2 =

=

λ

[ ] [ ] ),,,,,( to respect with minimized is

)1|(ˆ)(e(k)

functionobjectivethatsuch),,,,,,( Estimate

212121

3

2

3

2

212121

ccbbaa

kkyky

ccbbaaN

k

N

k∑∑==

−−==Ψ

Optimization formulation


Automation LabIIT Bombay2’nd Order ARMAX Model

0 50 100 150 200 250-1

0

1

y(k)

ARMAX(2,2,2,2): Measured and Simulated Ouputs

50 100 150 200 250

-0.05

0

0.05

e(k)

Time

46


Automation LabIIT BombayARMAX: Autocorrelation

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Auto Cotrelation for ARMAX(2,2,2,2)


Automation LabIIT BombayComparison of Model Predictions

0 50 100 150 200 250-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time

y(k)

Measured and simulated model output

ARMAX(2,2,2,2)ARX(6.6.2)PlantOE(2,2,2)

Best Fit (%)ARMAX : 76.45 ARX : 76.37 OE : 77.38

47


Automation LabIIT BombayComparison of Models

0 10 20 30 40 500

0.2

0.4

Sampling Instant

Step Response

y(k)

0 5 10 15 20 25 30 35 40 450

0.01

0.02

0.03

Sampling Instant

Impulse Response

y(k)

OE(2,2,2)ARX(6,6,2)ARMAX(2,2,2)



Comparison of Models: Nyquist Plots

-0.1 0 0.1 0.2 0.3 0.4

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

OE(2,2,2) ARX(6,6,2) ARMAX(2,2,2)

48


Automation LabIIT BombayPrediction Error Method

( )

[ ]

∑=

−

=

−−=

−+=−

+=

==

N

kN N

ZV

kkyky

kyqHkuqGqHkky

keqHkuqGky

Nk

1

2

1

N

)(k,1),(

function objective minimizes that FindMethod Error Prediction by nEstmimatio Parameter

),1|(ˆ)()(k, as defined is error prediction step One

)(),(1)(),(),()1|(ˆpredictor step-1 Optimal

)(),()(),()( Model

,....2,1:u(k)y(K),ZsetdataGiven

θεθ

θ

θθε

θθθ

θθ


Automation LabIIT BombayPEM: Parameter Estimation

∑=

=N

kN N

ZV1

2)(k,1),(

FunctionObjectiveofChoiceAlternate

θεθ

Typically, the resulting parameter estimationproblem is solved numerically using

(a) Nonlinear optimization (b) Gauss Newton Method

If it is desired to emphasize certain frequency of interest, then, we can minimize

filter a represents )((k))((k) where

)(k,1),(

1

1

2

1

−

−=

=

= ∑

qFqF

NZV

F

N

kFN

εε

θεθ

),(Minˆ

NN ZV θθ

θ =

49


Automation LabIIT BombayModel Order Selection

Model order determined by minimizing Akaike Information Criterion (AIC)

parameters model of Number :

2)ˆ,(1ln)ˆ(1

2

n

nkN

NAICN

kNN +⎥⎦

⎤⎢⎣⎡= ∑

=θεθ

AIC = Prediction Term + Model Order termPrediction Term: estimate of how ell the model fits data

Model Order Term: measure of model complexity required to obtain the fit

AIC strikes a balance between low residual variance and excessive number of model parameters, with smaller values

indicating more desirable models Basic Idea:

Penalize model complexity (measured by n) and obtain a model, which is reasonable w.r.t. variance errors and model

complexity


Automation LabIIT BombayARMAX: State Realization

[ ]

[ ] [ ]

[ ])1|()()()1|()|1( Observer State a as tionInterpreta

)()()(;

)()()(

0....01

...

...;......;

0....001....00...............0....100....01

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

11

22

11

2

1

2

1

2

1

−−+Γ+−Φ=+

+Φ−==ΓΦ−==

=⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=Γ

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

=Φ

+=+Γ+Φ=+

∞

∞−−

∞

−

−

∞

kkCxkyLkukkxkkx

ILqICqAqCqHqIC

qAqBqG

Cac

acac

L

b

bb

aa

aa

kekCxkykeLkukxkx

nnnn

n

50


Automation LabIIT BombayConnection with

Steady State Kalman Estimator

Steady state form of Kalman prediction estimator (for large time) is given as

TT

TT

CPLRPPCCPRCPL

Φ−+ΦΦ=

+Φ=

∞∞∞∞

−∞∞∞

1

12 )(

)1|(ˆ)()()()()1|(ˆ)|1(ˆ

−−=

+Γ+−Φ=+ ∞

kkxCkykekeLkukkxkkx

Thus, development of time series model can be viewed as identification of steady state Kalman estimator without

requiring explicit knowledge of noise covariance matrices (R1,R2 )

Steady state Kalman gain is parameterized through H(q) and estimated directly from data.

∞L


Automation LabIIT BombayMIMO System Identification

ARX Model:Method for ARX parameter identification can be extended to deal directly with multivariate data

OE / ARMAX / BJ Models: Typically, an n x m MIMO system is modeled as n MISO (Multi Input Single Output) systems

MISO models are combined to form a one MIMO model

Input excitation: Inputs can be perturbed sequentially or simultaneously

nikeqHkuqGkuqGky iimimii

,.....2,1)()()()(......)()()( 11

=+++=

51


Automation LabIIT BombayIdentification Experiments

on 4 Tank Setup

Input 1 Input 2

Output 1Output 2


Automation LabIIT Bombay4 Tank Setup: Input Excitations

0 200 400 600 800 1000 1200

-1

0

1

Inpu

t 1 (m

A)

Manipulated Input Sequence

0 200 400 600 800 1000 1200

-1

0

1

Inpu

t 2 (m

A)

Time (sec)

52


Automation LabIIT Bombay4 Tank Setup: Measured Output

0 200 400 600 800 1000 1200-5

0

5

Leve

l 1 (m

A)

Meaured Output

0 200 400 600 800 1000 1200-6

-4

-2

0

2

4

Leve

l 2 (m

A)

Time (sec)


Automation LabIIT BombaySplitting Data for

Identification and Validation

0 500 1000-5

0

5

y1

Input and output signals

0 500 1000

-0.50

0.51

Samples

u1

Identification Data Validation data

53


Automation LabIIT BombayMISO OE Model

MISO 2’nd Order Model

y1(t) = [B1(q)/A1(q)]u1(t) + [B2(q)/A2(q)]u2(t) + e1(t)

B1(q) = 0.1393 q-1 + 0.04704 q-2

B2(q) = 0.002375 q-1 + 0.01105 q-2

A1(q) = 1 - 0.2454 q-1 - 0.6571 q-2

A2(q) = 1 - 1.887 q-1 + 0.8903 q-2

Estimated using Prediction Error Method

Loss function 0.114719 Sampling interval: 3


Automation LabIIT BombayOE Model: Validation

1100 1150 1200 1250 1300 1350 1400

-3

-2

-1

0

1

2

3

Time

y1


oe221 Fits 87.07%Validation data

54



Residual Autocorrelation: ARX

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF) of arx20201 for y1


Automation LabIIT BombayResiduals Autocorrelations

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF) of armx441 for y1

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF) of bj33331 for y1

ARMAX4’th Order

B-J3’rd Order

55


Automation LabIIT BombayModel Validation

1100 1150 1200 1250 1300 1350 1400-6

-4

-2

0

2

4

Samples

y1


bj33331 89.89%armax4441 86.79%Validation dataarx20201 86.83%


Automation LabIIT BombayARMAX Model

ARMAX (4’th Order)

A(q)y1(t) = B1(q)u1(t) + B2(q)u2(t) + C(q)e1(t)

A(q) = 1 - 0.6236 q-1 - 0.8596 q-2 - 0.0758 q-3 + 0.568 q-4

B1(q) = 0.08324 q-1 + 0.02757 q-2 + 0.02681 q-3 - 0.1214 q-4

B2(q) = 0.004045 q-1 + 0.03261 q-2 - 0.01841 q-3 + 0.0201 q-4

C(q) = 1 - 0.4695 q-1 - 0.8017 q-2 - 0.1065 q-3 + 0.4855 q-4

Loss function 0.0243707

56


Automation LabIIT BombayBox-Jenkins Model

y(t) = [B(q)/F(q)]u(t) + [C(q)/D(q)]e(t)

B1(q) = 0.08196 q-1 + 0.1035 q-2 + 0.1323 q-3

B2(q) = 0.01197 q-1 + 0.001306 q-2 + 0.01304 q-3

C(q) = 1 - 1.976 q-1 + 1.126 q-2 - 0.1453 q-3

D(q) = 1 - 2.096 q-1 + 1.209 q-2 - 0.1128 q-3

F1(q) = 1 + 0.3058 q-1 - 0.5066 q-2 - 0.6204 q-3

F2(q) = 1 - 0.897 q-1 - 0.9828 q-2 + 0.8861 q-3

Loss function 0.0239039



x(k+1) = x(k) + x(k) + e(k)Y(k) = C x(k) + e(k) = [0.6236 1 0 0

0.8596 0 1 00.0758 0 0 1-0.5680 0 0 0 ]

= [ 0.0832 0.0040 = [ 0.15410.0276 0.0326 0.05790.0268 -0.0184 -0.0307

-0.1214 0.0201 ] -0.0826 ] ;C = [ 1 0 0 0 ]

ARMAX:State Realization

Φ

Γ ∞L

Φ Γ ∞L

57


Automation LabIIT BombayRecursive Parameter Estimation

[ ]

[ ] YAAAθ

eθAY

)()()()(ˆ estimation parameter square Least

)()()(e(N)

...e(3)e(2)

bbaa

)(........

)3()2(

y(N)...

1)y(ny(2)

form matrix in Arranging)3()2()2()1()(

)()()()()3()2()2()1()(

1dwithmodelARXordernd2' Consider

1

2

1

2

1

2121

NNNN

NNNN

kukukykykkekky

kekubkubkyakyaky

T

T

T

T

T

T

−=

+=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡+

−−−−−−=

+=

+−+−+−−−−==

ϕ

ϕϕ

ϕθϕ

N introduced asformal parameterto indicate that data up to instant N has been used


Automation LabIIT BombayRLS: Problem Formulation

[ ] )1()1()1()1()1(ˆ)1(ˆ

)1()(

)1(,)1(

)()1(

becomesAmatrix,line-onobtainedistmeasuremenadditionalanNhen

1++++=+

+

⎥⎦

⎤⎢⎣

⎡+

=+⎥⎦

⎤⎢⎣

⎡+

=+

− NYNANANAN

aswrittenbecanNestimateNew

NyNY

NYNNA

NA

TT

T

θ

θ

ϕ

Thus, A(N) keeps growing in size as new data arrives. Instead of inverting at every instat,Can we rearrange calculations at (N+1) so that solution at Instant N can be used to compute solution at instant (N+1)?

[ ][ ])1()1()()(

)1()1()()()1(ˆ 1

+++×

+++=+−

NyNNYNANNNANAN

T

TT

ϕϕϕθ

[ ])1()1( ++ NANAT

58


Automation LabIIT BombayRLS: Solution

[ ]

)(ˆ)1()|1(ˆ

)|1(ˆ)1()()(ˆ)1(ˆ

NNNNywhere

NNyNyNLNN

T θϕ

θθ

+=+

+−++=+

[ ] [ ]

equatiosns of set recursive followingbygivenisproblemabovetoSolution

entrearrangem some andBCDA

lemmainversionmatrixUsing1111111 −−−−−−− +−=+ DABDACBAA

[ ][ ] )()1()()1(

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

equations Riccati following solving bycomputedisL(N)matrixgainEstimator

1

NPNNLINPNNPNNNPNL

T

T

+−=+

++++=−

ϕϕϕϕ


Automation LabIIT BombayRLS: Initialization

[ ] increases N as )()(P(N) ensures choice This

0)0(ˆ i.e. estimate parameter initial the in trust not have we

that indicating )10 ( large chosen is where 0)0(ˆand)0(

matrix covariance initial thewithbegunareequationsrecursiveely,Alternativ

1

4

−→

=

≈

==

NANA

IP

T

θ

αθα

[ ]

0

0000

1000

00

0

NN for used be can nscanculatio Recursive)()()()(ˆ

)()()(

. rnonsingula is )()(that such choose to necessary is it

RLS,starttoconditionsinitialanobtain to order In

≥=

=

=

−

NYNANPNNANANP

NANANN

T

T

T

θ

59


Automation LabIIT BombayTime Varying Systems

[ ]

[ ][ ]

[ ]

20:95.0;1000:0.999)-1/(1 (ASL) Length Data Asymptotic

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

)(ˆ)1()1()()(ˆ)1(ˆby, given is system this for RLS

factor forgetting as called is

)()()(

function. loss the in weighting lexponentia an using achieved be can This data. old of influence the

eliminatetonecessaryisitsystems,varying time For

1

2

1

=====

+−=+

++++=

+−++=+

−=

−

=

−∑

ASLASL

kPkkKIkPkkPkkkPkK

kkkykKkk

kkyJ

T

T

T

N

k

TkN

λλλ

λϕϕϕλϕ

θϕθθ

λ

θϕλθ


Automation LabIIT BombayRLS Application to 4 Tank Data

)()2()1()2()1()2()1()(

2221

121121

kekudkudkubkubkyakyaky

+−+−+−+−+−−−−=

60


Automation LabIIT BombayModel Predictions


Automation LabIIT BombayParameter Variations

0 200 400 600 800 1000 1200

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Sampling Instant

Mod

el P

aram

eter

s

a1a2b1b2d1d2

61



Extended Recursive Formulations (ELS)

[ ])2()1()2(ˆ)1(ˆ)(vector regressor uses nFormulatio OE Recursive

)()()()3()2()2()1(ax(k)

ModelOE

2121

−−−−=

+=−+−+−+−=

kukukxkxk

kvkxkykubkubkxakx

ϕ

[ ]

instants previous at residuals model represent 2)-(kand 1)-(k where

)2()1()2()1()2()1()(vector regressor uses nFormulatio Recursive Extended

)2()1()()3()2()2()1(ax(k)

ModelARMAX

21

2121

εεεεϕ −−−−−−=

−+−++−+−+−+−=

kkkukukykyk

keckeckekubkubkxakx


Automation LabIIT BombayFrequency Domain Analysis

Time domain formulations of parameter estimation problem

Useful for carrying out parameter estimationDoes not provide any insight into internal working of optimization problem

Frequency domain (power spectrum) analysis Based on Fourier transform of auto-correlation and cross correlation function of signals Powerful tool for analysis (analogous to use of Laplace transforms in linear control theory) Provides insight into various aspects of optimization formulation Can be used for perturbation signal design and estimation error analysis

62


Automation LabIIT BombayStationary Process

).( time of tindependen is covariance-auto and mean zero has it since 'stationary' called is process stochastic Such

. and by defined uniquely is and of tindependen is )(R Covariance.00)( :Note

)()()()()(

)()()()()(

is covariance-auto and

0)()(

variance with process noise meanwhite zero a is e(k) where

)()( model noise Consider

2v

0 00

0 0

0

20

k

h(k)krifrh

ththjkjhth

jkt),ee(kEjhthkv(k),vER

ikeEhkvE

ikehkv

t tj

t jv

ii

ii

λτ

τλτδλ

τττ

λ

<=

−=−−=

−−−=−=

=−=

−=

∑ ∑∑

∑ ∑

∑

∑

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=


Automation LabIIT BombayQuasi-Stationary Process

process stationary a not is y(k) and )()(

have we ,0 Sincec)(stochasti tic)determinis(

)()()()()(

kuqGy(k)Ee(k)E

keqHkuqGky

==

+=+=

Quasi-stationary process

s(k) signal of function ncorrelatio-auto :)(

)(),(1),(),()()()(

)()( (i)to subjectisitifstationaryquasibetosaidis )( signalA

1

τ

ττ

s

s

N

ks

ss

ss

R

RkkRNN

LimCtkRtkRtsksEiikCkmkms(k)E

ks

=−∞→

≤=

∀≤=

∑=

63



Signal Spectrum and Cross Spectrum

ωωωω

τω

τω

ωτ

τ

ωτ

τ

of function valued complex general, in is, )( )( of function real a always is )(

exists. sum infinite the provided

)()(

as w(k) and s(k) between sprctrum cross and

)()(

s(k)signalofspectrumpowerdefine We

sw

s

jswsw

jss

eR

eR

ΦΦ

=Φ

=Φ

−∞

−∞=

−∞

−∞=

∑

∑

)()()(

)()()(

have we)(1)( Defining1

ττττ−=

−=

∞→= ∑

=

kwksERksksER

kfNN

LimkfE

sw

s

N

k


Automation LabIIT BombayPower Spectrum (Contd.)

[ ] )(21(0)R)(

Theorem) s(Parseval'transformFourierinverseofdefinition By

-ss

2 ∫Φ==π

πωω

πdksE

Note: Spectrum of signal s(t) represents Fourier transform of auto-covariance functionInverse Transform:

noise? white a being e(k) with spectrum same has H(q)e(k) v(k) process random the that such H(q) function transfer a find we Can

)(spectrumwithedisturbanca Given v

=

Φ ω

Fundamental Modeling Problem

64


Automation LabIIT BombaySpectral Factorization

If the residue signal v(k) is weakly stationary, i.e. cov[v(k),v(s)] is function of only (k-s) for any pair (k,s), then spectral factorization theorem states that such a

random sequence can be thought of being generated by a stable linear transfer function driven by white noise

22

iv

)()(

that such circle unit outside zeros and poles no with R(z), z,of function transfer rational monic a exists there Then).e (or

)cos(offunctionrationalais0)(that Suppose :Theorem

ω

ω

λω

ωω

iv eR=Φ

>Φ

Main Result


Automation LabIIT BombaySpectrum of White Noise

0 0 .2 0 .4 0 .6 0 .8 1

1 0 -0 .3

1 0 -0 .2

1 0 -0 .1

1 0 0

1 0 0 .1

P xx - X P o we r S p e c tral D e ns ity

F re q ue nc y

E s tim ate d

Theore tica l

⎩⎨⎧

±±==

=,....2,10

0)(2

ττλ

τfor

forree

1: 2 =λCase

White Noise:Uniform power spectrum density at all frequencies

65


Automation LabIIT BombayExample: Autocorrelation for AR

[ ] [ ] [ ]

[ ]

2

2

2

)5.0(34)(

)0(5.0)1(:1 For)1(5.0)0(:0 For

0 if

0 if 0)()()( But,)()1(5.0)(

)()()()1(5.0)()()()1(5.0)( :Example

λτ

τλττλ

ττττττ

τττ

τ=⇒

==

+==

==

>=−=

+−=−+−−=−

+−=

y

yy

yy

ye

yeyy

R

RRRR

tyteERRRR

tyteEtytyEtytyEtetyty

⎥⎦⎤

⎢⎣⎡ −

−= ∑

+=

N

tv tyty

NR

1)()(1)(ˆ

ττ

ττ

Numerical Estimate

Theoretical (with knowledge of )2λ



Example: Autocorrelation Function

)()1(5.0)( tetyty +−=

1: 2 =λCase

66



Spectrum of a Stochastic Process

Frequency Nyquist:/;]/,0[)(sin25.0)]cos(5.01[

1)exp(5.01

1)(

)(5.01

1)(

:)()(

variance with process noise white mean zero:)()()()(

processstochasticstationaryaConsider

222

22

1

22

2

1

TT

j

keq

ky

ExampleeH

kekeqHkv

y

iv

ππωωω

λ

ωλω

λω

λω

∈++

=

−−=Φ

−=

=Φ

=

−

−


Automation LabIIT BombaySpectrum: Colored Noise

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5Pxx - X Power Spectral Density

Frequency

TheoreticalEstimated from data

)()1(5.0)( tetyty +−=

1: 2 =λCase

67


Automation LabIIT BombaySpectra for Linear Systems

)()()(

)()()()(

variance with process noise white mean Zero :)(signal ticdeterminis and stationary-Quasi :)(

)()()()()(

222

2

ωω

λωω

λ

ω

ωω

ui

yu

iu

iy

eGeHeG

keku

keqHkuqGky

Φ=Φ

+Φ=Φ

+=

[ ] [ ] [ ]

[ ] 0)()()()(

)()()()()()()( spectrum with process stochastic Stationary :)(

)( spectrum with signal ticdeterminis and stationary-Quasi :)(v(k)u(k)s(k)

signalstochasticandticdeterminismixed of Spectrum

v

u

=−

+=−+−=−

ΦΦ

+=

τττ

τττω

ω

kvkuEasRR

kvkvEkukuEksksEkv

ku

vu


Automation LabIIT BombayErrors Analysis

[ ]( )[ ]

Error Variance Error Bias Estimation of

Error Total

)(k,1),(

minimizing by estimated Parameters)()()()(ˆ)()(ˆ

)()(ˆ)()(ˆ(k)

error Prediction)()(ˆ)()(ˆ

Model Identified / Proposed)()()()(

BehaviorTrue

1

2

1

1

+=⎭⎬⎫

⎩⎨⎧

=

+−=

−=

+=

+=

∑=

−

−

N

kN N

ZV

keqHkuqGqGqH

kuqGkyqH

keqHkuqGy(k)

keqHkuqGy(k)

θεθ

ε

68


Automation LabIIT BombayVariance Errors

[ ][ ]

Length Data :N Order Model :

)H(e)(eHVar

)e()e()(eGVar

arePEMusingestimatesofvariance Asymptotic

2ii

i

ii

nNnNn

u

v

ωω

ω

ωω

≅

ΦΦ≅

Implications:Variance errors can be reduced by

increasing the data length (N)choosing high signal to noise ratio

Models with large number of parameters require relatively larger data set for better parameter estimation.

Noise to Signal Ratio

)e()e((SNR) Ratio Noise to Signal i

i

ω

ω

v

u

ΦΦ=


Automation LabIIT BombayExample: Input Selection

0 50 100 150-1

0

1

u(k)

Random Binary Signal in frequency range [0,0.1]

0 50 100 150-1

0

1

Sampling Instant

u(k)

Random Binary Signal in frequency range [0.3,0.5]

Generated By MATLABCommand ‘idinput’

69


Automation LabIIT BombayInput Spectrum

0 0.2 0.4 0.6 0.8 110-2

10-1

100

101Pxx - X Power Spectral Density

Frequency

RBS in [0 0.1]RBS in [0.3 0.5]


Automation LabIIT BombayBias Error: Concept

Real systems are of very high order and model is always chosen of lower orderThus, bias errors are always present

in any identification exercise Classic Example in Process Control

se

ssG

36

8

1)(50s1(s)G : model FOPTD Identified

)110(1)( : Dynamics Process

−

+=

+=

70


Automation LabIIT BombayBias Errors: Concepts

Step Response

Time (sec)

Am

plitu

de

0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1

FOPTD Model Step ResponsePLant Step Response


Automation LabIIT BombayBias Error: Concept

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Model

Plant

71


Automation LabIIT BombayBias Errors

( )[ ]

[ ] [ ]

[ ]

2

2

1

2

1

2

1

)(ˆ1)()()(ˆ)()(

)(),(lim

Theorem sParseval' By

),(lim

)(k,1lim)()0(R

write can we large, is N length data When

)(k,1),(

minimizing by estimated Parameters)()()()(ˆ)()(ˆ(k)

errorPrediction

ω

ωω

π

π

ε

ωωω

ωωθ

θθεε

θεθ

ε

ε

ε

ivu

ii

N

N

N

k

N

kN

eHeGeG

dZVN

ZVNNN

kE

NZV

keqHkuqGqGqH

⎥⎦⎤

⎢⎣⎡ Φ+Φ−=Φ

Φ=∞→

∞→=⎥⎦

⎤⎢⎣⎡

∞→==

=

+−=

∫

∑

∑

−

=

=

−


Automation LabIIT BombayBias Error: Interpretations

Bias distribution of in frequency domain is weighted by Signal To Noise Ratio Input spectrum can be chosen intelligently to reduce variance errors in certain frequency regions of interestFor Output Error model (i.e. H(q)=1),

[ ] ∫−

⎥⎦⎤

⎢⎣⎡ Φ+Φ−=

∞→

π

π ω

ωω ωωωθ deH

eGeGN i

vuii

N 2

2

)(ˆ1)()()(ˆ)(

lim

2)(ˆ)( ωω ii eGeG −

[ ]

zedparameteri-under not is model if )()(ˆ Thus,

)()(ˆ)(lim 2

ωω

π

π

ωω ωωθ

ii

uii

N

eGeG

deGeGN

→

⎥⎦⎤

⎢⎣⎡ Φ−=

∞→ ∫−

72


Automation LabIIT BombaySummary

Grey box modelsBetter choice for representing system dynamics. Provide insight into internal working of the systemDevelopment process time consuming and difficult

Black Box Models Relatively easy to developProvide no insight into internal working of systems Limited extrapolation abilities.

Black Box Model Development Noise modeling is necessary to be able to extract the deterministic component of the model properly Prediction error method used for parameter estimation


Automation LabIIT BombaySummary

Black Box Model DevelopmentFIR and ARX models are relatively easy to develop but require large data set for reducing variance errorsOE, ARMAX or BJ models provide parsimonious description of model dynamics but require application nonlinear optimization for parameter estimationVariance errors are directly proportional to number of model parameters and inversely proportional to data length Frequency domain analysis provides insight into working of PEM. Variance errors can be reduced by appropriately selecting Signal to Noise RatioBias errors in certain frequency region of interest can be reduced by appropriately choosing the spectrum of perturbation input sequences

73


Automation LabIIT BombayReferences

Ljung, L., System Identification: Theory for Users, Prentice Hall, 1987. Sodderstorm and Stoica, System Identification, 1989.Astrom, K. J., and B. Wittenmark, Computer Controlled Systems, Prentice Hall India (1994).

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