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
1/18/2006 System Identification 3
Automation LabIIT Bombay
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
1/18/2006 System Identification 4
Automation LabIIT Bombay
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
1/18/2006 System Identification 5
Automation LabIIT Bombay
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
1/18/2006 System Identification 6
Automation LabIIT Bombay
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
1/18/2006 System Identification 7
Automation LabIIT Bombay
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
1/18/2006 System Identification 8
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
1/18/2006 System Identification 9
Automation LabIIT Bombay
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
−==
++=
=
−=
−=
=
1/18/2006 System Identification 10
Automation LabIIT Bombay
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
1/18/2006 System Identification 11
Automation LabIIT Bombay
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
1/18/2006 System Identification 12
Automation LabIIT Bombay
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
1/18/2006 System Identification 13
Automation LabIIT Bombay
Example: Stirred Tank Heater-Mixer
[ ]
)(;/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
1/18/2006 System Identification 14
Automation LabIIT Bombay
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
Example: Stirred Tank Heater-Mixer
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
1/18/2006 System Identification 15
Automation LabIIT Bombay
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)
1/18/2006 System Identification 16
Automation LabIIT Bombay
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
1/18/2006 System Identification 17
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
1/18/2006 System Identification 18
Automation LabIIT Bombay
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
1/18/2006 System Identification 19
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
1/18/2006 System Identification 20
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
1/18/2006 System Identification 21
Automation LabIIT Bombay
),,,,,(
),,,,,(
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
1/18/2006 System Identification 22
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
1/18/2006 System Identification 23
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
1/18/2006 System Identification 24
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
1/18/2006 System Identification 25
Automation LabIIT Bombay
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
1/18/2006 System Identification 26
Automation LabIIT Bombay
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
1/18/2006 System Identification 27
Automation LabIIT Bombay
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
1/18/2006 System Identification 28
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
1/18/2006 System Identification 29
Automation LabIIT Bombay
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 )()()(
1/18/2006 System Identification 30
Automation LabIIT Bombay
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
1/18/2006 System Identification 31
Automation LabIIT Bombay
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
1/18/2006 System Identification 32
Automation LabIIT Bombay
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
1/18/2006 System Identification 33
Automation LabIIT Bombay
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
1/18/2006 System Identification 34
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
1/18/2006 System Identification 35
Automation LabIIT Bombay
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
1/18/2006 System Identification 36
Automation LabIIT Bombay
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
1/18/2006 System Identification 37
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
1/18/2006 System Identification 38
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
1/18/2006 System Identification 39
Automation LabIIT Bombay
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
)
1/18/2006 System Identification 40
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
1/18/2006 System Identification 41
Automation LabIIT Bombay
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
1/18/2006 System Identification 42
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
1/18/2006 System Identification 43
Automation LabIIT Bombay
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
1/18/2006 System Identification 44
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
1/18/2006 System Identification 45
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
1/18/2006 System Identification 46
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
1/18/2006 System Identification 47
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
1/18/2006 System Identification 48
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
1/18/2006 System Identification 49
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
1/18/2006 System Identification 50
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
1/18/2006 System Identification 51
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
1/18/2006 System Identification 52
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
1/18/2006 System Identification 53
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
1/18/2006 System Identification 54
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
1/18/2006 System Identification 55
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
1/18/2006 System Identification 56
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
1/18/2006 System Identification 57
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
1/18/2006 System Identification 58
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
1/18/2006 System Identification 59
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
1/18/2006 System Identification 60
Automation LabIIT Bombay
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
1/18/2006 System Identification 61
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 +=
1/18/2006 System Identification 62
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
1/18/2006 System Identification 63
Automation LabIIT Bombay
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
1/18/2006 System Identification 64
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
1/18/2006 System Identification 65
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
1/18/2006 System Identification 66
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
1/18/2006 System Identification 67
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
1/18/2006 System Identification 68
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
1/18/2006 System Identification 69
Automation LabIIT BombayARX: Order Selection
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
1/18/2006 System Identification 70
Automation LabIIT BombayARX: Order Selection
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
1/18/2006 System Identification 71
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
1/18/2006 System Identification 72
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
1/18/2006 System Identification 73
Automation LabIIT Bombay
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
1/18/2006 System Identification 74
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
1/18/2006 System Identification 75
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
1/18/2006 System Identification 76
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
1/18/2006 System Identification 77
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
1/18/2006 System Identification 78
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
1/18/2006 System Identification 79
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
1/18/2006 System Identification 80
Automation LabIIT Bombay
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
1/18/2006 System Identification 81
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
1/18/2006 System Identification 82
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
1/18/2006 System Identification 83
Automation LabIIT Bombay
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
1/18/2006 System Identification 84
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
1/18/2006 System Identification 85
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
1/18/2006 System Identification 86
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
1/18/2006 System Identification 87
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
1/18/2006 System Identification 88
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
1/18/2006 System Identification 89
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
1/18/2006 System Identification 90
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
1/18/2006 System Identification 91
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)
1/18/2006 System Identification 92
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
1/18/2006 System Identification 93
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)
1/18/2006 System Identification 94
Automation LabIIT Bombay
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
1/18/2006 System Identification 95
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
1/18/2006 System Identification 96
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
1/18/2006 System Identification 97
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
1/18/2006 System Identification 98
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
1/18/2006 System Identification 99
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
1/18/2006 System Identification 100
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
1/18/2006 System Identification 101
Automation LabIIT BombayIdentification Experiments
on 4 Tank Setup
Input 1 Input 2
Output 1Output 2
1/18/2006 System Identification 102
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
1/18/2006 System Identification 103
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)
1/18/2006 System Identification 104
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
1/18/2006 System Identification 105
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
1/18/2006 System Identification 106
Automation LabIIT BombayOE Model: Validation
1100 1150 1200 1250 1300 1350 1400
-3
-2
-1
0
1
2
3
Time
y1
Measured and simulated model output
oe221 Fits 87.07%Validation data
54
1/18/2006 System Identification 107
Automation LabIIT Bombay
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
1/18/2006 System Identification 108
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
1/18/2006 System Identification 109
Automation LabIIT BombayModel Validation
1100 1150 1200 1250 1300 1350 1400-6
-4
-2
0
2
4
Samples
y1
Measured and simulated model output
bj33331 89.89%armax4441 86.79%Validation dataarx20201 86.83%
1/18/2006 System Identification 110
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
1/18/2006 System Identification 111
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
1/18/2006 System Identification 112
Automation LabIIT Bombay
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
1/18/2006 System Identification 113
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
1/18/2006 System Identification 114
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
1/18/2006 System Identification 115
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
1/18/2006 System Identification 116
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
1/18/2006 System Identification 117
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
1/18/2006 System Identification 118
Automation LabIIT BombayRLS Application to 4 Tank Data
)()2()1()2()1()2()1()(
2221
121121
kekudkudkubkubkyakyaky
+−+−+−+−+−−−−=
60
1/18/2006 System Identification 119
Automation LabIIT BombayModel Predictions
1/18/2006 System Identification 120
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
1/18/2006 System Identification 121
Automation LabIIT Bombay
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
1/18/2006 System Identification 122
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
1/18/2006 System Identification 123
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
<=
−=−−=
−−−=−=
=−=
−=
∑ ∑∑
∑ ∑
∑
∑
∞
=
∞
=
∞
=
∞
=
∞
=
∞
=
∞
=
1/18/2006 System Identification 124
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
1/18/2006 System Identification 125
Automation LabIIT Bombay
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
1/18/2006 System Identification 126
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
1/18/2006 System Identification 127
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
1/18/2006 System Identification 128
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
1/18/2006 System Identification 129
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
1/18/2006 System Identification 130
Automation LabIIT Bombay
Example: Autocorrelation Function
)()1(5.0)( tetyty +−=
1: 2 =λCase
66
1/18/2006 System Identification 131
Automation LabIIT Bombay
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
=
−
−
1/18/2006 System Identification 132
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
1/18/2006 System Identification 133
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
1/18/2006 System Identification 134
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
1/18/2006 System Identification 135
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=
1/18/2006 System Identification 136
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
1/18/2006 System Identification 137
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]
1/18/2006 System Identification 138
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
1/18/2006 System Identification 139
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
1/18/2006 System Identification 140
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
1/18/2006 System Identification 141
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=∞→
∞→=⎥⎦
⎤⎢⎣⎡
∞→==
=
+−=
∫
∑
∑
−
=
=
−
1/18/2006 System Identification 142
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
1/18/2006 System Identification 143
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
1/18/2006 System Identification 144
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
1/18/2006 System Identification 145
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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