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© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Daniel E. Rivera, Associate Professor
Control Systems Engineering Laboratory
Department of Chemical and Materials Engineering
Arizona State University [email protected] AZ 85287-
(480) 965-
© Copyright, 1998-2004 ChE 494/
Introduction to System
Identification
Course Objectives
- Provide lab exercises that will give students a working feel
Identification Toolbox) will be the program of choice.for the course topics. MATLAB (particularly the System
- Present fundamental background to allow students to
identification.make judicious choices of design variables in system
- Provide a glimpse of cutting-edge identification research at
ASU and other academic institutions around the world.
System Identification
test is equivalent.”a system within a specified class of systems, to which the system under “Identification is the determination, on the basis of input and output, of
- L. Zadeh, (1962)
SystemSystem
Inputs
Outputs
Disturbances
systems from experimental dataSystem identification focuses on the modeling of dynamical
Some System Identification Facts
- problem not exclusively associated with control design,
although it forms a significant part of control implementation
- often times, the system identification task is the most
implementationexpensive and time consuming part of advanced control
- broadly applicable technology with applications in many
diverse fields
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Shell Heavy Oil Fractionator Example
Top Draw
setpoint,endpoints attop and sideduty to maintainbottoms refluxside draw and/orManipulate top,
duties.intermediate refluxthe upper anddisturbances fromReject
above constraints.TemperatureRefluxKeep Bottoms
LC
A
T
T T
LC
LC
FEED BOTTOMS REFLUX
INTERMEDIATE REFLUX
UPPER REFLUX
TOP DRAW
SIDE DRAW BOTTOMS
SIDE
STRIPPER
FC FC
Q(F,T)
CONTROL
F
T
PC
T
A
T
Endpoint Top EndpointSide
Side Draw
Reflux DutyBottomsDutyIntermediate RefluxUpper Reflux Duty
Reflux TempBottoms
20
40
60
80
100
120
140
160
180
200
OUTPUT #
0 0
20
40
60
80
100
120
140
160
180
200
INPUT #
Distillation Column Data
- response of overhead temperature (top) to changes in reflux
flowrate (bottom)
Epi Reactor Temperature Control
- keep center, front, side and rear temperatures constant by
adjusting power to the lamp banks
solid: center; dashed: side; dotted:front; dash-dotted: rear
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-40-30-20- 0
10
Time [seconds]
Temp. Deviation [C]
solid:master; dashed:side; dotted:front; d-dotted:rear
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-8 -6 -4 - 0
Time [seconds]
Power [%]
solid:master; dashed:side; dotted:front; d-dotted:rear
Epi Reactor Identification Data
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Stages of System Identification
Experimental Design and Execution
Data Preprocessing
Model Structure Selection
Parameter Estimation
Model Validation
STAGES OF SYSTEM IDENTIFICATION
Start
Experimental Design
Model Validation"Identification"and Execution
meet validation criteria?Does the model
( Step, Pulse, or
PRBS-Generated Data)
( Linear Plant and Disturbance Models) step-response)cross- correlation,(Simulation, Residual auto and
No
End
Yes
a priori
process
information
- (^) Model Structure Determination - (^) Parameter Estimation• Data Preprocessing
Stages of System Identification - II
courtesy P. Lindskog, ISY, Linköping University, Sweden Prior system knowledge: physics, linguistics, first-hand, etc.
Experiment
design
Pre-treat
data
Choose model
structure
Choose
performance
criterion
Parameter estimation
Validate model
Not OK
revise!
OK
accept model!
Not OK
revise prior?
Controller Design & Commissioning
Keys to Successful System Identification
in Practice
tradeoffsassociated decision variables in terms of bias-varianceUnderstanding the various identification methods and
Effective use of
a priori
knowledge regarding the system
simulation, prediction, control)to be identified and the intended application (e.g.,
"the classical statistical approach," per Ljung...
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Skill-level issues
: many system identification methods
signal processing, discrete-time systems, and optimization.assume the user has extensive background in statistics,
Large number of design variables.
Process operating restrictions
make identification one of the
implementation projects.most time consuming tasks in advanced control
System Identification Challenges
(Disturbance)
(Input)
(Output)
CONTROLLER
of significant changes in the feed flowrate.Objective: Use fuel gas flow to keep outlet temperature under control, in spite
Furnace Control Example
The "Shower Problem"
Hot
Cold
shower temperature despite cold water fluctuations...Consider the problem of adjusting hot water flow to maintain
control problem...Makes this a difficultTransportation lag
Process Dynamics and Control, Wiley, 1989, Chapter 7. Many references for this technique, example: Seborg, Edgar, and Mellichamp, p(s) =
K e
θ s
τ
s + 1
,
Response of a first-order with deadtime model for a step input of magnitude A
Graphical System Identification Using Step Testing
θ
Time
KA
τ
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
LOOP OPEN
RESPONSE
Furnace example with PRBS input, PID with filter controller
IDENTIFICATION
DATA
CLOSED LOOP
RESPONSE
Input
OutputMeasured
From Identification to Controller Implementation
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
-15-
1015
Input
Time[Min]
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
10152025
Measured Output
Time[Min]
Course Outline
Signals and Systems Overview
Input Signal Design and Nonparametric Estimation
Parametric Model Estimation and Validation
Control-Relevant and Closed-Loop Identification
Multivariable Identification
Issues in nonlinear and semiphysical identification
Course Focus
Very broad subject
(Mostly) LINEAR
DISCRETE
- Parametric or nonparametric?
BOTH
- Time or frequency domain?
BOTH
Systems Representations
Parameter System^ Nonlinear Lumped
State-Space
Model
Linearization
s-domain
Transfer Function
Model
transforms Laplace
Step/
Impulse
Response
and
ResponseFrequency
z-domain
Transfer Function
Model
T
Sampling
(difference equation)
Discrete-
Step/time
Impulse
Response
and
ResponseFrequency
Discrete-time S-S Model
T
Sampling
Realization
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Discrete Model Representations
PLANT
Step
Response
PLANT
Impulse
Response
U(k)
Y(k+1)
D
EquationsDifference
G(z)
U(z)
Y(z)
Z-Transforms
Nonparametric
{
Parametric
{
Pulse Transfer Functions
computer control
algorithm
Zero-order
Hold
P(s)
computer
k
u
k
y
u
( t )
y ( t )
ZOH-equivalent Pulse Transfer Function
T
time
time
time
time
discrete input
continuous
input
continuous
output
discrete output
k
y
u
( t )
y ( t )
k
u
Examples
with DelayFirst-OrderRampIntegrating/LagFirst-OrderStep^ Impulse
s-domain
z-domain
δ time
( t )
s ( t )
=
t
≥
0 t
s 1
z
z
−
(^1)
K
τ s
(^1)
K
s
K
τ s
(^1)
exp(
θ
s )
θ =
NT
FunctionTransfer ZOH Pulse FunctionTransferZOH PulseFunctionTransferZOH Pulse
KT
z
−
1
K
exp(-
T
τ ))
z (^) - N
z
exp(-
T
τ )
K
exp(-
T
τ ))
z
exp(-
T
τ )
System Identification Structure
Random Signal
Input Signal
Output Signal
u
y
P(z)
a
+
+ H(z)
υυυυ
Disturbance Signal
transfer functionsP(z) and H(z) are discrete-time (z-domain)
y ( t ) =
p ( z ) u
( t ) +
H
z ) a ( t )
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
System Identification, Revisited
white noise signal
Deterministic)(Random orInput Signal
autocorrelated)Output Signal (random,
u
y
P(z)
a
+
+ H(z)
υυυυ
(random, autocorrelated) Disturbance Signal
u and y are
cross
correlated
a and y are
cross
correlated
andIf u and a are statistically independent, then u
ν
will be un
cross
correlated...
"Plant Friendly" Input Signal Design
be as short as possible
restrictionsnot take actuators to limits, or exceed move size
(i.e., low variance, small deviations from setpoint)cause minimum disruption to the controlled variables
A plant friendly input signal should:
Note that theoretical requirements may strongly conflict
with "plant-friendly" operation!
Pseudo-Random Binary Sequence
1
n r
Test Signal(Modulo 2 Adder) Exclusive OR
Shift Registers
generated using shift registers and Boolean algebra The PRBS is a periodic, deterministic input which can be
number of shift registers (nr), and signal amplitudeThe main design variables are switching time (Tsw),
PRBS, continued
magnitude = +/- 1.0. One cycle duration is 45 minutes long.PRBS design for Tsampl = 1, Tsw = 3, n (registers) = 4, and signal
0
5
10
15
20
25
30
35
40
45
-0.
0
1
One cycle of the PRBS time input signal
Time[Min]
10 0
10
10
10
10
0
Radians/Min
AR
Power Spectrum of the PRBS input
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Inputs to Consider
Step/Pulse Inputs
Gaussian White Noise
Random Binary Signal (RBS)
Pseudo-Random Binary Signal (PRBS)
multi-level Pseudo-Random Signals
crest factor)Multisine inputs (e.g., Schroeder-phased, minimum
Nonparametric Methods
identification data- direct estimation of impulse response coefficients fromCorrelation Analysis:
data- direct estimation of frequency response from identificationSpectral Analysis:
Correlation Analysis Results, Hairdryer Data
0
0
10
20
Covf for filtered y
-0.
0
1
0
10
20
Covf for prewhitened u
-0.
0
0.20.40.
0
10
20
Correlation from u to y (prewh)
-0.
0
0
10
20
Impulse response estimate
Wing Flutter Example, Spectral Analysis
4 5 6 7 8 9
10
11
-25-20-15- -5 0
Amplitude [dB]
Frequency [Hz]
Smoothed SPA model (solid). Raw ETFE (*).
4 5 6 7 8 9
10
11
0
50
100 150
Phase [degree]
Frequency [Hz]
Smoothed SPA model (solid). Raw ETFE (*).
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Model Validation Techniques
output from the model).Simulation (plot the measured output time series versus the predicted
loss function and select the minimum.parameter estimation; for a number of different model structures, plot theCrossvalidation (simulate on a data set different than the one used for
regarding process).Impulse, step, and frequency responses (compare with physical insight
resemble white noise).Scatter Plots/correlation analysis on the prediction errors (make sure they
Information criteria (Akaike or Rissanen's Maximum Description Length)
Modeling Requirements for Process
Control
Modeling Control
Modeling/ Control
START
"Decomposed"
"Integrated/Synergistic"
Same result is not obtained from both
approaches!
Control-Relevant Identification
Some general ideas behind control-relevant modeling
Design variables for control-relevant id
Control-relevant prefiltering
Control-relevant input signals
Brief comments on uncertainty estimation from id data
Integrated system id and PID controller design
Control-Relevant Prefiltering
Time
Overhead Temperature
Solid: Raw Data; Dashed: Prefiltered Data
20
40
60
80
100
120
140
160
180
200
Time
Reflux Flow
20
40
60
80
100
120
140
160
180
200
purposesinformation in the data most important for control The purpose of c-r prefiltering is to emphasize
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Problems in Closed-Loop Identification
C
P
P
C
d
F
+
+ +
+
+
+
+
-
r
y
u
d
u d
υυυυ
-
and input (u) as a result of the controlcrosscorrelation will exist between disturbance (d)
away" at excitationcontrol action will introduce additional bias by "eating
Refinery Debutanizer
FEED FLOW
REFLUX FLOW
REBOIL FEED TEMP
T
FC
FC
T
F
FEED TEMP
P
G
T
FUEL GAS SPECIFIC GRAVITY FUEL GAS FLOWBOTTOMS TEMP
BOTTOMS-TO-FEED DIFFERENTIAL PRESSURE
MPC loop between
Bottoms
Fuel Gas Flow SPTemperature and
-4 - 0 2 4 0
50
100
150
200
250
300
Output Series
time
-0.
-0.
0
0
50
100
150
200
250
300
PRBS Signal and Input Series
Debutanizer Closed-Loop Testing
Temperature Bottoms SetpointFuel Gas Flowrate
respectivelySetpoint; dashed line shows external signal (ud); solid lines show u and y,Closed-loop data set generated by signal injection at the Fuel Gas Flowrate
Multivariable System Identification
Motivation for multivariable identification
Multiple input extensions to:
PRBS, RBS design
ARX estimation
PEM
Brief overviews of Bayard
s, Zhu
s, and subspace methods
Overview of ASU
s MIMO control-relevant methodology
zippered
multisine signals
Illustrations from various applications
© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved
Mixing Tank Example, Continued
The first-principles model for this system is:
V
dtdc
=
q c c c −
( q c
q w )
c
Using a forward-difference approximation on the derivative leads to
c ( t (^) + 1)
−
(^) c ( t )
T
=
q c ( t ) c c ( t )
V
−
( q c ( t ) +
q w
( t ))
c ( t )
V
which solving for
c ( t (^) + 1) yields
c ( t
c ( t ) +
q c ( t )
c c ( t )
T
V
−
( q c ( t ) +
(^) q
w ( t ))
c ( t ) T
V
c Rearranging and consolidating terms leads to the semiphysical structure ( t ) =
θ 1 c ( t −
1)+
θ 2 q c ( t −
c c ( t −
1)+
θ 3 q c ( t −
c ( t −
1)+
θ 4 q w
( t −
c ( t −
θ 1 , θ 2 ,
θ 3 , and
θ 4
can be estimated via linear regression.
System Identification Toolbox (SITB)
Graphical User Interface (GUI)
