disturbances system identification daniel e. rivera, …csel.asu.edu/downloads/publications/sysid...

D.E. Rivera, Introduction to System Identification, ChE 494/598, January 20, 2004

© Copyright 1998-2004 by D.E. Rivera, All Rights Reserved

1

Daniel E. Rivera, Associate Professor

Control Systems Engineering LaboratoryDepartment of Chemical and Materials Engineering

Arizona State UniversityTempe AZ 85287-6006

[email protected](480) 965-9476

© Copyright, 1998-2004

ChE 494/598Introduction to System

Identification

Course Objectives

• Provide lab exercises that will give students a working feel for the course topics. MATLAB (particularly the System Identification Toolbox) will be the program of choice.

• Present fundamental background to allow students to make judicious choices of design variables in system identification.

• Provide a glimpse of cutting-edge identification research at ASU and other academic institutions around the world.

System Identification“Identification is the determination, on the basis of input and output, of a system within a specified class of systems, to which the system under test is equivalent.” - L. Zadeh, (1962)

SystemSystemInputs Outputs

Disturbances

System identification focuses on the modeling of dynamical systems from experimental data

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 expensive and time consuming part of advanced control implementation

• broadly applicable technology with applications in many diverse fields



2

Shell Heavy Oil Fractionator Example

Top Draw

• Manipulate top, side draw and/or bottoms reflux duty to maintain top and side endpoints at setpoint,

• Reject disturbances from the upper and intermediate reflux duties.

• Keep Bottoms Reflux Temperature above constraints.

LC

A

T

T

T

LC

LC

FEED

BOTTOMS REFLUX

INTERMEDIATE REFLUX

UPPER REFLUX

TOP DRAW

SIDE DRAW

BOTTOMS

SIDESTRIPPER

FC

FC

Q(F,T)CONTROL

F

T

PC

T

A

T

TopEndpoint

SideEndpoint

Side Draw

Upper Reflux Duty

Intermediate Reflux Duty

BottomsReflux Duty

BottomsReflux Temp

-2

0

2

4

0 20 40 60 80 100 120 140 160 180 200

OUTPUT #1

0

0 20 40 60 80 100 120 140 160 180 200

INPUT #1

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

-10

0

10

Time [seconds]

Tem

p. D

evia

tion

[C]

solid:master; dashed:side; dotted:front; d-dotted:rear

0 200 400 600 800 1000 1200 1400 1600 1800 2000-8

-6

-4

-2

0

Time [seconds]

Pow

er [%

]

solid:master; dashed:side; dotted:front; d-dotted:rear

Epi Reactor Identification Data



3

LT

ADI

SFGI

CW

LT

LT

Controller

Demand

Forecast

Real

D1D2D3t

ADI: Assembly-Die Inventory

SFGI: Semi-Finished Goods Inventory

CW: Component Warehouse

Fab/Sort starts

A/T starts

Shipments

Demand

A/T: Assembly/Test Facility

Semiconductor Mfg Supply Chain Managment Fab/Test Node Dynamic Response

Lo

adO

uts

Sta

rts

Time

Wing Flutter Example

0 1 2 3 4 5 6 7 8-10

-5

0

5

10

Exci

tatio

n

Time [s]

Filtered data used for modeling

0 1 2 3 4 5 6 7 8-5

0

5

Res

pons

e

Time [s]

Filtered data used for modeling

• artificial mechanical vibrations (top) introduced to a wing at certain flight conditions; responses shown on bottom

Other Challenging Application Areas

• Economic/financial systems

• modeling economic indicators such as the Dow Jones, S&P 500 indices

• Behavioral/social systems

• time-varying adaptive interventions for the prevention of chronic, relapsing disorders (such as alcoholism, smoking and drug abuse)



4

Stages of System Identification

• Experimental Design and Execution

• Data Preprocessing

• Model Structure Selection

• Parameter Estimation

• Model Validation

STAGES OF SYSTEM IDENTIFICATIONStart

Experimental Design and Execution

"Identification"

Model Validation

Does the model meet validation criteria?

( Step, Pulse, or PRBS-Generated Data)

( Linear Plant and Disturbance Models)

(Simulation, Residual auto and crosscorrelation,step-response)

No

End

Yes

a priori processinformation

• Model Structure Determination

• Parameter Estimation

• Data Preprocessing

Stages of System Identification - II

• courtesy P. Lindskog, ISY, Linköping University, Sweden

Prio

r sy

stem

kno

wle

dge:

phy

sics

, lin

guis

tics,

firs

t-ha

nd, e

tc.

Experimentdesign

Pre-treatdata

Choosemodel

structure Chooseperformance

criterion

Parameter estimation

Validatemodel Not OK revise!

OK accept model!Not OK revise prior?

Controller Design & Commissioning

Keys to Successful System Identification in Practice

• Understanding the various identification methods and associated decision variables in terms of bias-variance tradeoffs

• Effective use of a priori knowledge regarding the system to be identified and the intended application (e.g., simulation, prediction, control)

"the classical statistical approach," per Ljung...



5

• Skill-level issues: many system identification methods assume the user has extensive background in statistics, signal processing, discrete-time systems, and optimization.

• Large number of design variables.

• Process operating restrictions make identification one of the most time consuming tasks in advanced control implementation projects.

System Identification Challenges

(Disturbance)

(Input)

(Output)

CONTROLLER

Objective: Use fuel gas flow to keep outlet temperature under control, in spite of significant changes in the feed flowrate.

Furnace Control Example

The "Shower Problem"

Hot Cold

Consider the problem of adjusting hot water flow to maintain shower temperature despite cold water fluctuations...

Transportation lagMakes this a difficult control problem...

Many references for this technique, example: Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, 1989, Chapter 7.

p(s) = K e-θs

τ s + 1,

Response of a first-order with deadtime model for a step input of magnitude A

Graphical System Identification Using Step Testing

θTime

KA

τ



6

0 200 400 600 800 1000 1200 1400 1600 1800 20000

5

10

15

20

25Measured Output

Time[Min]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-1

-0.5

0

0.5

1Input

Time[Min]

Open-loop disturbance (no control)

2000 minutes

Fuel Gas Flow (Manipulated Variable)

Temperature(ControlledVariable)

Consider the application of step testing to a system subject to a drifting, nonstationary disturbance.

Furnace Example (Continued) Perils of Step Testing

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

0 5 10 15 20 25 30 35 40

Time[Min]

Compare Step Responses: FOPDT Model[--], PLANT Data[-], TRUE PLANT[-.]

40 minutes

Temperature Response to a Step Increase in Fuel Gas Flow

p(s) (est) = 0.748 exp(-6s)-------------------- 7.095 s + 1

p(s) (true) = exp(-5s) -------------- 10 s + 1

Design Variable Selection Issues• Input Signal Selection: Random Binary,

Pseudo-Random Binary, or Multiple Step/Pulse Inputs?

• Input Signal Parameters: Example: PRBS - number of shift registers, switching time, signal magnitude, and signal duration

• Data Preprocessing: Detrending, control-relevant prefiltering, outlier removal, etc.

• Model Structure Selection and Parameter Estimation: ARX, ARMAX, Output Error, Box-Jenkins

• Model Validation: Simulation, Crossvalidation, Correlation Analysis; Examination of Step, Impulse, Frequency Responses.

Principal Sources of Error in System Identification

• BIAS. Systematic errors caused by

- input signal characteristics (i.e., excitation) - choice of model structure

- mode of operation (i.e., closed-loop instead of open-loop)

• VARIANCE. Random errors introduced by the presence of noise in the data, which do not allow the model to exactly reproduce the plant output. It is affected by the following factors:

- number of model parameters

- duration of the identification test

- signal-to-noise ratio

ERROR = BIAS + VARIANCE



7

OPENLOOP

RESPONSE

Furnace example with PRBS input, PID with filter controller

IDENTIFICATION DATA

CLOSED LOOP

RESPONSE

Input

MeasuredOutput

From Identification to Controller Implementation

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-15

-10

-5

0

5

10

15Input

Time[Min]

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-5

0

5

10

15

20

25Measured 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

- Linear or Nonlinear?

(Mostly) LINEAR

- Continuous or Discrete?

DISCRETE

- Parametric or nonparametric?

BOTH

- Time or frequency domain?

BOTH

Systems Representations

Nonlinear Lumped Parameter System

State-SpaceModel

Linearization

s-domainTransfer Function

Model

Laplace transforms{Step/

ImpulseResponse

and FrequencyResponse z-domain

Transfer FunctionModel

T

Sampling

(difference equation)

}Discrete-timeStep/

Impulse Response

and FrequencyResponse

Discrete-time S-S Model

T

Sampling

Realization



8

Discrete Model Representations

PLANTStep

Response

PLANTImpulse

Response

U(k) Y(k+1)D

DifferenceEquations

G(z)U(z) Y(z)

Z-Transforms

Nonparametric

{

Parametric

{

Pulse Transfer Functionscomputer

controlalgorithm

Zero-orderHold P(s) computer

kukyu(t) y(t)

ZOH-equivalent Pulse Transfer Function

T

time

timetime

time

discreteinput

continuousinput

continuousoutput discrete

output

ky

u(t)y(t)

ku

Examples

Impulse

Step

First-OrderLag

Integrating/Ramp

First-Orderwith Delay

s-domain z-domain

1 1

time

δ (t)

s(t) =1 t ≥ 0

0 t < 0

1s

zz −1

Kτs +1

Ks

Kτs +1

exp(−θs)

θ = NT

ZOH PulseTransferFunctionZOH PulseTransferFunctionZOH PulseTransferFunction

KTz −1

K(1 - exp(-T /τ))z -Nz - exp(-T/τ)

K(1 - exp(-T /τ))z - exp(-T /τ)

System Identification Structure

Random Signal

Input Signal

Output Signal

u

yP(z)

a

++

H(z)

υυυυ Disturbance Signal

P(z) and H(z) are discrete-time (z-domain) transfer functions

y(t) = p(z)u(t) +H(z)a(t)



9

Signals Overview

• White versus autocorrelated signals

• Crosscorrelated versus uncorrelated signals

• Deterministic versus stochastic signals

• Stationary versus nonstationary signals

Mean, auto and cross-covariance, power and cross-spectra will be the measures/tools utilized here

Example 1: White-noise signal

tx = ta ta = N(0, a2σ )

1txta

0

0.5

1

1.5

0 1 2 3

Frequency [Radians]

phi

POWER SPECTRUM OF WHITE NOISE

0

0.5

1

1.5

0 2 4

Lag

rhok

AUTOCORRELATION OF WHITE NOISE

kρ = kγa2σ a

2σ = 1Φ(ω )

ω

0 200 400 600 800 1000 1200 1400 1600 1800 2000-1

-0.5

0

0.5

1

1.5

Sample Number

Signal

-20 0 20 40-0.5

0

0.5

1

Lag k

rhok

Autocorrelation Coefficients

10-2 10-1 10010-2

10-1

100

Frequency [Radians/Time]

Power Spectral Density

Example 1: White Noise Signal Analysis (From Sample Estimators)

0 200 400 600 800 1000 1200 1400 1600 1800 2000-3

-2

-1

0

1

2

3

Sample Number

Signal

-20 0 20 40-0.5

0

0.5

1

Lag k

rhok

Autocorrelation Coefficients

10-2 10-1 10010-2

10-1

100

101

Frequency [Radians/Time]

Power Spectral Density

Example 3: AR(1) signal



10

System Identification, Revisitedwhite noise signal

Input Signal (Random orDeterministic)

Output Signal (random, autocorrelated)

u

yP(z)

a

++

H(z)

υυυυDisturbance Signal (random, autocorrelated)

• u and y are crosscorrelated• a and y are crosscorrelated• If u and a are statistically independent, then u

and ν will be uncrosscorrelated...

"Plant Friendly" Input Signal Design

• be as short as possible

• not take actuators to limits, or exceed move size restrictions

• cause minimum disruption to the controlled variables (i.e., low variance, small deviations from setpoint)

A plant friendly input signal should:

Note that theoretical requirements may strongly conflict with "plant-friendly" operation!

Pseudo-Random Binary Sequence

1 0 1 1 0 1 1 0 1

1 nr

Exclusive OR(Modulo 2 Adder)

Test Signal

Shift Registers

The PRBS is a periodic, deterministic input which can be generated using shift registers and Boolean algebra

The main design variables are switching time (Tsw), number of shift registers (nr), and signal amplitude

PRBS, continued

PRBS design for Tsampl = 1, Tsw = 3, n (registers) = 4, and signal magnitude = +/- 1.0. One cycle duration is 45 minutes long.

0 5 10 15 20 25 30 35 40 45-1

-0.5

0

0.5

1

One cycle of the PRBS time input signal

Time[Min]

100

10-3

10-2

10-1

100

Radians/Min

AR

Power Spectrum of the PRBS input



11

Inputs to Consider• Step/Pulse Inputs

• Gaussian White Noise

• Random Binary Signal (RBS)

• Pseudo-Random Binary Signal (PRBS)

• multi-level Pseudo-Random Signals

• Multisine inputs (e.g., Schroeder-phased, minimum crest factor)

Nonparametric Methods

• Correlation Analysis:

- direct estimation of impulse response coefficients from identification data

• Spectral Analysis:

- direct estimation of frequency response from identification data

Correlation Analysis Results, Hairdryer Data

0

0.05

0.1

0.15

-20 -10 0 10 20

Covf for filtered y

-0.5

0

0.5

1

1.5

-20 -10 0 10 20

Covf for prewhitened u

-0.2

0

0.2

0.4

0.6

-20 -10 0 10 20

Correlation from u to y (prewh)

-0.05

0

0.05

0.1

0.15

-20 -10 0 10 20

Impulse response estimate

Wing Flutter Example, Spectral Analysis

4 5 6 7 8 9 10 11-25

-20

-15

-10

-5

0

Am

plitu

de [

dB]

Frequency [Hz]

Smoothed SPA model (solid). Raw ETFE (*).

4 5 6 7 8 9 10 110

50

100

150

Pha

se [

degr

ee]

Frequency [Hz]

Smoothed SPA model (solid). Raw ETFE (*).



12

Smoothing, Filtering, Prediction

• In the prediction problem, current and previous measurements from the plant are used to obtain estimates k+1 (or beyond) time steps in the future

A(z)y(t) =B(z)F (z)

u(t− nk) +C(z)D(z)

e(t)

A(z) = 1 + a1z−1 + . . .+ anaz

−na

B(z) = b1 + b2z−1 + . . .+ bnbz

−nb+1

C(z) = 1 + c1z−1 + . . .+ cncz

−nc

D(z) = 1 + d1z−1 + . . .+ dndz

−nd

F (z) = 1 + f1z−1 + . . .+ fnfz

−nf

In transfer function form:

y(t) = p(z)u(t) + pe(z)e(t)

p(z) =B(z)

A(z)F (z)z−nk pe(z) =

C(z)A(z)D(z)

Prediction-Error Model Structures

C(z)D(z)

1A(z)

e

yu B(z)F(z)

−nkz ++

Prediction-Error Family of Models

Popular PEM StructuresMethod p(z) pe(z)ARX B(z)

A(z)z−nk 1

A(z)

ARMAX B(z)A(z)z

−nk C(z)A(z)

FIR B(z)z−nk 1Box-Jenkins B(z)

F (z)z−nk C(z)

D(z)

Output Error B(z)F (z)z

−nk 1

A(z)y(t) =B(z)F (z)

u(t− nk) +C(z)D(z)

e(t)

y(t) = p(z)u(t) + pe(z)e(t)

ARX Parameter EstimationThe one-step ahead predictor for y

y(t|t−1) = −a1y(t−1)−. . .−anay(t−na)+b1u(t−nk)+. . .+bnbu(t−nk−nb+1)

can be expressed as a linear regression problem via

ϕ = [−y(t− 1) . . . −y(t− na) u(t− nk) . . . u(t− nk − nb + 1) ]T

and θ, the vector of parameter estimates:

θ = [ a1 . . . ana b1 . . . bnb ]T

Rewriting the objective (“loss”) function as

minθV = min

θ

1N

N∑i=1

[y − ϕT (t)θ

]2

leads to the well-established linear least-squares solution

θ = 1N

N∑t=1

ϕ(t)ϕT (t)−1 1N

N∑t=1

ϕ(t)y(t)



13

Model Validation Techniques• Simulation (plot the measured output time series versus the predicted

output from the model).

• Crossvalidation (simulate on a data set different than the one used for parameter estimation; for a number of different model structures, plot the loss function and select the minimum.

• Impulse, step, and frequency responses (compare with physical insight regarding process).

• Scatter Plots/correlation analysis on the prediction errors (make sure they resemble white noise).

• 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

Ove

rhea

d T

empe

ratu

re Solid: Raw Data; Dashed: Prefiltered Data

-2

0

2

4

0 20 40 60 80 100 120 140 160 180 200

Time

Ref

lux

Flow

-2

0

2

0 20 40 60 80 100 120 140 160 180 200

The purpose of c-r prefiltering is to emphasize information in the data most important for control purposes



14

Problems in Closed-Loop Identification

C P

PC dF

+++

+ + ++-

r yu

d

ud

υυυυ

-

• crosscorrelation will exist between disturbance (d) and input (u) as a result of the control

• control action will introduce additional bias by "eating away" at excitation

Refinery Debutanizer

FEED FLOW

REFLUX FLOW

REBOIL FEED TEMP

T

FC

FC

TF

FEED TEMP

P

G

T

FUEL GAS SPECIFIC GRAVITY

FUEL GAS FLOW

BOTTOMS TEMP

BOTTOMS-TO-FEED DIFFERENTIAL PRESSURE

MPC loop between Bottoms

Temperature and Fuel Gas Flow SP

-4

-2

0

2

4

0 50 100 150 200 250 300

Output Series

time

-0.1

-0.05

0

0.05

0.1

0 50 100 150 200 250 300

PRBS Signal and Input Series

Debutanizer Closed-Loop Testing

Bottoms

Temperature

Fuel Gas Flowrate

Setpoint

Closed-loop data set generated by signal injection at the Fuel Gas Flowrate Setpoint; dashed line shows external signal (ud); solid lines show u and y, respectively

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



15

MIMO PRBS Input Experimental Data (Noise free)

PRBS: Specifying τLdom = 5, τHdom = 33, αs = 2, βs = 3, Tmaxsettle = 165,

and Tsampl = 2 min leads to nr = 7, Tsw = 6 min, and D = 168 min.

Signal magnitude set at usat = 0.001.

0 100 200 300 400 500 600 700 800-0.02

0

0.02

y1[T

21]

MIMO PRBS Input Experiment Data

0 100 200 300 400 500 600 700 800-1

0

1x 10-3

u1[L

]

0 100 200 300 400 500 600 700 800-0.02

0

0.02

y2[T

7]

0 100 200 300 400 500 600 700 800-1

0

1x 10

-3

Time[Min]

u2[V

]

Control-Relevant Identification Methodology

Schroeder-Phased

PRBS

Random Binary Sequence

SIMO

SIMOMIMO

MIMOSIMO

High-Order ARX Estimation

DFTAnalysis Frequency-Weighted

Curvefittingand

Controller Design

“Plant-Friendly”Input Design

Nonparametric Estimation

Control-RelevantParameterEstimation

SIMO: Single-Input, Multi-Output

MIMO:Multi-Input, Multi-Output

Move Horizon OO

OO

OO

OO O

O

FuturePast

k k+1 k+2 k+M k+P

u(k)

y(k)O O

O

O^

Prediction Horizon

Model Predictive Control

Following prediction, the MPC controller solves the fol-lowing multiobjective optimization problem,

min[∆u(k),...,∆u(k+m)]

Keep controlled variables at setpoint︷︸︸︷p∑�=1

∥∥∥∥ΓY� (y(k + � | k)− r(k + �))∥∥∥∥2

+

Move suppression︷︸︸︷m∑�=1‖Γu�∆u(k + �− 1)‖2

Semiphysical ModelingBrine-Water Mixing Tank Example

Consider the dynamics of a tank mixing fresh and brine flow streams



16

Mixing Tank Example, ContinuedThe first-principles model for this system is:

Vdc

dt= qc cc − (qc + qw) c

Using a forward-difference approximation on the derivative leads to

c(t + 1)− c(t)T

=qc(t) cc(t)

V− (qc(t) + qw(t)) c(t)

Vwhich solving for c(t + 1) yields

c(t + 1) = c(t) +qc(t) cc(t) T

V− (qc(t) + qw(t)) c(t) T

VRearranging and consolidating terms leads to the semiphysical structure

c(t) = θ1c(t−1)+θ2qc(t−1) cc(t−1)+θ3qc(t−1) c(t−1)+θ4qw(t−1) c(t−1)

θ1, θ2, θ3, and θ4 can be estimated via linear regression.

System Identification Toolbox (SITB)Graphical User Interface (GUI)

disturbances system identification daniel e. rivera, …csel.asu.edu/downloads/publications/sysid...

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