Experimental Control Science1 Experimental Control Science Methodology, Algorithms, Solutions Zhiqiang Gao, Ph.D. Center for Advanced Control Technologies Cleveland State University

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Experimental ControlScience

Methodology, Algorithms, Solutions

Zhiqiang Gao, Ph.D. Center for Advanced Control Technologies

Cleveland State UniversityDecember 24, 2004

http://cact.csuohio.edu

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Outline• Introduction

• Questions

• Research Direction

• Methodology

• Active Disturbance Rejection

• Advanced Technologies

• Take Aways

• Open Problems

3

From Applied Researchto

Advanced Technologies

Center for Advanced Control Technologies

http://cact.csuohio.edu

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Center for Advanced Control Technologies

Zhiqiang Gao, Director

Sridhar Ungarala, Chemical Engineering

Daniel Simon, Embedded Control Systems, Electrical Engineering

Paul Lin, Mechanical Engineering.

Yongjian Fu, Software Engineering

Sally Shao, Mathematics

Jack Zeller, Engineering Technology

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Past Projects• Temperature Regulation• Intelligent CPAP/BiPAP• Motion Indexing• Truck Anti-lock Brake System• Web Tension Regulation• Turbine Engine Diagnostic• Computer Hard Disk Drive• Stepper Motor Field Control• 3D Vision Tire Measurement• Digitally Controlled Power Converter

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Sponsors• NASA• Rockwell Automation• Kollmorgen• ControlSoft• Federal Mogul• AlliedSignal Automotive• Invacare Co.• Energizer• Black and Decker• Nordson Co.• CAMP

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NASA Intelligent PMAD Project

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Web Tension Regulation

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Truck Anti-lockBrake System

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Turbofan engine

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A Non-isothermal CSTR

• CV: productconcentration CA

• MV: Coolant flowrate qc

• Difficulties:– Strong nonlinearity– Time varying

parameters: φc(t) φh(t)(catalyst deactivationand heat transferfouling)

11( )

0

0

( ) exp ( )

( ) exp ( )

1 exp ( )

AAf A A c

f A c

p

c pc

c h cf

p c pc

dC q EC C k C t

dt V RT

dT q H ET T k C t

dt V C RT

C hAq t T T

C V q C

!

!"

"!

" "

# $= % % %& '

( )

# $%* # $= % + %& ' & '& ' ( )( )

+ ,# $ # $+ % % %- .& ' & '& ' & '- .( ) ( )/ 0

Coolant

Feed

q c

Product, CA

AT

AC

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Nonlinear 3-Tank Fault Id. Problem

6 possible faults 2 inputs 3 outputs

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CACT Mission• Define, Articulate, Formulate

Fundamental Industrial Control Problems

• Solutions and Cutting Edge Technologies

• Performance and Transparency

• Synergy in Research and Practice

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Outline•• IntroductionIntroduction

• Questions

•• Research DirectionResearch Direction

•• MethodologyMethodology

•• Active Disturbance RejectionActive Disturbance Rejection

•• Advanced TechnologiesAdvanced Technologies

•• Take AwaysTake Aways

•• Open ProblemsOpen Problems

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Questions

• What is control & where does it belong?

• What is a good controller & how to find it?

• Does a theory-practice gap exist? Why?

• Can theoretical advance be driven by practice?

• What is the most fundamental control problem?

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How do we describe it?

• An Art of Practice?• Hidden Technology?• Mathematics?• Engineering Science?• Control Science?• Natural Science?

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Where does control belong?

• Electrical Engineering• Mechanical Engineering• Chemical Engineering• Aerospace Engineering• System Engineering• Mathematics• Biology?

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Is there a theory-practice gap?

Control Theory

⇓

Engineering Problem Solving

?

19

Can theory be driven by practice?

New Theory

⇑ ?

Engineering Problem Solving

20


•• QuestionsQuestions

• Research Direction






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Theory vs. Practice

A Historical Perspective

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Looking back

• PID (N. Minorsky) 1922• Nyquist 1932• Bode 1940• Kalman 1961 …• Ho 1982• Han 1989/1999

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Classical Control Era

ControlPractice

ControlResearch

ControlTheory

Mathematics

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Modern Control Era

ControlPractice

ControlResearch

ControlTheory

MathematicsResearch

Theory

unobservable

uncontrollable

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<The Structure of Scientific Revolutions>by Thomas S. Kuhn

Research:

• A strenuous and devotedattempt to force natureinto the conceptualboxes supplied byprofessional education

• Most scientists areengaged in mopping upoperations

Science:

• Suppresses fundamentalnovelties because theyare necessarilysubversive of its basiccommitments.

• Predicated on theassumption that thescientific communityknows what the world islike.

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• Methodology





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Control as an Experimental Science

• Y.C. Ho, IEEE AC, Dec. 1982

• “Control” as experimental science (the 3rd dimension w.r.t. the gap)

• Experiment vs. Application(detective vs. craftsman)

• “observation-conjecture-experiment-theory-validation”

• Carried out by BOTH theorists andexperimentalists

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Experiment Discover Theorize

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Reconnect

ControlPractice

ControlResearch

ControlTheory

Mathematics

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The Han Paradigm

• Is it a Theory of Control or a Theory of Model?

• Paradox of Robust Control

(Godel’s Incompleteness Theorem)

• An Alternative Design Paradigm

– Explore Error-Based Control Mechanisms

– Active Disturbance Rejection

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The Paradox of theRobust Control Problem

Making model-dependent controldesign independent of the model

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GÖdel’s Incompleteness Theorem

“Within any formal system of axioms,such as present day mathematics,questions always persist that canneither be proved or disproved on thebasis of the axioms that define thesystem.” --paraphrased by S. Hawking

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Is the solution to the robust control problemoutside the existing control theory?

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Problem Reformulation

reconnect theory to practice

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Making Problem Definition Realistic

• Assumptions on the plant:– What is the minimum info needed for design?– What info is available in practice?

• Design Objectives:– Absolute requirements– Criteria of optimality (judgment for comparison)

• Design Constraints:– Actuator/sensor/digital controller– Hard and soft constraints

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• Active Disturbance Rejection




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Questions

• What is control & where does it belong?

• What is a good controller & how to find it?

• Does a theory-practice gap exist? Why?

• Can theoretical advance be driven by practice?

• What is the most fundamental control problem?

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Uncertainty principle in control?

• Kalman Filter: uncertainty of measurement

• Industry Control: uncertainty of dynamics

• Disturbance: dynamics beyond the math model

• Disturbance ⇔ Uncertainty

• Control ⇔ Disturbance Rejection?

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Disturbance Rejection

• Modeling: Uncertainty ReductionExample: modeling ⇒ design ⇒ tuning

• Passive Disturbance RejectionExample: PID tuning

• Active Disturbance RejectionExample: Invariant Principle, ADRC (Han)

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A Motion Control Case Study

( , , )y f y y w u= +&& &

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Model-Based Method

( , , )y f y y w u= +&& &

Modeling: in analytical form

Design Goal:

Plant:

( , , )f y y w&

( , )y g y y=&& &

( , , ) ( , )u f y y w g y y= ! +& &

Examples: pole placement; feedback linearization

Control Law:

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Industry Practice

( , , ) ( , ) ( , )y f y y w l y y g y y= + !&& & & &

The PID example

With unknown,( , , )f y y w& ( , )u l y y= &

( , , , ) ( )p I Dy f t y y w K e K edt K e

e r y

= + + +

= !

"&& & &

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The Han Methods

• Beyond PID ⇒ Nonlinear PID ⇒ Time Optimal Control ⇒ Discrete Time Optimal Control ⇒ Find other error-based designs

• Find a way around modeling ( , , )f y y w&

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Getting around modeling

• Adding a sensor

• Estimating in real time

( , , )f y y w y u= !& &&

( , , )f y y w&

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Active Disturbance Rejection

1 2

2 3

3

1

,

x x

x x u

x f

y x

=!"

= +"#

="" =$

&

&

&&

Augmented plant in state space:

Extended State Observer (Han)

1 2 31 2 3 z x z x z x f! !! =

1 2 1 1 1

2 3 2 2 1

3 3 3 1

( )

( )

( )

z z g z y

z z g z y u

z g z y

!

!

!

= " "#$

= " " +%$ = " "&

&

&

&

1 2 3, , ( , , )x y x y x f y y w= = =& &

( , , ) y f y y w u= + !&& &

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Active disturbance compensation

1 2

2 0

1

x x

x u

y x

=!"

#$" =%

&

&

1 2

2

1

x x

x f u

y x

=!"

= +#" =$

&

&

0 3

3

u u z

z f

= !

"

1 2( , , )?( ) or f x x wf t

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Observer Comparison

Luenberge Observer Extended State Observer

Plant

y(t)

w(t)

Extended

State Observer

u(t)Plant

y(t)

w(t)

Luenberger

State Observer

u(t)

yy

y& y&

( , , )y f y y w u= +&& &

f

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Observer Comparison

Luenberger Observer

• Needs expression of f• Model-based• For LTI systems only

Extended State Observer

• Estimates y, dy/dt, and f• Model-independent• Linear or nonlinear• TI or TV• One-parameter tuning

( , , )y f y y w u= +&& &

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!

!

( ) ( , , )ny f y y w u= +&

0

ˆu f u= ! +

( )

0

ny u!

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Active Disturbance Rejection ControlADRC

• Generalized disturbance rejection:– Internal disturbance: system dynamics– External disturbance– Combined into f

• Easily tuned– Z. Gao, ACC2003

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Bandwidth-based Tuning

0 1 2 3 4 5 60

1

2position

y z1

0 1 2 3 4 5 6-1

0

1

2velocity

dy/dtz2

0 1 2 3 4 5 6-50

0

50disturbance and unknown dyanmics

time second

f z3

0 1 2 3 4 5 60

1

2transient profile and output

bandwidth: 4 rad/sec bandwidth: 20 rad/sectransient profile

0 1 2 3 4 5 60

0.5

1error

0 1 2 3 4 5 6-1

0

1

2control signal

time second

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Hardware Test: torque disturbance

0 2 4 6 8 10 120

0.5

1

1.5

0 2 4 6 8 10 12-0.1

0

0.1

0 2 4 6 8 10 12-5

0

5

Torque Disturbance Rejection Rev.

Rev.

Volts

Position

Position error

Control Command

ADRC

ADRC

ADRC

PID

PID

PID

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Performance of the disturbance observer

0 1 2 3 4 5-30

-20

-10

0

10

20

30

a(t)

z3(t)

Total disturbance and its estimation

Time (sec.)

f(t)

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Motion Control Demo

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• Advanced Technologies



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Algorithms

• Nonlinear PID• Discrete Time Optimal Control• Active Disturbance Rejection• Single Parameter Tuning• Wavelet Controller/Filter

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Nonlinear PID

• Error driven, not model-based• Nonlinear “proportional” term gp(e)

– Small error, large gain– Reduce the role of integrator

• Nonlinear integral control– Reduce phase lag– Maintain zero s.s. error and good disturbance rejection

• Nonlinear differentiator– Noise immunity

( ) ( ) ( )p p I i D du K g e K g e dt K g e= + +! &

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Discrete Time Optimal Control Law1 2

0

1 2

2

0

0

2 0

2 0

(( , , , )

;

8 | |

( ), | |2

/ , | |

( ), | |

, | |

u fst x x r h

d rh d hd

y x hx

a d r y

a dx sign y y d

a

x y h y d

r sign a a d

fst ar a dd

=

= =

= +

= +

!"+ >#

= $# + %&

>"#

= !$%#&

1

2

0

( 1) ( ) ( )

| ( ) |

1 0, ,

0 1

(0)

0

0f

x k Ax k Bu k

u k r

x hx A B

x h

x x

x

+ = +

!

= = =" # " # " #$ % $ % $ %& ' & '& '

=

" #= $ %& '

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Comparison of switching curves

Details

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• Manufacturing (Motion, Web Tension, CNC)

• Power Electronics (Motor, PMAD, Converters)

• Aircraft (Flight, Jet Engine)

• Process Control (CSTR)

• Biomedical (Ankle)

• Health/fault Monitoring (A benchmark prob.)

• Automobile (Truck ABS)

Technologies

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Take Aways

• Think outside “the box”

• Active disturbance rejection

• From problems to methods tomethodology

http://[email protected]

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Open Problems• Characteristics of ESO

– Convergence,– Rate of Convergence,– Boundedness– Bound of error– Order estimation– b0 estimation (Initial results)

• Practical Optimality (Initial results)• Reformulation of process control problems• Cybernetics

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A Research Alliance• Practitioners/Researchers/Mathematicians

• Discover (both practitioners and theoreticians)

• Theorize– Prove stability and convergence– Generalize a particular solution/method– Establish a new kind of theory

• Validate– Verify the new theory against other problems– Define the range of applicability

Documents

Experimental Control Science1 Experimental Control Science Methodology, Algorithms, Solutions Zhiqiang Gao, Ph.D. Center for Advanced Control Technologies Cleveland State University