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Bridging the theory-practice gap through problem reformulation:

a motion control case study

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

Cleveland State UniversityJune 24, 2004

www.cact.csuohio.edu

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Outline

• Introduction

• The Theory-Practice Gap

• An Experimental Science

Approach to Control Research

• Problem Reformulation

• Conclusions

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From Applied Research to

Advanced Technologies

Center for Advanced Control Technologies

www.cact.csuohio.edu

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CACT Mission• Define, Articulate, Formulate

Fundamental Industrial Control Problems

• Solutions and Cutting Edge Technologies

• Performance and Applicability

• Synergy in Research and Practice

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Center for Advanced Control Technologies

FACULTY:

Dr. Zhiqiang Gao, Director of CACT, ACRL and AERL

Dr. Daniel Simon, Director of the Embedded Systems Laboratory in Electrical Engineering.

Dr. Paul Lin, Director of the 3D Optical Measurement Laboratory in Mechanical Engineering.

Dr. Yongjian Fu, Software Engineering

Dr. Sally Shao, Mathematics

Prof. Jack Zeller, Engineering Technology, 40 yrs+ experience, P.E.

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Center for Advanced Control TechnologiesDoctoral Candidate Researchers:

Frank Goforth Robert Miklosovic Zhan Ping Wankun ZhouAaron Radke Chunming YangQing Zheng Sri Kiran Kosanam

Masters Candidate Researchers:Eric Dittmar Bharath EndurthiHrishikesh Godbole Sai Kiran GummaQing Guo Ivan JurcicSrujan Kusumba Mike GrayXiaolong Li Ramgopal MushiniNuha Nuwash Tong RenBhavinkumar Shah Chirayu ShahMadhura Shaligram

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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 • AlliedSignal Automotive• Invacare Co.• Energizer• Rockwell Automation• Kollmorgan• ControlSoft• Black and Decker• Nordson Co. • CAMP

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NASA Intelligent PMAD Project

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Networked Power Converters

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Case Study: Web Tension Regulation

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Case Study: Truck Anti-lock Brake System

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Case Study: Computer Hard Disk Drive

We build it, test it, and make it work.

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We apply our research using our partner’s products.

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We get results: Wavelet control for robust machines.

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We get results: Advanced motor field control reduces cost.

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We get results: Model Independent control design & tuning.

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We write the software.

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We have staff from the“School of Hard Knocks”.

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It All Comes Down To Mathematics

• Level of abstraction

• Clarity in thinking

• Theory and guidance

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Theory vs. Practice

A Historical Perspective

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The Classical Control Era

ControlPractice

ControlResearch

ControlTheory

Mathematics

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The Modern Control Era

ControlPractice

ControlResearch

ControlTheory

Mathematics

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The transition did go quietly

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4/1964 IEEE Trans. Automatic Control Editorial

“In recent years, there has been considerable discussion about the gap which appears to exist between control theory and its application…”

“It appears that the problem of the gap is a control problem in itself; it must be properly identified and optimized through proper action…”

AACC Theory and Applications Committee meeting, 3/24/1964“Bridging the Gap Between Theory and Practice”

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4/1965 IEEE Trans. Automatic Control Guest Editorial by Harold Chestnut

Proposed Solutions:• Company sponsored

Education

• Component Study by

Universities

• Promotion of Economic

Incentives

• Publication Policies

• Definition of characteristics of

systems and subsystems

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8/1967 IEEE Trans. Automatic Control Editorial by J.C. Lozier

On panel discussions:“These panels, staffed with leading theoreticians, have automatically assumed that theory is ahead of practice, and they conclude that the solution lies in reeducating the designers. The establishment has spoken.”

Suggestion in bridging the gap: stimulate– papers on general practice– papers on advanced engineering

practice– a more hospital atmosphere where

results are as important as methods

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2/1968 IEEE Trans. Automatic Control Announcement by John. B. Lewis

• Special two day conference preceding JACC meeting

• Responding to 65 and 67 editorials

• Each session is “a complete case history necessarily brings together theory and practice”

• “Demonstration of what can be achieved by applying control theory to major problems”

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12/1982 IEEE Trans. Automatic Control Editorial by Y.C. Ho

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

• Experiment vs. Application (detective vs. craftsman)

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

• Carried out by BOTH theorists and experimentalists

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The debate continues

• “On Control Theory and Practice”, G. John, AC, June 1970

• “Editorial: Some Thoughts on Research”, J. Mereditch, AC, Feb.

1980

• “Editorial: Theory and application: A common ground?” M. Sain,

June 1980.

• “An Industrial Point of View on Control Teaching and Theory”, E. H.

Bristol, CSM, Feb. 1986

• Special Issue on Theory and Practice Gap, CSM December 1999.

• Theory vs. Practice Forum, ACC 2004

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The ISA Initiative

• ISA (Instrument, systems, and automation) is the largest organization of instrument and control engineers in the world

• ISA is organizing a Theory vs. Practice Forum at ACC2004 (by Z. Gao and R. Rhinehart)

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Reflection on Control Research

What and Why?

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What is controls?

Controls: An instrument or a set of instruments used to operate, regulate, or guide a machine or vehicle

-the American Heritage Dictionary

Is it a branch of engineering, science, or mathematics?

-

r ye

ReferenceSpeed

uPlantController

Error Gas PedalPosition

VehicleSpeed

Sensor

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Control Engineering?

Engineering: The application of scientific principles to practical ends as the design, construction, and operation of efficient and economical structures, equipment, and systems. -the American Heritage Dictionary

If control is a branch of engineering, what are the scientific principles behind it?

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Control Science?

Science: The observation, identification, description, experimental investigation, and theoretical explanation of natural phenomena. -the American Heritage Dictionary

Can we learn from birds?

Should we?

Is Controls a natural

phenomena?Was Controls learnt?

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The Theory-Practice Divide

• Practitioners practice, improvise, experiment (experience counts in industry)

• Theorists theorize

Modern Control Theory is often viewed as a branch of Applied Mathematics

• Why the divide?

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Experimental Controls Research

Discover vs. Apply

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Experiment Discover Theorize

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• Observation: 95% of controllers used in Industry is PID

• Conjectures: • Error based design must have merits;• Solution to robust control is outside the realm of

modern control theory;• Better controllers can be found experimentally

p I Du K e K edt K e

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Experiment #1: A design not strictly based on the math model

( , , , )y f t y y w bu

( , , , ) ( )f t y y w f t

0( ( ) ) /u f t u b

0y u

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A unique disturbance estimator

1 2

2 3

3

1

,

x x

x x bu

x f

y x

Augmented plant in state space:

Extended State Observer

1 1 2 3 32 z x z x z x

1 2 1 1 1

2 3 2 2 1

3 3 3 1

( )

( )

( )

z z g z y

z z g z y bu

z g z y

1 2 3, , ( , , , )x y x y x f t y y w

( , , , ) y f t y y w bu

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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 bu

y x

0 3

3

( ) /u u z b

z f

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

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Active Disturbance Rejection Control

ProfileGenerator

NonlinearPD

Plant

ExtendedState Observer

(ESO)

1/b0 b0

+_

+_+_

r(t) v2(t)

v1(t)

u0(t) u(t) y(t)

z3(t)z2(t)

z1(t)

w(t)

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A Breakthrough in Motion Control

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

( , , , ) ( )y f t y y w bu t

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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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Extension to Higher Order MIMO Plants

( ) ( 1)( , , , , ( )) , , ,n n l l mY F Y Y Y d t U Y R U R d R

( 1)( ) ( , , , , ( ))nW t F Y Y Y d t Model of F(.) in the state space→ in the time domain：

( )U W t V ( )nY V

How to reconstruct the extended state

From U and Y ?

( 1)( ) ( ( ), ( ), , ( ), ( ))nW t F Y t Y t Y t d t

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Extended state( ) ( )nY W t U

0( )U W t U ( )

0nY U

( ) ( 1)( , , , , ( ))n nY F Y Y Y d t U ( 1)( ) ( , , , , ( ))nW t F Y Y Y d t

1 2 01 1 1

2 3 02 2 1

1, 0 1

1, 0 1 1 1

( )

( )

( )

( )

i i i i

i i i

ni n i n n i i i

n i n n i i

z z g z y

z z g z y

z z g z y u

z g z y

Extended state observer (ESO)

1

2

( 1)

1 ( )

nn

n

Z Y

Z Y

Z Y

Z W t

1 0nU Z U

Dynamic linearization and decoupling

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Successful Applications

• Motion Control (All manufacturing Industries)• Web Tension Regulation (paper, steel, printing..)• Machine Tools• Power Electronics (Motor, Converters …)• Aircraft Control (MIMO)• Process Control (with long transport delay)• Active Magnetic Bearing

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New Control Technologies “Discovered”

• Nonlinear PID

• Discrete Time Optimal Control

• Active Disturbance Rejection

• Single Parameter Auto-Tuning

• Wavelet Controller/Filter

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A Paradigm Shift

• Gao, Huang, Han, CDC2001

• Existing Paradigm: Model Based Design

• New Paradigm: Error (e=r-y) Based

• Same Objective: Desired error behavior

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The Paradox of the Robust Control Problem

Making the performance of the model-dependent control design 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 can neither be proved or disproved on the basis of the axioms that define the system.” --paraphrased by S. Hawking

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Is the solution to the robust control problem outside the existing control theory?

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Problem Reformulation

reconnect theory to practice

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Reconnect

ControlPractice

ControlResearch

ControlTheory

Mathematics

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Components of Problem Definition

• 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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A Motion Control Case Study

( , , , )y f t y y w bu

: continuous and differentiable

b, k1, and k2 are given

( , , , ) Ff t y y w

( , , , )f t y y w

1 2| ( , , , ) | ,| ( , , , ) |f t y y w k f t y y w k

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A Common Design Objective

Make y follow a reference signal, v, within a specified accuracy:

|v - y| < g(v,t), g(v,t) > 0 is given a special case: g(v,t) is a constant

0 1 2{ :| | || , | , | }V v v v vv v v

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Common Design Constraints

max max| || , |u uu u

u is “smooth” is “small”2

1

| ( ) |

t

t

u t dt

Hard constraints

Soft constraints

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• A dynamic system represented in s.s. as

• p is the parameter vector to be selected (tuned)

Controller C(p)

( , , , , )( , , , )

z p z u v v yu q z v v y

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Problem Formulation

• Does a controller C(p) exist that meets the

design objective subject to the constraints?

• If so, how to find it?

• If there is more than one solutions, what is the

optimal solution in practical sense?

• How to find such an optimal solution?

( , , , ) Ff t y y w Vv

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Where are we?

ObservationsConjecturesExperiments

• Theory?– formulate the problem

• Validation?

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Build A New Research Infrastructure• Practitioners/Researchers/Mathematicians

• Discover (both practitioners and theoreticians)

• Theorize– Prove stability and convergence– Generalized– Establish a new kind of theory

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

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Conclusions

• Think out of box:

controls as an experimental science

• Experiments lead to new methods

• From problems to methods to

theory

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A Nonlinear PID Application

Kd

SignalConditioning

TransientProfile

Kp

Ki

DC-DCConverter

2( 1)

s

s

s

1

OutputVoltagePWM

NPID CONTROLLER

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Load Disturbance Rejection Load increasing(3->36A)Load decreasing(36->3A)

PIPI

NPID NPID

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Output Voltage1.0V/div

Time: 2.0ms.div

PI

NPID

ADRC

TOADRC

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Discrete Time Optimal Control Law

1 2

0

1 2

20

02 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

ax y h y d

r sign a a dfst a

r 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

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73

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Other Nonlinear Feedbacks(not based on Lyapunov methods)

• Explore the use of nonlinear mechanisms

– Nonlinear feedback

– Nonlinear differentiation

– Nonlinear PID

– Discrete Time Optimal Control

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Nonlinear PID

• 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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Non-smooth feedback

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10 ( ),a

u e sign e e r y , 0,y w u w

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Non-smooth feedback

2 7

, 2 s i n ( 1 0 ) , 1 0 ( )a

y w u w t u e s i g n e

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Further gain experimentation

u = e

u = fal(e)

e

u

1 d -d

1

| | ( ), | | ,( ) 0

/ , | | ,

a

a

e sign e e dfal e d

e d e d

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A special case of NPD

• Time Optimal Control (TOC) of a double integral plant

• Solution obtained in the 60s for continuous plants

• Chattering problem

• Recent solution (DTOC) for discrete plant fundamentally resolved the chattering

• Used as a controller or a differentiator