Intelligent Robust Control Of Precision

Positioning Systems Using Adaptive Neuro

Fuzzy Inference System (ANFIS)

12/27/2011

By:

Ph.D. Candidate: Safanah M. Raafat

December 2011

1

Contents

Introduction

Literature Review

Objectives; Methodology, experimental results and Outcomes

Conclusions

The Contributions

Recommendations for Future Work

12/27/2011 2

Modern precision positioning systems. e.g. machine tools.

Systems requirements:

-Robust stability, -High speed ; high productivity, -High precision/accuracy, -High tracking performance -

Robustness to uncertainties.

Robust control strategies;

systematic way of dealing with uncertainties.

Conservative uncertainty weighting functions

Hybrid intelligent systems

In this work, the uncertainty model is developed via a specially constructed (ANFIS) that can estimate a non-conservative

uncertainty bound around a prefixed nominal model.

12/27/2011 3

Introduction

Problem Statement and Research Significance

Arbitrary uncertainty weighting function for synthesis

The uncertainty set is not validated

The performance weighting function is not automatically selected

The disturbance effect of crosstalk between the axes is not considered in the robust controller design

12/27/2011 4

The applied H∞ robust control for positioning

systems

Intelligent technique ANFIS

H∞ robust control Systematic

intelligent robust control scheme

Non-conservative uncertainty

bounds

Robust stability and performance

Validation of the uncertainty model

and robust controller

12/27/2011 5

Research Philosophy

Research Objectives

1

• Develop an intelligent identification of uncertainty bounds .

2

• Develop an optimized intelligent identification of the uncertainty bounds.

3 • Validate the resulted uncertainty set.

4

• Design a corresponding H∞ robust controller with further improvements.

5 • Investigate and experimentally verify the application of

the intelligent robust controller.

12/27/2011 6

Scope of the Research

Develop a Systematic practical methodology for intelligent estimation of uncertainty followed by H∞ robust controller design.

Develop a novel ANFIS based identification methodology of uncertainty bounds.

Accurate, non-conservative, relative simplicity of calculation, less computational time, and validated uncertainty weighting function.

Experimental application on different high precision positioning systems.

12/27/2011 7

Literature Review Controller type Specifications

Advantages Limitations

Robust controller

[e.g. Liu et. al., 2003;

Choi, Kim and Choi ,

2001; Liu, Luo and

Rahman , 2005; Sato,

Ishibe and Tsuruta

,2007; Larochea et.

al.(2004; Rijanto,

2000; Lee and

Salapaka, 2009]

Formulate the control

problem to include the

plant model

uncertainties :

QFT, H∞ loop shaping,

H∞ optimization and

LMI

1- Systematic treatment

of Uncertainties.

2- Suitable formulation

of the robust control

design problem.

1-The weighting

functions are selected

in a lengthy trial and

error procedure .

2-The uncertainty

weighting function

are not validated .

3-The interaction

between the axes was

not considered.

Intelligent robust

controller

[e.g. Yu and Tao, 2006;

Vagia, Nikolakopulos

and Tzes ,2006;

Yongjun et. al., 2008]

Evolve the sensitivity

and control weighting

functions,

Design an intelligent

pre-filter,

Neural observer

Large improvement in

tracking performance.

1- Long time of

evaluation.

2- Clear analysis of

the uncertainties and

resulted sensitivity,

robust stability are

not considered.

12/27/2011 linkviva.docx 8

Methodology

Implement the experimentally obtained data for estimation of uncertainty bound using ANFIS structure

Prepare ANFIS structure ; memberships, rules, clusters

Prepare experimental data using closed loop controlled system

Obtain a suitable nominal model for the system under study

12/27/2011 9

First Objective Develop an intelligent identification of uncertainty bounds for

robust controller design, using Adaptive Neuro Fuzzy Inference

System (ANFIS).

Outcomes from First Objective

Accurately reflects additive uncertainty associated with the identified model.

Considerably eliminate arbitrarily or time consuming trial and error procedure.

This method for identification gives accurate results for relatively small amplitude of input signals

First objective requirement is satisfied. 12/27/2011 10

First Objective

Experimental Results

10-4

10-3

10-2

10-1

-41

-40

-39

-38

-37

-36

-35

-34

-33

Ma

gn

itud

e (

dB

)

Bode Diagram

Frequency (rad/sec)

Anfis uncert. bound

Wa anfis

Methodology

12/27/2011

)),(),(()( 1 kkrfkeikfi jejGANFISjG

)j(G)j(GK)k(e kfkesrf )n,...,k(k 1

||)( erfrf eeJ

erf is the updating error

Stopping criteria

11

Second Objective Develop an optimized intelligent identification of the uncertainty bounds.

The purpose is to deal with more highly disturbed signals more efficiently

and easily.

Bode Diagram

Frequency (rad/sec)

10-4

10-3

10-2

10-1

-100

-80

-60

-40

-20

0

20

Ma

gn

itu

de

(d

B)

CIN

anfis 1

anfis 2

12/27/2011

Bode Diagram

Frequency (rad/sec)

10-4

10-3

10-2

10-1

-30

-20

-10

0

10

20

30

40

50

Ma

gn

itu

de

(d

B)

CIN

anfis 2

anfis 1

The identified intelligent weighting function Wa using square input signal

5V p-p, 1Hz for closed- loop identification,(a) without control, (b) with PV

control

(a) (b)

Second Objective

Experimental Results

12

Second Objective

Application of intelligent ANFIS estimation of Uncertainty on MIMO

System; Active Magnetic Bearings

12/27/2011

Learning

time (sec.)

v-gap Best objective

LMI

K-

LMI

Order -

Wa

N. o. I .No. o. I. LMI

CIN NNs 106.2426 0.707 1.998 [4,32] 4 100 47

ANFIS2 16.2154 0.707 1.997 [4,28] 3 20 44

10-5

10-4

10-3

10-2

10-1

100

-200

-190

-180

-170

-160

-150

-140

-130

-120

-110

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

wa1

wa2

wa3

wa4

• Similar accuracy to CIN neural network

•shorter learning time

•less number of iterations of training.

• lower order of the ANFIS estimated weighting function .

• lower order of the evaluated LMI robust controller. 13

Outcomes from Second Objective

Theorem

• Let ; and for a given model error estimation, which satisfies certain stopping criteria.

• Hence, the intelligent non-conservative uncertainty weighting function Wai obtained from the intelligent estimated uncertainty bounds is

• Optimized uncertainty bounds

The method can be efficiently implemented with highly disturbed signals.

The time is a bit longer than that of a simple ANFIS structure.

Second objective requirement is satisfied

12/27/2011 14

)(minmin krfrf ee ),...,1( nnkk ) ( minmin rffifi eGG

|G|W fiai min

Third Objective Validate the estimated uncertainty bound using v-gap metric

• The v-gap metric is the maximum distance between the

frequency response loci of two systems, a nominal model GN

and a perturbed model Gi, respectively, when plotted on the

surface of the Riemann sphere, whenever a certain winding

number/encirclement condition is satisfied (Vinnicombe,

1993).

12/27/2011 15

otherwise 1

0 if )G,G(W))e(G),e(G(kmax)G,G( iN

j

i

j

N

iNv

))e(C/),e(G(kminb jj

NC,GN

1

)G,G(b|C iNvC,GN

Outcomes From Third Objective -1

12/27/2011

v K,GNb

vK,GNb

Much smaller v-gap than in ANFIS1 identified

uncertainty bound. K,NGb

16

For ANFIS1 uncertainty set

Outcomes from Third Objective - 2

• The between v-gap metric and b is larger for ANFIS2.

Therefore, ANFIS2 is strongly preferred for robust control

design.

• Third objective requirement is satisfied.

12/27/2011

Square input

signal

20V p-p, 1Hz

Uncontrolled closed loop test signal Controlled closed loop test signal

v- gap b v- gap b

CIN 0.9986 0.4446 --- 0.6348 0.5379 ---

ANFIS 1 0.3193 0.3242 --- 0.2913 0.5359 0.2446

ANFIS2 0.0166 0.2117 0.1951 0.2896 0.6572 0.3676

vK,GNb

17

vK,GNb vK,GN

b

For ANFIS2 uncertainty set

Design an optimal H∞ controller that uses the developed

intelligent uncertainty weighting function.

+

u

y

r

z1

z2

Wu Wa

∆

GN - Wee

K1

N

u

eNee

a

GII

W

WGWW

W

)s(P00

00

The entire-connection of the

robustly -controlled system

1

TW

RW

SW

a

u

e

The performance

criterion

10-2

-70

-60

-50

-40

-30

-20

-10

Ma

gn

itu

de

(d

B)

Bode Diagram

Frequency (rad/sec)

Intelligent.unc.bnd

anfis 2

b

bs

es

MsW

/

bc

ubc

us

MsW

1

/

Fourth Objective

Methodology

12/27/2011

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

Time (sec.)

Dis

pla

ce

me

nt (d

eg

.)

Measured.sys.PEM

Simulated. sys.

Measured.sys.CIN

Measured. sys.Fuzzy

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

Time (sec.)

Dis

pla

ce

me

nt (d

eg

.)

Measured.PEM

Measured.Fuzzy

Measured .CIN

Reference

Fourth Objective

Experimental Results; ANFIS1

100

105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (rad/sec)

Mu

up

pe

r/lo

we

r b

ou

nd

s

Mu plot of robust stability margins (inverted scale)

Mu 1

Mu 2

19

Transient responses of

the controlled system

Robustness test

under load

µ Stability

Analysis

Fourth Objective

Experimental results, ANFIS2

12/27/2011

Closed loop step response of the motion

system using ANFIS2 from PV closed loop

controlled, and ANFIS2 from closed loop

uncontrolled system

0 1 2 3 4 50

5

10

15

20

25

30

35

Time (sec.)

Measu

red

Dis

pla

cem

en

t (d

eg

.)

Reference

Measur.disp.ANFIS2.sqr

Measur.disp.ANFIS2.sin

Measur.disp.ANFIS2.saw

0 1 2 3 4 50

5

10

15

20

25

30

35

Time(sec.)

Measu

red

Dis

pla

cem

en

t (d

eg

.)

Measur.disp. ANFIS2. contr.

Measur.disp. ANFIS2.uncontr.

Reference

Closed-loop step response of the motion

system using ANFIS2- Wa from closed-loop

uncontrolled system data.

20

12/27/2011

Bode Diagram

Frequency (rad/sec)

10-2

100

102

-100

-80

-60

-40

-20

0

20

Ma

gn

itud

e (

dB

)

1/|We(jw)|

|S| no control

|S| Hinf

Ta

Methodology

The constrained optimization problem:

Minimize f0 (x) = 1/|We (x)|

Subject to f1(x) ≤ Ta, , f2(x) ≤ σmax

Ta is the maximum allowable

tolerance between the sensitivity

function and the reciprocal of the

norm of the performance

weighting function. σmax is the maximum allowable

singular value of the closed-loop

controlled system.

b

bs

es

MsW

/

21

Fourth Objective –cont.

Formulate constrained optimization of the performance weighting

function

we

Fourth Objective- cont.

• Compensate for the effect of crosstalk between

the axes by developing an intelligent

disturbance weighting function.

12/27/2011

102

0

1

2

Mag

nit

ud

e (d

B)

102

0

1000

2000

3000

Frequency (Hz)

Ph

ase

(deg

rees

)

Gyx

Gxy

y

x

yydy

dxxx

y

x

u

u

GG

GG

y

y

22

Fourth Objective- cont.

Methodology

12/27/2011

ijj

i

di)j(U

)j(Y)j(G

ji,,j,,i 2121

The intelligent disturbance weighting function will be used

in a disturbance rejection - robust control synthesis of the X-

Y positioning system.

23

Intelligent estimation of the disturbance weighting function,

using ANFIS

|Gdi(jω)| ANFIS

|Yx(jω)|

|Uy(jω)|

|Gdi(jω)|

ANFIS

Yi(t)

Uj(t)

Frequency (jω)Time to

Frequency

Conversion

Time to

Frequency

Conversion

/ +

Fourth Objective

Experimental Results

12/27/2011

10-2

-90

-80

-70

-60

-50

-40

-30

Mag

nit

ud

e (d

B)

Bode Diagram

Frequency (rad/sec)

|Wdx(jw)|

ANFIS. bnd.

10-2

-70

-60

-50

-40

-30

-20

-10

0

10

Mag

nitu

de (d

B)

Bode Diagram

Frequency (rad/sec)

ANFIS bnd.

|Wdy(jw)|

Estimated ANFIS disturbance and evaluated

disturbance weighting function Wd

24

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

Time (sec.)

Pos

ition

(Mm

)

Wa ANFIS

We opt. added

0.1 0.15 0.2

420

440

460

480

500

520

Time (sec.)

Pos

ition

(Mm

)

Wa ANFIS

We opt. added

Transient Response of the robust

controlled system using optimized We

Outcomes From Fourth Objective

An H∞ robust controller is designed using the intelligently estimated uncertainty weighting function.

A new method based on constrained optimization is formulated to tune the performance weighting function.

The intelligent disturbance weighting functions accurately reflect the disturbing effect caused by crosstalk.

Fourth objective requirements are achieved.

12/27/2011 25

The Fifth Objective

Investigate and experimentally verify the robust stability and performance of two different positioning systems:

• The Single Axis Positioning System.

• Two Axes Positioning System.

In order to guarantee precise reference tracking two different control schemes are developed;

• A specially designed integral controller augmented with the closed-loop robust control system.

• Two Degree Of Freedom H∞ (2-DOF H∞) robust control configuration.

12/27/2011 26

12/27/2011 27

The Fifth Objective

Methodology

Proposed Algorithm for Intelligent Robust Control

Estimate and validate the

system’s nominal model

Select the control weighting

function Wu

Select the performance

weighting function We

Estimate the intelligent

uncertainty bound

Estimate the disturbance weighting

function Wd

Design of H∞ controller

Practical implementation Optimize We

Start

End

Validation of

uncertainty

bound

The Fifth Objective

Experimental Application of Intelligent Robust Control on

the Single Axis Positioning System

Target

PC

Host

PC

Single

Axis

Table

DC Servo

motor

Increment

al encoder

The nominal model is

J lp M

B1 B2

xRotational motion Linear motion

θω

Tm Flx.

)t(u(.)fxlKK

lBBx

lKK

MlJ

pTa

p

pTa

p

2

21

2

ssG

9401.29

0928.13920

12/27/2011 28

The Identified Transfer Function of the

Nominal Model (μm/V)

10-2

-140

-120

-100

-80

-60

-40

-20

Ma

gn

itud

e (

dB

)

Bode Diagram

Frequency (rad/sec)

ANFIS Unc.Bound

|Wa(jw)|

e u2 y

ei

r

s

K I

1K iG

+

+

--

]))(())((

[)( 12s

Kssy

s

KssrKsu II

H∞ can provide a robust stability and performance, but for the reference tracking there must be an additional control action.

s

GKKGKssS iIi 111

I

i

i

I

KsGK

GK

KssF

1

11

The Fifth Objective

Methodology

Modified Integral Robust Tracking Control Scheme

12/27/2011 29

The Fifth Objective

Methodology

2 DOF H∞ Controller Synthesis

1- Solve the stability

problem.

2- Design the second H∞

controller, based on the

stabilized system to get an

improved robust

performance.

12/27/2011 30

K2 K1- G(s)- -

r e1 u1 yu2e2

K1 Gs(s)- Ws(s)-

r e1 u1 yyp

The Fifth Objective

Experimental results using Integral- H∞ and 2-DOF

H∞ robust controllers.

0 2 4 6 8 10-200

0

200

400

600

800

1000

1200

Time (sec.)

Pos

itio

n (M

icro

.m)

2Hinf

Reference

I-Hinf

12/27/2011 1.6 1.8 2 2.2 2.4 2.6

840

860

880

900

920

940

960

980

1000

1020

1040

Time (sec.)

Posi

tion

(M

icro

.m)

2Hinf

Reference

I-Hinf

0 2 4 6 8 10-30

-20

-10

0

10

20

30

Time (sec.)

Po

siti

on

(M

icro

.m)

2Hinf.

I-Hinf

Tra

ckin

g E

rror

(µm

)

31

The Fifth Objective

Experimental results

2-DOF H∞ and integral- H∞ robust controllers.

0 2 4 6 8 10-1000

-500

0

500

1000

Time (sec.)

Posi

tion (

Mic

ro.m

)

Reference

2 Hinf

I-Hinf

12/27/2011 0.8 1 1.2 1.4 1.6 1.8

250

300

350

400

450

Time (sec.)

Reference

2 Hinf

I-Hinf

0 2 4 6 8 10-30

-20

-10

0

10

20

30

Time (sec.)

Pos

itio

n (M

icro

.m)

2Hinf

I-Hinf

Tra

ckin

g E

rror

(µm

)

32

The Fifth Objective

Experimental results

Intelligent Robust Control on X-Y Positioning System

X mechanism

Y mechanism

Parametric identification is applied

for each axis in order to obtain the

following nominal models :

).s(s

.Gxx 3

4

1035711

1083786

).s(s

.Gyy

2860154

1081317 3

y

x

yydy

dxxx

y

x

u

u

GG

GG

y

y

12/27/2011 33 10

-2.710

-2.510

-2.310

-2.1

-90

-80

-70

-60

-50

-40

Mag

nit

ud

e (d

B)

Bode Diagram

Frequency (rad/sec)10

-2

-70

-60

-50

-40

-30

-20

-10

Mag

nit

ud

e (d

B)

Bode Diagram

Frequency (rad/sec)

The Fifth Objective

Intelligent Robust control of two axes

positioning system

12/27/2011

+ y z1

z2

We Wa

∆

GN We

K

- e

r

Wd

+

u

34

Disturbance rejection - robust

control synthesis

12/27/2011 35

The Fifth Objective

Experimental Results

Intelligent Robust control of two axes positioning system

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

X-Axis Position (mm)

Y-A

xis

Po

siti

on

(m

m)

Tracking Rersponse of CircleContour

Circle Contoure

-5 0 5-5

0

5

X-Axis (mm)

Y-A

xis

(m

m)

Reference Contour

Tracking Response

The circle contour The rhombus shape

The Fifth Objective

Experimental results:

The circle contour, tracking and contour errors

0 1 2 3 4 5-2

-1

0

1

2x 10

-3

Time (sec.)

Tra

ckin

g E

rror

(mm

)

Ey

Ex

1 2 3 4 5-1

-0.5

0

0.5

1x 10

-3

Time (sec.)

Con

tour

Err

or (

mm

)

Tracking Errors Contour Error

12/27/2011 36

Outcomes from Fifth Objective

Experimental demonstrations validate the benefits of each robust control configuration

• The 2-DOF H∞ scheme can achieve less tracking error. (more sensitive and starting oscillations may be developed).

• The Integral-H∞ scheme can provides good tracking (proper selection of the integral gain).

Fifth objective requirements are achieved.

12/27/2011 37

Conclusions

Two ANFIS schemes within MEM framework.

Validation of the resulted intelligent uncertainty weighting function for robust controller design. Successful application of the proposed ANFIS estimation algorithm for more complicated MIMO system.

Optimized performance weighting function.

An intelligent disturbance weighting function .

Different control schemes were developed for practical applications.

Problem statements were addressed.

12/27/2011 38

The Contributions

The development of an efficient, systematic and practical estimation of reduced onservativeness uncertainty bound.

• ‘Intelligent robust control design of a precise positioning system’, International Journal of Control, Automation, and Systems, Vol.8, No.5, Oct., 2010, DOI 10.1007/s12555-010-0521-0 .

• ‘Improved intelligent identification of uncertainty bounds; design, model validation and stability analysis’, International Journal of Modelling, Identification, and Control-special issue: Neural network and fuzzy logic for modelling ad control of mechatronic system, in Press, 2010, ISSN (Online): 1746-6180 - ISSN (Print): 1746-6172.

The development of constrained optimization procedure.

• “Bounded Constrained Optimization of Performance Weighting Function for Precise Robust Positioning Control System “, 2011 4th International Conference on Mechatronics (ICOM), 17-19 May 2011, Kuala Lumpur, Malaysia.

• ‘Enhanced Servo Performance of a single Axis Positioning System in an Intelligent Robust Framework’, IEEE International Symposium on Intelligent Control, Yokohama, Japan, September 8-10,2010, p.2450-2455, ISBN:978-1-4244-5361-0.

12/27/2011 39

The Contributions- cont.

The development of the intelligent disturbance weighting function .

• ‘Intelligent Disturbance Rejection for Robust Tracking Performance of X-Y Positioning System’, Proc. Of the IEEE Int. Conf. on Mechatronics and Automation, August 4-7, 2010, Xi’an China, pp. 252-257, ISBN: 978-1-4244-5141-8.

The systematic approach of robust controller design is applied efficiently to a single axis and X-Y positioning systems.

• ‘Design of Robust H∞ Controller for Precise Positioning System’, submitted to journal of control theory and applications.

• ‘Intelligent Robust Control for Precise Tracking Performance of X-Y Positioning System’, submitted to Journal of Intelligent systems and robotics.

12/27/2011 40

Recommendations for Future Work

Including iterative Learning Control (ILC).

Application with other control methods, e.g. Sliding Mode control and nonlinear control.

The implementation of another type of actuators like piezoelectric actuator.

Exploring the identification of the whole nonlinear system using ANFIS or approximation using Takagi-Sugeno (T-S) Fuzzy modelling.

12/27/2011 41

Thanks For Listening

12/27/2011 42