ball and beam pid

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ISSN0280-5316

ISRNLUTFD2/TFRT--5736--SE

Modeling and Control of the

Ball and Beam Process

Marta Virseda

Department of AutomaticControl

Lund Institute of Technology

March2004



Department of Automatic Control

Lund Institute of Technology

Box 118

SE!!1 "" Lund S#eden

Document name

MASTER THESIS

Date of issue

March 2004

Document Number

ISRNLUTFD2/TFRT--5736--SE

Author(s)

Marta Virs!aSupervisor

R"#$ %"ha&ss"& a&! A&!rs R"'rtss"& at LTH i&L(&!) E&ri*( +a,&s at U&i) ! Va##a!"#i! i& S.ai&

Sponsoring organization

Title and subtitle

M"!#i& a&! "&tr"# "$ th +a## a&! +a1 r"css M"!##ri& si1(#ri& "ch r#ri& a (#a . '"1 .r"css&)

Abstract

One of the most difficult problems that an engineer who works with modeling

deals with, is the question about how to translate a physical phenomenon into a

set of equations. It is usually difficult to capture all dynamics and phenomena,

so one usually strives for a set of equations that describes the physical system

approximately and adequately with the accuracy for the purpose. In our case, we

model the dynamics relevant for control design.

The topic of this thesis was to do an in-depth study of the all and eam

process. Two different experimental implementation of the all and eamprocess have been considered, both available at the course lab at the

!epartment of "utomatic #ontrol, $und. The first step consisted of deriving

the equations of motion, that is, to do the mathematical modeling of the

process.

In order to implement this model %odelica has been used. %odelica, which is a

powerful language for modeling of physical systems, uses the tool !ymola.

"nother model was designed also with %odelica but with the help of the extension

of the multi body library, which uses a methodology based on ob&ect orientation

and symbolic manipulation of equations. 'ith this last model it was possible to

visuali(e an animation in real time )!.

The following step of the pro&ect was to do control design for the

different models. The obtained simulations were shown in !ymola and

*imulink.

+inally experiments on the real process were developed, based on vision feedback.

Keywords

M"!#i& c"&tr"# "'8ct-"ri&t! 1"!#i&

lassification system and!or inde" terms (if any)

Supplementary bibliographical information

#SSN and $ey title

0290-53:6 #S%N

&anguage

E&#ish Number of pages

47 'ecipients notes

Security classification

The report may be ordered from the Department of Automatic ontrol or borrowed through*niversity &ibrary+ %o" ,+ S-.//0 11 &und+ Sweden 2a" 345 45

/// 4/ 4,



Ac$no#ledgments

This ;"r has '& acc"1.#ish! at th D.art1&t "$ A(t"1atic "&tr"# at L(&!

I&stit(t "$ Tch&"#", i& S;!& 't;& <ct"'r 2003 a&! March 2004)

I&itia##, I ;"(#! #i t" =.rss 1, ratit(! t" Dr) R"#$ %"ha&ss"& $"r ii& th

."ssi'i#it, t" ;"r i& this !.art1&t (&!r his s(.risi"&)I ;ish t" =.rss 1, si&cr ratit(! t" A&!rs R"'rtss"& ;h" (i!! this thsis a&!

h#.! ;h&r I ;as i& &!) H h#.! 1 ;ith 1, !"('ts a&! *(sti"&s)

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1, ra&!.ar&ts ;h" ha '& &r"(s ;ith thir &c"(ra1&t)

This thsis is !!icat! t" th1)



TABLE %& C%'TE'TS

Introduction((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 1

Dymola and )odelica ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( !

!(1 Dymola ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( !2):): D,1"#a $at(rs )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 3

2):)2 Th 1"!#i& &ir"&1&t ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 4

!(! )odelica((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( *2)2): M"!#ica $at(rs)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 5

!(+ ,o# to learn Dymola and )odelica((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 8

Ball and Beam ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( -

+(1 .rocess Description ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( -

)athematical modeling (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 1!

*(1 Ball and Beam )odel(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 1!4):): N;t"&>s sc"&! #a; )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) :24):)2 Lara&ia& 1th"!s)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) :4

)odeling in )odelica ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 1-

/(1 )odels #ith )odelica language (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 1-5):): M"!#ica 1"!#? N;t"&>s sc"&! #a; )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 205):)2 M"!#ica 1"!#? Lara&ia& 1th"! ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 2:

/(! )odel #ith )odelica0s liraries (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( !+

Control Design and Simulations (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( !2

2(1 Control design ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( !26):): "&tr"# !si&? #i&ar "'srr-'as! c"&tr"# )))))))))))))))))))))))))))))))))))))))))))))))))))))) 26

6):)2 "&tr"# Dsi&? &"&#i&ar s,st1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 29

2(! Simulations (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( +16)2): Si1(#ati"&s i& Si1(#i&)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 3:

Experiments((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( ++

3(1 Experimental Setup (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( ++

3(! ,o# to do the experiments((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( +*

Conclusions (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( *!

8(1 Conclusions (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( *!



:

Chapter 1

Introduction

This thsis c"&c&trats "& th st(!, "$ th +a## a&! +a1 .r"css) Th 1"!#i& has '&!"& ;ith M"!#ica a&! D,1"#a)

D,1"#a is a& i&trat! &ir"&1&t $"r !#".i& 1"!#s i& th M"!#ica #a&(a a&!

a si1(#ati"& &ir"&1&t $"r .r$"r1i& =.ri1&ts)T" !" th c"&tr"# !si& thr ar 1a&, ."ssi'i#itis@ this .r"8ct .rs&ts a& "'srr-

'as! c"&tr"# !si& !#".1&t i& Mat#a' 'as! "& ."#-.#ac1&t) Th si1(#ati"&s ar 1a!

i& Si1(#i& a&! D,1"#a) Th =.ri1&ts ;ith th ra# +a## a&! +a1 .r"css ar !"& ;ith th

c"&tr"##r i1.#1&t! i& a ra#-ti1 =t&si"& t" Si1(#i& a&! ias(a# $!'ac $r"1 a ca1ras,st1)

A sh"rt !scri.ti"& "$ th thsis cha.tr is "(t#i&! '#";?

Chapter ! .rs&ts a& =.#a&ati"& "$ !i$$r&t t""#s (s! $"r th 1"!#i&)

Chapter + !scri's th st(!, "$ th +a## a&! +a1 .r"css)

Chapter * .rs&ts th 1ath1atica# 1"!#i& "$ th .h,sica# s,st1) Chapter

/ is th 1"!#i& i& M"!#ica "$ th 'a1 a&! 'a## .r"css( Chapter 2

c"&tai&s c"&tr"# !si& "$ th "'tai&! 1"!#s a&! th c"rrs."&!i&si1(#ati"&s)

Chapter 3 c"&tai&s th =.ri1&ts !"&)

Chapter 8 .rs&ts th s(11ari! c"&c#(si"&s "$ this thsis)



Chapter !

Dymola and )odelica

This cha.tr is a sh"rt i&tr"!(cti"& t" th !i$$r&t t""#s (s! $"r th 1"!#i& i& th

thsis)

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I& :CC6 D,&asi1 t"" th i&itiati t" !si& a &; (&i$i! "'8ct-"ri&t! #a&(a $"r

.h,sica# s,st1s 1"!#i& ca##! M"!#ica) F"(&!! i& :CC2 D,&asi1 is th #a!i& !#".r

a&! i1.#1&t "$ "'8ct-"ri&t! 1(#ti-&i&ri& 1"!#i& tch&"#", a&! th M"!#ica#a&(a 2)

D,1"#a - D,&a1ic M"!#i& La'"rat"r, - is a t""# $"r 1(#ti-!"1ai& 1"!#i& a&!si1(#ati"&) D,&asi1 !#".! it) D,1"#a s(.."rts th 1"!#i& #a&(a M"!#ica $"r ;hich

thr ar a &(1'r "$ $r a&! c"11rcia# 1"!# #i'raris that i&c#(! 1"!#s "$ 1cha&ica#

thr1a# h,!ra(#ic a&! thr1"!,&a1ic a&! c"&tr"# s,st1s) 4

D,1"#a (ss a &; 1"!#i& 1th"!"#", 'as! "& "'8ct "ri&tati"& a&! s,1'"#ic

1a&i.(#ati"& "$ *(ati"&s) Th (s(a# &! $"r th 1a&(a# c"&rsi"& "$ *(ati"&s t" a '#"c !iara1 is r1"! ', th (s "$ a(t"1atic $"r1(#a 1a&i.(#ati"&)



ha.tr 2) D,1"#a a&! M"!#ica

3

!(1(1 Dymola features4

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A(t"1atic ca(sa#it, a&a#,sis

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¾ Ha&!#s #ar c"1.#= 1(#ti-!"1ai& 1"!#s

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"&&ct ra.hica##, +(i#! ic"&s - ."#,"&s circ#s t=t c"#"r tcG T=t !it"r $"r !c#arati"&s a&! *(ati"&s A(t"1atic HTML 1"!# !"c(1&tati"&

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¾ A&i1ati"&

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¾ Si1(#at"r

Ha&!#s <DE a&! DAE 1"!#s Stat "r th art &(1rica# i&trati"& F#=i'# i&itia#iati"& I&tr$ac t" =tr&a# -$(&cti"&s Mat#a'/Si1(#i& = a& ! SAE i&tr$acs DDE i&tr$ac Ra# ti1 si1(#ati"& :



!(1(! The modeling en5ironment

D,1"#a is a& i&trat! &ir"&1&t $"r !#".i& 1"!#s i& th M"!#ica #a&(a a&!

a si1(#ati"& &ir"&1&t $"r .r$"r1i& =.ri1&ts)

Th D,1"#a 1"!#i& &ir"&1&t is !ii!! i& t;" 1ai& .arts?

¾ At th 1"!#i& 1"! ca##! M"!# E!it"r ;hr th 1"!#s ar c"1."s! $r"1

#i'rar, c"1."&&ts $r"1 th M"!#ica sta&!ar! #i'rar, "thr #i'raris a&!

c"11rcia# a&! .r".ritar, #i'raris as ;## as 1"!#s !#".! ', th (sr) M"!#sar ithr c"1."s! "$ "thr 1"r .ri1iti c"1."&&ts "r !scri'! ', *(ati"&s at

th #";st ##)

This 1"! a##";s?

M"!# c"1."siti"& D$a(#t .ara1tr stti&s D$i&iti"& "$ *(ati"&s

¾ At th si1(#ati"& 1"! D,1"#a tra&s$"r1s a !c#arati *(ati"& 'as! 1"!#!scri.ti"& i&t" $$ici&t c"!) I& this ## it is ."ssi'# sh"; th rs(#ts as a&i1ati"&s

"r (si& .#"t ;i&!";s $"r is(a#iati"& "$ si1(#ati"& rs(#ts)

D,1"#a .r"i!s a c"1.#t si1(#ati"& &ir"&1&t '(t ca& a#s" =."rt c"! $"r

si1(#ati"& i& Si1(#i&) I& a!!iti"& t" th (s(a# "$$#i& si1(#ati"& D,1"#a ca& &rat c"! $"r

s.cia#i! har!;ar-i&-#"". s,st1s s(ch as !SAE = a&! "thrs 5)

D,1"#a (ss M"!#ica ;hich is a& "'8ct "ri&tat! 1"!#i& #a&(a ;hich s(.."rts

hirarchica# str(ct(ri& r(s a&! "#(ti"& "$ #ar a&! c"1.#= 1"!#s i&!.&!&t $r"1 th

a..#icati"&) It (ss aca(sa# 1"!#i& 'as! "& !i$$r&tia# a&! a#'raic *(ati"&s)

!(! )odelica

M"!#ica 3 is a .";r$(# #a&(a $"r 1"!#i& "$ .h,sica# s,st1s !si&! t" s(.."rt$$cti #i'rar, !#".1&t a&! 1"!# =cha&) It is a 1"!r& #a&(a '(i#t "& aca(sa#

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1"!#i& &";#!)

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;ith &";#! a'"(t 1"!#i& #a&(as a&! 1"!#s ;ith !i$$r&tia# a#'raic *(ati"&s

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%a&(ar, 2002 a &; rsi"& ;as r#as!)



!(!(1 )odelica features

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c"&t=t)

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thr1"!,&a1ic h,!ra(#ic 'i"#"ica# a&! c"&tr"# a..#icati"&s ca& ' !scri'! a&!c"&&ct!)

¾ Is a& "'8ct-"ri&t! #a&(a ;ith a &ra# c#ass c"&c.t that (&i$is c#asss

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"$ c"1."&&ts a&! "#(ti"& "$ 1"!#s)

¾ Has a str"& s"$t;ar c"1."&&t 1"!# ;ith c"&str(cts $"r crati& a&! c"&&cti&c"1."&&ts) Th(s th #a&(a is i!a##, s(it! as a& architct(ra# !scri.ti"&

#a&(a $"r c"1.#= .h,sica# s,st1s a&! t" s"1 =t&! $"r s"$t;ar s,st1s)

M"!#s a&! s('-1"!#s ar !c#ar! as c#asss ;ith c"&&cti"& i&tr$acs ca##!

connectors) This c"&&cti"& ca.a'i#it, a##";s th (s "$ 1"!# #i'raris t" c"1."s c"1.#=

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that 1as #i$ $"r th 1"!# '(i#!rs c"&si!ra'#, asir a&! 1"r .r"!(cti)

F"##";i& ar !scri'! th &ra# $at(rs "$ M"!#ica)

,ierarchical modeling

M"!#ica s(.."rts '"th hih ##s 1"!#i& ', c"1."siti"& a&! !tai#! #i'rar,

c"1."&&t 1"!#i& ', *(ati"&s) M"!#s "$ sta&!ar! c"1."&&ts ar t,.ica##, aai#a'# i&

1"!# #i'raris) Usi& a ra.hica# 1"!# !it"r a 1"!# ca& ' !$i&! ', !ra;i& ac"1."siti"& !iara1 ', ."siti"&i& ic"&s that r.rs&t th 1"!#s "$ th c"1."&&ts !ra;i&

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a&&"tati"&s i& M"!#ica 1a ic"&s a&! c"1."siti"& !iara1s ."rta'# 't;& !i$$r&t t""#s2)



A& =a1.# "$ th t=t(a# r.rs&tati"& "$ a si1.# 1"t"r !ri s,st1 is?

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c"&&ctar)$#a&Ja 1"t"r)$#a&J'@

c"&&ctar)$#a&J' i&rtia)$#a&Ja@&! M"t"rDri@

Th c"1."siti"& !iara1 "$ th 1"!# c#ass M"t"r is sh";& '#";?

Fi(r 2):? M"t"r !ri ;ith M"!#ica #i'raris



Structure model liraries

A .";r$(# .aca c"&c.t is aai#a'# t" str(ct(r #ar 1"!# #i'raris a&! t" $i&! a

c"1."&&t i& a $i# s,st1 ii& its hirarchica# M"!#ica c#ass &a1)

,yrid modeling

A (&i*( $at(r "$ M"!#ica is th ha&!#i& "$ !isc"&ti&("(s a&! aria'# str(ct(rc"1."&&ts s(ch as r#a,s s;itchs 'ari& $ricti"& c#(tchs 'ras i1.act sa1.#! !ata

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a si1(#at"r t" i&tr"!(c $$ici&t ha&!#i& "$ &ts &!! i& s(ch cass)

Array

M"!#ica s(.."rts arra,s (ti#ii& a Mat#a' #i s,&ta=) Th #1&ts "$ arra,s 1a, '

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Class parameters

+si!s "r!i&ar, &(1ric .ara1trs M"!#ica a##";s 1"!# c#ass .ara1trs) As a&

=a1.# ass(1 that a& a(t"-t(&i& c"&tr"##r sh"(#! r.#ac a I c"&tr"##r) It is "$ c"(rs

."ssi'# t" 8(st r.#ac th c"&tr"##r i& a ra.hica# (sr &ir"&1&t i)) t" crat a &; 1"!#)Th .r"'#1 ;ith this s"#(ti"& is that t;" 1"!#s 1(st ' 1ai&tai&!) M"!#ica has th

ca.a'i#it, t" i&sta! s('stit(t a 1"!# c"1."&&t s" "&#, "& rsi"& "$ th rst "$ th 1"!# is

&!!)

E6uations

M"!#s i& M"!#ica ar 1ath1atica##, !scri'! ', !i$$r&tia# a#'raic a&! !iscrt

*(ati"&s) N" .artic(#ar aria'# &!s t" ' s"#! 1a&(a##,) A M"!#ica t""# ;i## ha &"(h

i&$"r1ati"& t" !ci! it a(t"1atica##,) M"!#ica is !si&! s(ch that aai#a'# s.cia#i!a#"rith1s ca& ' (ti#i! t" &a'# $$ici&t ha&!#i& "$ #ar 1"!#s hai& 1"r tha& h(&!r!

th"(sa&! *(ati"&s) M"!#ica is s(it! a&! (s! $"r har!;ar-i&-th-#"". si1(#ati"&s a&! $"r 1'!!! c"&tr"# s,st1s)



!(+ ,o# to learn Dymola and )odelica

Th $irst tas "$ this thsis ;as #ar&i& D,1"#a a&! th M"!#ica #a&(a) I& 1,

c"1.(tr rsi"& 5): "$ D,1"#a ;as i&sta##! aai#a'# as $r s"(rc c"!) M"st "$ th

=isti& t(t"ria#s ar 'as! "& =a1.#s 'si!s I (s! sra# t(t"ria#s a&! '""s s"1 "$ th1;r?

¾ Th (i! Ktti& Start! ;ith D,1"#a : i&c#(!! i& D,1"#a (sr>s 1a&(a# ;hich

.r"i!s s"1 =a1.#s i& "r!r t" (i! ,"( thr"(h D,1"#a) F"r !tai#!i&$"r1ati"& a'"(t th .r"ra1 ,"( ca& c"&s(#t th "&-#i& !"c(1&tati"& aai#a'# i&

th H#. 1&( a$tr s#cti& D"c(1&tati"&)

¾ Th '"" KI&tr"!(cti"& t" h,sica# M"!#i& ;ith M"!#ica 6 ;hich i&tr"!(cs t"

th M"!#ica 1"!#i& #a&(a a&! sh";s th (s th .";r$(# $at(rs "$ thM"!#ica #a&(a)

¾ A t(t"ria# .aca ca##! K+i&&r t(t"ria# 7 ;hich c"&tai&s =rciss) Ths=rciss ar ""! t" #ar& t" ;rit si1.# 1"!#s th (s th M"!#ica sta&!ar!

#i'rar, tcG

¾ 6odelica tutorial document a&! 6odelica specification document+ '"th !$i& th

M"!#ica #a&(a 9)

¾ Advanced Tutorial is a c"&ti&(ati"& "$ +i&&r t(t"ria# C)



C

Chapter +

Ball and Beam

This cha.tr is a& i&tr"!(cti"& t" th .r"css "$ th 'a## r"##i& "& th 'a1)

+(1 .rocess Description

Th 'a## a&! 'a1 s,st1 is "& "$ th 1"st &!(ri&#, .".(#ar a&! i1."rta&t #a'"rat"r,1"!#s $"r tachi& c"&tr"# s,st1s &i&ri&) Th 'a## a&! 'a1 is ;i!#, (s! 'ca(s it is

r, si1.# t" (&!rsta&! as a s,st1 a&! ,t th c"&tr"# tch&i*(s that ca& ' st(!i! it c"r

1a&, i1."rta&t c#assica# a&! 1"!r& !si& 1th"!s) It has a r, i1."rta&t .r".rt,@ it is ".&

#"". (&sta'#)

Fi(r 3):? Diara1 "$ +a## a&! +a1



ha.tr 3) +a## a&! +a1

Th s,st1 is r, si1.# a st# 'a## r"##i& "& th t". "$ a #"& 'a1) Th 'a1 is

1"(&t! "& th "(t.(t sha$t "$ a& #ctrica# 1"t"r a&! s" th 'a1 ca& ' ti#t! a'"(t its c&tr

a=is ', a..#,i& a& #ctrica# c"&tr"# si&a# t" th 1"t"r a1.#i$ir)

Th c"&tr"# 8"' is t" a(t"1atica##, r(#at th ."siti"& "$ th 'a## "& th 'a1 ',cha&i& th a&# "$ th 'a1) This is a !i$$ic(#t c"&tr"# tas 'ca(s th 'a## !"s &"t sta, i&

"& .#ac "& th 'a1 '(t 1"s ;ith acc#rati"& that is a..r"=i1at#, .r"."rti"&a# t" th ti#t"$ th 'a1) I& c"&tr"# tr1i&"#", th s,st1 is ".& #"". (&sta'# 'ca(s th s,st1 "(t.(t

th 'a## ."siti"& i&crass ;ith"(t #i1it $"r a $i=! i&.(t 'a1 a&#) F!'ac c"&tr"# 1(st

' (s! t" sta'i#i th s,st1 a&! t" . th 'a## i& a !sir! ."siti"& "& th 'a1)

T;" !i$$r&t i1.#1&tati"&s "$ th +a## a&! +a1 .r"css ha '& c"&si!r! '"th

aai#a'# at th c"(rs #a' "$ th D.art1&t "$ A(t"1atic "&tr"# s $i(r 3)2 a&! 3)3)

Fi(r 3)2? +a## a&! +a1 c"&&ct! ia ar'"=

Th 'a1 is c"&&ct! t" a #"cit, c"&tr"##! D-1"t"r ia a ar'"=) This 1as it

1(ch asir t" c"&tr"# th .r"css a&! thr is r, #itt# i&$#(&c "$ th cr"ss-c"(.#i&s $r"1th 'a## t" th 'a1)



Fi(r 3)3? +a## a&! +a1 !irct#, c"&&ct! t" th a=is "$ a D-1"t"r

Dirct !ri& 'a1? Th 'a1 is !irct#, c"&&ct! t" th a=is "$ a D-1"t"r)

Th 'a1 a&# is 1as(r! '(t &"t th a&(#ar #"cit,) Th 'a## ."siti"& &!s a#s" t" '

1as(r! a&! i& th sta&!ar! c"&$i(rati"& "$ th .r"css at th D.art1&t "$ A(t"1atic"&tr"# th 1ta# 'a## c#"ss a& #ctrica# circ(it a#"& th rai# "$ th 'a1)

I& "(r =.ri1&ta# st(. s cha.tr 7 ; ha (s! a isi"& s,st1 $"r !tr1i&i& th

'a## ."siti"&) Th s.! "$ th 'a## &!s t" ' sti1at!)



:2

Chapter *

)athematical modeling

This cha.tr !scri's h"; t" "'tai& th *(ati"&s "$ 1"ti"& $"r th +a## a&! +a1

.r"css)

*(1 Ball and Beam )odel

Th *(ati"&s $"r "'tai&i& th 1"!# i& M"!#ica ha '& ra#i! ', 1a&s "$ t;"

!i$$r&t 1th"!s) <& "$ th1 is !ri! ;ith si1.# 1ath1atic *(ati"&s (si& N;t"&>s

sc"&! #a; a&! a&"thr thr"(h th (s "$ th Lara&ia& Mth"!)

*(1(1 'e#ton0s second la#

I& this 1"!# ; $irst c"&si!r "&#, th r#ati"& 't;& th 'a1 a&# a&! th ."siti"& "$ th 'a##)

α =



ha.tr 4? Math1atica# 1"!#i&

Th .ara1trs "$ th 'a## a&! 'a1 ar !$i&! as $"##";s?

α Î +a1 a&# c""r!i&atL Î +a1 L&th

1Î Mass "$ th 'a##RÎ Ra!i(s "$ th 'a##% Î +a##>s 1"1&t "$ i&rtiaÎ raitati"&a#acc#rati"& = Î "siti"&"$ th 'a##

N#cti& $ricti"&a# $"rcs th t;" $"rcs i&$#(&ci& th 1"ti"& "$ th 'a## ar?

- Ft= F"rc !( t" tra&s#ati"&a# 1"ti"&

- Fr= F"rc !( t" 'a## r"tati"&

Tra&s#ati"&a#?

d2 "

M "M =dt

2

2 t"

= m ⋅

M "M

(4):)

T"r*( !( t" th 'a## r"tati"& is?! ⎜

v '⎟ ! 2 ⎜

"⎟

⎛ ⎞ ⎛ ⎞T F R %⋅ ⋅

dw ' % ⋅

⎝ ' % ⋅

' ⎠

7=⋅

r

2 r"

r"dt

= 7

⋅

M "M '

2

⎠ ⎝dt dt

2 '

(4)2)



Th 1"1&t i&rtia "$ th 'a## is s.hr?

7 =2

⋅ m ⋅ '2

5

(4)3)

S('stit(ti& th *(ati"& 4)3 i&t" th *(ati"& 4)2 ; t?

2 r"

= 2⋅ m ⋅ M "M

5

N;t"&>s sc"&! #a; a#"& th i&c#i&ati"&?

2 r"

+ 2 t"

= m ⋅ g ⋅ si&α (4)4)

S('stit(ti& i& th *(ati"& 4)4?

2⋅ m ⋅ M "M + m ⋅ M "M = m ⋅ g ⋅ si&α

5

I& c"&c#(si"& ; ha?

M "M =5

⋅ g ⋅ si&α7

I& this !riati"& ; ha ta& i&t" c"&si!rati"& h"; th 'a## ."siti"& a$$cts th a&#)

*(1(! Lagrangian methods

Th c"&tr"# "'8cti is t" c"&tr"# th t"r*( τ a..#i! at th .i"t "$ th 'a1 s(chthat th 'a## ca& r"## "& th 'a1 a&! trac a !sir! tra8ct"r,) Th t"r*( ca(ss th(s acha& "$ th 'a1 a&# a&! a 1"1&t i& th ."siti"& "$ th 'a##)

τ O '

r '



Th .ara1trs "$ th 'a## a&! 'a1 ar !$i&! as $"##";s?

I ' Î +a1>s 1"1&t "$ i&rtia

O 'Î +a1 a&#1s Î Mass "$ th 'a##Is Î +a##>s 1"1&t "$ i&rtiar s Î Ra!i(s "$ th s.hrr ' Î "siti"& "$ th 'a##

Th i&tic &r, "$ th s,st1 is?

T = T beam + T

sphere

T beam

= T beam

frame

+ T beam

beam

body

center

T sphere = T

sphere frame + T spherebeam body

center

Each &r, is?

T beam

frame = 0

¾ Th r"tati"&a# i&tic &r, "$ th 'a1 is?

: 2T beamcenter

= ⋅ #b

⋅θMb

2

¾ Th 'a## has i&tic &r, ;ith rs.ct t" th $ra1 i& '"th ra!ia# a&! circ(#ar 1"ti"&?

: ( 2 ) 2 : 2T sphere frame

= ⋅ m s

⋅ r b

2⋅θM

b

+ ⋅ m s

⋅ r Mb

2

¾ Th r"tati"&a# i&tic &r, "$ th s.hr a'"(t its '"!, c&tr

: 2T spherecenter

= ⋅ # s

⋅θM s

2

(4)5)

Th r"tati"&a# i&rtia "$ th r"##i& s.hr is

# =2

⋅ m s5

s⋅ r

2

(4))

(4)6) s



a&! a..#,i& a r"##i& ;ith"(t s#i..i&?4) (4)7)

r Mb = r

s ⋅θM

s

S('stit(ti& th *(ati"&s 4)6 a&! 4)7 i&t" th *(ati"& 4)5 ; t?

: ⎡ 2 ⎤ ⎡ : ⎤ : ⎡ 2 ⎤

T spherecenter =

2⋅ ⎢

5⋅ (m ⋅ r

2 ) ⋅

r2

⋅ r M 2

⎥ =

2⎢ 5

⋅ m s

⋅ r M 2

⎥ ⎣ ⎦ ⎣ s ⎦

⎣ ⎦

Th i&tic &r, "$ th s,st1 is?(4))

T =:

⋅ ⎡ # ⋅θM 2

+

(m⋅ r

2 )⋅θM 2+

m

⋅ r M 2

+

2⋅ m ⋅ r M

2

⎤

(4))

4)9

2 ⎣ b b s b b s b 5 s b ⎦

Si1.#i$,i& th *(ati"& 4)9 ; ha?

T =:

⋅ ⎡( # + m ⋅ r

2 )⋅θM 2

+

7⋅ m ⋅ r M

2 ⎤

2 ⎣

b s b b5

s b

⎦

Th r"##i& 'a## a#"& =hi'its th ."t&tia# &r, "$ th s,st1?

8 = m s

⋅ g ⋅ r b

⋅ si&θb

T" s(1 (.?

¾ 7I'ETIC E'E9:

T =:

⋅ ⎡( # + m ⋅ r

2 )⋅θM 2

+

7⋅ m ⋅ r M

2

⎤

(4))

(4)C)

2 ⎣

b s b b5

s b

⎦

¾ .%TE'TIAL E'E9:

8 = m s

⋅ g ⋅ r b

⋅

si&θb

(4))

(4):0)

s ⎥ ⎢

b b b

⎢( )

⎥

⎢⎥

⎢ ⎥



T" ;rit th *(ati"&s "$ 1"ti"& ; !$i& th Lara&ia& L t" ' th !i$$r&c 't;&

th i&tic a&! ."t&tia# &r, "$ th s,st1)

&(9 9M ) = T (9 9M )− 8

(9 )

(4)::)

;hr * !&"t th s"-ca##! &ra#i! c""r!i&ats "$ th s,st1 T is th i&tic &r,

"$ th s,st1 a&! V is th ."t&tia# &r, "$ th s,st1)

S('stit(ti& th *(ati"&s 4)C a&! 4):0 i&t" th *(ati"& 4):: ; t?

& =:

⋅ ⎡ # ⋅θM 2

+

(m⋅ r

2 )⋅θM 2+

m

⋅ r M 2

+2

⋅

m

⋅ r M 2 ⎤

−

m

⋅ g ⋅

r

⋅ si&θ

2b b s b b s b

5 s b s b b

Lara&>s *(ati"&s "$ 1"ti"& ar $"r1! $r"1?

⎡ ∂ ⎤d & ∂ &

⎢ ⎥ − =dt ⎣∂9Mi ⎦ ∂9i

2 9i

;hr F*i is th =tr&a# $"rc i& this cas F*i isτ)

*iÎ

i& this cas arθ ' a&! r M

b

∂ &

∂θMb

=:

⋅ # + m ⋅ r2 ⋅ 2 ⋅θM

2b s b b

∂ &=

:⋅

7⋅ 2 ⋅

m

⋅ r

∂r Mb

2 5

I$ ; !ri '"th =.rssi"&s ;ith rs.ct t" ti1 t ; t?

⎡ ∂ ⎤ &d ⎢ ⎥ ⎣ ∂θ

Mb ⎦

= # dt

b

+ m s ⋅ r

2

⋅θM

+ 2 ⋅

m s

⋅

r b

⋅

r Mb

⋅θMb

⎡ ∂ ⎤ &d ⎢ ⎥

⎣ ∂r Mb ⎦

=7

m ⋅ Mr M

dt 5 s b

s b

b b



Th "thrs *(ati"&s ;hich ; &! ar?

∂ &

∂θb

= −m s

⋅ g ⋅ r b

⋅ c"sθb

∂ & = :⋅ [2 ⋅

m

⋅

r b

⋅θM 2 ]− m

⋅ g ⋅ si&θb

∂r b

2

E;<ATI%'S %& )%TI%'4

2 M M #b + m s

⋅ r b

⋅θb

+ 2 ⋅ m s⋅ r

b⋅ r M

b⋅θ

b+ m s

⋅ g ⋅ r b

⋅ c"sθb

= τ

Mr

M

+5

⋅ g

⋅

si&θ

− r

⋅θM 2

=

0b

7b b b

;hr τ is th t"r*( a..#i! t" th 'a1)

b s s



:C

Chapter /

)odeling in )odelica

I& this cha.tr th t;" 1"!#s ar ;ritt& i& th M"!#ica #a&(a thr"(h *(ati"&s a&!

a&"thr 1"!# is crat! ;ith th M"!#ica>s #i'raris) P a#s" .rs&t si1(#ati"& rs(#ts (si&D,1"#a $"r th !i$$r&t 1"!#s)

/(1 )odels #ith the )odelica language

M"!#ica is (&#i 1"st &ra#-.(r."s #a&(as &"t .ri1ari#, 'as! "& a#"rith1s '(t

(ss *(ati"&s i&sta! that is "& !"s &"t &! t" r$"r1(#at th !i$$r&tia# *(ati"& i&t" th

sta&!ar! $"r1 =.#icit

d"=

dt

f " u t

'(t ca& ha &ra# *(ati"&s ;hr th ti1 !riatis a..ar i1.#icit#,) F"r r, 1"!# th

.r"ra11r ca& !$i& a &(1'r "$ *(ati"&s !scri'i& th .r".rtis "$ th 1"!#) Th

*(ati"&s !$i& th r#ati"& 't;& th !i$$r&t *(a&titis i& th si1(#ati"&)

Th 1ai& ras"& ;h, M"!#ica (ss *(ati"& is that r, si1(#ati"& .r"'#1 i& $act is a1ath1atica# .r"'#1) It a#s" i th #a&(a a hih a'stracti"& ## 'ca(s a& *(ati"& is

"$t& 1"r i&t(iti tha& a& a#"rith1)

D,1"#a .r"i!s a c"1.#t si1(#ati"& &ir"&1&t) D,1"#a tra&s$"r1s a !c#arati*(ati"& 'as! 1"!# !scri.ti"& i&t" $$ici&t c"!) A$tr a 1"!# has '& tra&s#at! i&t"

si1(#ati"& c"! th si1(#ati"& r(& is st (.) ara1trs a&! i&itia# c"&!iti"&s ar ' !$i&! as

;## as th !(rati"& "$ th si1(#ati"&)



ha.tr 5) M"!# i& M"!#ica

D,1"#a is (s! t" is(a#i !,&a1ic 'hai"r) #"ts "$ th ti1-rs."&s "$ aria'#s ar

"$t& har! t" i&tr.rt a&! a 1"r ra#istic ra.hica# i; is &!!) D,1"#a s(.."rts a 3-

!i1&si"&a# a&i1at! 1"!# i; i& a!!iti"& t" !,&a1ic 1"!#i& ;ith a #i'rar, "$ ra.hica#

"'8cts) Ph& a 1"!# is !scri'! i& D,1"#a ;ith *(ati"&s a&! s('1"!#s it is a#s" ."ssi'#

t" !$i& its is(a# a..ara&c) This is !"& i&c#(!i& .r!$i&! ra.hica# "'8cts "$ ari"(ssha.s)

Th t;" "'tai&! 1"!#s i& th .ri"(s cha.tr ca& ' ;ritt& i& M"!#ica thr"(h thir

*(ati"&s "$ 1"ti"&) Ths t;" .r"ra1s ar sh";& '#"; ;ith th rs.cti si1(#ati"&s !"&

;ith D,1"#a)

/(1(1 )odelica model4 'e#ton0s second la#

1"!# 'a##J'a1J:i1."rt M"!#ica)SI(&its@

// "&sta&ts

c"&sta&t Ra# M"!#ica)"&sta&ts)J& Qraitati"&a# Acc#rati"&Q@

// ara1trs

.ara1tr Ra# :)0 QM"t"r c"&sta&tQ@

// Varia'#s

Ra# a#.ha Q+a1 a&#Q@ Ra#

QV#"cit, "$ th 'a##Q@ Ra# aQAcc#rati"& "$ th 'a##Q@ Ra# =

Q"siti"& "$ th 'a##Q@

Ra# ( Q"&tr"# si&a#Q@*(ati"&

!r=@a !r@

// Math1atica# 1"!#i&a 5/7M"!#ica)Math)si&a#.ha@

!ra#.ha (@

// "&tr"# si&a#( 0@

&! 'a##J'a1J:@



Fi(r 5):? D,1"#a si1(#ati"&? Rs."&s "$ 'a1 a&# ."siti"& "$ th 'a## a&! #"cit, ".&-#"". rs."&s $"r 1"!# 'as "& N;t"&>s sc"&! #a;

This 1"!# $its ;## th 'a1 ;hich is !ri& ia a ar'"= that is th 'a## ."siti"& a&!

th t"r*( th 'a## ca(ss !"s &"t a$$ct th a&# "$ th 'a1)

/(1(! )odelica model4 Lagrangian method

1"!# 'a##J'a1J2

i1."rt M"!#ica)SI(&its@

// ara1trs

.ara1tr SI(&its)Mass 10): QMass "$ 'a##Q@

.ara1tr SI(&its)L&th L:)0 Q+a1 #&thQ@ .ara1tr SI(&its)Ra!i(s r0)0:5 QRa!i(s "$ th 'a##Q@ .ara1tr SI(&its)M"1&t<$I&rtia %2/51r2 QS.hrBs 1"1&t "$ I&rtiaQ@

.ara1tr SI(&its)M"1&t<$I&rtia I:7 Q+a1 I&rtia a#(Q@

// "&sta&ts

c"&sta&t Ra# M"!#ica)"&sta&ts)J& Qraitati"&a# Acc#rati"&Q@

// Varia'#s

Ra# thta Q+a1 A&#Q@

Ra# ; QA&(#ar V#"cit,Q@Ra# a#.ha QA&(#ar Acc#rati"&Q@Ra# =start0)0 Q"siti"& th 'a## i& th 'a1Q@

Ra# QV#"cit, "$ th 'a##Q@

Ra# a QAcc#rati"& "$ th 'a##Q@Ra# ta(start0)0 QT"r*(Q@

*(ati"&



!r=@a !r@

; !rthta@

a#.ha !r;@

// Lara&ia& 1th"!

I 1=2a#.ha 21=; 1=M"!#ica)Math)c"sthta ta(@a 5/7M"!#ica)Math)si&thta - =;2 0@

// "&tr"# #a;ta( 0@

&! 'a##J'a1J2@

Fi(r 5)2? D,1"#a si1(#ati"&? Rs."&s "$ 'a1 a&# ."siti"& "$ th 'a## a&! #"cit, <.&-

#"". rs."&s $"r th Lara&ia& 1"!#

This 1"!# is a..r".riat $"r a 'a1 ;hich is !irct-!ri& $r"1 th 1"t"r &" ar'"=

a&! as, t" 1" ', ha&!) Thr$"r th 'a## ."siti"& a&! th c"rrs."&!i& t"r*( !( t"rait, ;i## a$$ct th 'a1 a&# 1(ch 1"r)

+"th 1"!#s ca& ' =."rt! t" Si1(#i& $"r ithr si1(#ati"& "r c"&tr"# !si& ;hich is

#atr t" ' (s! $"r th c"&tr"# "$ th 'a1 a&! 'a## .r"css thr"(h $!'ac $r"1 a isi"&s,st1) Pith this .r"ra1 r(&&i& "& a&"thr c"1.(tr ; ca& !tr1i& th 'a## ."siti"&

;hich ar th& s&t t" th c"&tr"##r ia a &t;"r c"&&cti"&)

Th =."rt "$ a #i&ari! 1"!# t" Si1(#i& ca& ' ra#i! $r"1 1"! Si1(#ati"& "$

D,1"#a) I& "r!r t" !" it ,"( ca& (s th c"11a&! Li&ari i&c#(!! i& th Si1(#ati"& M&()

This c"11a&! ca#c(#ats a #i&ari! 1"!# at s"1 !tr1i&! i&itia# a#(s) Th #i&ari!1"!# is st"r! i& Mat#a' $"r1at a&! ca& ' #"a!! i&t" Mat#a' ;ith th 1-$i# t#"a!#i&)

A&"thr ;a, t" !" th =."rtati"& $r"1 D,1"#a t" Si1(#i& ca& ' $"(&! i& th



Mastr Thsis r"8ct ', air a##ir C "r Fra&csc" a#(i :0)

/(! )odel #ith )odelica0s liraries

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Th #i'rar, .r"i!s 'asic 1"!# c#asss $"r rii! '"!is 8"i&ts $"rcs 1as(r1&t a&!

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F"r a (sr it is as, t" i&tr"!(c &; c"1."&&ts "r c"., a&! 1"!i$, =isti& "&s)

A (&i*( $at(r "$ th #i'rar, is th .r".rt, that 8"i&ts ca& ha a aria'# str(ct(r) That

is r, !r "$ $r!"1 "$ a 8"i&t ca& ' #"c! a&! (&#"c! !(ri& 1"1&t ;ith"(t

!&rati& $$ici&c,)

This #i'rar, "$$rs a& a#tr&ati t" th M"!#icaA!!iti"&s)M(#ti+"!, #i'rar,)

Th "'tai&! 1"!# ;ith M"!#ica #i'raris is sh";& '#";?

1"!# TstS.hr

.ara1tr SI)Acc#rati"& C)9:@

i&&r .ara1tr SI)Acc#rati"&3 rait,0-0W@

// E##i.s"i! s1i-!ia1trsi&&r .ara1tr SI)L&th a::@

i&&r .ara1tr SI)L&th '::@i&&r .ara1tr SI)L&th c::@

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+as +as: a&&"tati"& =t&t-60 0@ -40 20@

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Fi(r 5)3? Architct(r "$ "'tai&! 1"!# ;ith M"!#ica #i'raris

<& sh"(#! c"&si!r a## c"&&ct"rs (s! a'" as 'i!irct! "&s) Arr";s i& Fi(r 5)3 ar

(s! t" sh"; th s1a&tics "$ i&tracti"&s) It>s c#ar that th ;h"# c"&str(cti"& c"&si!r!a'" is a irt(a# "&) "&str(cti& th 1"!# th c"1.i#r =tracts a## *(ati"&s $r"1 th

"'8cts a&! ass1'#s th1 c"1."si& th DAE s,st1 ".ti1i! $"r a &(1ric i&trat"r)

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A 'asic $at(r is that a## c"1."&&ts ha a&i1ati"& i&$"r1ati"& ;ith a..r".riat !$a(#t

sis a&! c"#"rs) A t,.ica# scr&sh"t "$ th a&i1ati"& "$ 'a1 a&! 'a## is sh";& i& th Fi(r5)4 '#";)

Fi(r 5)4? A&i1ati"& "$ "'tai&! 1"!# ;ith M"!#ica #i'raris



26

Chapter 2

Control Design and Simulations

This cha.tr !tai#s th c"&tr"# !si& "$ th t;" "'tai&! 1"!#s ;ith th M"!#ica

#a&(a) Pith th N;t"&>s sc"&! #a; 1"!# ; !" th st(!, thr"(h a& "'srr !si& a&!;ith th Lara&ia& 1"!# ; ;"r ;ith a #i&ari! 1"!# a&! a& "'srr t" !" th st(!,) I&

this cha.tr ar a#s" sh";& th "'tai&! si1(#ati"&s i& Si1(#i&)

2(1 Control design

2(1(1 Control design through oser5er

A## stats ar &"t aai#a'# $"r $!'ac i& 1a&, cass a&! "& &!s t" sti1at

(&aai#a'# stat aria'#s) Esti1ati"& "$ (&1as(ra'# stat aria'#s is c"11"&#, ca##!"'srati"&) A !ic "r a c"1.(tr .r"ra1 that sti1ats "r "'srs th stat is ca##! a

stat-"'srr "r si1.#, a& "'srr) I$ th stat-"'srr "'srs a## stat aria'#s "$ th

s,st1 rar!#ss "$ ;hthr s"1 stat aria'#s ar aai#a'# $"r !irct 1as(r1&t it isca##! a $(##-"r!r stat "'srr)

A& "'srr that sti1ats #ss tha& th !i1&si"& "$ th stat-ct"r is ca##! a r!(c!-"r!r

stat-"'srr "r si1.#, a r!(c!-"r!r "'srr)

Th ras"& $"r i&tr"!(ci& th "'srr is that ."# .#ac1&t ;ith $(## stat $!'ac is

&"t r, .ractica#) First $"r a& &-!i1&si"&a# s,st1 it r*(irs & 1as(r1&ts ;hich i& t(r&

1a&s & tra&s!(crs) S(ch a c"&tr"##r ;"(#! ' '"th =.&si a&! '(#,) F(rthr t" 'i1.#1&ta'# a## th stats ;"(#! ha t" ' 1as(ra'#) E& i$ s(ch a stat 1"!#

$"r1(#ati"& c"(#! ' "'tai&! it 1iht &"t ' a .r$rr! $"r1(#ati"&)

26



ha.tr 6? "&tr"# Dsi& a&! Si1(#ati"&s

+asica##, thr ar t;" $"r1s "$ th i1.#1&tati"& "$ a& sti1at"r as ".& #"". a&!

c#"s!-#"".) Th !i$$r&c 't;& ths t;" is a c"rrcti"& tr1 i&"#i& th sti1ati"&

rr"r (s! t" a!8(st th rs."&s "$ th sti1at"r) A c#"s!-#"". sti1at"r is r$rr! t" as a&

"'srr)

+ca(s "$ "(r +a## a&! +a1 1"!# is ".& #"". (&sta'# is &cssar, t" ha s"1 i&!

"$ 1as(r1&t "$ th 'a## #"cit,) Th c#assica# .r"."rti"&a# D-c"&tr"##r ts a #"cit,1as(r ', !i$$r&tiati& th 'a## ."siti"&) Th c#assica# .has #a! c"1.&sat"r !"s

s"1thi& r, si1i#ar) A&"thr ;a, "$ !"i& this is t" (s a& "'srr 'as! (."& a 1"!# "$

th 'a## a&! 'a1 t" sti1at th s,st1s stats a&! (s th stat sti1ats "$ 'a## ."siti"& a&!#"cit, i& a stat $!'ac c"&tr"##r)

P !si& a $(##-"r!r stat "'srr t" sti1at th"s stats that ar &"t 1as(ra'#) P

;a&t t" !" "(r "'srr as $ast as it is ."ssi'# ;ith"(t a1.#i$,i& t"" 1(ch hih $r*(&c,!ist(r'a&cs) Thr$"r ; .#ac th ."#s "$ th "'srr i& th ra& "$ t;" t" $i ti1s #arr

i& 1a&it(! tha& th c"&tr"##r ."#s)Pith "(r c"&tr"##r ; ;a&t t" t that th s,st1 is sta'#) +#"; a sch1atic "$ "(r

.#a&t-"'srr a&! c"&tr"##r s,st1 is sh";&)

Fi(r 6):? Si1(#i&? N;t"&Bs sc"&! #a; 1"!#@ .#a&t-"'srr a&! c"&tr"##r



2(1(! Control Design4 nonlinear system

"&si!ri& th "'tai&! *(ati"&s "$ 1"ti"& i& scti"& 4):)2 ; !" a st(!, "$ c"&tr"#

!si& thr"(h a #i&ari! stat 1"!# "$ th s,st1 ar"(&! a *(i#i'ri(1 ."siti"&) P a#s""'tai& a sta'i#ii& c"&tr"##r $"r th #i&ar 1"!# ;h& th a&(#ar ."siti"& is 1as(r!)

Th *(ati"&s "$ 1"ti"& ;r?

2 M M #b + m s

⋅ r b

⋅θb

+ 2 ⋅ m s⋅ r

b⋅ r M

b⋅θ

b+ m s

⋅ g ⋅ r b

⋅ c"sθ b= τ

Mr M +5

⋅ g ⋅

si&θ

− r ⋅θM 2 = 0

b7

b b b

Th &"&#i&ar stats ar?θ ' θMb r a&! r M

a&! th stat *(ati"&s 'c"1?

⎧ d θ⎪ dt ⎪⎪ d θM

= θM

=

f :

⎛ 2 ⋅ ms

⋅ r ' ⎞ ⎛ ms

⋅ g ⋅ r '

⋅ c"sθ '

⎞ ⎛ : ⎞

⎪ dt

= θM = −⎜ ⎟ ⋅θM

⋅ r Mb

− ⎜ # + m ⋅ r2

⎟ + ⎜ ⎟ ⋅τ = f

+ m⋅

r2 2

⎨⎪ dr ⎪ dt ⎪

⎝ b

= r M = f3

s b ⎠ ⎝ b s b ⎠ ⎝ b s b ⎠

⎪ dr M = Mr M =

5⋅

r

⋅θM 2−

5⋅ g ⋅ si&θ = f

dt 7b b

7b 4

T" ha a !sir! *(i#i'ri(1 i&?

θe

= 0

r e

= :

θM

e

= 0

r Me

= 0

; th(s !$i& th stats a

":

= θ −θe

= θ

"2

= θM −θM

e = θM

"3

= r − r e

= r −:

"4

= r M − r Me = r M



a&! th c"&tr"# i&.(t

u = τ −τ e

+, s('stit(ti& a#(s i& $ 2 ; "'tai&?

τ e

= m s

⋅ g ⋅ r e

;"ri& ;ith ths &; a#(s i& th stat *(ati"&s a&! c"&si!ri& th *(i#i'ri(1=:=2=3=4(0 ; th(s ha th #i&ari! stat 1"!#?

"M = A ⋅ " + % ⋅ u y = ⋅ "

;hr = is a ct"r r.rs&ti& th stat ( is a sca#ar r.rs&ti& th i&.(t a&! , is a

sca#ar r.rs&ti& th "(t.(t) Th 1atrics A + a&! !tr1i& th r#ati"&shi.s 't;& th stat a&! i&.(t a&! "(t.(t aria'#s)I& "(r s,st1 th 1atrics A + a&! ha th $"##";i& r.rs&tati"&?

⎡ 0 : 0 0⎤ ⎡ 0 ⎤

⎢ −m s

⋅ g ⎥ ⎢ : ⎥ ⎢

0 0 # + m

⎥ ⎢ ⎥ =

⎡: 0 0 0⎤ A = ⎢ b s ⎥ % = ⎢ #

'+ ms

⎥ ⎢

0 0 : 0⎥

⎢ 0 0 0 :⎥ ⎣ ⎦

⎢ 0 ⎥⎢ − 5

⋅ g 0 0 0⎥⎢⎣ 7 ⎥⎦ ⎣⎢ 0 ⎥⎦

Th &=t st. is t" chc th c"&tr"##a'i#it, a&! th "'sra'i#it, "$ th s,st1) F"r th

s,st1 t" ' c"1.#t#, stat c"&tr"##a'# th c"&tr"##a'i#it, 1atri=

: = [ % A% A

2 %

A3 %]

1(st ha th ra& "$ 4) Th ra& "$ th 1atri= is th &(1'r "$ i&!.&!&t r";s "r c"#(1&s) I& th sa1 ;a, $"r th s,st1 t" ' c"1.#t#, stat "'sra'# th

"'sra'i#it, 1atri=

⎡ ⎤ ⎢ ⎥

: = ⎢ A ⎥o ⎢A

2 ⎥ ⎢ 3 ⎥

⎣A ⎦

0

c



1(st a#s" ha ra& "$ 4)

T" "'tai& a sta'i#ii& c"&tr"##r $"r th #i&ari! 1"!# ; &! t" ha a sta'#

charactristic ."#,&"1ia# $"r th c#"s! #"". s,st1 r""ts strict#, i& th #$t ha#$-.#a&) I&

this =a1.# ; !tr1i& a stat-$!'ac ai& s(ch that th c#"s!-#"". ."#s ar ats-: "r!r 4) Th charactristic ."#,&"1ia# is?

αcss:4s

44s

36s

24s:

αcAA44A

36A

24AI

Th(s ', Acr1a&&Xs $"r1(#a

$ = [0 0 0 :]⋅ :−:

⋅α

A

;hr Yc is th c"&tr"##a'i#it, 1atri= .ri"(s#, !$i&!)

P .#ac th ."#s "$ th "'srr i& s-5 that is $i ti1 'ir tha& "(r c"&tr"##r

."#s ;ith it ; t that "(r s,st1 is $astr) Th(s ; ha?

α"ss54s

420s

3:50s

2500s625

α"AA420A

3:50A

25004A625I

Th(s ', Acr1a&&Xs $"r1(#a

⎡0⎤ ⎢ ⎥

& = α A ⋅ :−:

⋅ ⎢0⎥

o o ⎢0⎥ ⎢ ⎥

⎣:⎦

;hr Y" is th "'sra'i#it, 1atri= !$i&! '$"r)

Th(s ; ha th "'srr $!'ac c"&tr"##r

d"Z= 3 A − % ⋅ $ − & ⋅ ⋅ "Z + & ⋅ y dt

u = −$ ⋅ "Z

c c



Th $"##";i& Si1(#i& 1"!# ca& ' (s! t" si1(#at th s,st1)

Fi(r 6)2? "&tr"##r "$ #i&ari! Lara&ia& 1"!# i& Si1(#i&

2(! Simulations

2(!(1 Simulations in Simulin$

Si1(#i& is a& i&tracti &ir"&1&t i&trat! i& Mat#a' $"r 1"!#i& si1(#ati"& a&!

a&a#,i& !,&a1ic s,st1s) Si1(#i& .r"i!s a ra.hica# (sr i&tr$ac $"r c"&str(cti& '#"c !iara1 1"!#s ia !ra a&! !r". ".rati"&s) M"!#s ca& ' r"(.! i&t" hirarchis t" crat a

si1.#i$i! i; "$ c"1."&&ts "r s('s,st1s) Thr$"r a 1"!# '(i#t (. i& Si1(#i& c"&sists "$ '#"cs that c"rrs."&! t" s('s,st1s "$ th 1"!#) I& ach '#"c thr ar 1ath1atica#

r#ati"&shi.s that !scri' th .h,sica# 'hai"r "$ th s,st1) I$ th s('s,st1s a$$ct ach

"thr i&$"r1ati"& 't;& th '#"cs has t" ' =cha&! i& "r!r t" a#(at th r#ati"&shi.s)

Th '&$it ;ith Si1(#i& is th as, t" (s !si& a&! si&c th s"$t;ar is i&trat!

;ith Mat#a' it 1as it r, .";r$(# ;ith a #"t "$ (s$(# a&a#,sis a&! !si& a..#icati"&s)



A&"thr '&$it is th as, ;a, t" crat .#"ts "$ si&a#s) F(rthr th a'i#it, t" asi#, a&!*(ic#, si1(#at a 1"!# '(i#t i& Si1(#i& is a&"thr '&$it ;ith this 1"!#i& .r"ra1)

H";r it #acs th '&$its "$ M"!#ica t" !$i& i1.#icit *(ati"&s a&! r#ati"&shi.s)

¾ "&tr"# !si& thr"(h "'srr

Fi(r 6)3? Si1(#ati"&s "'tai&! ;ith th N;t"&>s sc"&! #a; 1"!# a&! "'srr

I& Fi(r 6)3 ; ca& s th 'hai"r ;ith th ca#c(#at! c"&tr"##r a&! "'srr) Th

c#"s! #"". s,st1 'has ;##)P ca& "'srr h"; th ."siti"& a&! th ."siti"& sti1at "$ th 'a## ar a#1"st th sa1)

+"th rs."&ss ar sta'i#i! at : #i th r$r&c a&! 'si!s ; ca& s that th 'a1 a&#

a&! th #"cit, sti1at ar sta'i#i! i& $; sc"&!s)



33

Chapter 3

Experiments

This cha.tr !scri's th =.ri1&ta# st(. $"r th +a## a&! +a1 =.ri1&ts) I&itia#

=.ri1&ts ar .r$"r1! t" i!&ti$, .r"css .ara1trs a&! $i&a##, th c"&tr"# =.ri1&ts ar .r$"r1!) A& a#tr&ati i1.#1&tati"& t" th "& !scri'! i& ha.tr 6 is .rs&t!)

3(1 Experimental Setup

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1"t"r a&! th 'a1 a&# ca& ' 1as(r!) Th 'a## ."siti"& is sti1at! $r"1 a isi"& s,st1;ritt& ', T"1as <#ss"& hD st(!&t at th D.art1&t "$ A(t"1atic "&tr"# a&! (ss a Fir-i

ca1ra c"&&ct! t" a r(&&i& a K'a## !tcti"& .r"ra1) Th 'a## c""r!i&ats ar s&t ia a

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Fi(r 7):)



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Si1(#i& ct r# [ Li&( = 0

Fi(r 7):? Stch "$ th c"&&cti"&s 't;& (s! 'a## a&! 'a1 isi"& s,st1 a&! >s

3(! ,o# to do the experiments

Th $irst tas i& this ;"r c"&sists "$ !tr1i&i& ;hich is th c"rrct 1"!# "$ th ra#

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'#"; a&! th c"&tr"##r is i1.#1&t! i& Mat#a'/Si1(#i& ;ith a ra#-ti1 =t&si"& ;hr a

Si1(#i& '#"c r.rs&ts th c"&&cti"& t" a&a#" i&.(ts a&! a&a#" "(t.(ts s Fis) 7)2 a&!7)3)

Fi(r 7)2? +a## a&! +a1 (s!

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Th .ri"! ti1 "$ th "sci##ati"& is a..r"=i1at#, 3 sc"&!s thr$"r th $r*(&c,; ≈ 2ra!/s) N"; that ; ha "(r $r*(&c, ; ca& $i&! ;hich ar "(r !a1.i& a&! "(r "rsh""t)

Pith ths a#(s a&! (si& t""#s "$ Mat#a' ; ca& "'srr th i1.(#s rs."&s "$ th

s,st1 'i& this sh";& i& th Fi 7)6?

Fi(r 7)6? I1.(#s Rs."&s "$ th +a## a&! +a1 .r"css

P ca& "'srr $r"1 th rs."&s that thr is a& a!!iti"&a# !a1.i& a&! a& "sci##at"r,

'hai"r i& c"1.aris"& ;hat ; ha !isc(ss! .ri"(s#,)

Fr"1 Fi(r 7)6 ; ca& sti1at th i1.(#s rs."&s c"rrs."&! t" a s,st1 ;hichtra&s$r $(&cti"& has a ai& ≈ 22) This ai& has '& ca#c(#at! thr"(h "$ th Fi(r 7)5)

; = 22

s2

+ 0)6 ⋅ s + 4

Thr"(h this tra&s$r $(&cti"& ; ca& r.rs&t th s,st1 i& stat-s.ac $"r1 !"i& itth $"##";i& ;a,?

( ,

22

s 2 + 0)6 ⋅ s + 4

s2

⋅ y + 0)6 ⋅ s ⋅ y + 4 ⋅ y = 22 ⋅ u



; ha =:, th& th stat *(ati"&s ar?

"M: = "

2

"M2

= M yM

s('stit(ti&?

"M2

= M yM = −0)6 ⋅ yM − 4 ⋅ y + 22 ⋅ u

"M2

= −0)6 ⋅ "2

− 4 ⋅ ":

+ 22 ⋅ u

Th stat-s.ac r.rs&tati"& is sh";& '#";?

⎧⎡ "M:

⎤ ⎡ 0 : ⎤ ⎡ ":⎤ ⎡0 ⎤

⎪ ⎢ ⎥ = ⎢ ⎥ ⋅ ⎢ ⎥ + ⎢ ⎥ ⋅ u

⎪⎣ "M2 ⎦⎨ ⎣− 4 − 0)6⎦ ⎣ "2 ⎦ ⎣22⎦ (7):) ⎪ ⎡ "

: ⎤⎪ y = [: 0]⋅ ⎢ ⎥

⎩ ⎣ "2 ⎦

T" !" th st(!, "$ th s,st1 ; ;rit a& 1-$i# i& Mat#a') I& this 1-$i# ; i&c#(! th

1atrics "$ stat-s.ac s,st1 a&! th "'tai&! a#(s "$ !sir! $r*(&c, a&! !a1.i&) Pith

ths a#(s ; ca& ca#c(#at th !sir! r""ts "$ "(r s,st1)

N"; ; ha "(r ."#s ; ca& (s Mat#a' t" $i&! a c"&tr"##r 1atri= ', (si& th place c"11a&! s ha.tr 6)

T" i1.#1&t "(r c"&tr"##r ; ha t" $i&! a& "'srr ;hich ca& !" a& sti1ati"& "$

th (&1as(ra'# stat aria'#s) N"r1a##, th ."#s "$ th "'srr ar $astr tha& th"s "$ thc"&tr"##r) P .#ac th ."#s t;" ti1s $astr tha& "(r c"&tr"##r) Th "'srr is 'asica##, a

c"., "$ th .#a&t@ it has th sa1 i&.(t a&! a#1"st th sa1 !i$$r&tia# *(ati"&) A& =tra tr1

c"1.ars th act(a# 1as(r! "(t.(t , t" th sti1at! "(t.(t yZ @ this ;i## ca(s th

sti1at!

stats "Z t" a..r"ach th a#(s "$ th act(a# stats =) Th rr"r !,&a1ics "$ th "'srr ar

i& ', th ."#s "$ A-L) I& th sa1 ;a, as '$"r ;ith th c"11a&! place ; $i&! th

"'srr L 1atri=)

Th &=t st. is t" !" th Si1(#i& 1"!# ;ith th .#a&t "$ th ra# .r"css a&! th

"'tai&! "'srr a&! c"&tr"##r) This Si1(#i& 1"!# is sh";& i& th Fi 7)7)

37



Fi(r 7)7? Si1(#i& 1"!# "$ "'srr 'as! c"&tr"# "$ 'a1 a&#

P ca& &"; si1(#at th c#"s!-#"". rs."&s a&! ; ca& chc i$ th $"(&! 1"!# is

c"rrct) ha&i& th a#(s th ; ; ca& "'srr i$ th s,st1 is $astr "r s#";r a&! ; ca&s i$ its 'hai"r is riht) P ca& a#s" cha& th a#(s "$ th i&.(ts th chc i$ th rs."&s

"$ th ra# .r"css is c"rrct i& r, 1"1&t)

A !si& .r"'#1 !"s &"t &cssari#, ha a (&i*( a&s;r) Usi& this 1th"! "r a&,

"thr 1a, rs(#t i& 1a&, !i$$r&t c"1.&sat"rs)

F"r th i1.#1&tati"& t" c"&tr"# th ra# .r"css ; (s a !iscrt-ti1 $!'ac $r"1 a&

"'srr 'as! "& a sa1.#! rsi"& "$ th s,st1 7):)

P ha (s! th Mat#a' c"11a&! c2! t" c"&rt 't;& c"&ti&("(s a&! !iscrt ti1)

Th sa1.#i& ti1 $"r th Si1(#i& c"&tr"##r ;as st t" :01s) Th ."#s "$ th !iscrt-ti1!si& (s! i& place ;as tra&s$"r1! $r"1 c"&ti&("(s-ti1 s.ci$icati"&s acc"r!i& t"!iscJ."#=.c"&tJ."#) tsa1.)

39



%all controller

F"r th ra#-ti1 i1.#1&tati"& "$ th 'a## c"&tr"##r ; ha ch"s& t" !" a casca!-!si& r-(si& th 'a1-c"&tr"##r $r"1 th .ri"(s s('scti"& $"r th i&&r #"". i&sta! "$

th c"&tr"##! .rs&t! i& ha.tr 6)

Th c""r!i&ats "$ th 'a## $r"1 th isi"& s,st1s ar (.!at! ;ith 30H) H";r th

ca1ra .ict(rs 1a, ' !#a,! \ 301s i& th c"&tr"##r a&! th ca1ra is &"t s,&chr"&i!

;ith th rst "$ th c"&tr"##r) T" ha&!# this (&crtai&t, ; ha ch"s& t" r(& a $astr i&&r #"". a&! a& "(tr-#"". ;ith a s#";r sa1.#i& rat)

Fr"1 ha.tr 4 ; ha a& a..r"=i1at 1"!# "$ a !"('# i&trat"r $r"1 'a1 a&# t" 'a## ."siti"& a&! ; start t" !" a si1.# =.ri1&t t" $i&! "(t th ai& ;ith rs.ct t" th

."siti"& .r"i!! ', th isi"& s,st1) P ca& is(a#i it i& th $i(r 7)9)

Fi(r 7)9? ra.hic t" $i&! "(t th ai& ;ith rs.ct t" th ."siti"& .r"i!! ', th isi"& s,st1

A i!" s*(&c "$ th =.ri1&t ca& ' $"(&! at :3)

3C



Fi(r 7)C? Si1(#i& 1"!# t" !" th 'a1 c"&tr"#

40



Fi(r 7):0? Si1(#i& 1"!# t" !" th 'a## ."siti"& c"&tr"#s tr"(h th isi"& s,st1



Chapter 8

Conclusions

I& this cha.tr th c"&c#(si"&s "$ this ;"r ar .rs&t! $"##";! ', s(sti"&s "& h";

t" c"&ti&( th !#".1&t)

8(1 Conclusions

Th 1ai& "a# "$ this thsis ;as t" !" th 1"!#i& a&! c"&tr"# "$ th +a## a&! +a1

.r"css) This "'8cti has '& $(#$i##! ', th .rs&tati"& "$ sra# i&!s "$ c"&tr"# ;hich

ha '& !"& thr"(h M"!#ica a&! Mat#a' a&! $i&a##, ', =.ri1&t "& th ra# .r"css)

<r this thsis ; ca& is(a#i h"; c"&tr"# th"r, r#ats t" ra# s,st1s) "1.(tr

si1(#ati"&s ca& h#. '(t si1(#ati"&s a#"& ar #i1it! ', h"; ;## th s,st1 i& *(sti"& has

'& 1"!#!) It is 1"r &#iht&i& i$ th rs(#ts "'tai&! th"rtica##, ar act(a##, a..#i! t"a .h,sica# s,st1) A..#,i& th th"rtica##, ca#c(#at! rs(#ts t" a ra# s,st1 h#.s (s &"t "&#,

is(a#i '(t t" a#(at h"; ;## th 1"!# ;as a'# t" .r!ict th s,st1 .r$"r1a&c)

Th 'st ;a, t" #ar& a'"(t c"&tr"# s,st1s is t" !si& a c"&tr"##r a..#, it t" th s,st1

a&! th& ;atch th s,st1 i& ".rati"&) A s,st1 is 1"!#! "& a c"1.(tr a&! ;ith th h#.

ra.hics th s,st1 ca& ' s& i& acti"&) H";r th s,st1 'i& "'sr! i& th si1(#ati"&

is i& ra#it, 8(st a 1"!# "$ th tr( s,st1) Th 1"!# 1(st a#;a,s ' a si1.#i$i!r.rs&tati"& "$ th s,st1 a&! ca&&"t r.r"!(c a## as.cts "$ s,st1 'hai"r) S(ch $$cts

ar !i$$ic(#t t" si1(#at a&! ar 'st (&!rst""! $r"1 ha&!s-"& =.ri&c ;ith th .h,sica#

s,st1)

42



List of &igures

Chapter !

Fi(r 2):? M"t"r !ri ;ith M"!#ica #i'raris )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 6

Chapter +

Fi(r 3):? Diara1 "$ +a## a&! +a1 ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) C

Fi(r 3)2? +a## a&! +a1 c"&&ct! ia ar'"= )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) :0

Fi(r 3)3? +a## a&! +a1 !irct#, c"&&ct! t" th a=is "$ a D-1"t"r ))))))))))))))))))))))))))))) ::

Chapter /

Fi(r 5):? D,1"#a si1(#ati"&? Rs."&s "$ 'a1 a&# ."siti"& "$ th 'a## a&!#"cit, ".&-#"". rs."&s $"r 1"!# 'as "& N;t"&>s sc"&! #a; )))))))))))))))))))))))))))))))) 2:

Fi(r 5)2? D,1"#a si1(#ati"&? Rs."&s "$ 'a1 a&# ."siti"& "$ th 'a## a&!

#"cit, <.& #"". rs."&s $"r th Lara&ia& 1"!# ))))))))))))))))))))))))))))))))))))))))))))))))))))))) 22

Fi(r 5)3? Architct(r "$ "'tai&! 1"!# ;ith M"!#ica #i'raris )))))))))))))))))))))))))))))))))))) 24

Fi(r 5)4? A&i1ati"& "$ "'tai&! 1"!# ;ith M"!#ica #i'raris )))))))))))))))))))))))))))))))))))))) 25

Chapter 2

Fi(r 6):? Si1(#i&? N;t"&Bs sc"&! #a; 1"!#@ .#a&t-"'srr a&! c"&tr"##r ))))))))))))))) 27



Fi(r 6)2? "&tr"##r "$ #i&ar Lara&ia& 1"!# i& Si1(#i& )))))))))))))))))))))))))))))))))))))))))))) 3:

Fi(r 6)3? Si1(#ati"&s "'tai&! ;ith th N;t"&>s sc"&! #a; 1"!# a&! "'srr ))))))))) 32

Chapter 3

Fi(r 7):? Stch "$ th c"&&cti"&s 't;& (s! 'a## a&! 'a1 isi"& s,st1 a&!

>s)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 34

Fi(r 7)2? +a## a&! +a1 (s! )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 34

Fi(r 7)3? "&tr"##r "$ th +a## a&! +a1 .r"css )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 35

Fi(r 7)4? +a## a&! 'a1 ;ith !ista&c! ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 35

Fi(r 7)5? Rs."&s "$ th 'a1 ;ith a..#i! i1.(#s )))))))))))))))))))))))))))))))))))))))))))))))))))))) 35

Fi(r 7)6? I1.(#s Rs."&s "$ th +a## a&! +a1 .r"css ))))))))))))))))))))))))))))))))))))))))))))))) 36

Fi(r 7)7? Si1(#i& 1"!# "$ "'srr 'as! c"&tr"# "$ 'a1 a&# ))))))))))))))))))))))))))))))))) 39

Fi(r 7)9? ra.hic t" $i&! "(t th ai& ;ith rs.ct t" th ."siti"& .r"i!! ', th

isi"& s,st1 ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 3C

Fi(r 7)C? Si1(#i& 1"!# t" !" th 'a1 c"&tr"# ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 40

Fi(r 7):0? Si1(#i& 1"!# t" !" th 'a## ."siti"& c"&tr"#s tr"(h th isi"& s,st1 )))))) 4:



Biliography

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http://www.modelica.org/Conference2002/BeginnersTutorial.zip

http://www.modelica.org/documents.shtml

http://www.modelica.org/Conference2002/AdvancedTutorial.zip

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http://www.modelica.org/Conference2002/BeginnersTutorial.zip

http://www.modelica.org/documents.shtml

http://www.modelica.org/Conference2002/AdvancedTutorial.zip



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http://www.control.lth.se/education/masterthesis/BaB_video.npg

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