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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
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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 $ey title
0290-53:6 #S%N
&anguage
E&#ish Number of pages
47 'ecipients notes
Security classification
The report may be ordered from the Department of Automatic ontrol or borrowed through*niversity &ibrary+ %o" ,+ S-.//0 11 &und+ Sweden 2a" 345 45
/// 4/ 4,
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Ac$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)
I ;ish t" tha& a## 1, S.a&ish $ri&!s $"r thir i&s.irati"&a# a&! 1"ra# s(.."rt a&!s.cia##, t" 1, $ri&!s Ra(# a&! Ta&ia ;h" h#.! 1 i& 1, ar#ir !a,s ;h& I arri! t"
S;!&)
Last '(t crtai&#, &"t #ast I ;"(#! #i t" sh"; 1, ratit(! t" 1, .ar&ts 1, sistr a&!
1, ra&!.ar&ts ;h" ha '& &r"(s ;ith thir &c"(ra1&t)
This thsis is !!icat! t" th1)
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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 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( *!
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:
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$$r&t t""#s (s! $"r th 1"!#i&)
Chapter + !scri's th st(!, "$ th +a## a&! +a1 .r"css)
Chapter * .rs&ts th 1ath1atica# 1"!#i& "$ th .h,sica# s,st1) Chapter
/ is th 1"!#i& i& M"!#ica "$ th 'a1 a&! 'a## .r"css( Chapter 2
c"&tai&s c"&tr"# !si& "$ th "'tai&! 1"!#s a&! th c"rrs."&!i&si1(#ati"&s)
Chapter 3 c"&tai&s th =.ri1&ts !"&)
Chapter 8 .rs&ts th s(11ari! c"&c#(si"&s "$ this thsis)
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Chapter !
Dymola and )odelica
This cha.tr is a sh"rt i&tr"!(cti"& t" th !i$$r&t t""#s (s! $"r th 1"!#i& i& th
thsis)
!(1 Dymola
Th c"1.a&, D,&asi1Bs 1issi"& is t" !#". th s"$t;ar t""#s that i&!(str, &!s $"r
s"#i& !1a&!i& 1"!#i& si1(#ati"& a&! !si& .r"'#1s) Th 1.hasis is "& ha&!#i&
#ar 1(#ti-&i&ri& s,st1s $$ici&t#,) D,1"#a .i"&r! th "'8ct-"ri&t! .h,sica#1"!#i& 1th"!"#", a&! th acc"1.a&,i& si1(#ati"& tch&"#",)
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"&)
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ha.tr 2) D,1"#a a&! M"!#ica
3
!(1(1 Dymola features4
¾ M"!# Tra&s#at"r
A(t"1atic ca(sa#it, a&a#,sis
S,1'"#ic s"#(ti"& "$ *(ati"&s a&! i&!= r!(cti"& S"#(ti"& "$ a#'raic #"".s A(t"1atic ha&!#i& "$ ti1 a&! stat &ts &rat -c"!
¾ Ha&!#s #ar c"1.#= 1(#ti-!"1ai& 1"!#s
¾ Fastr 1"!#i& ', ra.hica# 1"!# c"1."siti"&
ra.hica# 1"!# !it"r a&! 'r";sr Dra s('1"!#s $r"1 #i'raris ara1tr $"r1s
"&&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"&
¾ N" 1a&(a# *(ati"& 1a&i.(#ati"& &!!
¾ Fastr si1(#ati"& - s,1'"#ic .r-.r"cssi&
¾ <.& $"r (sr !$i&! 1"!# c"1."&&ts
¾ <.& i&tr$ac t" "thr .r"ra1s
¾ A&i1ati"&
Ra# ti1 3D a&i1ati"& +"=s s.hrs c,#i&!rs tc - .r!$i&! is(a# c#asss I1."rt "$ DF a&! STL $i#s Hi!!& s(r$ac r1"a# sha!i& #"tti&
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"& :
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!(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$$cti #i'rar, !#".1&t a&! 1"!# =cha&) It is a 1"!r& #a&(a '(i#t "& aca(sa#
1"!#i& ;ith 1ath1atica# *(ati"&s a&! "'8ct-"ri&t! c"&str(cts t" $aci#itat r(s "$
1"!#i& &";#!)
Th ;"r ;ith M"!#ica start! i& S.t1'r :CC6 ;ith a r"(. "$ a'"(t $i$t& .rs"&s
;ith &";#! a'"(t 1"!#i& #a&(as a&! 1"!#s ;ith !i$$r&tia# a#'raic *(ati"&s
DAE)Th $irst rsi"& "$ th M"!#ica #a&(a !$i&iti"& ;as $i&ish! i& S.t1'r :CC7) I&
%a&(ar, 2002 a &; rsi"& ;as r#as!)
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!(!(1 )odelica features
Th $"(r 1"st i1."rta&t $at(rs "$ M"!#ica ar?
¾ Is 'as! "& *(ati"&s i&sta! "$ assi&1&t stat1&ts) This .r1its aca(sa#
1"!#i& that is 'ttr r(s "$ c#asss si&c *(ati"&s !" &"t s.ci$, a crtai&!ata $#"; !ircti"&) Th(s a M"!#ica c#ass ca& a!a.t t" 1"r tha& "& !ata $#";
c"&t=t)
¾ Has 1(#ti-!"1ai& ca.a'i#it, 1a&i& that 1"!# c"1."&&ts c"rrs."&!i& t"
.h,sica# "'8cts $r"1 sra# !i$$r&t !"1ai&s s(ch as )) #ctrica# 1cha&ica#
thr1"!,&a1ic h,!ra(#ic 'i"#"ica# a&! c"&tr"# a..#icati"&s ca& ' !scri'! a&!c"&&ct!)
¾ Is a& "'8ct-"ri&t! #a&(a ;ith a &ra# c#ass c"&c.t that (&i$is c#asss
&rics a&! &ra# s('t,.i& i&t" a si&# #a&(a c"&str(ct) This $aci#itats r(s
"$ c"1."&&ts a&! "#(ti"& "$ 1"!#s)
¾ Has a str"& s"$t;ar c"1."&&t 1"!# ;ith c"&str(cts $"r crati& a&! c"&&cti&c"1."&&ts) Th(s th #a&(a is i!a##, s(it! as a& architct(ra# !scri.ti"&
#a&(a $"r c"1.#= .h,sica# s,st1s a&! t" s"1 =t&! $"r s"$t;ar s,st1s)
M"!#s a&! s('-1"!#s ar !c#ar! as c#asss ;ith c"&&cti"& i&tr$acs ca##!
connectors) This c"&&cti"& ca.a'i#it, a##";s th (s "$ 1"!# #i'raris t" c"1."s c"1.#=
1"!#s ;ith th !ra a&! !r". a&! c"&&cti"& !ra;i& $aci#itis "$ 1"!r& ra.hica# !it"rs)
Th "a# "$ th M"!#ica !si& $$"rt is t" !si& .h,sica# s,st1s 1"!#i& #a&(a
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&
c"&&cti"&s a&! ii& .ara1tr a#(s i& !ia#"( '"=s) "&str(cts $"r i&c#(!i& ra.hica#
a&&"tati"&s i& M"!#ica 1a ic"&s a&! c"1."siti"& !iara1s ."rta'# 't;& !i$$r&t t""#s2)
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A& =a1.# "$ th t=t(a# r.rs&tati"& "$ a si1.# 1"t"r !ri s,st1 is?
1"!# M"t"rDri
ID c"&tr"##r?M"t"r 1"t"r@ar'"= ar &:00@
I&rtia i&rtia %:0@
*(ati"&c"&&ctc"&tr"##r)"(t"rt 1"t"r)i&"rt@
c"&&ctc"&tr"##r)i&"rt2 1"t"r)"(t"rt@
c"&&ctar)$#a&Ja 1"t"r)$#a&J'@
c"&&ctar)$#a&J' i&rtia)$#a&Ja@&! M"t"rDri@
Th c"1."siti"& !iara1 "$ th 1"!# c#ass M"t"r is sh";& '#";?
Fi(r 2):? M"t"r !ri ;ith M"!#ica #i'raris
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Structure model liraries
A .";r$(# .aca c"&c.t is aai#a'# t" str(ct(r #ar 1"!# #i'raris a&! t" $i&! a
c"1."&&t i& a $i# s,st1 ii& its hirarchica# M"!#ica c#ass &a1)
,yrid modeling
A (&i*( $at(r "$ M"!#ica is th ha&!#i& "$ !isc"&ti&("(s a&! aria'# str(ct(rc"1."&&ts s(ch as r#a,s s;itchs 'ari& $ricti"& c#(tchs 'ras i1.act sa1.#! !ata
s,st1s a(t"1atic ar'"=s tc) M"!#ica has i&tr"!(c! s.cia# #a&(a c"&str(cts a##";i&
a si1(#at"r t" i&tr"!(c $$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, '
"$ th 'asic !ata t,.s Ra# I&tr +""#a& Stri& "r c"1."&&t 1"!#s) This a##";sc"&&i&t !iscrtiati"& "$ si1.# .artia# !i$$r&tia# *(ati"&s)
Class parameters
+si!s "r!i&ar, &(1ric .ara1trs M"!#ica a##";s 1"!# c#ass .ara1trs) As a&
=a1.# ass(1 that a& a(t"-t(&i& c"&tr"##r sh"(#! r.#ac a I c"&tr"##r) It is "$ c"(rs
."ssi'# t" 8(st r.#ac th c"&tr"##r i& a ra.hica# (sr &ir"&1&t i)) t" crat a &; 1"!#)Th .r"'#1 ;ith this s"#(ti"& is that t;" 1"!#s 1(st ' 1ai&tai&!) M"!#ica has th
ca.a'i#it, t" i&sta! s('stit(t a 1"!# c"1."&&t s" "&#, "& rsi"& "$ th rst "$ th 1"!# is
&!!)
E6uations
M"!#s i& M"!#ica ar 1ath1atica##, !scri'! ', !i$$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)
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!(+ ,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)
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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
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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$$r&t i1.#1&tati"&s "$ th +a## a&! +a1 .r"css ha '& c"&si!r! '"th
aai#a'# at th c"(rs #a' "$ th D.art1&t "$ A(t"1atic "&tr"# s $i(r 3)2 a&! 3)3)
Fi(r 3)2? +a## a&! +a1 c"&&ct! ia ar'"=
Th 'a1 is c"&&ct! t" a #"cit, c"&tr"##! D-1"t"r ia a ar'"=) This 1as it
1(ch asir t" c"&tr"# th .r"css a&! thr is r, #itt# i&$#(&c "$ th cr"ss-c"(.#i&s $r"1th 'a## t" th 'a1)
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Fi(r 3)3? +a## a&! +a1 !irct#, c"&&ct! t" th a=is "$ a D-1"t"r
Dirct !ri& 'a1? Th 'a1 is !irct#, c"&&ct! t" th a=is "$ a D-1"t"r)
Th 'a1 a&# is 1as(r! '(t &"t th a&(#ar #"cit,) Th 'a## ."siti"& &!s a#s" t" '
1as(r! a&! i& th sta&!ar! c"&$i(rati"& "$ th .r"css at th D.art1&t "$ A(t"1atic"&tr"# th 1ta# 'a## c#"ss a& #ctrica# circ(it a#"& th rai# "$ th 'a1)
I& "(r =.ri1&ta# st(. s cha.tr 7 ; ha (s! a isi"& s,st1 $"r !tr1i&i& th
'a## ."siti"&) Th s.! "$ th 'a## &!s t" ' sti1at!)
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:2
Chapter *
)athematical modeling
This cha.tr !scri's h"; t" "'tai& th *(ati"&s "$ 1"ti"& $"r th +a## a&! +a1
.r"css)
*(1 Ball and Beam )odel
Th *(ati"&s $"r "'tai&i& th 1"!# i& M"!#ica ha '& ra#i! ', 1a&s "$ t;"
!i$$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 =
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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)
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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$$cts th a&#)
*(1(! Lagrangian methods
Th c"&tr"# "'8cti is t" c"&tr"# th t"r*( τ a..#i! at th .i"t "$ th 'a1 s(chthat th 'a## ca& r"## "& th 'a1 a&! trac a !sir! tra8ct"r,) Th t"r*( ca(ss th(s acha& "$ th 'a1 a&# a&! a 1"1&t i& th ."siti"& "$ th 'a##)
τ O '
r '
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Th .ara1trs "$ th 'a## a&! 'a1 ar !$i&! as $"##";s?
I ' Î +a1>s 1"1&t "$ i&rtia
O 'Î +a1 as Î Mass "$ th 'a##Is Î +a##>s 1"1&t "$ i&rtiar s Î Ra!i(s "$ th s.hrr ' Î "siti"& "$ th 'a##
Th i&tic &r, "$ th s,st1 is?
T = T beam + T
sphere
T beam
= T beam
frame
+ T beam
beam
body
center
T sphere = T
sphere frame + T spherebeam body
center
Each &r, is?
T beam
frame = 0
¾ Th r"tati"&a# i&tic &r, "$ th 'a1 is?
: 2T beamcenter
= ⋅ #b
⋅θMb
2
¾ Th 'a## has i&tic &r, ;ith rs.ct t" th $ra1 i& '"th ra!ia# a&! circ(#ar 1"ti"&?
: ( 2 ) 2 : 2T sphere frame
= ⋅ m s
⋅ r b
2⋅θM
b
+ ⋅ m s
⋅ r Mb
2
¾ Th r"tati"&a# i&tic &r, "$ th s.hr a'"(t its '"!, c&tr
: 2T spherecenter
= ⋅ # s
⋅θM s
2
(4)5)
Th r"tati"&a# i&rtia "$ th r"##i& s.hr is
# =2
⋅ m s5
s⋅ r
2
(4))
(4)6) s
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a&! a..#,i& a r"##i& ;ith"(t s#i..i&?4) (4)7)
r Mb = r
s ⋅θM
s
S('stit(ti& th *(ati"&s 4)6 a&! 4)7 i&t" th *(ati"& 4)5 ; t?
: ⎡ 2 ⎤ ⎡ : ⎤ : ⎡ 2 ⎤
T spherecenter =
2⋅ ⎢
5⋅ (m ⋅ r
2 ) ⋅
r2
⋅ r M 2
⎥ =
2⎢ 5
⋅ m s
⋅ r M 2
⎥ ⎣ ⎦ ⎣ s ⎦
⎣ ⎦
Th i&tic &r, "$ th s,st1 is?(4))
T =:
⋅ ⎡ # ⋅θM 2
+
(m⋅ r
2 )⋅θM 2+
m
⋅ r M 2
+
2⋅ m ⋅ r M
2
⎤
(4))
4)9
2 ⎣ b b s b b s b 5 s b ⎦
Si1.#i$,i& th *(ati"& 4)9 ; ha?
T =:
⋅ ⎡( # + m ⋅ r
2 )⋅θM 2
+
7⋅ m ⋅ r M
2 ⎤
2 ⎣
b s b b5
s b
⎦
Th r"##i& 'a## a#"& =hi'its th ."t&tia# &r, "$ th s,st1?
8 = m s
⋅ g ⋅ r b
⋅ si&θb
T" s(1 (.?
¾ 7I'ETIC E'E9:
T =:
⋅ ⎡( # + m ⋅ r
2 )⋅θM 2
+
7⋅ m ⋅ r M
2
⎤
(4))
(4)C)
2 ⎣
b s b b5
s b
⎦
¾ .%TE'TIAL E'E9:
8 = m s
⋅ g ⋅ r b
⋅
si&θb
(4))
(4):0)
s ⎥ ⎢
b b b
⎢( )
⎥
⎢⎥
⎢ ⎥
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T" ;rit th *(ati"&s "$ 1"ti"& ; !$i& th Lara&ia& L t" ' th !i$$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
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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
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: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"&)
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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:@
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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$$ct th a&# "$ th 'a1)
/(1(! )odelica model4 Lagrangian method
1"!# 'a##J'a1J2
i1."rt M"!#ica)SI(&its@
// ara1trs
.ara1tr SI(&its)Mass 10): QMass "$ 'a##Q@
.ara1tr SI(&its)L&th L:)0 Q+a1 #&thQ@ .ara1tr SI(&its)Ra!i(s r0)0:5 QRa!i(s "$ th 'a##Q@ .ara1tr SI(&its)M"1&t<$I&rtia %2/51r2 QS.hrBs 1"1&t "$ I&rtiaQ@
.ara1tr SI(&its)M"1&t<$I&rtia I:7 Q+a1 I&rtia a#(Q@
// "&sta&ts
c"&sta&t Ra# M"!#ica)"&sta&ts)J& Qraitati"&a# Acc#rati"&Q@
// Varia'#s
Ra# thta Q+a1 A&#Q@
Ra# ; QA&(#ar V#"cit,Q@Ra# a#.ha QA&(#ar Acc#rati"&Q@Ra# =start0)0 Q"siti"& th 'a## i& th 'a1Q@
Ra# QV#"cit, "$ th 'a##Q@
Ra# a QAcc#rati"& "$ th 'a##Q@Ra# ta(start0)0 QT"r*(Q@
*(ati"&
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!r=@a !r@
; !rthta@
a#.ha !r;@
// Lara&ia& 1th"!
I 1=2a#.ha 21=; 1=M"!#ica)Math)c"sthta ta(@a 5/7M"!#ica)Math)si&thta - =;2 0@
// "&tr"# #a;ta( 0@
&! 'a##J'a1J2@
Fi(r 5)2? D,1"#a si1(#ati"&? Rs."&s "$ 'a1 a&# ."siti"& "$ th 'a## a&! #"cit, <.&-
#"". rs."&s $"r th Lara&ia& 1"!#
This 1"!# is a..r".riat $"r a 'a1 ;hich is !irct-!ri& $r"1 th 1"t"r &" ar'"=
a&! as, t" 1" ', ha&!) Thr$"r th 'a## ."siti"& a&! th c"rrs."&!i& t"r*( !( t"rait, ;i## a$$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& ' $"(&! i& th
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Mastr Thsis r"8ct ', air a##ir C "r Fra&csc" a#(i :0)
/(! )odel #ith )odelica0s liraries
This scti"& =.#ai&s a&"thr a#tr&ati t" 1"!# r"##i& "& a s(r$ac)
Th $r M(#ti+"!, #i'rar, c"&tai&s 3-!i1&si"&a# 1cha&ica# c"1."&&ts t" 1"!# rii!
1(#ti-'"!, s,st1s s(ch as r"'"ts sat##its "r hic#s)
Th #i'rar, .r"i!s 'asic 1"!# c#asss $"r rii! '"!is 8"i&ts $"rcs 1as(r1&t a&!
a&i1ati"& #1&ts) R"#(t .ris1atic a&! "thr i!a# 8"i&ts c"&&ct '"!is) i&1atic #"".sca& ' ha&!#! ', (si& c(t-8"i&ts t" 'ra th #"".s)
F"r a (sr it is as, t" i&tr"!(c &; c"1."&&ts "r c"., a&! 1"!i$, =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::@
i&&r .ara1tr SI)A&# !#taM"!#ica)"&sta&ts).i/:0@
+as +as: a&&"tati"& =t&t-60 0@ -40 20@
R"##i&+"!, R"##i&+"!,:*start:000Wrstart0:0WI: 0 0@ 0 : 0@ 0 0 :start0)0500W"1astart-:-:-0)05W a&&"tati"& =t&t20 0@ 40 20@E##i.s"i!J"&J#a& E##i.s"i!J"&J#a&: a&&"tati"& =t&t-20 0@ 0 20@
*(ati"&
c"&&ct+as:)I&"rtR"## E##i.s"i!J"&J#a&:)<(t"rtAa&&"tati"& ."i&ts-50 :C@ -50 30@ -:3 30@ -:3 :C@
c"&&ct+as:)<(t"rt E##i.s"i!J"&J#a&:)I&"rtA
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a&&"tati"& ."i&ts-50 :@ -50 -:0@ -:3 -:0@ -:3 :@
c"&&ctE##i.s"i!J"&J#a&:)I&"rt+ R"##i&+"!,:)<(t"rt
a&&"tati"& ."i&ts-7 :@ -7 -:0@ 30 -:0@ 30 :@c"&&ctE##i.s"i!J"&J#a&:)<(t"rt+ R"##i&+"!,:)I&"rtR"##
a&&"tati"& ."i&ts-7 :C@ -7 30@ 30 30@ 30 :C@
&! TstS.hr@
F"r 1"r !tai#s .#as r$r t" th ;' .a?htt.?//;;;)1"!#ica)"r/"&$r&c2003/.a.rs)sht 1 # :2
Th architct(r "$ i&$"r1ati"& i&tracti"&s is sh";& '#";?
Fi(r 5)3? Architct(r "$ "'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
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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
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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
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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
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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, $"r th s,st1 t" ' c"1.#t#, stat "'sra'# th
"'sra'i#it, 1atri=
⎡ ⎤ ⎢ ⎥
: = ⎢ A ⎥o ⎢A
2 ⎥ ⎢ 3 ⎥
⎣A ⎦
0
c
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1(st a#s" ha ra& "$ 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
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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)
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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)
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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
P (s th 'a## a&! 'a1 .r"css sh";& i& Fi(r 7)2) Th 'a1 is c"&tr"##! ', a D-
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
&t;"r c"&&cti"& t" a Li&(= r(&&i& th c"&tr"# i1.#1&tati"& i& a ra#-ti1 =t&si"& "$ Mat#a'/Si1(#i& !#".! ', A)+#"1!##) A stch "$ th c"&&cti"&s ca& ' is(a#i! i&
Fi(r 7):)
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ha.tr 7) E=.ri1&ts
Fir-i
Visi"& [ P i& ! " ;s 0
1atc"11
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#
.r"css "$ th 'a## a&! 'a1 i&c#(!i& $i&!i& a..r".riat .ara1trs) T" $i&! this c"rrct1"!# ; !" s"1 =.ri1&ts ;ith th ra# .r"css) Th K'a## a&! 'a1-.r"css is sh";&
'#"; 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
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; 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
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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
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%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
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Fi(r 7)C? Si1(#i& 1"!# t" !" th 'a1 c"&tr"#
40
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Fi(r 7):0? Si1(#i& 1"!# t" !" th 'a## ."siti"& c"&tr"#s tr"(h th isi"& s,st1
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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""! $r"1 ha&!s-"& =.ri&c ;ith th .h,sica#
s,st1)
42
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List of &igures
Chapter !
Fi(r 2):? M"t"r !ri ;ith M"!#ica #i'raris )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 6
Chapter +
Fi(r 3):? Diara1 "$ +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
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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" $i&! "(t th ai& ;ith rs.ct t" th ."siti"& .r"i!! ', th
isi"& s,st1 ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 3C
Fi(r 7)C? Si1(#i& 1"!# t" !" th 'a1 c"&tr"# ))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 40
Fi(r 7):0? Si1(#i& 1"!# t" !" th 'a## ."siti"& c"&tr"#s tr"(h th isi"& s,st1 )))))) 4:
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Biliography
: Hi#!i& E#1*ist Da +r]c S& Eri Mattss"& Ha&s <#ss"& a&! Marti& <ttr Dymola Dynamic 6odelling &aboratory *ser<s 6anual+ rsi"& 5 S.2002
2 D,&asi1 A+ Rsarch ar I!"& SE-22370 L(&! SPEDENhtt.?//;;;)D,&asi1)s
3 M"!#ica H"1.a?
htt.?//;;;)1"!#ica)"r
4 D,1"#a H"1.a?htt.?//;;;)!, 1"#a)c"1
5 M"!#ica Ass"ciati"& 2002?
=6odelica . A *nified >b?ect.>rientec &anguage for @hysical Systems
6odeling=+ &anguage Specification+ 8ersion /1+ ,1 7anuary /11/
6 Micha# Ti##r #ntroduction @hysical 6odeling with 6odelica 200:
#(;r Aca!1ic ('#ishrs +"st"& IS+N 0-7C23-7367-7
7 +i&&r t(t"ria#?htt.?//;;;)1"!#ica)"r/"&$r&c2002/+i&&rsT(t"ria#)i.
9 M"!#ica !"c(1&ts?htt.?//;;;)1"!#ica)"r/!"c(1&ts)sht1#
C A!a&c! t(t"ria#?
htt.?//;;;)1"!#ica)"r/"&$r&c2002/A!a&c!T(t"ria#)i.
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:0 Mastr Thsis r"8ct?
Bavier allier ) D.art1&t "$ A(t"1atic "&tr"#) L(&! I&stit(t "$ Tch&"#",)
A((st 2002 ^ %a&(ar, 2003
:: Mastr Thsis r"8ct?
2rancesco alugi) D.art1&t "$ A(t"1atic "&tr"#) L(&! I&stit(t "$
Tch&"#",) <ct"'r 200: ^ A.ri# 2002
:2 htt.?//;;;)1"!#ica)"r/"&$r&c2003/.a.rs)sht 1 #
Sssi"& 9AMchatr"&ic S,st1s - II
2C7
Ia& I) "ss&" a&! Maia S) Star"saia M"sc"; Stat U&irsit, "$ thSric R(ssia? Cow >ne an Simulate Dynamics of 'olling %odies via Dymola Approach to 6odel 6ultibody System Dynamics *sing 6odelica
:3 htt.?//;;;)c"&tr"#)#th)s/!(cati"&/ 1 astrthsis/+a+Ji!")&.
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