INVERTED PENDULA SIMULATION AND MODELING - A GENERALIZED APPROACH

Transcription

1 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc INVERTED PENDULA SIMULATION AND MODELING - A GENERALIZED APPROACH JADLOVSKÁ SLÁVKA, JADLOVSKÁ ANNA Deparme of Cyberec ad Arfcal Iellgece, Faculy of Elecrcal Egeerg ad Iformac, Techcal Uvery of Košce, Leá 9, 4 Košce, Slovak Republc, e-mal: jadlovka@gmal.com, Aa.Jadlovka@uke.k Abrac: The am of h paper o provde a complex vew of he modelg ad mulao of vered pedula yem. Crucal modelg procedure uch a he dervao of dffereal moo equao for vered pedula yem ad ymbolc learzao wh repec o a gve equlbrum po are preeed form of ymbolc MATLAB algorhm, geeralzed for a yem of vered pedula. The algorhm are he ued o derve accurae mahemacal model of a gle ad double vered pedulum a well a her lear approxmao. The paper he pree Ivered Pedula Modelg ad Corol, a hemac Smulk block lbrary deged by he auhor. A par of he lbrary, he ope-loop dyamc of he model aalyzed a ere of mulao experme ad a e of uable ae-pace corol algorhm ha ablze he pedulum he vered poo deged ad uppored by a e of lbrary block. Keyword: olear dyamcal yem, yem of vered pedula, geeralzed modelg, aepace corol, MATLAB/Smulk block lbrary INTRODUCTION Ivered pedula yem repree a gfca group of mechacal yem ued corol educao wh a varey of praccal applcao, cludg (ee [Jadlovká, 9], [Sula, 4]): mulao of he uable yem of a huma or roboc upper lmb f he ceer of preure placed below ceer of gravy modelg a huma or a robo adg uprgh mulao of a pace hule or a rocke akg off mle gudace, f hru acuaed a he boom of a all vehcle Thorough phycal aaly geerally requred o oba mahemacal expreo ha model he real yem dyamc wh uch accuracy ha poble o ue hem a ubue cae a laboraory model uavalable. Depe he fac ha vered pedula aalycal defcao codered a well-explored maer ad he equao of moo of h ype of a yem are adardly cluded a umber of ource (e.g. [Schlegel e al., 5] provde he equao for a gle, double ad rple pedulum model), a geeral algorhm, whch would oupu he equao of moo for ay gve umber of pedula, ha o ye bee roduced. Sce he force ummao mehod baed o Newo law of moo ed o be error-proe ad cao be ealy raformed o a algorhm, h paper ue he Lagrage approach o perform he dervao of he moo equao. I order o oba a prece a approxmao of he real model dyamc a poble, Raylegh dpao fuco ha decrbe he vcou yem dampg ad frco wa egraed he adard Euler-Lagrage equao. The bgge advaage of he Lagrage mechac employme ha ca be ealy algorhmzed o a MATLAB fuco (m-fle). I a mlar geeralzed way, a learzed model, whch eceary for ay lear feedback coroller deg, creaed. The ue of lear coroller upo olear yem jufed by he ealy verfable aumpo ha he behavor of a lear approxmao ear o he equlbrum po how lle error compared o he olear orgal. Block lbrare repree he objec-oreed, eve-flow-drve, uer-fredly problem-olvg approach wh he MATLAB/Smulk evrome. Through he Smulk Lbrary Brower, a umber of pre-alled (Toolbox) lbrare ca be acceed ad ued o olve or mulae varou cefc ad echcal ue by mea of block ercoecg. Furhermore, o provde a wder varey of problem wh uch uer flexbly, cuom maked block may be creaed ad grouped o C4a

2 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc uer-deged lbrare. The fal eco of he paper decrbe a block lbrary whch wa developed o provde uch ofware uppor for he mulao ad corol of vered pedula yem. MATHEMATICAL MODELING. Moo Equao Dervao Geeral Procedure The olear mechacal SIMO yem of vered pedula o a car compoed of homogeou, oropc rod whch are jo-boud ogeher ad aached o a able movg bae. The pu of uch a yem he force acg upo he car; he mulple,.e. + oupu are repreeed by car poo [ m ] ad pedula agle [ rad ]. Oly he car poo drecly affeced by he pu force, herefore ay yem of vered pedula codered o be uder-acuaed. Th eco oule he heorecal backgroud, aumpo ad derved formula ha led o he creao of a algorhm whch derve he equao of moo for ay gve umber of pedula aached o a car. The algorhm wa mplemeed o MATLAB uder he ame of vpederv.m. The umber of pedula eed o be pecfed a he fuco parameer. Throughou he dervao proce, we aume ha all moo boud o he xy plae wh he car movg alog he le decal o x ax, whch a he ame me repree he projeco of he zero poeal eergy level o he xy plae he value of every agle deermed clockwe from he uprgh poo all parameer are dexed he followg maer: aged o he car, o repree he dvdual pedula arg wh he pedulum rod aached drecly o he car Le u fr roduce a vecor of geeralzed coordae, whch correpod o he yem + degree of freedom,.e. oupu: ( ) = ( θ ( ) θ ( ) K θ ) T θ () The Euler-Lagrage equao repree he yem degree of freedom each ad he codeed vecor form hey appear a: d L L D + = Q * d θ () & θ θ& where Lagrage fuco (Lagraga) defed a he dfferece bewee he yem kec ad poeal eergy E θ = E θ θ& L K, P () Raylegh (dpave) fuco decrbe he vcou (frco) force wh he yem D = Dθ & (4) * ad Q he vecor of geeralzed exeral force acg upo he yem. From ow o, we wll ue x, y o deoe he coordae of poo ad v x, wll deoe he veloce he dreco of he axe. The omeclaure of he umercal parameer ha decrbe he yem wll obey he adard coveo: m kg ma of he car ( = ) ad he pedula ( = o = ) [ ] [ m] l legh of -h pedulum [ m ] [ ] [ ] kg, kgm g gravaoal accelerao ( g = 9,8m wll be ued) δ frco coeffce of he car aga he urface ( = ) / dampg coa relaed o he pvo po of -h pedulum ( = o = ) Coruco of Lagrage moo equao: Ou of all codered ubyem, oly he dyamc of he car drecly affeced by he exeral force acg upo he yem. Therefore, he vecor of exeral force ha he followg form: * = ( F K ) T Q (5) v y C4a

3 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc Sce he oal eergy of a mul-body yem gve a he um of eerge ha bef he dvdual bode, he relao ha characerze a yem of vered pedula o a car are: E K = E, E = E, D ) = D = K P = P ( (6) The ue of Lagrage equao herefore raform he proce of dervg he moo equao o he deermao of kec, poeal ad dpao eerge for he car ad all pedula. Thee eed o be expreed erm of he geeralzed coordae. Eergec balace of he car: Aumg he car moo o be lear, we ca decrbe mahemacally ug a gle =. paal dmeo. Idefyg a he x ax, he oly poo coordae deoed a x θ Therefore, he poeal eergy of he car equal E P = (ee aumpo.). The kec eergy ad he dpao fuco boh deped o he car velocy: E D K m v = m & θ = = x (7) = x (8) δ v = δ & θ Eergec balace of -h pedulum: Le u uppoe ha he whole ma of a pedulum rod coceraed ceer of gravy (CoG) whch decal o he geomercal ceer of he rod he dace of l from he pvo po. The coordae of he CoG of -h pedulum rod are hece expreed a: l θ + lk θk θ x k = = y l l k coθk coθk k = whle he veloce he dreco of he axe equal: & l θ + l & kθk coθ & k θ coθ vx k = = vy l l & + kθk θ & k θ θ k = The poeal eergy of -h pedulum defed by he hegh of he CoG above he x ax: l E P = m gy = m g lk coθ k coθk () k = ad he kec eergy of each pedulum a um of wo expreo ha decrbe he pedulum ralaoal ad roary moo: EK = mv + J & Tθ () where J T = ml he pedulum mome of era wh repec o he ceer of gravy ad v v + v = he magude of -h pedulum ralaoal velocy. x y The dpao propere of he -h pedulum deped quadracally o he agular veloce of pedulum marked a ad : (9) () C4a

4 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc D = δ θ & θ & () whch yeld D = δ & θ f =. The fuco of MATLAB Symbolc Mah Toolbox eabled u o clude he abovemeoed phycal relaohp he vpederv.m ad, furhermore, o geerae a mplfed ad rearraged form of he equao, equvale o he mo lkely form obaed by maual dervao. A example of he commad wdow oupu produced by vpederv.m led [Jadlovká e al., 9]. I addo o he eveual ymbolc moo equao he prey form, all phycally gfca ep of he dervao proce are dplayed ad ca be racked. Fg. Sgle ad double vered pedulum o a car cheme ad bac omeclaure. Sgle ad Double Ivered Pedulum By eg o, we oba a yem of a gle vered pedulum (Fg., lef), whch compoed of a pedulum rod aached o he car. The eq=vpederv() commad wa ued o oba he model whch co of ecod-order olear dffereal equao ha decrbe he car ubyem: ( m + m )& θ + δ & θ + m l (&& θ coθ & θ θ ) = F (4) ad he pedulum ubyem: J & θ + δ & θ + ml & θ coθ mgl θ = (5) where J = m l ad for he pedulum mome of era wh repec o he pvo. Coecg a couple of rgd rod a jo ad aachg oe of hee o a car produce a yem of a double vered pedulum (Fg., rgh). Aalogcally o he gle vered pedulum yem, eq=vpederv() reur he ecod-order olear dffereal equao ha decrbe he car ubyem: ( m + m + m )&& θ + δ & θ + m l + m l && θ coθ & θ θ + ml he lower pedulum ubyem: (&& θ coθ & θ θ ) = F C4a 4 ( J + m l )&& θ + ( δ + δ )& θ δ & θ + m l + m l && θ coθ mll (&& θ co( θ θ ) + & θ ( θ θ )) m + m gl θ = + + (6) (7)

5 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc ad he upper pedulum ubyem: J && θ + δ (& θ & θ ) + ml && θ coθ + (8) + mll ( && θ co( θ θ ) & θ ( θ θ )) mgl θ = where J = ml, J = ml ad for he mome of era of he lower ad upper pedulum wh repec o her pvo po. The auomacally geeraed olear dffereal equao were, boh cae, decal o he equao derved maually (compare (5),(6) o [Roubal, ], [Schlegel e al., 5], (7)-(9) o [Bogdaov, 4], [Demrc, 4]), whch cofrm he valdy of he algorhm. I worh og ha he complee dervao procedure he cae of a double ad rple vered pedulum, a wa cluded e.g. [Schlegel e al., 5], eed o have bee doe. The geeral phycal relaohp led. ead mply ha oce he gle vered pedulum model ha bee derved, all eergec balace relaed o he car ad he lower pedulum of a double vered pedulum yem are already kow ad oly hoe whch decrbe he upper pedulum eed o be compued. LINEAR APPROXIMATION OF INVERTED PENDULA SYSTEMS We wll ex focu o he aaly of vered pedula yem baed o he ae-pace heory of couou dyamcal yem, whch decrbe a olear SIMO yem wh ue of a dffereal ae equao ad a algebrac oupu equao: x& = f ( x, u, ), (9) where y( ) = g( x, u, ) x he ae vecor, u he calar pu value, y () he oupu vecor. Amog he cocluo draw from he aaly above wa ha he order of a yem of vered pedula o a car +. Therefore, a ae vecor he followg form wa roduced o decrbe he yem: T = x x x = θ θ T x &... + () ad he force acg upo he car wa logcally defed a he oly pu of he yem: F u = () The oupu equao defed uch a way o ha vecor y () would eher repree he vecor of T geeralzed coordae θ, or he whole ae vecor (9), f eceary. To deerme he ae equao x & = θ& & θ, he Lagrage moo equao eed above all o be rewre o he ocalled mmal ODE form M ( θ )& θ + Nθ (, θ& ) θ& + Pθ ( ) = V () whch make poble o olae he dervave of ae-pace vecor: θ& θ& = && θ ( Mθ ( )) ( V Nθ (, θ& ) θ& + Pθ (& ) whch every eleme of θ, θ & ca be ubued by () x couerpar. The fac ha a olear auoomou dyamc yem ha o edecy o chage ae f equlbrum correpod o equale x & = ad u = u = S (4) By olvg hee, we oba he p equlbrum po of he yem ( j =,,..., p ): T j x S = x S xs... x( + ) S (5) C4a 5

6 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc Wh ue of he Taylor ere, we ca ow creae he lear approxmao o he whole ae equao f, u f * x, u : by ubug ( x ) by ( ) f ( ) ( x, u ) f ( ) ( x u ) x, u f xs, us + xk xks + ( u us ) (6) x u * +, k = k x S, us x S, us f T For he cae of he uprgh poo, he equlbrum x = x = ad ce u = ad he ae-pace decrpo of he phycally realzable learzed yem gve a x& = Ax + bu = Cx S u, y The decrbed raformao of he Lagrage mahemacal model o a ae-pace marx form wa mplemeed o MATLAB. Sce he complexy of ymbolc marce creae grealy wh creag order of he yem, oly he ae-pace marce of he gle vered pedulum model he uprgh poo, produced by marce_gle.m, are dplayed here a a example. A = 6 l mg 4δ 6δ 4 (8) = b = C 4m + m 4m + m l 4m + m 4m + m g( m + m ) 6δ δ( m + m ) 6 4m + m l 4m + m m l 4m + m l ( 4m + m ) Oce aga, he commad wdow oupu of he fuco ca be prevewed [Jadlovká e al., 9]. The geeraed ae-pace marce were proved o be accurae (compare (8) o [Schlegel, 5]). 4 IPMAC INVERTED PENDULA MODELING AND CONTROL (SIMULINK BLOCK LIBRARY) A rucured Smulk block lbrary uder he ame of Ivered Pedula Modelg ad Corol (IPMaC) wa deged o provde ofware uppor for he aaly ad yhe of vered pedula yem. The IPMaC ca be fully egraed o he Smulk Lbrary Brower ad ued decally o he pre-alled Smulk block lbrare. The followg eco provde a bref gh o he lbrary fucoaly. = S (7) Fg. Smulk block of vered pedula model cluded he IPMaC lbrary 4. Ope-Loop Dyamcal Aaly The mahemacal model of a gle ad double vered pedulum were mplemeed o he programmg evrome of MATLAB/Smulk form of aomc lbrary block: Sgle Ivered Pedulum o a Car ad Double Ivered Pedulum o a Car; boh wh her ow co ad a dyamcal paramerc block mak. The block mak of each mplemeed yem make poble o C4a 6

7 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc chage he yem parameer, pecfy he al codo (whch eable he al defleco aaly), eable or dable he pu force ad adju he umber of oupu, whch equvale o equppg a real model wh eor. The block hemelve have a cell rucure,.e. each olear equao ha par of he yem mahemacal model correpod o a ubyem block ercoeced wh he oher wh repec o her muual phycal relao. A a llurao, Fg. ad Fg. 4 depc he er rucure of ubyem block Car ad Pedulum wh he Sgle Ivered Pedulum o a Car block, boh of whch repree a exac raformao of olear equao o Smulk block dagram. F 4 df /d f df /d (m*l)/ d/d cle d/d co /(m+m) df d df d () x o Iegraor df d df d -C- -C () x o Iegraor f f dela df /d df /d Fg. The Car ubyem wh he fuco block Sgle Ivered Pedulum o a Car df /d co -Kcoe fuco (m*l)/ Do Produc e fuco (m*g*l)/ /J df d df d x o (4) Iegraor df d dfd -C- -C () x o Iegraor f f f dela df /d df /d Fg. 4 The Pedulum ubyem wh he fuco block Sgle Ivered Pedulum o a Car Creag complee ad fucoal mulao model of vered pedula o a car form of a aomc co allow for dealed obervao of her dyamc wh o addoal modelg apar from pu/oupu block afflao. The aalye of he ope-loop dyamcal behavor of boh he gle ad double vered pedulum yem were performed a a repoe o a gal coraed erm of me ad amplude, cluded he IPMaC a he Impule block. To vew he gal geeraed durg mulao, Scope ad Scope raddeg block were ued, he laer dplayg he gal degree raher ha rada. Boh cheme ca be ru from he Demo Smulao eco of he IPMaC, whch bacally a colleco of lk o mulao cheme whoe purpoe o olve varou aaly- ad yhe-relaed problem. The cheme are compoed early excluvely of he IPMaC block. The dyamc of gle vered pedulum yem wa aalyzed for wo group of parameer: group I: m =. kg, m =. 75kg, l =. 5m, δ =. kg, δ =. 48kgm group II: m =. kg, m = kg, l =. 8m, δ =. kg, δ =. kgm ad he umerc value ued he double vered pedulum yem mulao were: m =. kg, m =. 75kg, m =. 75kg, l =. 5m, l =. 5m, δ =. kg, δ =. kgm, δ =. kgm If we ake a cloer look a he mulao reul (Fg. 5, Fg. 6), everal cocluo ca be draw depedely of he umber of pedula aached o a car: From he mome he car ar o move a a repoe o he me-coraed pu force mpule, velocy decreae hrough me ad gradually come dow o zero becaue of he pree frco. All pedula fall couerdreco o he car ( Newo Law of era), pag hrough C4a 7

8 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc ocllaory rae ae before fally ablzg hemelve he able equlbrum po whch hey are all pog dowward. The backward mpac of each pedulum o he car, whch creae wh he wegh of he load aached, alo vble. Sce uch ope-loop behavor correc compared o geerally kow emprcal obervao of pedula behavor, ca be cocluded ha boh model dplay accepable overall performace..4 Car Poo Aaly - Sgle Ivered Pedulum Pole Agle Aaly - Sgle Ivered Pedulum. Car Poo - ParamGroup Car Poo - ParamGroup -5 Pole Agle - ParamGroup Pole Agle - ParamGroup Car Poo [m] Smulao me [] Smulao me [] Fg. 5 Sgle Ivered Pedulum o a Car - car poo ad pedulum agle.4 Car Poo Aaly - Double Ivered Pedulum 5 Pole Agle (Lower & Upper) Aaly - Double Ivered Pedulum. Car Poo Pole Agle (Lower) Pole Agle (Upper) -5 Car Poo [m] Smulao me [] Smulao me [] Fg. 6 Double Ivered Pedulum o a Car car poo, upper ad lower pedula agle 4. Verfcao of Sae-Space Corol Algorhm The couou lear feedback mehod were employed o demorae he corollably propere of he mulao model of gle vered pedulum. The corol objecve wa o ablze T he pedulum he uprgh (vered).e. uable poo,.e. o maa he equaly x =, whle he dvdual approached problem were: al defleco of he pedulum (ozero al codo) compeao of a me-coraed durbace pu gal rackg a requred poo of he car or a combao of he hree. The pedulum had o be kep uprgh ay cae. I kow (e.g. from [Jadlovká, 9], [Demrc, 4], [Jadlovká, ]) ha f a x, brg he yem o he org of feedback ga k appled o a meaurable full ae vecor he ae pace. A pecfed ozero requred value w requre a addoal, epo ga k v. The corol law wa herefore coruced form of he followg um: u = u f + u v + d u = k x + k v w + d u (9) where = kx he feedback compoe, = k w he epo compoe ad u f u v he umeaured durbace pu. A uch, he corol law evaluaed wh he Sae Space Coroller (SSC) block from he IPMaC. The block dyamc mak allow he uer o pck he mehod o deerme he feedback ga vecork : he pole-placeme algorhm or he lear quadrac regulao (LQR) opmal corol mehod are avalable, boh of whch are uppored by Corol w ad Toolbox form of bul- fuco (acker/place, lqr). The ozero epo pu durbace pu d u may opoally be eabled or dabled o a o adju he block appearace o mach he corol objecve. v d u C4a 8

9 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc I cae we uppoe ha meaureme lmao make mpoble o rereve he ae pace vecor a a whole, a emaor cluded he corol loop o provde he approxmaed (recoruced) ae vecor xˆ. The prcple of Lueberger ae emaor le he gradual mmzao of he emao error ~ x = x xˆ. To keep he me behavor of he error depede of yem parameer,.e. o maa: ~ x & = ( A LC) ~ x () where L he emaor ga marx, he emaor creae a model of he orgal yem he form: x & ˆ = Axˆ + bu + L( y Cxˆ ) () whch he relao evaluaed wh he Sae Emaor block. Oce aga, he block mak allow he ga o be deermed aleravely hrough pole-placeme or lear quadrac corol mehod, accordg o he uer choce. xe xe'=axe+bu +L(y-ye) Sae Emaor u y Impule Sep x du w Sae Space Coroller Sae Space Coroller u x' = Ax+Bu y = Cx+Du Sae-Space Scope Scope raddeg xe xe'=axe+bu +L(y-ye) u y Sae Emaor Coa f car poo Scope x Sae Space Coroller u F exeral force o he car Sgal w Sgal Bulder Sae Space Coroller f pole agle Scope raddeg Sgle Ivered Pedulum o a Car Fg. 7 Example of corol mulao cheme The example cheme above llurae wo way of ercoecg he block ha reul a corol cheme for he gle vered pedulum yem. A mulao cheme of a learzed ad a olear model how, wh he Sae Space Coroller ad Sae Emaor block a par of boh cheme. The adjume of he umber of block pu alo demoraed. A wa he cae wh ope-loop aaly, hee cheme ca be locaed he Demo Smulao eco of he IPMaC. Fg. 8 ad Fg. 9 docume he me-depede behavor for boh he car poo ad he pedulum agle of he olear gle vered pedulum yem cae he corol objecve o maa he dered car poo whle keepg he pedulum uprgh; o durbace pu wa codered. I order o ue he lear mehod of yhe, he learzao of he olear vered pedulum yem wa performed by callg he marce_gle.m fuco. Ug he parameer from group I (eco 4.), he followg lear ae-pace marce were obaed: A = , b = , C = () C4a 9

10 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc I Fg. 8, he effec of elecg ( ) how. The feedback ga vecor k of ( ) for he deged pole of he yem wa compued ad appled o he yem. Fg. 9 depc he reul of a LQR-baed coroller deg. I he T LQ + mmzed quadrac fucoal J x Qx u Ru d = dagoal ad choe o be dag( 5 ) f f, boh weghg marce are Q =, R =. The prcple of eparably allow he feedback ad emaor ga o be deermed depedely of each oher,.e. ug a dffere mehod. Seg he emao error marx pole o ( ), whch made hem abou me faer ha he coroller pole, reuled he followg emaor ga marx: L = T..5 Car Poo - Referece Trajecory (Pole Placeme) Car Poo Sepo [m] Car Poo [m] 4 Pole Agle Sablzao - Pole Placeme Pole Agle Sepo [deg]. Car Poo [m] Smulao me [] Smulao me [] Fg. 8 Sgle Ivered Pedulum o a Car mulao reul for pole-placeme corol whou emaor (car poo, pedulum agle). Car Poo - Referece Trajecory (LQR + Pole-Placeme) 5 Pole Agle Sablzao - LQR + Pole Placeme Car Poo Sepo [m] Car Poo [m] 4 Pole Agle Sepo [deg].5. Car Poo [m] Smulao me [] Smulao me [] Fg. 9 Sgle Ivered Pedulum o a Car mulao reul for LQR corol wh pole-placemedeged emaor (car poo rackg, pedulum agle ablzao) The mulao reul reveal ha boh corol block do reaoably well. The ably of he deged block o corol he yem wh repec o all above-preeed requreme ha bee demoraed for boh mehod, alhough LQR corol produce lghly beer reul depe he eed for a emaor. Overally, he mulao reul jufy he ue of lear corol mehod o corol olear yem. CONCLUSION The purpoe of h paper wa o propoe a orgal cocepo of olvg he ak of modelg ad corol of vered pedula dyamcal yem. I focued o demorag he aalogy whch foud whe we derve mahemacal model for yem of vered pedula o a car for a chagg. Praccal mporace of ymbolc mahemacal ofware wa poed ou a Symbolc Mah Toolbox wa ued he proce of developme of geeral ymbolc procedure ha eher yeld he equao of moo of vered pedula yem ad hece auomaze he mahemacal modelg, C4a

11 9 h Ieraoal Coferece PROCESS CONTROL Jue 7,, Kouy ad Deou, Czech Republc or perform he ymbolc lear raformao of a pecfed olear yem. Such approach hould elmae all facual or umerc error ha hould are durg he proce of mahemacal modelg. The raformao of he derved equao of gle ad double vered pedulum yem o Smulk block cheme wa he ba of he creao of vered pedula mulao model whch were egraed he IPMaC, a rucured Smulk block lbrary deged by he auhor of h paper. The core of he lbrary herefore repreeed by he dyamc-maked mulao model, pre-prepared for ue ope-loop aaly a well a ae-pace coroller deg. The IPMaC block lbrary wa roduced [Jadlovká, 9] ad gog hrough coa mproveme proce. Ug he auomac mahemacal model dervao, furher expao of he modelg eco hould be raghforward. A eco o roary vered pedula, where he bae movg a plae raher ha a gle coordae, beg deged o erve a ofware uppor for a ewly-purchaed laboraory model of gle roary vered pedulum. To eable furher verfcao of he corollably propere of vered pedula yem, all cluded yem hould alo become corol pla. Fally, a wa hed [Jadlovká e al., 9], he releae of he ex vero of he IPMaC hould coa a oably expaded corol eco,.e. a wder varey of coroller block ad corol cheme addo o he already cluded feedback corol algorhm. I ummary, we beleve ha he dea of creag a hemac Smulk lbrary, whch would group accurae mulao model of mechacal yem ogeher wh ueful pu/oupu block, uable coroller block ad demorao mulao, could fd ue for a umber of ype of dyamcal yem. Lbrare of hydraulc or elecrcal yem could follow he ep of he IPMaC, whch we coder a a corbuo o modelg ad corol educao a echcal uvere. ACKNOWLEDGEMENTS Th reearch wa uppored by he Scefc Gra Agecy of Slovak Republc uder he Vega projec Mulage Nework Corol Syem wh Auomac Recofgurao (No./67/8), a well a by he Agecy for he EU Srucural Fud of he Mry of Educao of Slovak Republc uder he projec: Cere of Iformao ad Commucao Techologe for Kowledge Syem (projec umber: 6). REFERENCES JADLOVSKÁ, S. 9. Ivered Pedula Modelg ad Corol [ Slovak]. Bachelor he. Košce: FEI TU Košce, 64 p. JADLOVSKÁ, S.; JADLOVSKÁ, A. 9. A Smulk Lbrary for Ivered Pedula Modelg ad Smulao. I: 7h Aual Coferece Proceedg of he Ieraoal Scefc Coferece - Techcal Compug Prague, November 9, 9. [CD-ROM] SULTAN, K. 4. Ivered Pedulum, Aaly, Deg ad Implemeao, from hp:// c ROUBAL, J.. Nolear Pedulum Corol [ Czech]. Dploma he. Prague: Faculy of Elecrcal Egeerg,. Czech Techcal Uvery Prague SCHLEGEL, M.; MEŠŤÁNEK, J. 7. Lmao o he Ivered Pedula Sablzably Accordg o Seor Placeme. I: Proceedg of he 6h Ieraoal Coferece o Proce Corol, Šrbké Pleo, Jue -4, 7. [CD-ROM] BOGDANOV, A. 4. Opmal Corol of a Double Ivered Pedulum o he Car. Techcal Repor CSE-4-6, OGI School of Scece ad Egeerg, OHSU DEMIRCI, M. 4. Deg of Feedback Coroller for a Lear Syem wh Applcao o Corol of a Double-Ivered Pedulum. Ieraoal Joural of Compuaoal Cogo, Vol., No., p JADLOVSKÁ, A.. Modelg ad Corol of Dyamc Procee Ug Neural Nework [ Slovak]. Košce: Edo of Scefc Docume, FEI TU, Iformaech C4a