Slovak University of Technology in Bratislava Institute of Information Engineering, Automation, and Mathematics PROCEEDINGS

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1 Slovak Unvery of echnology n Bralava Inue of Informaon Engneerng, Auomaon, and Mahemac PROCEEDINGS 17 h Inernaonal Conference on Proce Conrol 9 Hoel Baník, Šrbké Pleo, Slovaka, June 9 1, 9 ISBN hp:// Edor: M. Fkar and M. Kvanca Jadlovká, A., Dolnký, K., Lonščák, R.: Applcaon of Degned Program Module n C# Language for Smulaon of Model of Dynamc Syem, Edor: Fkar, M., Kvanca, M., In Proceedng of he 17h Inernaonal Conference on Proce Conrol 9, Šrbké Pleo, Slovaka, , 9. Full paper onlne: hp://

2 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka APPLICAION OF DESIGNED PROGRAM MODULES IN C# LANGUAGE FOR SIMULAION OF MODELS OF DYNAMIC SYSEMS A. Jadlovká, K. Dolnký, R. Lonščák Deparmen of Cybernec and Arfcal Inelgence, Faculy of Elecrcal Engneerng, echncal Unvery of Košce, Lená 9/A, 4 1 Košce e-mal: Anna.Jadlovká@uke.k, kaml@plazma.k, Rchard.Loncak@uke.k Abrac: he purpoe of h paper o gve a bref lluraon of poble of programng language C# n feld of modelng and clacal conrol heory. We decrbe mplemenaon of algorhm ha perform ranformaon of connuou lnear dynamc yem no her dcree equvvalen, connuou PID conrol no dcree form and employ hee algorhm n dcree cloed loop conrol. We alo how how mplemened numercal mehod can be ued o olve yem of dfferenal equaon whch are ued n modelng of nonlnear dynamc yem. All requred relaed funcon are mplemened and negraed no program module whch are ued ogeher n wo mlar applcaon degned o mulae conrol of lnear and nonlnear dynamc yem Ball & Plae hu verfyng robune of he PID conroller. Keyword: Velocy algorhm, dgal conroller, Ball & Plae, C# n modelng 1 INRODUCION Feld of modelng and conrol enjoy ubanal aenon. I requred however o know and maer rgh algorhm, programmng echnque, mahemac mehod and conrol procedure o employ correcly n developed applcaon. h paper dedcaed o modelng and conrol of dynamc yem wh employmen of compuer. Hgher programmng language provde an excellen oluon for olvng problem ha nclude mulaon of conrol, denfcaon and lkewe. Some cae hough requre dfferen approach. When man am peed or ably of applcaon be o ue language a C or C++. here were a lo of new programmng language n la pa year. Que a lo of aenon ha been acqured by a one called C#. here are numerou publcaon encompang algorhm wren n C# for compuer cence and arfcal nellgence. Ye publcaon ha are ryng o ue h language for clac heory of conrol are very rare. herefore we decded o conrbue no h feld wh conen of h paper and alo demonrae poble of h language. MODELING OF DYNAMICAL SYSEMS WIH PROGRAM MODULES CREAED IN C# Before we can ar wh conrol we need o defne he dynamc yem n conrol rucure. here are numerou mehod for bu bacally we ue yem of dfferenal equaon n ome form. For h parcular ak we o ue a yem of dfferenal equaon n ae pace a eay o ue n erave algorhm. In general we can decrbe SISO lnear dynamc yem wh followng equaon: where: x() u() y() ɺx ( ) = Ax( ) + B u( ) (1) y( ) = Cx( ) + D u( ) () - vecor of nernal ae, - yem npu, - yem oupu, A, B are conan marxe decrbng nernal dynamc of yem, 534

3 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka C, D are conan marxe ha decrbe how are oupu bound wh nernal ae of yem and nal condon. A mpoble for a compuer o proce daa n every nan we need o mplfy our model and ranform no a dcree equvalen. x(( + 1) ) = Fx( ) + Gu( ) y( ) = Hx( ) + Ju( ) (3) F, G, H, J - marxe decrbng he yem n dcree ae pace. Marxe F, G, H, and J are acqured by ranformaon of marxe A, B, C, D, whch characerze dynamc yem n connuou ae pace. x( ) - vecor of nernal ae varable n me x(( + 1) ) - vecor of nernal ae varable n me ( + 1) u( ) y( ) - npu n me - oupu n me Such ranformaon handled by Connuou o Dcree yem ranformaon module (CD). I approprae o analyze he yem repone o varou npu before we ll apply any conrol a all. If he cae SISO (Sngle Inpu Sngle Oupu) lnear dynamc yem we can ealy apply Syem Analy module (SA) whch come wh nuve uer nerface. Afer provded wh marxe F, G, H, J wll compue and plo yem repone o chooen npu. o conrol he yem we are ung connuou PID algorhm ranformed o dcree equvalen a PSD conrol. h ranformaon managed by PID o PSD ranformaon module (PIDPSD). Cloed loop conrol of he yem can be mulaed ung Cloed Loop Smulaon module (CLS). I we o verfy robune of he conroller. h poble o accomplh by everal way. One of hem caung a durbance n conrol. If we ponder he fac ha every real dynamc yem nonlnear and han f we have a nonlnear model of he yem beer o verfy he robune of he conroller by applyng o he orgnal nonlnear model whch wa mplfed o lnear by lnearzaon. Fr of all a wh lnear dynamc yem alo nonlnear dynamc yem need o be defned. Moly decrbed by yem of nonlnear dfferenal equaon. hee yem are uually olved by numercal mehod. For example by Runge-Kua mehod whch ued n Nonlnear Dfferenal Syem Solvng module (NDSS). In followng chaper our aenon wll be focued on decpron of ndvdual program module han on a demonraon of hee module by an applcaon whch combne hem ogeher. We ll um up reul and n he concluon ae of h paper wll be evaluaed..1 Connuou o dcreee yem ranformaonmodule CD Module npu: - marxe A, B, C, D ha are decrbng connuou lnear dynamc yem - amplng perod Module oupu: - marxe F, G, H, J ha are decrbng dcree lnear dynamc yem.1.1 Module analy Dervaon of dcree ae pace followng a requred for acqurng neceary formula for ranformaon from dcree o connuou ae pace. Le aume we have a lnear dynamc yem decrbed by (1) and (): Le modfy eqaon (1): A dx( ) A A e e Ax( ) = e B u( ) d A ( A d e x( )) = e B u( ) d Av Av d( e x( v)) = e B u( v) dv A A Av e x( ) e x( ) = e B u( v) dv A A Av e x( ) = e x( ) + e B u( v) dv A( ) A( v) x( ) e x( ) e B u( v) dv, = + where x( ) nal value of ae vecor. Under he condon ha amplng ekvdand wh amplng rae, for oluon n nerval v <,( + 1) >, ( = ), whle npu npu n h nerval mee he condon u( v) = u( ). And f we nroduce a ubuon: v = τ, dv = dτ v = τ =, v = τ = and f we re conderng only dcree value of me, where N = N + {}. =, = ( + 1), hen we oban a dcree ae pace decrpon: A Aτ x(( + 1) ) = e x( ) + e dτ B u( ) 535

4 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka where: hen: Snce: F = e A (4) G = e Aτ dτ B (5) x(( + 1) ) = Fx( ) + G u( ) (6) y( ) = Hx( ) + J u( ). (7) H = C, J = D e A no aylor ar- I poble o unroll funcon ray: 3 n A A A A 3 A n e = lm I (8) n 1!! 3! n! where I reprezen unarly marx whch ha he ame dmenon a marx A. Afer negraon (5) we oban accordng o (Krokavec 6): G = [ A e ] B = A ( e I) B = A ( F I ) B 1 A 1 A 1 n 1 1 A A n G = A ( F I) B = A lm I I B n 1! n! n A A 3 A n G = lm I B (9) n 1!! 3! n!.1. Module mplemenaon We can begn wh equaon (8) a (9) whch are fundamenal formula for h module. Par of he program ha handle ranformaon of he marxe can be decrbed n followng ep. ranformaon of marx A no marx F: 1. Inalzaon of varable A,, e(), ε from uer nerface (value are up o uer),.. F I 3. Savng e( 1) (error from prevou ep). 4. A A 5. Compuaon of facoral. 6. A F F +! 7. Compuaon of e( ) for marx F. 8. Connung wh ep 3 and + 1, f condon e( ) e( 1) < ε no me. Oherwe end. ranformaon of marx B no marx G: 1. Inalzaon of varable A, B,, e(), ε from uer nerface (value are up o uer),.. G I 1! 3. Savng e( 1). 4. A A 5. Compuaon of facoral. 6. A G G +! 7. Compuaon of e( ) for marx G. 8. Connung wh ep 3 and + 1, f condon e( ) e( 1) < ε no me. Oherwe end. Compuaon of e( ) for marx F: 7.1. Inalzaon of varable F, Y, r,. 7.. Y F 7.3. Y YF r r +Y [, ] Connung wh ep 4 unl coun of row n marx Y n reached r r e( ) - error n ep, ε - accuracy r - reul, Y - nrumenal varable Compuaon of e( ) for marx G can be done by analogy.in followng we ll be ung horened noaon = k, ( + 1) = k + 1, ec. o mplfy noaon of eqaon. And for amplng perod.. Syem Analy module SA Module npu: - marxe F, G, H, J decrbng dcree lnear dynamc yem - amplng perod - mulaon me pan, f - vecor of npu value u Module oupu: - vecor of oupu value y 536

5 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka..1 Module analy A we know from dcree yem heory value of ae and oupu varable can be compued for nex ep n an erave compuaon ung eqaon (3). We can analyze he yem from graphcal reprezenaon of yem repone... Module mplemenaon Par of he module ha dedcaed o compue y(k) n an erave compuaon can be decrbed n followng hree ep. 1. Generaon of npu gnal.. Compuaon baed on eqaon (3) of ae vecor x(k +1) and oupu y(k) n cycle for each value of dcree npu gnal. 3. Dplay of he reul..3 PID o PSD ranformaon module PIDPSD Module npu: - parameer of PID conroller K, I, d, µ - amplng perod - conrol npu u( k 1) - conrol conran u max ( u mn ) - conrol error e( k) and e( k 1) Module oupu: - conrol npu u( k ).3.1 Module analy We can ar wh eqaon decrbng deal PID conroler: where: K I d e() u() 1 de( ) u( ) = K[ e( ) + e( τ ) dτ d ] + (1) d I - proporonal gan, - negral me conan, - dervaonal me conan, - conrol error, - conrol npu. Equaon (1) can be ranformed no dcree equvalen accordng o (Harány 1998): e() + e( k) u( k) = K[ e( k) + ( + k 1 = 1 I d e( k)) + ( e( k) e( k 1))] (11) where negraon wa ubued by rapezod approxmaon and dervaon by fr dfference. If we wan (11) o copy (1) wh afyng precon we have o chooe approprae amplng perod. Equaon (11) repreen o called poonal algorhm for PSD conroller. From effcency pon of vew beer o ue velocy algorhm where ncreae of conrol nervenon mee condon: where u( k) = u( k) u( k 1) = = q e( k) + q e( k 1) + q e( k ) 1 (1) D q = K(1 + ) + (13) I D S q = 1 K(1 + ) (14) q S I D = K. (15) Hence conrol nervenon can be compued ung erave formula: u( k) = u( k 1) + u( k) (16) o he velocy algorhm for PSD conroller : u( k) = u( k 1) + q e( k) + q e( k 1) + q e( k ) 1 characer of PID conroler reman preerved f followng condon wll be me: q >, q1 q <, ( q + q1 ) < q < q. Whle we are ang only equaon for PSD conrol. We can rewre (16) no followng form where u( k) = u ( k) + u ( k) + u ( k), (17) p d u ( k) = ( q q ) e( k), p u ( k) = u ( k 1) + ( q + q + q ) e( k 1), 1 u ( k) = q ( e( k) e( k 1)). d o preven wnd-up effec, whch occur n conrol when conrol npu are bounded o nerval < umn, umax >, we wll modfy velocy algorhm n followng way. If hen and u( k) >= umax (or u( k) umn u ( k) = u ( k 1) u( k) >= ), = umax (or u( k) umn = ). o preven oclaon and udden change n conrol gnal u( k) we can ue dervave fler whch 537

6 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka ued n (Melchar 8) and demonraed on Fg. 1. y r (k) Where e(k) K K I µ /(3 ), K I K D d u(k) Fp( )= B( ) A( ) K d µ +1 Fg. 1: Schema of dervave fler. - negral gan, - dervaonal gan, Fp ( ) - ranfer funcon of conrolled yem. Conrol rule for PID conroller wh dervave fler : KI KD U ( ) = K + E( ) Y ( ). µ + 1 Afer ubuon for PSD algorhm: y(k) ( z 1) = we oban conrol rule ( z + 1) 1 1 de + d1 ez + dez E( z) + d y + d1yz + d yz Y ( z) U ( z) = c + c z + c z 1 1 n Z ranformaon. h afer backward ranformaon gve followng recurve conrol rule: 1 u( k) = ( c1u ( k 1) cu( k ) + dee( k) c + d e( k 1) + d e( k ) + d y( k) (18) where 1e e y + d y( k 1) + d y( k )) 1y y c = 4µ +, c 1 = 8µ, c = 4µ, d = 4K µ + K + K µ + K, e I I d1 = 8K µ + K, (19) e I d = 4K µ K K µ + K, d e I I = K, d1 = 8K, d = 4K y 4 D are parameer of (18)..3. Module mplemenaon y D We can mplemen an algorhm whch can be employed n dcrezaon of PID conroller o PSD form. Par of he module ha handle h pecfc ak can be ummarzed no nex ep: y D 1. Acquon of PID conroler parameer K, I, D, amplng perod (and fler parameer µ ).. Parameer q, q 1, q compuaon baed on (13), (14), (15) or compuaon of parameer for PSD wh dervave fler baed on (19). And ung (17) we can compue conrol npu u( k ). h par of module can be decrbed n followng ep: 1. Acquon of conrol npu u( k 1) and conrol conran u max e( k 1) from anoher module. ( u mn ) and conrol error e( k ) and. Compuaon of conrol npu u( k) baed on (17). 3. If u( k) >= umax (or u( k) >= umn ), hen u ( k) = u ( k 1) and u( k) = umax (or u( k) = umn ). Or f we are ung PSD wh dervave fler we can compue u( k) ung (18) wh followng mplemenaon: 1. Acquon of conrol npu u( k 1), u( k ), conrol error e( k ), e( k 1), e( k ) and oupu y( k ), y( k 1), y( k ) from anoher module.. Compuaon of conrol npu u( k) baed on (18). If we wan o mplemen dcrezaon of P, PI, PD conroller we can do by analogy dervng from mplemenaon of dcrezaon of PID conroler..4 Cloed Loop Smulaon module CLS Module npu: - marxe A, B, C, D ha decrbe connuo lnear dynamc yem and amplng perod - conroller parameer K, I, d, µ - vecor of reference value y r - mulaon me pan, f - conrol npu u( k ) Module oupu: - vecor of conrol npu u - vecor of conrol error e - vecor of oupu value y.4.1 Module analy A we know from auomac conrol heory, ome dynamc yem can be conrolled by PID conrol algorhm. For ucceful PID conrol we need o oban a e of parameer for hee algorhm by 538

7 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka ung convenen mehod of ynhe. he man elemen of he conrol algorhm are repreened by (6) and (7). Ung hee formula we can compue value of ae and oupu varable for each ep n an erave compuaon. Smlarly ung (17) or (18) we can oban conrol nervenon of he conroller for each ep k. Schema of cloed loop conrol rucure demonraed on Fg.. y r(k) y r (k) e(k) e(k) y o(k) Dgal conroller Senor z (k) u(k) z 1 (k) Conrolled yem Fg. : Schema of cloed loop conrol - dered value - conrol error u(k) - conrol npu aqured ung (17) or (18) z 1 (k) y(k) y o (k) z (k) - perurbaon caued by oude world y(k) - yem oupu whch can be meaured aqured ung (3) - oberved yem oupu - noe ha affecng he enor In h module we wll employ CD and PIDPSD whch we decrbed earler. We can for example ue Naln or Graham-Lanhrop mehod for conroler ynhe aed n (Madaráz 7), (Mkleš 1986)..4. Module mplemenaon Implemenaon of velocy algorhm, whch ung eleced mehod of ynhe (Naln and Graham- Lanhrop) and wa verfyed on model of dynamc yem (mple mechanc ocllaor and vagon e n (Dolnký 8) n cloed loop conrol ung language C#, can be decrbed n followng ep. 1. Acquon of yem model parameer (marxe A, B, C, D and amplng perod ).. ranformaon of marxe A, B, C, D o her dcree equvalen F, G, H, J. 3. Acquon of conroller parameer (conroller ype and uben parameer). 4. Acquon of reference, perurbaon and noe gnal parameer (decrpon n uer manual (Dolnký 8). 5. Reference gnal generaon. 6. Perurbaon gnal generaon (f wa defned). 7. Noe gnal generaon (f wa defned). 8. Recuren compuaon of repone of conrolled yem o reference gnal. Implemenaon of recuren compuaon of repone of conrolled yem o reference gnal (ep number 8) decrbed n followng ep k 8.. e( k) y ( k) y ( k) r o 8.3. Conrol nervenon compuaon accordng o (17) (or (18) f we are ung dervave fler) ung PIDPSD module Sumng perubaon and conrol nervenon (f perurbaon wa defned) Compuaon of repone of conrolled yem o conrol nervenon wh perubaon accordng o (3) Oberved conrolled yem oupu compuaon (value of conrolled yem oupu ummed wh noe) Savng e( k 1) (and f we are ung dervave fler avng u( k 1) and y( k 1) ) Savng e(k) and u(k) (and f we are ung dervave fler y( k ) ) If number of ample of reference gnal no exceeded connue wh ep and k k + 1. Oherwe end..5 Nonlnear Dfferenal Syem Solvng module NDSS Module npu: - arng condon f (, x ), - vecor of nonlnear funcon f (, x ), - me pan, f, - nal ep h mn and mnmal ep h. Module oupu: - vecor of fnal value f ( f ).5.1 Module analy h module enable u o mulae nonlnear model of dynamc yem decrbed by yem of nonlnear dfferenal equaon. 539

8 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka dx1 = f1(, x1,..., xn ) d dx = f (, x1,..., xn ) d... dxn = fn (, x1,..., xn ) d Module mplemen numercal mehod Runge Kua of 4 h order. Accordng o (Buša e al. 6) h mehod baed on approxmaon expreed n followng form. where: x k + k + k + k 1 3 4, j + 1 = x, j + () f ( ) - -h nonlnear funcon ( = 1,,..., n ) f ( ) - nal value of -h funcon f ( ) - fnal value of -h funcon f x, j - value of -h ae varable n ep j, x, j value of -h ae varable n ep j + 1. Value k k are: k = h. f (, x ) 1, j j, j 1 6 k = h. f ( + h /, x + k / ) (1) k = h. f ( + h /, x + k / ) 3 j, j k = h. f ( + h, x + k ) 4 j, j 3 5. Acual me ncreaed by ep h. 6. Compuaon of value x, j + 1 accordng o (). 7. If acual me leer han fnal me we ll connue wh ep oherwe end. 3 APPLICAION OF DESIGNED PROGRAM MODULES O PHYSICAL SYSEM BALL & PLAE 3.1 Applcaon analy Whle mplemenng our applcaon we have o repec ha he real model nonlnear and ha o our benef o mulae boh lnear and nonlnear model behavour. herefore requred o creae wo applcaon ha wll negrae and conrol mplemened program module a dfferen module wll be ued for mulaon and conrol n each cae. 3. Analy of dynamc yem Ball & Plae h model repreened by a ball rollng on a plae. h plae conrolled by a couple of ep moor. Poon of he ball caned by a camera. Image are analyed by a compuer whch wll deermne he ball locaon and conecuvely wll deermne and end ou approprae volage o ep moor. hee moor wll lean he plae o dered angle and force he ball o move n approprae drecon o he dered reul would be acheved. Schema from (Humuof: CE151 Ball and Plae Apparau Educaonal Manual ) hown on Fg. 3, block chema on Fg. 4. o check f he ep mall enough we can ue followng. ( k k ) /( k k ) () 3 1 If value () approxmaely equal o.5 we wll preerve la ued ep. If () conderable greaer han.5 we wll reduce ep h..5. Module mplemenaon 1. Inalzaon of varable (nal me, fnal me, ep, mnmal ep, nal condon, collecon of dferenal equaon).. If ep doen mee he precon condon and greaer han mnmal ep han we ll reduce ep by half. 3. Compuaon of parameer k k accordng o (1). Fg. 3: Schema of Ball & Plae and apparau 4. If ep doen mee he precon condon and greaer han mnmal ep han we ll connue wh ep oherwe we ll connue wh ep 5. 54

9 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka yx() yy() Compuer ua() ub() Servo yem a() b() Ball & plae Fg. 4: Block chema of Ball & Plae Varable decrpon: y x (), y y () u a (), u b () a(), b() x() y() - ball locaon on plae deermned from acqured mage [m, m] - volage conrollng ndvdual epng moor [V, V] - angle repreenng nclnaon of plae [rad, rad] x(), y() - ball real poon [m, m] A here no feedback beween ball poon and plae nclnaon poble o dvde he model no wo eparae par ervomechanm and ball freely rollng on he plae. Maemacal model whou ervomechanm wa derved n (Humuof: CE151 Ball and Plae Apparau Educaonal Manual ) from bac Euler - Lagrange equaon. where q qɺ V Q d V + = Q d qɺ q q - he -h generalzed coordnae - he fr dervaon of he -h generalzed coordnae by me - knec energy of he yem - poenal energy of he yem - he -h generalzed force Dealed dervaon n (Humuof: CE151 Ball and Plae Apparau Educaonal Manual ). Fnally we ge a yem of four dfferenal equaon of he econd grade. where I x m ɺɺ x mɺ ɺ y ɺ x mg (3) b : ( + ) ( αβ + α ) + nα = rb I y m ɺɺ y mɺ αβɺ x ɺ β y mg β (4) b : ( + ) ( + ) + n = rb I + I + mx ɺɺ α + mɺɺ βxy + ɺ βxy ɺ + ɺ βxyɺ + ɺ αxx ɺ ( p b ) ( ) + mgx co α = ɺɺ F d α co α ( I p + Ib + mx ) β + m( αxy + αxy + αxy + βxx) + mgx co β = ɺɺ ɺɺ ɺ ɺ ɺɺ F d β co β x, y - ball coordnae on he plae [m] (5) (6) r b - ball radu [m] ω - vecor of ball angular velocy [rad/] α, β - angle of plae nclnaon [rad] I b - ball nera [kg.m ] I p - plae nera [kg.m ] m - ball ma [kg] g - gravonal acceleraon [m - ] F α F β - force nfluencng he plae n he drecon of ax x [N] -force nfluencng he plae n he drecon of ax y [N] Ball moon decrbed by (3) and (4) whch decrbe dependence of ball acceleraon from angle and angular velocy of he plae nclnaon. Eqaon (5) and (6) are decrbng how he plae nclnaon dynamc nfluenced by he exernal drvng force and he poon and peed of he ball. Accordng o he B&P manual poble o mplfy he dynamc of B&P. Ung he aumpon from manual we ll fnally oban followng model. d x d d y d 3..1 Servo yem 5 = g n α Kbα 7 5 = g n β Kbβ 7 Block chema of he ervo yem from (Humuof: CE151 Ball and Plae Apparau Educaonal Manual ) on Fg. 5. Due o lmaon of he yem h par conan everal nonlnear componen. Fg. 5: Block chema of ervo yem he fr nonlnear componen a fler called rae lmer whch rercng he cope of peed of change (dervaon) of npu volage. h fler olve he problem of ofware drver. he problem dwell n followng. Sack n whch we ore waned value can be acualzed only afer reachng he dered poon. In oher word we canno exceed nomnal peed of eppng moor whch deermned by frequency of mpule whch are upplyed by drvng card of 541

10 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka eppng moor. We can expe maemacally n followng aemen. where u α u αf k R uα ( k) uα ( k 1) rae = u ( k) = + R + u ( k 1) rae < +R α f α f u ( k) = R + u ( k 1) rae < -R α f α f u ( k) = u oherwe. α f α - fler npu - fler oupu - ep - amplng perod - rae rehold Addonal nonlnear dynamc are caued by a auraon fler called amplude lmer whch enure ha maxmum lope of he plae canno be exceeded. Inpu gnal lmed no nerval <-1,1> o afer mulplyng by ac gan f no plae nclnaon lm. Fnally we have o repec he fac ha he eppng moor have conan peed of eppng. Hence we need o add an elemen ha wll be nenve o ceran range of value and beyond h nerval oupu wll be a pove or negave conan value hu modelng he moon of he moor upward or downward. Syem parameer Parameer of dynamc yem can be meaured drecly or are known from (Humuof: CE151 Ball and Plae Apparau Educaonal Manual ). Normalzed parameer are followng. K overall yem gan [ - ] ω nomnal peed of ervo yem [ -1 ] m me conan of ervoyem [] K α ac gan [rad/mu] K b B&P yem gan [m - /rad] K x ball poon enor conan [MU/ m] Lnear model Sae vecor: x1 yx ball ' poon [ ] 1 x = x = yɺ x ball ' velocy [ ] x 3 α plae' nclnaon [ ] Inpu u α... dered plae angle en ou from Malab u α < 1, + 1 > Oupu y x... ball poon read o Malab y < 1, + 1 > x Sae equaon ɺx = A x + B 1 ɺx = K x + uα (7) 1 1 m m y x = = Lnear model ranfer funcon: u α [ 1 ] C x x (8) Y K x α Kβ K x K F( ) = = = U ( + 1) ( + 1) Syem characerc α m m. (9) he B&P model a 3rd order yem wh he nd order of aam Characerc nonlneare: - rae lmer - auraon - me conan dependen on he magnude and frequency of an npu gnal unmodelled propere : - frcon - defec n he ball and/or plae urface 3.3 Lnear Ball & Plae Conrol Smulaon module LBPC Module npu: - PID parameer K, d,, µ - amplng perod, - mulaon me pan, f, - noe and perurbaon parameer. Module oupu: - vecor of conrol npu for x and y ax u x, u y, - vecor of poon for x and y ax y x, y y, - vecor of oberved poon for x and y ax y ox, y oy, - vecor of conrol error for x and y ax e x, e y Module analy When we are aumng lnear model of Ball & Plae we can compue yem ae and oupu ung (7) and (8). Alhough Ball & Plae a MIMO (Mul Inpu Mul Oupu) dynamc yem poble o dvde no wo SISO dynamc yem (ball moon on he plae n drecon of ax x and moon n drec- 54

11 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka on of ax y). herefore we ll conduc analy only for one coordnae (rule for he econd coordae are he ame). z 1 (k) q = 3.3, q 1 = , q = 1.5. Conrol rule for PSD conroller followng: u( k) = u( k 1) e( k) e( k 1) e( k ). y r (k) e(k) yo(k) Dgal conroller Senor z (k) u(k) LI B&P model Fg. 6: Clooed loop conrol of lnear Ball & Plae model y r (k) e(k) y(k) - conrol npu(reference rajecory for one ax) - conrol error u(k) - conrol npu whch compued from (17) or (18)(volage uppled o eppng moor whch conrol nclnaon of plae n drecon of one ax) z 1 (k) y(k) y o (k) z (k) - perurbaon ha affec ball poon - oupu (ball poon) whch compued from (3) - oberved oupu (ball poon nfluenced by noe) - noe affecng he enor I mgh appear ha perurbaon and noe are he ame and can be condered a one gnal bu here dfference n characer of hoe gnal. Characer of perurbaon gnal deermnc whle characer of noe peudorandom alhough amplude lmed. We can ue perurbaon o mulae jumpng and ldng of he ball on he plae a hoe phenomena are preen n real model. Noe can be ued o mulae behavour of camera and mperfecon of mage recognon algorhm. Synhe of conrol baed on ranfer funcon (9). PID conroller degned by Naln mehod ha followng parameer K =.15, d = 4, = 8. Correpondng parameer of PSD conroller when amplng perod =.1 are: q = 5.157, q 1 = -1.14, q = 5. Conrol rule for PSD conroller followng: u( k) = u( k 1) e( k) e( k 1) + 5 e( k ). PID conroller degned by Graham-Lanhrop mehod ha followng parameer K = 1.743, d =.494, = Correpondng parameer of PSD conroller when amplng perod =.4 are: 3.4 Module mplemenaon o mulae cloed loop crcu for boh axe we can ue module CLS whch wa decrbed earler. Bu we have o modfy he mplemenaon of recurve compuaon of repone of conrolled yem o followng form k 7.. e( k) y ( k) y ( k) r o 7.3. Conrol nervenon compuaon accordng o (17) (or (18) f we are ung dervave fler) ung PIDPSD module Compuaon of ae vecor accordng o he fr equaon of (3), 7.5. Summng perurbaon and ball poon (f perurbaon wa defned) Lmaon of he ball poon Compuaon of repone of he yem accordng o econd equaon n (3) Compuaon of oberved oupu (value of oupu ummed wh noe), 7.9. Savng e( k 1) (and f we are ung dervave fler avng u( k 1) and y( k 1) ) Savng e(k) and u(k) (and f we are ung dervave fler yo ( k ) ) If number of ample of reference gnal no exceeded connue wh ep and k k + 1. Oherwe end. 3.5 Nonlnear B&P Conrol Smulaon Module NBPCSM Module npu: - PID parameer K, d,, µ, - Samplng perod, - Smulaon me pan, f, - noe and perurbaon parameer Module oupu: - vecor of conrol npu for x and y ax u x, u y, - vecor of poon for x and y ax y x, y y, - vecor of oberved poon for x and y ax y ox, y oy, - vecor of conrol error for x and y ax e x, e y. 543

12 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka Module analy When we are aumng nonlnear model of Ball & Plae we have o compue yem ae and oupu ung (3) a (4) whch are decrbng behavour of he nonlnear dynamc yem. A no he force F α and F β bu drecly he angle α and β are yem npu. h due o he fac ha he frequency of a epper below he acceleraon lm. No ep can be lo and he magnude of load momen canno affec he moor poon. h aumpon reul n omng he equaon (5) and (6) a aed n (Humuof: CE151 Ball and Plae Apparau Educaonal Manual ). Bu of coure we ll have o repec lm of dynamc yem. y rx (k) y ry (k) y ox (k) e x (k) e y (k) y oy (k) z x (k) Senor for x ax locaon Dgal conroller for ax x Dgal conroller for ax y Senor for y ax locaon z y (k) u x (k) u y (k) Nonlnear B&P model z 1x (k) z 1y (k) y x (k) y y (k) Fg. 7: Clooed loop conrol of nonlear Ball & Plae y rx (k) e x (k) - conrol npu (reference rajecory for ax x) - conrol error for ax x u x (k) - acon npu whch compued from (17) or (18) (volage uppled o eppng moor whch conrol nclnaon of plae n drecon of x ax) z 1x (k) y x (k) y ox (k) z x (k) - perurbaon ha affec ball poon - oupu for ax x (x coordnae of ball poon) whch compued from (3) - oberved oupu for ax x (x coordnae of ball poon nfluenced by noe) - noe affecng he enor n coordnae x Decrpon for y ax varable can be done by analogy. o conrol nonlnear dynamc yem Ball & Plae we ued conroller degned ung ranfer funcon of lnear model of Ball & Plae Module mplemenaon 1. Acquon of conroller parameer (conroller ype and uben parameer).. Acquon of reference, perurbaon and noe gnal parameer and ample me. 3. Reference gnal generaon. 4. Perurbaon gnal generaon (f wa defned). 5. Noe gnal generaon (f wa defned). 6. Recuren compuaon of repone of conrolled yem o reference gnal. Implemenaon of recuren compuaon of repone of conrolled yem o reference npu (ep number 6) decrbed n followng ep k 6.. e ( k) y ( k) y ( k) x rx ox e ( k) y ( k) y ( k) y ry oy 6.3. Compuaon and rae and amplude lmaon of volage for each moor Compuaon of plae nclnaon angle, her lmaon, compuaon of dfference beween preen and la value and flraon (baed on envy) Numercal negraon of angle Compuaon of ball poon and dervaon of ball poon Summng perurbaon and ball poon (f perurbaon wa defned) Compuaon of ball poon affeced wh noe and lmaon of ball locaon Savng ex ( k 1), ey ( k 1), ex ( k ), ey ( k ), ux ( k ), uy ( k ) (f we are ung dervave fler avng alo yox ( k ), yoy ( k ), yox ( k 1), yoy ( k 1), ux ( k 1), uy ( k 1) ) and plae nclnaon ( α ( k), β ( k) ) If number of ample of reference npu no exceeded connue wh ep and k k + 1. Oherwe end. 3.6 Applcaon reul For mulaon of conrol of he dynamc yem Ball & Plae we ued followng gnal. A a conrolled npu we ued poon of he ball and a a conrol npu we ued ball dered poon or dered rajecory. Reul depend on employed conroller, dered rajecory, me pan provded o cover he rajecory, amplng perod, perurbaon and noe. A a perurbaon gnal we ue a gnal ha affec ball poon. hu we can approxmae real condon where ball n ceran momen looe conac wh he plae or ldng. Noe n h yem conderable a mage proceng by camera deermnng he locaon 544

13 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka of he ball affeced by lgh condon and ball color. In general reul of conrol of lnear model are beer. Followng pcure llurae reul of lnear and nonlnear model. Smulaon me wa 6 econd for qare and crcle rajecory, 1 econd for ar and helx and 3 econd for poon. For conrol of lnear model we ued conroller degned by Graham- Lanhrop mehod (K = 1.743, d =.494, = 1.98) wh dervave fler ( µ = ), a we acqured beer 1 reul. For conrol of nonlnear model we ued conroller degned by Naln mehod (K =.15, d = 4, = 8) whou dervave fler. d Fg. 11: rackng of qare rajecory (nonlnear model) Fg. 1: rackng of crcle rajecory (lnear model) Fg. 8 : Conrol o dered poon (lnear model) Fg. 13: rackng of crcle rajecory (nonlnear model) Fg. 9: Conrol o dered poon (nonlnear model) Fg. 14: rackng of ar rajecory (lnear model) Fg. 1: rackng of qare rajecory (lnear model) 545

14 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka Fg. 15: rackng of ar rajecory (nonlnear model) Fg. 19: rackng of qare rajecory (ax x) wh fler. Fg. 16: rackng of helx rajecory (lnear model) Fg. : rackng of qare rajecory (ax y) whou fler. Fg. 17: rackng of helx rajecory (nonlnear model) Fg. 1: rackng of qare rajecory (ax y) wh fler. Fg. 18: rackng of qare rajecory (ax x) whou fler. Fg. : rackng wh qare rajecory whou fler. Fg. 1 can be compared wh Fg. o demonrae effec of dervave fler when reference rajecory changng very eeply. 546

15 17h Inernaonal Conference on Proce Conrol 9 June 9 1, 9, Šrbké Pleo, Slovaka 4 CONCLUSION We creaed and verfed requred module ha allow u o ranform connuou lnear dynamc yem o her dcree equvalen and analyze behavour of hee dynamc yem. hen we connued wh ranformaon of PID algorhm o dcree form. We ued PSD algorhm wh an-wndup and dervave fler n mulaon of conrol n cloed loop crcu. A we waned o evaluae robune of conrol we creaed a program module ha enable u o olve nonlnear yem of dfferenal equaon. All hee module were ued n an applcaon ha mulae behavour of real phycal model Ball and Plae. We creaed wo applcaon one for mulaon of conrol of lnear model and econd for nonlnear model of dynamc yem Ball and Plae. hu we verfed he robune of degned conrol. Alo we howed ha whle C# doen drecly provde funcon or procedure for modelng or conrol of dynamc yem, we can creae hem by ourelve. Conderng he fac ha ynax of C# very eay and horough we can quckly mplemen requred funcon no program module whch can be ued n an applcaon n dered way. Alo we are no bound o hgher programmng language a Malab. Moreover poble o ue rch poble ha plaform.ne provde. D. Krokavec, A. Flaová. (6). Dkréne yémy. Košce: Elfa. 3. ISBN L. Madaráz, M. Bučko, L. Fozo. (7). Základy auomackého radena - 1. Elfa. Košce. ISBN J. Mkleš, V. Hula. (1986) eóra auomackého radena. Alfa. Bralava. ISBN J.Melchar. (8) Lneární Syémy (Učební ex). ZČU Plzeň (avalable on nerne). 5 ACKNOWLEDGMENS h reearch ha been uppored by he Scenfc Gran Agency of Slovak republc under projec Vega No. 1/617/8 Mulagen Nework Conrol Syem wh Auomac Reconfguraon. h uppor very graefully acknowledged. 6 REFERENCES J. Buša, V. Prč, Š. Schroer. (6). Numercké meódy, pravdepodobnoť a maemacká šaka. Košce: U-FEI, ISBN K Dolnký. (8). Degn and Realzaon of Program Module for Model of Dynamcal Syem. Bachelor he. (upervor: ao. prof. Ing. A. Jadlovká, PhD, Ing. R. Lonščák), FEI U, Košce, (n Slovak). L. Harány, J. Murgaš, D. Ronová, A. Kozáková. (1998). eóra auomackého radena. Bralava: Slovenká echncká Unverza Bralava, Fakula elekroechnky a nformaky. 16. ISBN Humuof: CE151 Ball and Plae Apparau Educaonal Manual. (1996-4). 547