The Application of Euler Lagrange Method of Optimization for Electromechanical Motion Control

Transcription

1 HE ANNAS OF "DUNAREA DE OS" UNIVERSIY OF GAAI FASCICE III, ISSN -454X EECROECHNICS, EECRONICS, AUOMAIC CONRO, INFORMAICS he Application o Euler agrange Metho o Optiization or Electroechanical Motion Control Ion BIVO, Cristian VASIACHE Universitatea "Dunarea e os" Galati Doneasca, 6-Galati Roania e-ail: Ion.Bivol@ugal.ro Abstract: Inustrial an non-inustrial processes such as prouction plans, robots, pups, copressors, hoe applications, transportation o people an goos etc., require soe ins o otion control. he ain unctions o electroechanical rives are to ajust these processes by controlling the torque, spee or position. he objective o this paper is to peror the control o otion while iniizing power losses, that is?ri, in process conversion o electrical energy to echanical energy. he optial control laws or our proble is in using the Euler agrange principle. We consier three types o controlle rives: torque, spee an position. Each o the has ierent control laws. By ipleentation o these controls with Borlan C++ an Matlab environent, substantial energy savings are obtaine. Keywors: Motion control, sensor less, energy saving, ac rives syste. INRODUCION he electrical rives have a wie range o practice in inustry an transportation. It is possible to istinguish three ierent requireents with regar to the ain state control: orque controlle rives Spee controlle rives Position controlle rives Depening on the application there are ierent eans or accuracy or ynaic response. With increasing accuracy the cost an coplexity rise steeply. In the classical approach, the iniization inex to be perore is the accuracy an the tie o the ynaic response. hat is, the syste coponents ust have the capability to ove in iniu tie with the best accuracy. But when the otion tie gets shorter acceleration increases by / an consequently the current an power issipation increase an otor overheats. In this paper another choice o esign objectives is use: peror a ove within tie, while iniising the otor teperature. here are ierent solutions o this proble, epening o the rive targets: torque, spee or position control.. DYNAMICS OF A MECHANICA DRIVE he equations escribing the ynaic behavior o a echanical rive with constant inertia are: his paper was recoene or publication by E. Rosu 5

2 HE ANNAS OF "DUNAREA DE OS" UNIVERSIY OF GAAI FASCICE III, ISSN -454X ε () (,, um, (,, u, α ε y y(x)? (x,y ) where is the angular velocity an a is angular coorinate. (x,y ) y (x) In orer to gain better insight the otor an loa torque, are consiere as unctions o,e an o the controls inputs u M an u to the actuators on the otor an loa sie. Hence, by neglecting the ynaics o the otor an loa, we arrive at a secon orer syste escribe by the two state equation (). - Motor ynaic /s oa ynaic Power supply Fig.. Bloc iagra or ynaics o echanical rive Figure shows a bloc iagra escribing the otion o the rive syste. We have assue that all oving parts o a rive can be cobine to or one eective inertia. However, or a ore etaile analyses o a echanical ynaic eects it ay be necessary to consier the istribution o the asses an the linage between then. he two blocs with ouble rae escribes the non linear unctions an ay contain aitional ynaic states escribe by ierential equations. 3. DESIGN SOUIONS he well nown optial regulator esign proble is to eterinate the optial control law u ( x, which can transer the syste ro its initial state to the inal state such that given perorance inex is iniize. he perorance inex is selecte to give the best trae-o between perorance an cost o control. he perorance inex that is wiely use in optial control esign is nown as the quaratic perorance inex an is base on iniu-error an iniuenergy issipation criteria. u u /s α Fig.. wo paths or transer ro (x, y ) state to (x, y ) state. Consier the syste escribe by () x &( Ax( + Bu( he quaratic perorance inex is: t (3) I ( x Qx + u Ru) t Subject to the ynaic syste equation (). In (3) Q is a positive sei-einite atrix on R is a real syetric atrix. Q is positive seieinite i all its principal inors are non-negative. he choice o the eleents o Q an R allows the relative weighting o iniviual state variable an iniviual inputs. he integrals variations ethos (the Potriagin's axiu principle or the Hailtonian etho) or the optial control low optiization give the iiculties in practical ipleentation, because either initial conitions or inal conitions are not copletely nown. In this paper, the application o agrangean etho o optiization to our proble in consiere. I we consier the unconstraint unctional: it ust in ( x) I. x (4) I g( x, y, y& ) y x x which peror the iniization o he proble's solution can be oun by eans o the Euler-agrange equation. g (5) y x y& I the target point is oving on trajectory C, the transversality conition is neee (a generalize ortogonality). x 6

3 HE ANNAS OF "DUNAREA DE OS" UNIVERSIY OF GAAI FASCICE III, ISSN -454X (6) [ ( y & g ) x + g y] g y & y& he optiisation proble or ynaic systes base on the Euler-agrange principle starts ro the general criteria: t (7) I ( x(, u(, + M ( x, t, x, t where an M are unctions eine in X U t R, respectively in X R Usually there are three types o optiisation proble: he agrange proble, when an M ; he Mayer proble when an M ; he Bolza proble when an M ; I the general state equation o the syste is: (8) x &( ( x(, u(. he unction Φ is eine as: & (9) Φ ( x x, ( x, x &, For agrange proble the unctional criteria eine as: () I ( x, + λ Φ( x, x, & where λ represents the agrange ultipliers. he Euler-agrange equation or the new unctional criteria are: () I x I u I λ I x& I u& I λ& By eans o Euler-agrange equations we can in the optial control laws, irst or echanical otion syste, an then or electrical rive syste. 4. MOION CONRO AWS WHICH MINIMISING HE MOOR EMPERAURE. he equations escribing the otions o rive with constant inertia an constant loa torque are: (3) α& & he perorance easure o energy optiisation leas to the syste is: (4) I R i. he otion torque equation is: where: (5) i ; or c otor, φ or ac otor with iel orientation 3 φ + σ where φ an φ are the lux o the achines. Fro equations (3,4), results: R (6) I 4.. Spee controlle rive In this case the proble is to oiy the state o the syste ro initial spee i to inal spee,with iniisation o energy issipation in electroechanical power conversion syste. In this conition the unctional criteria () is: (7) ( ) R + λ & + + λ I he Euler agrange equations with results ro (6) are : (8) R λ& λ& λ with initial an inal conitions: t i ; t λ Fro (7) results the optial control : & () 7

4 HE ANNAS OF "DUNAREA DE OS" UNIVERSIY OF GAAI FASCICE III, ISSN -454X where i + (9) () is the angular acceleration i ε he cost perorance inex is: () R I + [ ε ] 4.. Position controlle rive: In this case the proble is to achieve an ipose angular position a in tie with iniu cost. he perorance inex () is: () I R λ + λ ( α& ) + λ & an the Euler-agrange equations are: (3) + [ & + ] 3 R λ λ& λ λ& λ& + λ& 3 he optial control law results ro these equation by integration: (4) ( C C R where C an C are integration constants. With initial an inal conitions t +, a t - aa ; results: (5) α t t 6 α t 3 t 3 + t t t he equations (5) escribes the optial otion o the syste in tie when the ipose oveent angle is a. he evaluation o otion cost can be eectuate by eans o the relation (5): R (6) I ( ) or by echanical wor calculations: α (7) α Finally, the perorance inex (5) or position rive syste is: (8) I R 3 α + he optial control laws or spee an position rives are ierent. In the spee rive the ais is to transer the spee o the syste ro I to with inial isplaceent a. In the position rive, the ai is to transer the syste ro initial position to inally position a in tie with conition: ax. he perorance inex or position controlle rive with usual control law with constant ynaic torque or spee rive is: (9) R I + 6 α which is uch 3 greater than optial perorance inex (7). 8

5 HE ANNAS OF "DUNAREA DE OS" UNIVERSIY OF GAAI FASCICE III, ISSN -454X 6a / 4a / / / ax,5a / ax a / t t Fig. 3a Aa Fig. 3b Aa Fig. 3 orque an spee proiles or optial position control (a) an respectively sub optial control (b) I, ) C 4 C orque Controller q u q Inuction Machine i u,u q,i,i q C 3 ) i i (-) Flux Controller u I.M. i q i K, ax, α C K Fig.4 Bloc iagra o the control syste or inuction achine position rive syste 9

6 HE ANNAS OF "DUNAREA DE OS" UNIVERSIY OF GAAI FASCICE III, ISSN -454X 5. EXPERIMENA RESUS We consier the position rive syste with asynchronous achine. he ipose values or the syste are : the tie o execution, the axiu spee ax,the inal position α an the estiate loa torque. he coan syste calculate the optial trajectory or spee (C) an torque (C ), ig.4. hen the control syste in the appropriate u, u achine supply tensions coponents an q their pulsation. In igure 4 the bloc iagra o the control syste is presente. here are two PID controllers or torque an spee. he energy conversion process is nonlinear.for the copensation o the nonlinearites the unction an q are use. he pulsation are calculate in the rotor iel orientation theory anner, that is: Fig.5 Ipose values an experientals results or position spee an torque with optial velocity proile + (3) 3R pφ where φ, R are the rotor lux an rotor resistance, respectively.because the rive is not sensore with spee sensor the easure spee in (9) is replace by the calculate spee (4). he inuction otor rate paraeters use or experiental tests are : 4W,38V,5rp,5Hz. In igure 5 are represente the results or the ollowing test conitions : sec, ax 7s (3) α 46 ra, ; (3) i,9e3ws Fig.6 Ipose values an experientals results or position spee an torque with sub optial velocity proile he optial velocity proile is a parabola, which are quite well ollowe by the rive syste without spee sensor. I the velocity proile is triangular (ig.6) (33) ax 9s i 5,6e3ws; α 46ra

7 HE ANNAS OF "DUNAREA DE OS" UNIVERSIY OF GAAI FASCICE III, ISSN -454X 6. CONCUSIONS 7.REFERENCES he control o otion with iniizing power loses in process conversion o electrical energy to echanical by Euler-agrange etho o optiization is a new way in the iel o electrical rives. A sensor less control syste with PWM supply inverter or inuction otor was evelope. In this case the ipose inal position α is well reache without spee sensor. here are an iportant ierence between the cost inex in optial control case an others sub optial controls. However the ain question is the proper estiation o the loa torque..eonhar,w.,control o Electrical Drives, Springer, 996.Bolea,N.,Nasar,S.A.-Vector Control o AC Drives, CRC Press,99 3.Bivol,I.,-Moern Drives Systes, Acaeica, 999