SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE TECHNIQUE

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1 SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE 4 Jurnal Teknologi, 49(D) Dis. 8: 4 6 Universii Teknologi Malaysia SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE TECHNIQUE MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI Absrac. This paper presens servomoor posiion conrol using direc digial conrol and sae-space echnique. A mahemaical model of he sysem is derived and verified by SIMULINK/ MATLAB. Full-sae feedback conroller boh wih and wihou inegral conrol are proposed for he conroller srucures. The performances beween hese wo srucures are analyzed. Simulaion resuls are firs given. Nex, a program is developed via Microsof Visual Basic 6. (VB6.) and inerfaced wih he servomoor for experimenal evaluaion and validaion. Keywords: Full-sae feedback conrol; Bass and Gura s approach; graphical user inerface; saespace; servomoor; direc digial conrol Absrak. Keras kerja penyelidikan ini menyenuh mengenai eknik keadaan ruang dan kawalan digial erus dalam aplikasi sisem kawalan kedudukan moor servo. Permodelan maemaik sisem kawalan ersebu dierbikan dan model ersebu dienusahkan dengan menggunakan perisian MATLAB/SIMULINK. Pengawal suap balik keadaan anpa kawalan kamiran dan bersama kawalan kamiran dicadangkan unuk srukur pengawal. Presasi kedua-dua pengawal dianalisis dan kepuusan simulasi diserakan. Seerusnya aur cara kompuer direka benuk dengan menggunakan Microsof Visual Basic 6. dan dianaramukakan dengan moor servo unuk ujuan penilaian dan penenusahan uji kaji. Kaa kunci: Kawalan suap balik keadaan; pendekaan Bass dan Gura s; anaramuka pengguna grafik; keadaan ruang; moor servo; kawalan digial erus. INTRODUCTION Auomaic conrol is one of oday s mos significan areas of science and echnology. This aribued o he fac ha auomaion is linked wih he developmen of almos every form of echnology. Direc digial conrol (DDC) is one form of he auomaic conrol where DDC is ermed as using a digial compuer o inerface direcly wih a plan or sysem as he conrol device []. The dispariy beween DDC and indirec digial conrol or supervisory conrol is ha i does no require any addiional hardware Conrol & Insrumenaion Deparmen, Faculy of Elecrical Engineering, Universii Teknologi Malaysia, 83 UTM Skudai, Johor Bahru, Malaysia Tel: 7-39, Fax: fuaad@fke.um.my Faculy of Elecrical and Elecronics Engineering, Universiy College of Engineering and Technical Malaysia, Gambang, Pahang Darul Makmur Tel: syakirin@ump.edu.my

2 46 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI o implemen he conroller. Everyhing from conroller s algorihms or srucures in erms of codes and scrips can be programmed using compuer sofware. The objecives of his projec is o creae and develop a graphical user inerface (GUI) using Microsof Visual Basic 6. ha implemens DDC and sae-space echnique in conrolling servomoor shaf posiion. Conroller design srucure will use full-sae feedback wih and wihou inegral conrol [] where pole-placemen design via Bass and Gura s approach [3] is proposed. The full-sae feedback conroller via pole-placemen is chosen since i has he bes performance compared o oher conrollers in erms of oscillaion and seling ime [4]. Moreover, he pole-placemen design could also handle sysems wih ime-varying sae space represenaion [], or sysems wih muliple operaing condiions [6], as well as sysems wih muli-inpumuli-oupu (MIMO) signals requiremen [7]. Normally, conroller design for linear ime-varying differenial sysems is generally a difficul problem, because of he fundamenal problems relaed o he analysis of such sysems []. For simpliciy, he servomoor sysem will be analyzed as a linear ime-invarian (LTI) sysem, wih only single-inpu and single oupu condiion. This is due o he fac ha he mehod has he properies of he flexibiliy of shaping he dynamics of he closed-loop sysem o mee he desired specificaions [8]. The paper is organized as follows. In secion, a mahemaical model of servomoor is discussed where he sysem is considered as a second order sysem. In secion 3, sae-space modeling for coninuous and discree ime sysem is presened. Subsequenly in secion 4, full-sae feedback conroller using pole-placemen design is derived. Three case sudies are provided in secion where simulaion and experimenal analysis are horoughly discussed. Conclusions are given in secion 6.. MODEL OF SERVOMOTOR SYSTEM Fixed field R a L a e a () Armaure circui v b () Roor T m () i a () θ m () Figure Schemaic of servomoor sysem

3 SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE 47 E a (s) I a (s) τ (s) k sl a R a s J m sd m θ (s) sk b Figure Time domain block diagram represenaion of servomoor sysem Figure shows he schemaic of he servo moor, meanwhile Figure is he equivalen block diagram []. Based on he block diagram of Figure, by ignoring he armaure inducance of he sysem, he open loop ransfer funcion for a second order armaure conrolled servomoor sysem is given as follows: θ E () s = () s m a m a k RJ kk b s s Dm J m R a where J m = equivalen ineria by he moor = 3 6 kgm k b = back-emf consan = 6 3 Vsrad k = moor orque consan = 7 3 NmA R a = armaure resisance = 3. Ω D m = equivalen viscous densiy by he moor = small (canno be quoed) (.) The armaure inducance is ignored since is value is comparaively small (L m 8 mh). By subsiuing all he parameers ino he Equaion (.), he open loop ransfer funcion for he second order armaure conrolled servomoor sysem could be obained as follows: θm () s G() s = = (.) E s s.6s a () 3. STATE-SPACE MODELING 3. Modeling in Coninuous Time Based on he block diagram of he servomoor as shown in Figure, he sae variables for second order sysem are defined as

4 48 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI x () = θ m () = angular displacemen of he moor shaf dθm () x () = = angular velociy of he moor shaf d Meanwhile, he sae inpu and sae oupu for he second order sysem are defined as u() = e a () = inpu signal ino he servomoor y() = x () = oupu signal from he servomoor Le Now le () θm () i dx d x() = = = x() d d () θm () i dx d x() = = d d By manipulaing equaion (.), i could be obained ha i kk b k x() = Dm x ea J m R a RaJm () () (3.) (3.) (3.3) Therefore, he sae-space represenaion of servomoor in space marix could be expressed in his form: and i x() x() = kk b k e i Dm x() x() J m R a RJ a m x() = x() () [ ] y a () (3.4) (3.) 3. Modeling in Discree Time From equaion (3.4), i could be re-wrien ino () () e M x N i x x = i ( ) x() a () (3.6)

5 SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE 49 where M and N are defined as: DmRa kbk M = (3.7) J R k N = (3.8) J m R a Le he dynamic of he linear coninuous-ime servomoor sysem be represened by he following sae and oupu equaions respecively: where i m () = () () x Fx Gr a (3.9) y() = Cx() Dr() (3.) F = G M = N C = D = [ ] The sysem marix A (T ) and oupu marix B (T ) for discree ime servomoor sysem can be easily deermined as follows [9]: ( ) ( ) F = T = T AT = e = L SI F (3.) T F ( ) = τ = ( ) T = τ A( τ) Gdτ (3.) T B T e Gd L SI F Gdτ = Therefore, he sae-space represenaion of servomoor in discree ime space marix could be expressed in his form: ( MT ) ( ) ( e ) N N ( MT ) ( ) T ( e x ) k x k M = M ( ) ( M ea k (3.3) x k ) ( ) ( ) x k MT N ( MT ) ( e e ) M ( ) [ ] y k ( k) ( k) x = x (3.4)

6 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI 4. FULL-STATE FEEDBACK CONTROLLER DESIGN 4. Sae Feedback wihou Inegral Conrol The concep of feed-backing all he sae variables o he inpu of he sysem hrough a suiable feedback marix in he conrol sraegy is known as he full-sae variable feedback conrol echnique. Using his approach, he desired locaion of he closedloop eigenvalues (poles) of he sysem will be specified. Thus, he aim is o design a feedback conroller ha will move some or all of he open-loop poles of he measured sysem o he desired closed-loop pole locaion as specified. Hence, his approach is ofen known as he pole-placemen conrol design. In his paper, pole-placemen echnique via Bass and Gura s formula is proposed. In order o perform he pole-placemen design echnique, he sysem mus be a compleely sae conrollable sysem. In oher words, i mus be possible o move all he of sysem s open-loop poles by sae variable feedback, o any arbirary closedloop locaions. Therefore, before designing he conroller, a es has o be performed on he sysem marix by checking is rank where he rank mus be equal o he number of he column vecor. Then i can be concluded ha he sysem is compleely sae conrollable. Oherwise, anoher conroller design has o be performed. Figure 3 shows deailed block diagram of a sysem wih sae feedback conrol. D r() _ u() G dx() d x() C y() F K Figure 3 Deailed block diagram of a sysem wih sae feedback conrol The general sae space equaion for he block diagram in he Figure 3 is derived as: i () = [ ] () () x F GK x Gr (4.) Bass and Gura s formula o deermine he sae feedback gain marix is given as follows [3]:

7 where SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE K = Sae feedback gain marix S = [ G FG Fn G] a = a = W = ( ) T T T K = W S a a (4.) = Conrollabiliy marix a a an = Coefficiens of he desired characerisic equaion a a a n = Coefficiens of he sysem characerisic equaion a a a n n 4. Sae Feedback wih Inegral Conrol Normally, designing he sae feedback conroller by using only he pole-placemen design will give one major disadvanage where a large seady-sae error will be inroduced. In order o compensae his problem, an inegral conrol is added where i will eliminae he seady-sae error in responding o a sep inpu. Figure 4 shows he block diagram of he sysem wih he inegral conrol added. Inside he dashed box is he sae feedback conroller which has been designed before. A feedback pah from he oupu has been added o he error, e, which is fed forward o he conroller via an inegraor. The main funcion of adding an inegraor is o increase he sysem ype hus reduces he previous finie seady-sae error o zero. Therefore, a design for zero seady sae error for a sep inpu can be obained. From he block diagram in Figure 4, he sysem marix wih inegral conrol could be given as []: i x() F x() G = u i C xn () xn () () r() (4.3)

8 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI r() e x N Ke u() _ G dx() d x() C y() F K Figure 4 Block diagram of he sysem wih sae feedback and inegral conrol x y() = [ C ] x N However, from Figure 4, we can realize ha (4.4) x u() = Kx() KexN = [ K Ke] x (4.) N Therefore, he final derivaion for he sysem marix wih he inegral conrol is as follows: i x() ( F GK) GKe x() r = i C xn () xn () () r() (4.6) x y() = [ C ] x (4.7) N In order o implemen he Bass and Gura s formula o find he sae feedback gain marix for he sae-space sysem wih inegral conrol, some equaion modificaion has o be performed. Le F G F = and G = C (4.8) & (4.9) The conrollabiliy marix, S, can be obained using he F and G above. Now, he sae feedback gain marix can be obained by rewriing equaion (4.) as follows:

9 SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE 3 Therefore, ( ) T T T K = W S a a (4.) [ n ] K = K K K (4.) and he gain inegral can be obained as: K = K (4.) e. SIMULATION AND EXPERIMENTAL ANALYSIS Figure shows he coninuous ime hardware realizaion for he servomoor posiion conrol sysem wih he: (a) sae feedback conroller wihou inegral conrol (b) sae feedback conroller wih inegral conrol. Simulaions are performed for boh conrollers srucure where a uni sep inpu (-vols sep) signal is used as he reference signal. To accomplish one of he design requiremens, i is desired ha he oupu signal follows he given reference signal. Through simulaion, he mahemaical modeling for he servomoor is verified and he performances for boh conrollers srucures are analyzed. n Sep Inpu R(s) u (s) x =AxBu y=cxdu Sae-Space Servomoor Sysem hea (s) Oupu Scope K π u Sae-feeback gain (a) Sep Inpu R(s) Kn s InegraoInegral Gain u (s) x =AxBu y=cxdu Sae-Space Servomoor Sysem hea (s) Oupu Scope K π u [ ] x u Sae-feeback gain (b) Figure Sae feedback conroller (a) wihou inegral conrol (b) wih inegral conrol

4 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI Figure 6 Inpu and oupu plo from GUI of servomoor posiion conrol Figures 6 and 7 show he developed GUIs for he servomoor posiion conrol sysem

Thus his program provides more freedom and enables users o conrol he servomoor so ha i will perform jus he way hey wan i o be.

10 4 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI Figure 6 Inpu and oupu plo from GUI of servomoor posiion conrol Figures 6 and 7 show he developed GUIs for he servomoor posiion conrol sysem where he sae feedback conroller srucures are implemened. Basically, users will have he abiliy o specify he parameers for heir desired maximum percenage overshoo and seling ime. Thus his program provides more freedom and enables users o conrol he servomoor so ha i will perform jus he way hey wan i o be. Besides ha, users are also able o observe he responses of all signals online since hey are ploed immediaely as he sysem sars running. One example on how hese signals are ploed is shown in Figure 7. Figure 7 Sae variable responses from GUI of servomoor posiion conrol

11 SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE. Simulaion and Experimenal Resuls In order o analyze he performances of he proposed conrollers, he sysem is simulaed using MATLAB/SIMULINK. Nex, a compuer program is developed using Microsof Visual Basic 6. for experimen and resuls validaion. For his paper, here are hree differen case sudies presened for discussion. (a) Sae feedback conroller wih inegral conrol Case sudy : % maximum overshoo, seconds seling ime, vol iniial condiion 6 Simulaion Resul Sae Variable X 6 Experimenal Resul Sae Variable X Sae Variable X Sae Variable X Inpu signal ino servomoor, Va Inpu signal ino servomoor, Va Figure 8 Comparison beween simulaion and experimenal resuls for case sudy For he firs case sudy, i is desired o obain he oupu signal wih percen maximum overshoo and seconds seling ime. In his case, he iniial condiion is se o be vol. Plos on he lef-hand side of Figure 8 are he simulaion resuls while on he righ-hand side shows he plos of experimenal resul. Noice ha he x signal from experimen approximaely resembles wih he x signal from simulaion. The same observaion can be seen for x and he inpu signals. However, here are some noises associaed wih he signals from he experimen.

12 6 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI Case sudy : % maximum overshoo, 3 seconds seling ime, 9 vols iniial condiion Simulaion Resul Sae Variable X Experimenal Resuls Sae Variable X Sae Variable X Sae Variable X Inpu signal ino servomoor, Va Inpu signal ino servomoor, Va Figure 9 Comparison beween simulaion and experimenal resuls for case sudy In he second case sudy, i is desired o obain he response signal wih percen maximum overshoo and 3 seconds seling ime. In order o observe how he sysem responds o iniial condiion higher han he reference value, he iniial condiion is se o 9 vols. The same observaion as ha in he previous case is obained where he x signal from experimen approximaely resembles he signal from he simulaion. However, i can be observed ha ime delays and noises also exis in he experimenal resul as can be clearly seen in x and he inpu signals. Case sudy 3: % maximum overshoo, 4 seconds seling ime, 3 vols iniial condiion In he hird case sudy, i is desired o obain he response signal wih percen maximum overshoo and 4 seconds seling ime. In his case sudy, he iniial condiion is se o 3 vols. By comparing he simulaion and experimenal resuls, i can be observed ha he x signal from experimen is almos idenical o he simulaion resul bu has a small hump. As observed in previous cases, he x and inpu signals also resemble he simulaion resul wih he exisence of noise.

13 SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE 7 6 Simulaion Resul Sae Variable X 6 Experimenal Resuls Sae Variable X Sae Variable X Sae Variable X Inpu signal ino servomoor, Va Inpu signal ino servomoor, Va Figure Comparison beween simulaion and experimenal resuls for case sudy 3 (b) Sae feedback conroller wihou inegral conrol Case sudy : % maximum overshoo, seconds seling ime, vol iniial condiion Sae Variable X Sae Variable X Inpu signal ino servomoor, Va Figure Simulaion resul for case sudy

14 8 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI In he firs case sudy, i is desired o obain he response signal wih percen maximum overshoo and seconds seling ime wih vol iniial condiion. By referring o x signal, i can be observed ha he seady-sae error of he oupu response is approximaely 7.36 vols. Case sudy : % maximum overshoo, 3 seconds seling ime, 9 vols iniial condiion Sae Variable X Sae Variable X Inpu signal ino servomoor, Va Figure Simulaion resul for case sudy For he second case sudy, i is desired o obain he response signal wih percen maximum overshoo and 3 seconds seling ime. The iniial condiion is se o 9 vols. The same observaion can be obained for x signal where large seadysae error (approximaely 8. vols) is produced by he oupu response. Case sudy 3: % maximum overshoo, 4 seconds seling ime, 3 vols iniial condiion In he hird case sudy, i is desired o obain he response signal wih percen maximum overshoo and 4 seconds seling ime. For his case sudy, he iniial condiion is se o 3 vols. By referring o x signal in he Figure 3, he seady-sae error of he oupu response is approximaed o be 79. vols.

15 SERVOMOTOR CONTROL USING DIRECT DIGITAL CONTROL AND STATE-SPACE 9 3 Sae Variable X Sae Variable X Inpu signal ino servomoor, Va Figure 3 Simulaion resul for case sudy 3. Discussion Figures 8 o show he simulaion and experimenal resuls for he performances of hree differen case sudies for sae-feedback conroller wih he inegral conrol. While Figures o 3 show he performances of hree differen case sudies for sae-feedback conroller wihou he inegral conrol. From Figures 8 o, i can be observed ha he experimenal resuls for all hree cases equivalen wih he simulaion resuls. I can clearly be shown ha he sae variable x signals (which are he desired oupu responses) from experimens is approximaely same wih he sae variable x signals from he simulaion. The sae variable x and he inpu signals, on he conrary, resembled he simulaion resuls bu in he presence of accepable range of noise. The exisence of noise is pracically due o he characerisics and dynamics of a real sysem. Figures o 3 show only he simulaion resuls for sae-feedback conroller wihou he inegral conrol. From hese plos, i can be observed ha he seadysae values were far off han he reference inpu ( vols uni sep inpu) hus lead o unaccepably large seady-sae errors. In engineering poin of view, by having very large seady-sae error does reflec o insabiliy of he designed sysem. Pracically, he experimenal procedures can no be performed using he sae feedback conroller design wihou he inegral conrol due o signal limiaion of he hardware componens. Furhermore, expeced insabiliy sysem from simulaion has already proven ha here will no be possible o obain he desired oupu response by experimen.

16 6 MOHD FUA AD RAHMAT & MOHD SYAKIRIN RAMLI 6. CONCLUSION A graphical user inerface (GUI) using Microsof Visual Basic 6. is successfully developed where sae-feedback conroller wih and wihou inegral conroller srucures have been implemened. Through experimen, i can be observed ha he experimenal resuls for he sae-feedback conroller wih inegral conrol agreed wih he simulaion resuls. While, he simulaion resuls for he sae-feedback conroller wihou he inegral conrol show ha his design leads o unaccepably large seady-sae errors on he oupu responses (x ). Therefore, experimen canno be performed using he GUI o implemen his conroller srucure. REFERENCES [] Tee, K. G.. Posiion Conrol of DC Servomoor Using Direc Digial Conrol and Sae Space Technique. Tesis Sarjana Muda UTM. [] Nise, N. S. 4. Conrol Sysems Engineering. John Wiley & Sons. 4 h Ediion [3] Paraskevopoulos, P. N Digial Conrol Sysem. Prenice Hall [4] Sub, G., D. S. Hyun, J. Park, K. D. Lee and S. G. Lee.. Design of a Pole Placemen Conroller for Reducing Oscillaion and Seling Time in a Two-Ineria Moor Sysem. 7 h Annual Conference of he IEEE Indusrial Elecronics Sociey. [] Zenger, K. and R. Ylinen.. Pole Placemen of Time-Varying Sae Space Represenaions. 44 h IEEE Conference on Decision and Conrol, and he European Conrol Conference. [6] Chow, J. H. 99. A Pole Placemen Design Approach for Sysems wih Muliple Operaing Condiions. IEEE Transacions on Auomaic Conrol. [7] Nordsrom, K. and H. Norlander On he Muli-Inpu Pole Placemen Conrol Problem. Proceedings of he 36 h Conference On Decision & Conrol. [8] Shyu, K. K., C. K. Lai and J. Y. Hung.. Toally Invarian Sae Feedback Conroller for Posiion Conrol of Synchronous Relucance Moor. IEEE Transacions on Indusrial Elecronics. [9] Paraskevopoulos, P. N Digial Conrol Sysem. Prenice Hall. 8-9.

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