Position control of DC motor using fractional order controller

Transcription

1 ARCHIVES OF ELECTRICAL ENGINEERING VOL. 6() (1) DOI 1.478/ee-1-41 Position control of DC motor using frctionl order controller ANDRZEJ RUSZEWSKI ANDRZEJ SOBOLEWSKI Fculty of Electricl Engineering Biłystok University of Technology e-mil: (Received: revised: 9.1.1) Abstrct: The er resents the roblem of osition control of DC motor with rted voltge 4 V loded by flywheel. The frctionl order PD controller imlemented in Ntionl Instruments NI ELVIS II rogrmmed in LbView is used for controlling. The simle method for determining stbility regions in the controller rmeters sce is given. Knowledge of these regions ermits tuning of the controller nd ensures required the hse mrgin of the system. Key words: osition DC motor controller frctionl order 1. Introduction In recent yers considerble ttention hs been id to frctionl clculus nd its liction in mny res in science nd engineering (see e.g. [ ]). In control system frctionl order controllers re used to imrove the erformnce of the feedbck control loo. The frctionl PID controllers nmely PI λ D controllers including n integrtor of λ order nd differentitor of order were roosed in [15 16]. The roblem of tuning the PI λ D controllers hve been resented in the literture (see e.g. [ ]). The symtotic stbility is the bsic requirement of closed-loo system. Some methods for determining the symtotic stbility regions in the frctionl controller rmeter sce were roosed in [8 17]. Gin nd hse mrgins re mesures of reltive stbility for feedbck system. For exmle in er [18] simle method of determining the stbility region (stisfying the conditions of gin nd hse mrgins) in the rmeter sce of frctionl order inertil lnt with time dely nd frctionl order PID controller ws given. In order to imlement (digitl reliztion) of the frctionl order controller the roximte time rtionl trnsfer function should be determined. Some methods of obtining the discrete trnsfer function roximting the continuous frctionl order trnsfer function hve been roosed in the literture [1-14 -]. Generlly the digitl trnsfer function is bsed on the cnonicl form of the infinite imulse resonse filter (IIR filter). Such n lgorithm cn be Unuthenticted Downlod Dte 7/5/18 6:4 AM

2 56 A. Ruszewski A. Sobolewski Arch. Elect. Eng. directly imlemented into ny rocessor bsed devices s for instnce rogrmmble logic controller (PLC) [1 14]. In this work the roblem of osition control of DC motor loded by flywheel is resented. The considered DC-motor is comonent of the system QUANSER DC Motor Position Control (DCMPC). The frctionl order PD controller imlemented in the device Ntionl Instruments NI ELVIS II rogrmmed in the grhicl environment LbView is used. The rmetric synthesis of the controller is crried out using the roosed method for determining the stbility regions in the controller rmeter sce for the secified hse mrgin of the system. The controlled object is modeled s the trnsfer function of the inerti with integrtion. The comrtive studies of this control system re resented.. Controlled system The DC Motor Position Control system shown in Figure 1 consists of direct-current motor with n inerti wheel on the motor shft with rotor inerti gcm. The nominl outut ower of the motor is 1.98 W nd the mximl rottionl seed reches 75 rm with seed constnt 685 rm/v. The ngulr osition of the motor shft is mesured by n oticl encoder with resolution of roximtely.879. Due to the resolution of the encoder nd the use of the 4 bit counter the rnge of vlues tht cn be red by the encoder is roximtely ±496 π. The motor is driven using ulse-width modulted (PWM) ower mlifier. The mximum outut voltge of the mlifier is 4 V. Fig. 1. DCMPC system nd controlled object: 1) DC motor ) encoder ) motor metl chmber 4) Inertil lod The ste-test method is used for ccurte lnt modeling for control system nlysis nd design. This model is comred with the mesured resonse by running the simultion nd ctul system in rllel. The model rmeters re tuned for better fit. The identifiction of the motor model rmeters is erformed using V ste inut in n oen-loo. Figure shows the comrison of the model nd the motor resonse chrcteristics which re crried out for smling time 5 ms. Unuthenticted Downlod Dte 7/5/18 6:4 AM

3 Vol. 6(1) Position control of DC motor using frctionl order controller 57 The resulting model is trnsfer function from voltge to motor shft osition K G ( s) K > τ > (1) s( sτ 1) where K is the stedy-stte gin nd J is the time constnt. Fig.. Ste resonses of lnt nd its model with ste inut V The exerimentlly obtined ste resonse of the system is roximted to the ste resonse of the model (1) with the rmeters K 5 [rd/sv] nd J.15 [s].. Frctionl order PID controller The frctionl order PI λ D controller [15 16] ws roosed s generliztion of the PID controller [] with integrtor of rel order λ nd differentitor of rel order. The trnsfer function of such tye controller in the Llce domin tkes the form λ C( s) k ki s k d s λ > > () where k k i i k d denote the roortionl integrl nd differentil gins of the controller resectively. k k i nd k d rmeters usully tke non-negtive vlues. Tking λ 1 nd 1 we obtin clssicl PID controller. If k d or k i we obtin PI λ controller or PD controller resectively. All these tyes of controllers re rticulr cses of the PI λ D controller. In this er we use the PD controller whose trnsfer function hs the form C( s) k k s. () The block digrm of the considered feedbck control system is shown in Figure where G(s) hs form (1) nd C(s) hs form (). d Unuthenticted Downlod Dte 7/5/18 6:4 AM

4 58 A. Ruszewski A. Sobolewski Arch. Elect. Eng. r e C ( s ) u G(s) y - Fig.. Feedbck control system structure.1. Prmetric synthesis of the controller The symtotic stbility is the bsic requirement of closed-loo system. Gin nd hse mrgins re mesures of reltive stbility for feedbck system therefore the synthesis of control systems is very often bsed on them. In tyicl control systems the hse mrgin is from to 6 wheres the gin mrgin is from 5 db to 1 db. In this er we tuning the controller for the secified hse mrgin becuse hse mrgin is closely relted to trnsient resonse i.e. overshoot. For this urose we use the hse mrgin tester ex(jφ) where φ is hse mrgin. This tester does not exist in the rel control system it is only used for the controller tuning. Tking into ccount the hse mrgin tester ex(jφ) in the min th of control lnt (1) nd controller () we obtin the chrcteristic olynomil of the closed-loo system jφ w( s) e K ( k k s ) s( sτ 1). d (4) Using the clssicl D-rtition method the stbility region in the rmeter lne (k d k ) my be determined nd the rmeters cn be secified. The stbility boundries re curves on which ech oint corresonds to olynomil (4) hving zeros on the imginry xis. It my be the rel zero boundry or the comlex zero boundry. It is esy to see tht olynomil (4) hs zero s if k (the rel zero boundry). The comlex zero boundry corresonds to the ure imginry zeros of (4). We obtin this boundry by solving the eqution jφ w( jω) e K ( k k ( jω) ) jω( jωτ 1) d (5) which we get by substituting s jω in olynomil (4) nd equting to. The comlex Eqution (5) cn be rewritten s set of rel equtions in the form Re[ w ( jω)] Im[ w( jω)]. (6) where Re[w(jω)] nd Im[w(jω)] denote the rel nd the imginry rts of (5) resectively. Finlly by solving the Equtions (6) we obtin 1 ωτ sin φ cosφ k ω d (7) π K sin π π ωτ sin φ cos φ k ω. (8) π K sin Unuthenticted Downlod Dte 7/5/18 6:4 AM

5 Vol. 6(1) Position control of DC motor using frctionl order controller 59 Equtions (7) nd (8) determine the comlex zero boundry s function of ω. The rel zero boundry nd the comlex zero boundry for ω decomose lne (k d k ) into some regions. The stbility region is chosen by testing n rbitrry oint from ech region nd checking the stbility of olynomil (4). In this er only the stbility region in the rmeter lne of olynomil (4) is resented. When the stbility regions re known the tuning of the frctionl controller cn be crried out. On comuting comlex nd rel zero boundries by the roosed method we obtin the stbility regions in the controller rmeter lne (k d k ). The stbility regions for φ 6.6 re shown in Figure. The comlex boundry of stbility regions is clculted for trnsfer function (1) with K 5 nd τ.15. The stbility region lies between line k (the rel zero boundry) nd the curve ssigned to secified hse mrgin φ (the comlex zero boundry). On choosing ny oint from the stbility region we obtin the controller rmeter vlues rovided the hse mrgin of this system is greter thn tht secified for drwing the comlex boundry. Any oint from the stbility region rovides hse mrgin of this system greter thn 6. The vlue of.6 is chosen s n exmle Fig. 4. Stbility regions for φ k k d The controller rmeters nd stbility mrgins of the control system for ll oints mrked in Figure 4 re listed in Tble 1. It is shown tht the stbility mrgin vlues re greter thn tht secified for drwing the comlex boundry of the stbility region. Tble 1 confirms tht oints from the stbility regions stisfy the hse mrgin requirements. Tble 1. Controller rmeters nd hse mrgins Point Controller rmeters Phse mrgin [ ] 1 k d. k k d.6 k k d 1. k Unuthenticted Downlod Dte 7/5/18 6:4 AM

6 51 A. Ruszewski A. Sobolewski Arch. Elect. Eng... Digitl reliztion of the frctionl order controller Trnsfer functions () nd () of the controller re the irrtionl functions in the vrible s. Therefore ccurte hysicl reliztion of the differentition nd integrtion of frctionl order is not ossible. In order to imlement the frctionl order controller the roximte time rtionl trnsfer function should be determined. The roximte time rtionl trnsfer function my be in the form of discrete trnsfer function with integrl orders. Such trnsfer function cn be directly imlemented into ny rocessor bsed devices. Some methods of obtining the discrete trnsfer function roximting the continuous frctionl order trnsfer function hve been roosed in the literture [1-14 -]. In the er [19] the reliztion of frctionl order controller imlemented in Ntionl Instruments sbrio-961 controller rogrmmed in LbView ws resented. In order to digitlly relize the frctionl order controller trnsfer function oertor-bsed continuous frction exnsion (CFE) scheme ws lied. In this er the sme scheme is used however the reliztion of frctionl order PD controller is imlemented in Ntionl Instruments NI Elvis II device. Trnsfer function () corresonds in the discrete domin to the discrete trnsfer function hs the form 1 ( ϖ ( )) C( z ) k k d z (9) 1 where the roximte trnsfer function of the differentitor of rel order hs the form [1] 1 T 1 ( ϖ ( z )) P( z Q( z T ) 1 ) T 1 z CFE 1 z q q 1 1z 1 1z 1 1 q z z q L L q m z q m z. (1) CFE{.} is the continued frction exnsion P nd Q re olynomils of degree nd q resectively is rtio term nd T is smling time. Deending on the vlue of { 1/7 1} we obtin Euler rule Al-Aloui rule nd Tustin rule resectively. Coefficients of P nd Q olynomils deend to the order of the roximte model nd we cn clculte them from the following forms for q 1 q q 1 1 (11) wheres for q Unuthenticted Downlod Dte 7/5/18 6:4 AM

7 Vol. 6(1) Position control of DC motor using frctionl order controller q q q (1) The rcticl frctionl controller reliztion is imlemented in NI Elvis II device using the NI LbView grhicl rogrmming lnguge. The rogrm window for entering the vlues of controller rmeters nd roximtion rmeters is shown in Figure 5. Fig. 5. Progrm window for setting u frctionl control coefficients One of the liction feture is set of controller rmeters like k k i k d λ nd roximtion rmeters T. All those dt re used to determine coefficients of ro- Unuthenticted Downlod Dte 7/5/18 6:4 AM

8 51 A. Ruszewski A. Sobolewski Arch. Elect. Eng. ximting discrete trnsfer function. For exmle if the controller rmeters re k d 1 k.1.6 (oint in Tb. 1) nd roximting coefficients re T.5. n 1 thn the discrete trnsfer function roximting the continuous frctionl order trnsfer function () hs the form z C ( z ) (1) 1 15 z wheres for n T.5 s. nd the sme vlues of controller rmeters (oint in Tble 1) the discrete trnsfer function (9) hs the form z 6.4z C ( z ). (14) z z The comrison of Bode chrcteristics for the continuous trnsfer function of controller () with k d 1 k.1.6 nd the discrete trnsfer functions (1) nd (14) is shown in Figure 6. Mgnitude (db) 4 () (1) 1 (14) Frequency (rd/s) 9 Phse (deg) Frequency (rd/s) Fig. 6. Comrison of Bode lots of trnsfer functions (1) (14) nd () with k d 1 k Exeriments The exeriments mde it ossible to observe how the osition control loo oertes s result of the set oint chnging from π to π rdins. All exeriments were crried out for smling time T.5 s. The controller rmeter vlues re chosen ccording to Tble 1 which resents hse mrgin secifictions (oints mrked in Fig. 4). Ste resonses of the system controlled by rcticl imlemented controller with n 1 nd n re shown in Figures 7 nd 8 resectively. From these figures we cn see tht the order of roximtion ffects the chrcter of ste resonses. When n 1 ll resonses re chrcterized by over- Unuthenticted Downlod Dte 7/5/18 6:4 AM

9 Vol. 6(1) Position control of DC motor using frctionl order controller 51 shoot wheres ll ste resonses obtined for n re eriodic. The system resonse for the control rmeters corresonding the oint 1 of Tble 1 hs the lrgest overshoot (Fig. 7). Fig. 7. Ste resonses of the closedloo system with the PD controller (roximtion order n 1) Fig. 8. Ste resonses of the closedloo system with the PD controller (roximtion order n ) Fig. 9. Ste resonses of the closedloo system with the PD controller relted to (n ) Unuthenticted Downlod Dte 7/5/18 6:4 AM

10 514 A. Ruszewski A. Sobolewski Arch. Elect. Eng. In lbortory test bench is lso exmined the imct of vlue coefficient on ste resonses. The controller rmeters k d nd k re tken in ccordnce with oint of Tble 1 nd the vlue of is vried. Exemlry time resonses obtined for..5.7 re shown in Figure 9. We cn see the trend of chnges in osition chrcteristics versus vlues of coefficient. The ste resonse with the shortest settling time the zero of stedy stte error nd the lck of overshoot is obtined for.5. The figure shows tht for smller vlues of (e.g..) ste resonses re chrcterized by the overshoot. Fig. 1. Ste resonses of the closedloo system with the PD controller relted to (.5 n ) Another exmintion refers to the influence of the roximtion coefficient on ste resonses. The controller rmeters k d nd k re tken ccording to oint of Tble 1 with the vlue of.5. Figure 1 shows the comrison of chrcteristics obtined for three vlues of the rmeter i.e..1. nd.9. We cn see tht coefficient hs the significnt effect on the overshoot nd stedy stte error. The incresing vlue of results in lrger oscilltions wheres the decresing cuses lrger stedy stte error. From Figures 9 nd 1 we cn see tht the vlues of coefficient nd order hve significnt effect on the overshoot nd stdy stte error. 5. Conclusion In this er the osition control of DC motor shft loded by flywheel is given. The frctionl order PD controller imlemented into Ntionl Instruments NI ELVIS II device rogrmmed in LbVIEW is used. The rmetric synthesis of the frctionl order PD controller ws erformed using the roosed method for determining the stbility regions in the rmeter sce of the controller. Knowledge of these regions ermits tuning of the controller nd ensures required the hse mrgin of the system. The results of exerimentl reserch confirm tht the roosed imlementtion technique simlifies the rocess of determining digitl reliztion of the continous frctionl order con- Unuthenticted Downlod Dte 7/5/18 6:4 AM

11 Vol. 6(1) Position control of DC motor using frctionl order controller 515 troller. Thnks to grhicl interfce user is ble to introduce frctionl order controller gins nd utomticlly determine the discrete trnsfer function roximting the continuous frctionl order trnsfer function. Acknowledgments The work ws suorted by the Ntionl Centre of Science in Polnd under grnt No. N N References [1] Al-Aloui M.A. Filling the g between the biliner nd the bckwrd difference Trnsforms: n interctive design roch. Int. J. Elect. Eng. Edu. 4(4): 1-7 (1997). [] Astrom K.J. Hgglund T. PID Controllers: Theory Design nd Tuning. nd ed. Reserch Tringle Prk NC: Instrument Society of Americ (1995). [] Bisws A. Ds S Abrhm A. Dsgut S. Design of frctionl-order PI λ D controllers with n imroved differentil evolution. Engineering Alictions of Artificil Intelligence (): 4-5 (9). [4] Busłowicz M. Selected roblems of continuous-time liner systems of non-integer order. Mesurement Automtion nd Robotics : (1) (in Polish). [5] Conetto R. Dongol G. Fortun L. Gllo A. New results on the synthesis of FO-PID controllers. Communictions in Nonliner Science nd Numericl Simultion 15(4): (1). [6] Cstillo J. Feliu V. Rivs R. Snchez L. Design of clss of frctionl controllers from frequency secifictions with gurnteed time domin behvior Comuters nd Mthemtics with Alictions 59(5): (1). [7] Ds S. Functionl frctionl clculus for system identifiction nd controls. Sringer Berlin (8). [8] Hmmci S.E. An lgorithm for stbiliztion of frctionl-order time dely systems using frctionl-order PID controllers IEEE Trns. on Automtic Control 5: (7). [9] Kczorek T. Selected Problems of Frctionl Systems Theory. Sringer Berlin (11). [1] Luo Y. Chen Y.Q. Frctionl order [roortionl derivtive] controller for clss of frctionl order systems. Automtic 45(1): (9). [11] Monje C.A. Vingre B.M. Feliu V. Chen Y. Tuning nd uto-tuning of frctionl order controllers for industry lictions. Control Engineering Prctice 16: (8). [1] Ostlczyk P. Eitome of the Frctionl Clculus Theory nd its Alictions in Automtics. Publishing Dertment of Technicl University of Łódź (8) (in Polish). [1] Petrs I. Frctionl-order feedbck control of DC motor. Journl of Electricl Engineering 6(): (9). [14] Petrs I. Reliztion of frctionl-order controller bsed on PLC nd its utiliztion to temerture control. Trnsfer inovcii 14: 4-8 (9). [15] Podlubny I. Frctionl differentil equtions. Acdemic Press Sn Diego (1999). [16] Podlubny I. Frctionl-order systems nd PI λ D controllers. IEEE Trns. on Automtic Control 44: 8-14 (1999). [17] Ruszewski A. Stbility regions of closed loo system with time dely inertil lnt of frctionl order nd frctionl order PI controller. Bull. Pol. Ac.: Sci. Tech. 56(4): 9- (8). [18] Ruszewski A. Stbiliztion of frctionl-order inertil lnts with time dely using frctionl PID controllers. Mesurement Automtion nd Robotics : (9) (in Polish). [19] Ruszewski A. Sobolewski A. Comrtive studies of control systems with frctionl controllers. Przegląd Elektrotechniczny 88(4b): 4-8 (1). [] Tenreiro M. Glhno A.M. Oliveir A.M. Tr J.K. Aroximting frctionl derivtives through the generlized men. Communictions in Nonliner Science nd Numericl Simultion 14(11): 7-7 (9). Unuthenticted Downlod Dte 7/5/18 6:4 AM

12 516 A. Ruszewski A. Sobolewski Arch. Elect. Eng. [1] Vingre B.M. Podlubny I. Hernndez A. Feliu V. Some roximtions of frctionl order oertors used in control theory nd lictions. Frctionl Clculus nd Alied Anlysis (): 1-48 (). [] Vingre B.M. Chen Y.Q. Petrs I. Two direct Tustin discretiztion methods for frctionl order differentitor/integrtor. Journl of the Frnklin Institute: Engineering nd Alied Mthemtics 4: 49-6 (). [] Zho C. Xue D. Chen Y.Q. A frctionl order PID tuning lgorithm for clss of frctionl order lnts. [In:] Proc. of the IEEE Interntionl Conference on Mechtronics & Automtion Nigr Flls Cnd (5). Unuthenticted Downlod Dte 7/5/18 6:4 AM