Fractional Order Controller for PMSM Speed Servo System Based on Bode s Ideal Transfer Function

Transcription

1 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp. -7 Sensors & Transduers 24 by IFSA Publishing, S. L. Frational Order Controller for PMSM Speed Servo Syste Based on Bode s Ideal Transfer Funtion, 2 You-Bin ZHANG, You-Guo PI College of Autoation Siene and Engineering, South China University of Tehnology, Guangdong, 564, China 2 Departent of Autoation, Guangdong University of Petroheial Tehnology, Guangdong, 525, China Tel.: E-ail: 668zyb@63.o Reeived: 8 April 24 /Aepted: 3 May 24 /Published: 3 June 24 Abstrat: The researh fouses on the frational order ontroller of the peranent agnet synhronous otor (PMSM) speed servo syste. In view of the fat that the algorith of frational order PID ontroller paraeter tuning is oplex and its alulation aount is large, a frational order ontroller based on Bode s ideal transfer funtion and its tuning algorith are put forward. This algorith is using the harater of Bode s ideal transfer funtion to tuning the frational ontroller paraeters. The tuning algorith is siple and easy. The plant is the atheatial odel of the PMSM speed servo syste whih the urrent loop is siplified. The frational ontroller is designed with this plant, and it is opared with the integer order PI speed loop ontroller by both siulation and experient. The siulation and experient results show that the frational ontroller an keep the syste overshoot while plant gain hanging. It eans that the frational ontroller has strong gain robustness on overshoot. And at the sae tie on the ontrol perforane index, the frational order ontroller is obviously superior to the integer order PI ontrol. Copyright 24 IFSA Publishing, S. L. Keywords: Bode s ideal transfer funtion, Frational order ontroller, PMSM, Speed servo syste, Robustness.. Introdution Peranent agnet synhronous otor (PMSM) has the advantages of high effiieny, high power density, high torque urrent ratio, good reliability, strong adaptability to environent, so it is widely used in industrial field, suh as robot ontrol, nuerial ontrol ahine tool, inforation tehnology, eletri vehile, et. []. The PMSM drive syste is widely adopted by double losed-loop integer order PI ontroller [2]. But PMSM is ultivariable, nonlinear and tie-varying syste [3]. The integer order PI ontrol () has great liitations, it ouldn t adapt to the hanges of syste paraeters [4]. In reent years, with the ontinuous iproveent of the frational alulus nuerial ethod and its algorith, all kinds of frational order analysis ethod, ontrol strategy and the frational order ontroller designs are put forward [5-7]. These proote the appliation of frational order ontrol theory [8]. Espeially, the frational order PID (FOPID) ontroller whih is put forward by Podlubny extended the integer order PID ontroller [9]; it an ahieve better ontrol perforane, but its ontrol paraeters as any as five and the five paraeters are diffiult to tuning. Soe sholars have put forward the tuning ethods of the frational order PID [-2], but these tuning ethods are oplex

2 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp. -7 and the alulations are large, so they are diffiult to proote. The Bode s ideal transfer funtion was put forward by Bode when he was designing a feedbak aplifier; it is fored by gain aplifier and integrator in series [3]. Beause the order of integer order alulus is integer, and the order of Bode s ideal transfer funtion is frational, the appliation of Bode s ideal transfer funtion in integer order ontroller is greatly liited. With the developent of the frational order ontrol theory, it is possible to adjust the ontrol syste as Bode s ideal for by a frational ontroller, and the design of the fration order ontrol is easier than the frational order PID ontroller. The Bode s ideal transfer funtion is only two paraeters, and the two paraeters an be tuned for speifi ontrol indiators and ontrol requireents. So the paraeters tuning is siple and reliable. In this paper, based on the frequeny doain harateristis of Bode s ideal transfer funtion and the losed-loop syste perforanes, a frational ontroller is put forward. When the ut-off frequeny and phase angle argin are given, the Bode s ideal transfer funtion an be got and it will be regarded as the open-loop transfer funtion of syste; then the frational order ontroller paraeters an be deterined obined with the plant. The realization ethod of this frational order ontroller is siple and easy. 2. Methodology 2.. Bode s Ideal Transfer Funtion The Bode s ideal transfer funtion is expressed as Eq.. ω (), Gs = R, () s where ω is the ut-off frequeny, that is G( jω ) =. is the order, its value an be arbitrary, > for integrator, < for differentiator. Frational ontroller design ethod and steps: For the plant Ps () and frational order ontroller Cs (), we an order the Bode s ideal transfer funtion Gs () as Eq. (2) Gs () CsPs () () =, (2) So the ontroller an be got by Eq. (3). Gs () ω Cs () = =, (3) Ps () sps () If the plant odel is known and the ontrol perforane requireents are given, the paraeters of the ontroller an be got. Therefore the frational ontroller design steps are as follows: ) Given ω and ; 2) Get ontrol plant odel and paraeters; 3) Get the frational order ontroller Cs () aording to Eq. (3). Disrete of frational order alulus: Beause there is no digital tool or ethod to reappear the frational order alulus opletely at present, people usually use the integer order approxiation ethod to realize the frational order alulus, and then it use integer order digital realization ethod to disperse the frational order alulus, suh as Euler ethod, Tustin ethod, Al-Alaoui ethod, FIR filter, IIR filter and so on. Oustaloup filter that belongs to IIR filter is a very good frequeny fitting [4]. Its priniple is as follow: At the seleted frequeny range ( ωb, ω h), for H() s = s ν, a rational funtion an be get to approxiate it. where N s + ω j H() s HN () s = K, (4) s + ω j= N j ν /2 N ω ω h j K = ωb j= Nωj ω h ωj = ωb ω b ω h ω j = ωb ω b j+ N+ ( + ν )/2 2N + j+ N+ ( ν )/2 2N +, (5) N is the approxiate order and it is usually no ore than The Model of PMSM Speed Servo Syste Aording to PMSM vetor transfer theory, the PMSM eletroagneti torque T e an be expressed as Eq. (6) 3 Te = npψ fiq 2 id =, (6)

3 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp. -7 where n p is the nuber of pole-pairs; ψ f is the rotor flux linkage; i d and i q are the arature urrent oponent in dq oordinate. The Blok diagra of PMSM speed servo ontrol syste is shown in Fig.. It is ainly inludes PMSM, Spae Vetor PWM (SVPWM), Current ontroller (ACR) and Speed ontroller (/) and inverter (Wang Rui-Ping and Pi You-Guo, 22). It is a double losed loop syste that onsists of urrent loop and speed loop. The PMSM speed ontrol syste an be regard as a DC speed ontrol syste and the dynai struture diagra is shown as Fig. 2 [5, 6]. In Fig. 2, WASR is speed ontroller whih is required to design. The WACR is urrent ontroller using integer order PI ontroller and the urrent loop is adjusted to typial I type syste. Then after redued order proessing, the syste is siplified as Fig. 3. So after urrent loop siplified, the plant an be shown as Eq. (7) where K p K p Ps () = sts ( + ), (7) Rα / β CT =, T = T Σ n. e Fig.. Blok diagra of PMSM servo ontrol syste. Fig. 2. The speed servo syste dynai struture. Fig. 3. The syste dynai struture diagra after urrent loop siplified. 2

4 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp Experient Setup The experient syste onsists of PMSM, PMSM driver board, siulator, enoder and the ontrol software is shown in Fig. 4. The PMSM is PB82BXS; PMSM driver board is designed by us; the ontrol hip uses DSPTMSF282. The ontrol software is the Ti Code Coposer Studio (CCS). When =, the syste overshoot is zero, and the phase angle is 9. The syste satisfies the requireent of phase angle stability when the syste has no overshoot. At the sae tie, phase angle urve is a horizontal line. It also an satisfy the aplitude argin. So when >, the syste ust be stable. Magnitude (db) Bode Diagra =. =.5 =.9 Fig. 4. The experient setup. Phase (deg) Results and Disussion Frequeny (rad/se) 3.. Bode s Ideal Transfer Funtion Perforanes For Eq. (), Bode s ideal transfer funtion an be regarded as a syste that is in a series with an integrator and a gain aplifier. It is only two paraeters. Bode s ideal transfer funtion of the frequeny doain is expressed as Eq. 8. G( jω) = ω ( jω) -π -π = ωω os( )+ j sin( ) 2 2 When ω = ( rad / s), (8), the aplitude frequeny harateristis and phase frequeny harateristis are shown in Fig. 5. Fig. 5 shows that its phase angle is independent of the frequeny; its phase angle is a onstant phase angle π /2, and the phase argin φ is π π /2. The aplitude-frequeny harateristi urve is a straight line and the line slope is 2. The step response urve of the Bode s ideal transfer funtion unit feedbak syste is shown in Fig. 6. Fig. 6(a) shows that the response speed is inreased when ω is inreased, and syste overshoot doesn't hange with ω, naely overshoot is not sensitive to the gain hanging. This eans the Bode s ideal transfer funtion syste has strong robustness to gain. It an be seen fro Fig. 6(b) that the syste overshoot inreases with the inrease of integral order, and the phase argin is π / Fig. 5. Bode Diagra of Bode s ideal transfer. ω = ω =25 ω =5 ω = Tie (se) (a) Tie (se) (b) =. =.2 =.4 =.6 =.8 Fig. 6. The step response of unit feedbak syste. 3

5 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp. -7 Aording to the Bode s ideal transfer funtion with exellent properties, it an be onsidered as a goal to design the ontroller. The designed ontroller adjusts the syste open-loop transfer funtion as sae as the Bode s ideal transfer funtion, so that the designed ontroller will has good ontrol perforane. Bode s ideal transfer funtion unit feedbak syste oposed of only two paraeters, and the effet of eah paraeter as desribed above. When and ω are speified, the syste perforane an be known. In general, the frequeny doain indexes of ontrol syste ain onern for the ut-off frequeny, the phase angle argin and aplitude argin. When speify these indexes, andω an be obtained diretly, so it is very siple to ake sure the ontroller paraeters Designed and Given the following onditions: ) Plant inertial tie:t =.25; 2) Plant gain: K p =2.3; 3) Cut-off frequeny: ω =22dB/rad; 4) Phase argin: φ =7, so =.222. So the plant is: K p 2.3 Ps () = = st ( s+ ) s(.25s+ ), (9) Σn The based on Bode s ideal transfer funtion an be got as Eq. (). Gs () ω sts ( + ) Cs () = = Ps () skp (.25 s s ) 22 = + 2.3, () The traditional speed ontrol often uses integer order PI (IOPI) ontrol strategy. Using the integral tie absolute error (ITAE) perforane index, given T and K p as above, the designed by this paper is shown as Eq. (). Kpn( τ ns+ ) WASR () s = τ ns, () 4.89(.545s + ) =.545s 3.3. Siulation Researh ( Coparison with the two Controllers) The step response using the two ontrollers is shown in Fig. 7. In Fig. 7, using the IOPI ontroller, its ontrol perforanes are following: %=2.8% σ (overshoot), t r =.745 (rise tie), t s (2%) =.93 (setting tie); using the, its ontrol perforanes are: σ %=8.62%, t r =.683, t s (2%) =.356. The siulation results show that is better than IOPI ontroller on the ontrol perforanes. It an be seen fro Fig. 5 that the designed following the proposed ethod in this paper is effetive Tie (se) Fig. 7. Siulation. Step responses oparison with the two ontrollers Siulation Researh 2 (The influene of plant gain hanging) In the atual otion ontrol syste, due to the environent hanging or other reasons, it ay ause the hange of the plant's gain. In Fig. 6, the unit step responses are plotted with the open-loop plant gain ±2% variations for value 2.3. It an be seen fro Fig. 8(a) that the enables the syste overshoot varied with plant gain hanging; the overshoot inreases with inrease of the plant gain. But in Fig. 8(b) the syste overshoot doesn t vary while plant gain hanges. It shows that the designed by this paper has strong gain robustness Siulation Researh 3 (The influene of the paraeters) When hoosing different phase angle argin and the ut-off frequeny, the ontrol of the paraeters will be hange. When the phase argin is set at 7, the step response of syste is shown in Fig. 9(a) and the step response data is listed in Table. When the ut-off frequeny is set at 5 (rad/s), the step response of syste is shown in Fig. 9(b) and the step response data is listed in Table 2. It an be seen fro Table and Table 2 that the akes the syste has the basi harateristi of the Bode ideal funtion. It an be a referene to adjust the two paraeters of. 4

6 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp Kp-2% Kp=2.3 Kp+2%.4.2 Kp-2% Kp=2.3 Kp+2% Tie (se) Tie (se) (a) (b) Fig. 8. Siulation. Step responses using ontroller with different plant gain ω =5 ω = ω = φ =5 φ =6 φ = Tie (se) Tie (se) (a) φ = 7 o (b) ω = 5 Fig. 9. Siulation. The step response with different paraeter. (rad/s) Table. The step response data with different ω. ω /( rad / s) σ (%) ts / s (2%) tr / s Table 2. The step response date with differentφ. φ (o) σ (%) t / s(2%) t / s Experient Researh Experient syste uses SVPWM ontrol strategy and respetively uses the sae and as that s r in siulation researh. The experiental data are exported fro CCS. The step response of syste is shown in Fig. 8 and the ontrol perforanes are shown in Table 3. Table 3. The step response experient data. σ (%) t / s(2%) t / s IOPI FO Fro Fig. and Table 3, it an be seen obviously that at the sae rise tie the overshoot and the set tie with the are saller than that with the IOPI ontroller, so we an see the outperfors the IOPI ontroller. The traking response and the error of and are shown in Fig. and Fig. 2.Fro Fig. s r 5

7 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp. -7 and Fig. 2, it an be seen that the traking perforane is better then that of the..6.4 input.4.2 Spped(pu) Tie (se) Fig.. Experient. Step responses oparison with two ontrollers Tie(se)..8.6 (a) Traking responses Error oparing Spped(pu) input Error(pu) Tie(se) (b) Error Error Tie(se) Fig. (a). Sine wave traking responses and error with the two ontrollers: traking responses Error oparing Tie(se) Fig. (b). Sine wave traking responses and error with the two ontrollers: error. Fig. 2. Triangular wave traking responses and error with the two ontrollers. The experient results further show that designed base on Bode s ideal transfer funtion is feasible and effetive, and it an better iprove the syste ontrol perforane. 4. Conlusions In this paper, the harateristis of the Bode s ideal transfer funtion have been studied. The study results point that the Bode s ideal transfer funtion unit feedbak syste has strong robustness of plant gain and phase angle stability. The designed based on Bode s ideal transfer funtion is feasible and effetive in pratie. The paraeter tuning is siple, onvenient and easy. Siulation and experiental results show that oparing with, to peranent agnet synhronous otor speed servo syste, an get better ontrol perforane; and the has strong gain robustness on overshoot. 6

8 Sensors & Transduers, Vol. 73, Issue 6, June 24, pp. -7 Referenes []. Wang Song, Study on the paraeters identifiation and ontrol strategies of peranent agnet synhronous otor, Beijing Jiaotong University, Beijing, 2. [2]. Zhang Hui, Study on robust ontrol of peranent agnet synhronous ahine drive syste, China University of Mining and Tehnology, Jiangsu, 2. [3]. Y. Z. Zhou, J. Zhong, J. H. Zhou, A novel sliding ode variable struture diret torque ontrol for peranent agnet synhronous otor drive syste, Journal of Power Supply, Vol., 2, pp [4]. C. Zhu and Y. Zou, Suary of researh on frational-order ontrol, Control and Deision, Vol. 2, Issue 24, 29, pp [5]. D. Matignon, В. d Andrea-Novel, Soe results on ontrol ability and observability of finite-diensional frational differential systes, in Proeedings of the Multionferene on Coputational Engineering in Systes and Appliation, Lille, IMACS IEEE-SMC, 996, pp [6]. D. Matignon, Stability results for frational differential equations with appliations to ontrol proessing, in Proeedings of the Multionferene on Coputational Engineering in Systes and Appliation, Lille, IMACS IEEE-SMC, 996, pp [7]. I. Podlubny, Frational order systes and ontrollers, IEEE Transations on Autoati Control, Vol., Issue 44, 999, pp [8]. H. S. Li, Y. Luo, Y. Q. Chen, A frational order proportional and derivative (FOPD) otion ontroller: tuning rule and experients, IEEE Transations on Control Systes Tehnology, Vol. 2, Issue 8, 2, pp [9]. V. Feliu-Batlle, R. R. Perez, F. G. C. Gara, Sith preditor based robust frational order ontrol: Appliation to water distribution in a ain irrigation anal pool, Journal of Proess Control, Vol. 3, Issue 9, 29, pp []. F. Merrikh-Bayat, M. Afshar and M. Karii- Gharteani, Extension of the root-lous ethod to a ertain lass of frational-order systes, ISA Transations, Vol. 48, Issue, 29, pp []. N. M. Qi, Z. G. Song and C. M. Qin, Optial Oustaloup digital realization of frational order systes, Control and Deision, Vol., Issue 25, 2, pp [2]. C. Y. Wang, Study on frational order PID ontroller paraeter tuning ethod and design, Jilin University, Jilin, 23. [3]. Z. Gao and X. Z. Liao, Iproved Oustaloup approxiation of frational-order operators using adaptive haoti partile swar optiization, Journal of Systes Engineering and Eletronis, Vol., Issue 23, 22, pp [4]. R. S. Barbosa, J. A. Tenreiro Mahado and I. M. Ferreira, Tuning of PID ontrollers based on Bode s ideal transfer funtion, Non-linear Dynais, Vol. 38, Issue /2, 24, pp [5]. R. P. Wang and Y. G. Pi, Frational-order PI speed ontroller for peranent agnet synhronous otor, Transations of China Eletrotehnial Soiety, Vol. 2, 22, pp [6]. A. Oustaloup, J. Sabatier and P. Lanusse, Fro frational robustness to CRONE ontrol, Frational Calulus Appliation, Vol., Issue 2, 999, pp Copyright, International Frequeny Sensor Assoiation (IFSA) Publishing, S. L. All rights reserved. ( 7