Modeling and Simulation of a Two-Mass Resonant System with Speed Controller

Transcription

1 International ournal of Information and Electronic Engineering, Vol. 3, No. 5, eptember 203 odeling and imulation of a Two-a Reonant ytem with peed ontroller Ghazanfar hahgholian, ember, IAIT Abtract The dynamic analyi and application of a PID controller with compenation to a two-pa reonant ytem i preented. A complete tate-pace mathematical model for ytm i firt developed. Then, a PID controller with feedbac compenation i deigned to regulate the peed of the ytem. The parameter of the controller deign by coefficient diagram method. Index Term PID controller, peed control, cofficient diagram method. I. INTRODUTION In ome indutrial application, lie a teel mill drive which ha a long haft and large load ide ma or a robot arm which ha flexible coupling, the mechanical part of the ytem ha very low reonant frequency. The dynamic of uch a ytem mut hence be modeled a a two-ma or multi-ma ytem [], [2]. A typical configuration of teel rolling mil ytem i how in Fig.. control [4], adaptive peed control [5], reonant ratio control [6] and tate robut control [7]. A predictive trategy bae on tate feedbac with a reduced order extended alman filter for the peed control of a two-ma ytem driven by a permanent magnet ynchronou motor i preented in [8]. A nonlinear approach for wind power capture optimization while limiting tranient load on the drive-train component uing a two-ma model and a wind peed etimator preented in [9]. In [0] propoed a alman filter and Q baed peed control with an integrator including load torque compenation, which alman filter ha been ued to etimate the load peed, haft torque and load torque. A neural networ etimator ha been ued to etimate the mechanical tate variable uch a the torional torque and load-ide peed of the two-ma drive ytem in []. The objective of thi paper i to develop an algorithm to deign a peed control trategy of a two-ma reonant ytem by a PID controller. It i een that the control ytem ha fat peed repone and robut tability. Thi paper decribe a tudy on the PID controller deign of two-ma reonant ytem conidering the repone frequency and repone tep. Fig.. A drive train with long pindle In the deign of a linear time-invariant control ytem uing the claical control theory, the firt tep i to chooe the controller type. In mot cae, the cot of control ytem increae with it complexity. Due to their merit uch a imple tructure, high efficiency and eay implementation, the conventional PID i widely ued in mot ervo application uch a actuation, robotic, machine tool, and o on. They are eay to adjut and configure, in addition to providing poibilitie of improvement in operation and control. The ta for the control of the two-ma reonant ytem are uppre the haft torional vibration, reject the effect of the load diturbance torque and tracing the load peed of the peed reference without overhoot [3]. The dynamic performance of peed and poition controlled multi-ma driving ytem can be deteriorated epecially due to the elatic coupling, non-linear friction and baclah. any method have been preented in the literature for the torional vibration control of two-ma ytem uch a liding mode anucript received eptember 20, 202; revied October 30, 202. Gh. hahgholian i with the Department of Electrical Engineering, Najafabad ranch, Ilamic Azad Univerity, Efahan, Iran ( hahgholian@iaun.ac.ir). II. ATEATIA ODE Fig. 2 how a chematic of a two-ma reonant ytem coniting of two lumped inertia and, repreenting the motor and load, repectively, coupled via a haft of finite tiffne, that i ubject to torional torque T and excited by a combination of electromagnetic torque T and load-torque perturbation T. The motor angular velocity i denoted and the load velocity i denoted [2]. The nominal parameter of the ytem how in Table I. Fig. 2. Two-ma drive ytem TAE I: NOINA PARAETER OF TE TWO-A PANT ymbol Quantity Value haft tiffne 30 Nm/rad haft damping coefficient 0.2 Nm/rad/ motor inertia 0.02 g.m 2 motor vicoity coefficient Nm/rad/ load inertia 0.65 g.m 2 load vicoity coefficient Nm/rad/ The tate equation of two-ma reonant ytem i a follow [3]: DOI: /IIEE.203.V

2 International ournal of Information and Electronic Engineering, Vol. 3, No. 5, eptember 203 d = T + T () dt d dt T = ( ) ( ) ( + ) T + T + T () d = + T T dt (2) The tranfer function load i denoted G () and the tranfer function motor i denoted G (). Fig. 3 provide a dynamic bloc diagram repreentation of the two-ma mechanical ytem which the electromagnetic torque produced by the ervo machine. The machine rotor angular velocity, load angular velocity and torional torque can be derived, uch a: R = A + (8) An increae in the motor inertia contant decreae the R without affecting the A. onverely, increaing the coefficient increae both R and A. Therefore, in the large ytem, which have large inertia, generally produce low natural frequencie. The dominant eigenvalue for open-loop ytem are P = and p 2,3 = ±j The reonant frequency i R =4.0 rad/ and anti-reonant frequency i A =3.5 rad/. The damping factor of the original plant without the controller i η= ince all eigenvalue of the ytem are on the left hand of the plane, the ytem i table but highly damped. The repone frequency of tranfer function relative to motor torque how in Fig. 4. Fig. 3. loc diagram of two-ma drive ytem + = = T Δ() T = 0 + ( + ) + = = T Δ() T = 0 T = 0 2 T ( + )( + ) = = T Δ() where the characteritic equation of the open loop ytem i given a: (3) (4) (5) Δ = + ( + ) ( + ) + ( + ) + + ( ) ( + ) + (6) ince damping loe uually conidered being relatively low, they are neglected without ignificantly affecting the accuracy of the forgoing analyi. The reonance frequency ( R ) and anti-reonance ( A ) frequency are defined a: A = (7) Fig. 4. The repone frequency of tranfer function relative to motor III. ONTROER DEIGN The oefficient Diagram ethod (D) i an indirect pole placement method to deign an appropriate characteritic polynomial. The D can give a controller deign which i both table and robut, and it ha the deired ytem repone peed. D need ome deign parameter with repect to the characteritic polynomial coefficient which are the equivalent time contant (τ) and the tability index (γ ) [4, 5]. A general form of characteritic polynomial of a cloed-loop control ytem i: n n n n n 0 = Δ ( ) = a + a a + a = a (9) There i a relationhip among the coefficient of the controlled ytem cloed-loop characteritic polynomial: a a τ = (0) γ γ γ o

3 International ournal of Information and Electronic Engineering, Vol. 3, No. 5, eptember 203 In thi ection, deign conideration of the propoed controller combined with feedbac compenation are given. Fig. 5 how a cloe-loop control with PID controller, G V ()= P + I /+ D, and feedbac compenation, G F ()= PF + IF / + DF. The motor peed, haft torque and motor torque are repreented a: Fig. 5. ontrol tructure of a two-ma drive ytem in imulin/atlab = T () VF TF T = T (2) R T T = T (3) R T The cloed-loop tranfer function from the command peed ( ) to the motor peed and the from the command peed to the diturbance torque are: VF = T = 0 GV () = (4) + [ G + G ()] TF V F = T = 0 = (5) + [ G + G ()] V F The cloed-loop tranfer function from the to the motor torque and from the to the haft torque i: TR T = T = 0 GV = (6) + [ G ( ) + G ( )] ( ) R V F T = T = 0 ()[ + GF ()] = (7) + [ G + G ()] V F The cloed loop characteritic polynomial in cloe-loop with controller i given by: Δ = ( + + ) PID O D DF 4 + ( + ) + ( ) R I IF D A DF A ( PA PFA) ( IA IFA) (8) Therefore, the cloed-loop characteritic equation of the P PF 3 ytem with only PID controller i imilar to that of the ytem with PID controller and compenation. The zero of the VF () depend on not only anti-reonance but alo the variation of forward controller gain, while the zero of tranfer function are independent of compenation gain. ut the zero of VT () are independent on the controller gain and compenation gain. In I-PD controller, the controller tructure conit of an integral controller in forward ( D =0, P =0) and proportional derivative compenation ( IF =0). If E V i output ignal of integral controller, the tate-pace equation of the ytem with I-PD controller are given by: d E V = I( ) (9) dt d = [ EV T ( + P) ] (20) dt + D d = + T T (2) dt d ( + P) T = [ ] + [ + ] dt + D ( + ) T + T (22) + D With compared the polynomial coefficient of characteritic equation in PID controller with compenation and D the equivalent time contant and PID controller gain can be found a following: γ γ τ A γ 3 I = I + IF = γγγ 2 3 γ γ3 = + γ = γγγ γ γ 2 = (23) D D DF γ γ γ P = P + PF = γ γ γ γ γ (24) (25) (26) In order to achieve ytem tability on bae Routh-urwitz algorithm, the condition for tability index for a four order ytem i: γ > 2 γ + γ (27) 3 IV. IUATION REUT In thi ection, we verify the validity of the propoed method. All imulation i executed by atlab. Fig. 6 how a cloe-loop control with PID controller and feedbac compenation, tructure in imulin/atlab. The parameter of the controller obtained from (9)-(22) are ummarized in Table II. The eigenvalue of the cloe-loop ytem are lited in Table III. Fig. 7 and 8 how the tep repone of motor 450

4 International ournal of Information and Electronic Engineering, Vol. 3, No. 5, eptember 203 peed and frequency repone for change in tability indexe with PID controller. The imulation reult of motor peed and load peed due to unit-tep command peed when the two-ma ytem i compenated by the conventional PID controller, I-PD controller and PID-P controller how in Fig. 9 and 0, repectively. haft torional torque and 4, repectively. It i hown that the controller yield a robut tability ytem. PID compenation PID controller motor peed load command peed PID.+ motor.+ haft.+ load diturbance torque load peed motor torque Fig. 6. ontrol tructure of a two-ma drive ytem in imulin/atlab TAE II: ONTROER PARAETER Fig. 9. tep repone of motor peed with PID controller and compenation tability index P I D τ γ =4, γ 2 =2.5, γ 3 = γ =4, γ 2 =2.5, γ 3 = γ =4, γ 2 =2.5, γ 3 = TAE III: EIGENVAUE OF TE OE-OOP YTE tability index Eigenvalue γ =4, γ 2 =2.5, γ 3 = ±j2.08, -4.07, -3. γ =4, γ 2 =2.5, γ 3 = ±j4.99, -6.79±j3.29 γ =4, γ 2 =2.5, γ 3 = ±j3.63, -5.2, Fig. 0. tep repone of load peed with PID controller and compenation Fig.. tep repone of haft torque with PID controller and compenation Fig. 7. The parameter effect of PID controller on motor peed Fig. 2. tep repone of motor torque with PID controller and compenation Fig. 8. The parameter effect of PID controller on frequency repone The imulation reult of haft torque and motor torque due to unit-tep command peed when the two-ma ytem i compenated by the conventional PID controller, I-PD controller and PID-P controller how in Fig. and 2, repectively. The imulation reult of haft torque and motor peed due to unit-tep command peed uing I-PD controller in the two-ma ytem with load inertia deviation how in Fig. 3 Fig. 3. tep repone of haft torque uing I-PD controller with load inertia deviation 45

5 International ournal of Information and Electronic Engineering, Vol. 3, No. 5, eptember 203 Fig. 4. tep repone of motor peed uing I-PD controller with load inertia deviation V. ONUION Vibration uppreion and attainment of robutne in motion control ytem i an important problem in indutry application. The repone of the ytem i tudied for command peed change. Analyi and imulation reult are illutrated to verify good performance obtained uing the propoed controller. REFERENE []. Wang, Y. Zhang,. Xu, Y. ing, and. Zhang, Torional vibration uppreion of rolling mill with contrained model predictive control, IEEE/ VVIA, vol. 2, pp , une [2] G. hahgholian and P. hafaghi, "imple analytical and robut controller deign for two-ma reonant ytem," IEEE/IEE, pp , Dec [3] Y.. im,.. im,.. im,.. Yoo', and..im, "Two degree of freedom peed control of induction motor having two ma reonant ytem, IEEE/IEON, vol. 2, pp , Aug [4]. Erentur, Nonlinear two-ma ytem control with liding-mode and optimized proportional-integral derivative controller combined with a grey etimator, IET ontr. The. Apll., vol. 2, no. 7, pp , [5]. chuter,. Wetermaier, and D. chroder, Non-identifier-baed adaptive peed control for a two-ma flexible ervo ytem: conideration of tability and teady tate accuracy, IEEE/ED, pp. -6, [6] T.. O ullivan,.. ingham, and N. chofield, igh performance control of dual-inertia ervo drive ytem uing low cot integrated AW torque tranducer, IEEE Tran. on Ind. Elec., vol. 55, no. 4, pp , Aug [7]. Peter, I. choling, and. Orli, Robut output feedbac control with a nonlinear oberver for a two-ma ytem, IEEE Tran. Ind. Appl., vol. 6, no. 2, pp , 2003 [8] E.. Fuente,. A. ilva, and. I. Yuz, "Predictive peed ontrol of a Two-a ytem Driven by a Permanent agnet ynchronou otor, IEEE Tran. on Ind. Ele., vol. 59, no. 7, pp , 202. [9]. ouhezzar and. iguerdidjane, "Nonlinear control of a variablepeed wind turbine uing a two-ma model," IEEE Tran. on Ene. on., vol. 26, no., pp , 20. [0].. i and.. ul, alman filter and Q baed peed controller for torional vibration uppreion in a 2-ma motor drive ytem, IEEE Tran. Ind. Ele., vol. IE-43, no. 6, pp , Dec [] T. O. owala and. zabat, Neural networ application for mechanical variable etimation of a two-ma drive ytem, IEEE Tran. Ind. Electron., vol. 54, no. 3, pp , [2] O. Ocal,. T. Oylemez, and A. ir, Robut pole aignment uing coefficient diagram method, AE, Dec [3] G. hahgholian,. Zinali, P. hafaghi, and.. Rezaei, Analyi and deign of torque control trategy for two-ma reonant ytem only by a PID controller, IEEDT, Oct./Nov [4] G. hahgholian, P. hafaghi,. Zinali, and. oalem, "tate pace analyi and control deign of two-ma reonant ytem," IEEE/IEE, pp , Dec [5] G. hahgholian and. Faiz, An analytical approach to ynthei and modeling of torque control trategy for two-ma reonant ytem, Int. Revi. of Auto. ont. (IREAO), vol. 2, no. 4, pp , uly Ghazanfar hahgholian wa born in Efahan, Iran, on Dec. 7, 968. e graduated in electrical engineering from Ifahan Univerity of Technology (IUT), Efahan, Iran, in 992. e received the.c and PhD in electrical engineering from Univerity Tabriz, Tabriz, Iran in 994 and cience and Reearch ranch, Ilamic Azad Univerity, Tehran, Iran, in 2006, repectively. e i now an aociate profeor at Department of Electrical Engineering, Faculty of Engineering, Ilamic Azad Univerity Najafabad ranch. e i the author of 00 publication in international journal and conference proceeding. i teaching and reearch interet include application of control theory to power ytem dynamic, power electronic and power ytem imulation. 452