Gray-Box Modeling of a Pneumatic Servo-Valve

Transcription

1 Acta Polytechnica Hungarica Vol. 7, No. 5, 010 Gray-Box Modeling of a Pneumatic Servo-Valve Farhad Toorani, Hoein Farahmandzad Moghtader Engineering Company Tehran, Iran toorani.farhad@gmail.com, h.farahmandzad@gmail.com Mojtaba Aghamiralim Department of Electrical Engineering Amirkabir Univerity of Technology Tehran, Iran miralim@aut.ac.ir Abtract: Thi paper preent the evaluation a well a the combined analytical and experimental identification (gray box identification) of a ervo-valve torque motor a the directional valve applied in a pneumatic actuator. Baed on analytical modelling, a imple linear parametric model with tranfer function and block diagram i developed. Next, the tatic and dynamic characteritic of the torque motor are obtained from experimental obervation. The characteritic confirm the deired linear behaviour of the torque motor. Hence, linearized coefficient from a bet curve fitting of tatic characteritic can be derived. Claical method of identification are applied on the frequency and tep repone obtained from a et of tet on the torque motor. Obtained tet reult baed on parimony principle and model order of analytical invetigation are then implemented to derive the bet identified tranfer function to decribe the performance of the ervo-valve torque motor. Deign parameter are etimated with the comparion of the experimental and analytical model. Thee parameter can be implemented with acceptable accuracy for ervo control tudie of pneumatic actuator. Keyword: pneumatic actuator; ervo-valve; limited angle torque motor; gray box modeling 1 Introduction Several advantage of pneumatic actuation ytem over hydraulic and electromechanical actuation ytem in poitioning application have caued their 19

2 F. Toorani et al. Gray-Box Modeling of a Pneumatic Servo-Valve wide application in indutrial automation, motion control of material and part handling, packaging machine, and in robotic. The advantage include low cot, cleanline, eae of maintenance, and high power to weight ratio. The drawback i due to the compreible nature aociated with gae and the high peed, which make it more difficult to control. An electro-pneumatic control valve play an important role in a pneumatic actuator. It characteritic influence remarkably the performance of the actuator. Thi power element with mall-amplitude, the low power electrical ignal provide a high repone modulation in pneumatic power. The dynamic repone, null hift, threhold, and hyterei are the mot critical parameter that trongly influence the dynamic and tatic characteritic of the pneumatic actuator [1]. Limited angle torque motor (LATM) are one of the electromechanical tranducer which, when integrated with nozzle, can be ued a ervo-control valve in either pneumatic or hydraulic actuator. They can erve a proportional diplacement, force output, or both. Hydraulic ervo-valve a a vital part of hydraulic actuation ytem have been tudied in literature. In [, 3], the influence of magnetic fluid on the dynamic characteritic of a hydraulic ervo-valve ha been imulated and teted. LATM are implemented a limited-motion rotary actuator in many area of meaurement, control, and energy converion application through an angle of le than ±180º. In [4], [6] a form of LATM, which i baed on Law relay principle [5], ha been evaluated and analyzed. The performance of thi kind of torque motor i baed on polarized reluctance principle in which torque and EMF are produced baed on flux-witching action of rotor aliencie on a permanent magnet flux. Many complex problem of Law relay in rotary and rectilinear verion [7, 8], have been retricted to deign problem which contain tatic and dynamic characteritic, e.g. tiffne of permanent magnet and torque enitivity with repect to current. The deign and control of a two-pole toroidally wound armature bruhle directcurrent LATM with application in the fuel control of ga-turbine engine ha been preented in [9]. To achieve a robut poition control of the motor, a PID controller with a cloed-loop fuel controller ha been implemented uing DSP. The contruction of ervo-valve tudied in our reearch i a type of Law relay which contain two winding et up on two bobbin placed around the rotor a control coil. Thi tructure lower the phyical ize, and the bobbin play the role of poition limiter. The tator contain two eparated oft iron yoke and two permanent magnet with high flux denitie. The main characteritic of the motor are it miniature ize, the capability of the rotor to revolve only up to ±3º, hort dynamic repone time due to it low inertia and the high magnetic tiffne that i deirable in ervo-valve, high reliability due to it imple contruction and the high quality of magnetic material. 130

3 Acta Polytechnica Hungarica Vol. 7, No. 5, 010 Thi paper preent a combined analytical (white box) and experimental (black box) modeling of a pneumatic ervo-valve torque motor, baed on an approximate mathematical model developed from deign data and experimental meaurement. The black box modeling approach i the bai of ytem identification which etimate dynamic repone and tranfer function parameter. Thi approach i impler than olving the nonlinear partial differential equation of the tructure, and i epecially uitable for deign tudie of the control loop of the pneumatic actuator when ued a an accurate poition tracker. Baed on claical method of identification, the frequency and tep repone of the teted motor have been fitted by a continuou tranfer-function repone. The unknown parameter of a mathematical model are derived according to the preented model. The experimental reult how that the propoed model provide an accurate prediction of the behavior of ervo-valve torque motor. Law Relay Principle Figure 1 The chematic of Law relay ervo-valve A chematic of the teted torque motor i hown in Figure 1. It contain armature, two parallel coil, two permanent magnet, and yoke. Four core teeth overlap with armature and caue four variable air-gap reluctance, which make the motor act a a witch reluctance type. The two magnet tend to keep the armature in a null poition (θ = 0). With no external excitation, the flux denity under each of the four teeth i equal and the low inertia of the rotor bridge pole caue a magnetic balance in the four air-gap. By connecting a power ource that can upply either unidirectional or bipolar current to the motor, an unbalance of flux denity in the air-gap i produced, which reult in a new equilibrium of rotor 131

4 F. Toorani et al. Gray-Box Modeling of a Pneumatic Servo-Valve poition. Inherently, the high tiffne of the magnet make it like a pring with a tiffne torque proportional to the angular poition. Since the torque produced by the flux of the control winding i proportional to the current (under non-aturated condition), thee two component of the torque make the motor uitable for ervo control ytem and accurate poitioning of a load connected to an actuator. A pneumatic nozzle i linked to the armature of the torque motor and the whole ytem act a a directional-proportional valve that guide compreed air to cylinder chamber. 3 Mathematical Model of a LATM The mathematical model of the Law relay ytem i baed on the dynamic equation of motor which are decribed by the following differential equation. λ(, i θ) dθ λ(, i θ) di v Ri= 0 θ dt i dt 1 λ(, i θ) λ(, i θ) dθ it () = v Ri. dt i θ dt d θ dθ Tm = J + K ( ) f + T θ dt dt λ(, i θ) dθ λ(, i θ) di v Ri= 0 θ dt i dt 1 λ(, i θ) λ(, i θ) dθ it () = v Ri. dt i θ dt dθ 1 nt () nt () = ( J ( Tm K fn T). ) dt dt () Here, flux linkage, λ(, i θ ), i a nonlinear function of current i and angular poition θ. T m, T and K f denote the motor developed torque, the tiffne torque, and the frictional coefficient of air, repectively. nt () i the motor peed, and finally J i the moment of inertia of bundle of rotor and nozzle, []. Becaue of the mall range of operation of θ (±3º), it can be aumed that T m i directly proportional to current i, tiffne torque T i directly proportional to (1) 13

5 Acta Polytechnica Hungarica Vol. 7, No. 5, 010 poitionθ, and the differential term λ θ i approximately contant. If the other differential term λ i i approximated to a contant value, then an acceptable linear model can be defined by the following equation: dθ di v Kb L Ri= 0 (3) dt dt d θ dθ Tm = J + K f + T (4) dt dt Where, T = K θ (5) T m = K i (6) t Subtituting T and Tm from eqn. (5) and (6) into eqn. (4) give d θ dθ Ki t = J + K f + Kθ (7) dt dt By ubtituting i from eqn. (7) into eqn. (3), the following equation i obtained: θ θ K θ K v K L + + θ d d J d f d b dt dt Kt dt Kt dt Kt J d θ K f dθ K θ 0 t t t R + + = K dt K dt K Conequently, the parametric tranfer function of Law relay a a ytem input ytem output (SISO) can be derived in a third order form a in the following: θ () G () = V() Kt = JL 3 K f R ( RK f + KbKt + KL) RK S + S S + J L + + JL JL (8) (9) 133

6 F. Toorani et al. Gray-Box Modeling of a Pneumatic Servo-Valve 4 Torque Motor Performance Becaue of the low working frequency of 1 Hz for thi ervo-valve, the tatic characteritic have more important effect on dynamic performance compared to the typical one ued in aeropace equipment and optical canning ytem. The meaured magnitude of the magnetic tiffne torque for each poition of haft deflection in the range of ±3º i hown in Figure. It i obviou that thi characteritic with a good approximation ha a linear behavior, which confirm the linear coefficient of K. From the bet linear curve fitting, the following value i obtained for thi coefficient: K = 1.1Nm rad = Nm deg Stiffne Torque (N.m) haft deflection (deg) Figure The meaured tiffne torque characteritic and the bet fit curve The torque-current characteritic of the motor hown in Figure 3 ha been obtained by exciting it coil with a variable DC ource. The figure how a good linear trend which again confirm the linear coefficient of K t. From the bet linear curve fitting, the following value i obtained for thi coefficient: Kt = 6.43 Nm A (10) (11) 134

7 Acta Polytechnica Hungarica Vol. 7, No. 5, 010 Torque (N.m) motor current (ma) Figure 3 The meaured torque-current characteritic and the bet fit curve Figure 4 preent the teady-tate characteritic of the experimental LATM operating in the open loop condition. The voltage-poition characteritic demontrate that with a deirable linear performance, the motor can be implemented in ervo control ytem. 4 3 Shaft poition (deg) applied voltage (V) Figure 4 The meaured voltage-poition characteritic and the bet fit curve 135

8 F. Toorani et al. Gray-Box Modeling of a Pneumatic Servo-Valve The approximately ideal characteritic of the deigned parameter in Figure 3 and 4 preent a good linear model for the controller deign procedure. The meaured parameter are ummarized in Table 1. Table 1 Meaured parameter of available LATM Torque contant Kt ( Nm A ) 6.43 Back EMF contant (. ) K V rad 0.08 b Winding reitance R ( Ω ) 3630 Winding inductance L( H ) 34 Rotor inertia (. ) J kg m Frictional coefficient (. ) K Nm rad f Magnetic tiffne K ( Nm rad ) 1.1 A block diagram repreentation of the torque motor i hown in Figure 5 to illutrate the dynamic performance of it linear model. In thi figure, the reitance R i meaured directly from voltage and current meaurement, the moment of inertia J i calculated accurately by SolidWork oftware, and the winding inductance i approximately calculated baed on analyi of the current flow of motor when excited with low frequency inuoidal voltage. Figure 6 how the inuoidal input voltage and time-repone waveform of current and the poition of motor at a frequency of 10 Hz. Figure 4 Block diagram of a LATM 136

9 Acta Polytechnica Hungarica Vol. 7, No. 5, 010 (a) (b) Figure 6 (a) 10Hz inuoidal input, 40V peak (b) Time-repone waveform of current (c) Time-repone waveform of poition (c) The frictional contant K f, and back EMF coefficient K b are difficult to meaure directly. Hence, they are etimated from the predicted tranfer function in the next ection. The experimental tet ytem, coniting of the torque motor, a function generator with it output ignal amplified by a voltage amplifier circuit, current enor, and an arc type potentiometer fed by ±500 mv voltage with it wiper (contact arm) oldered to an arm linked to the motor haft i hown in Figure

10 F. Toorani et al. Gray-Box Modeling of a Pneumatic Servo-Valve Figure 7 The experimental meaurement ytem 5 Claical Identification In linear time-invariant SISO dynamic ytem, it i poible to derive the linear tranfer function model by analyzing the repone of the ytem to any forcing input. The tep and frequency repone tet are two common practical method in claical identification [10]. The tep repone of a third-order ytem cloely reemble to that of a firt-order ytem with jut tracking wing. According to the parimony principle of model order determination [11], the impler model can be choen between two model with order n 1 and n ( n 1 > n ), if it adequately and accurately decribe the ytem. However, in thi cae, becaue we conider the combined analytical and experimental model, it i neceary to chooe the third-order model. Hence, by imultaneou analyi of tep and frequency repone combined with trial and error procedure, the bet third-order tranfer function i obtained. 138

11 Acta Polytechnica Hungarica Vol. 7, No. 5, 010 Figure 8 how the variation of the rotor poition in repone to a tep input voltage of 50 V in order to weep the entire range of haft deflection. Figure 8 Variation of haft deflection to a tep input voltage of 50 V peak Figure 9 and 10 how the meaured characteritic of frequency and normalized tep repone in comparion to the repone of the etimated tranfer function Gain (db) Etimated repone Meaured repone Frequency (rad/ec) Figure 9 Meaured and etimated frequency repone chracteritic with input 40 V peak 139

12 F. Toorani et al. Gray-Box Modeling of a Pneumatic Servo-Valve Normalized Step Repone Meaured repone Etimated repone Exerted tep Time (ec) Figure 10 Normalized Meaured and etimated tep repone chracteritic with input 40 V peak Taking into account the repone delay, the etimated tranfer function i obtained a follow: S θ ( ) e G () = = V() S+ 10 S S = ( )( ) e S 3 ( S + 49S S ) Aume that K, J and R are predetermined parameter. The remaining parameter, which are difficult to determine experimentally, are obtained by the comparion of the etimated tranfer function of eqn. 1 with the parametric one from eqn. 9, and by olving the four linear equation. The etimated value of thee parameter are hown in Table. Table Etimated value of the parameter by claical identification t ( ) K Nm A 5.3 (. ) K V rad 0.08 b L( H ) 9.9 (. ) K Nm rad f (1) 140

13 Acta Polytechnica Hungarica Vol. 7, No. 5, 010 A i hown in Figure 11, the dicrepancy between frequency repone of the ytem modeled by the etimated parameter and the meaured parameter i adequately acceptable. Error (db) Frequency (rad/ec) Figure 11 The error between the frequency repone of the ytem modeled by etimated and meaured parameter Concluion From a erie of linear mathematical equation, a parametric third-order tranfer function wa achieved in order to model a pneumatic ervo valve a a linear timeinvariant SISO ytem. The tatic and dynamic performance characteritic of the torque motor were obtained by a tet ytem. The reult howed a remarkably good linear trend, which made it uitable for ervo application. Baed on claical method of identification, including frequency and tep repone, an acceptable etimated model wa fitted to the meaured reult. Finally, with the comparion of etimated and parametric third-order tranfer function, the deign parameter of the ervo-valve were obtained. The reult were very cloe to the deign pecification of the manufactured ervo-valve, which validated the gray-box modeling. Reference [1] Igor L. Krivt, G. V. Krejnin: Pneumatic Actuating Sytem for Automatic Equipment Structure and Deign CRC Pre, 006 [] S. Li, Y. Song: Dynamic Repone of a Hydraulic Servo-Valve Torque Motor with Magnetic Fluid, Mechatronic, Vol. 17, pp , 007 [3] Songjing Lia, W. Bao, "Influence of Magnetic Fluid on the Dynamic Characteritic of a Hydraulic Servo-Valve Torque Motor," Mechanical Sytem and Signal Proceing, Vol., pp , 008 [4] I. R. Smith, J. G. Kettleborough, Y. Zhang: Simplified Modelling and Dynamic Analyi of a Law' Relay Actuator, Mechatronic, Vol. 9, pp ,

14 F. Toorani et al. Gray-Box Modeling of a Pneumatic Servo-Valve [5] Y. Zhang, I. R. Smith, J. G. Kettleborough: Performance Evaluation for a Limited-Angle Torque Motor, IEEE/ASME Tranaction on Mechatronic, Vol. 4, pp , September 1999 [6] H. R. Bolton, Y. Shakweh: Performance Prediction of Law' Relay Actuator, in IEE Conference on Electric Power Application, Vol. 137, Iue 1, 1990, pp [7] S. A. Evan, I. R. Smith, J. G. Kettleborough: Static and Dynamic Analyi of a Permanent Magnet Rectilinear Law' Relay Actuator, in IEE Conference on Electric Power Application, Vol. 146, Iue 1, 1999, pp [8] S. A. Evan, I. R. Smith, J. G. Kettleborough: Permanent-Magnet Linear Actuator for Static and Reciprocating Short-Stroke Electromechanical Sytem, IEEE/ASME Tranaction on Mechatronic, Vol. 6, pp. 36-4, March 001 [9] Ching Chih Tai, Shui-Chun Lin, Hu-Chih Huang, Yu-Ming Cheng: Deign and Control of a Bruhle DC Limited-Angle Torque Motor with it Application to Fuel Control of Small-Scale Ga Turbine Engine, Mechatronic, Vol. 19, Iue 1, 009, pp [10] Ngoc Dung Vuong, Marcelo H. Ang Jr: Dynamic Model Identification for Indutrial Robot, Acta Polytechnica Hungarica, Journal of Applied Science, Hungary, Vol. 6, Iue 5, pp , 009 [11] J. P. Norton: An Introduction to Identification Academic Pre Inc.,