Mathematical and Intelligent Modeling of Electropneumatic Servo Actuator Systems

Transcription

1 Autralian Journal of Baic and Alied Science, 3(4): , 2009 ISSN Mathematical and Intelligent Modeling of Electroneumatic Servo Actuator Sytem Hazem I. Ali, Samul Bahari B Mohd Noor, S.M. Bahi, Mohammad Hamiruce Marhaban Deartment of Electrical & Electronic Engineering, Faculty of Engineering, Univerity Putra Malayia, Malayia Abtract: The neumatic actuator rereent the main force control oerator in many indutrial alication, where it tatic and dynamic characteritic lay an imortant role in the overall behavior of the control ytem. Therefore, obtaining of accurate aroach for modeling the neumatic actuator i of rime interet to control ytem deigner. In thi aer a different methodologie for deriving and imulating the model of the neumatic ervo actuator controlled with roortional valve are reented. The model include cylinder dynamic, ayload motion, friction and valve characteritic. Key word: Nonlinear Sytem, Pneumatic actuator, Modeling, Friction modeling. INTRODUCTION Pneumatic actuator are widely emloyed in oition and eed control alication when chea, clean, imle, and afe oerating condition are required. In recent year, low cot microroceor and neumatic comonent became available in the market, which made it oible to adot more ohiticated control trategie in neumatic ytem control (Jihong et al., 2007). Alo the neumatic cylinder can offer a better alternative to electrical or hydraulic actuator for certain tye of alication and the neumatic actuator rovide the reviouly enumerated qualitie at low cot. They are alo uitable for clean environment and afer and eaier to work with. However, oition and force control of thee actuator in alication that require high bandwidth i ome how difficult, becaue of comreibility of air and highly nonlinear flow through neumatic ytem comonent (Edmod and Yildirim, 2001). A tyical neumatic ytem include a force element (neumatic cylinder), a command device (valve), connecting tube, and iton, reure and force enor. The external load conit of the ma of external mechanical element connected to the iton and erha a force roduced by environmental interaction. A chematic diagram of the neumatic actuator ytem i hown in Figure 1. The final deciion on the bet tye and deign configuration for neumatic actuator can be made only in relation to the requirement of a articular alication. The neumatic actuator ha mot often been of the iton cylinder tye becaue of it low cot and imlicity (Tablin and Gregory, 1963). The neumatic ower i converted to traight line recirocating and rotary motion by neumatic cylinder and neumatic motor. The neumatic oition ervo ytem are ued in numerou alication becaue of their ability to oition load with high dynamic reone and to augment the force required moving the load. Pneumatic ytem are alo very reliable (Clement and Len., 1985). Pneumatic ytem have many attribute that make them attractive for ue in difficult environment. The imortant attribute are: (i) gae are not ubected to the temerature limitation of hydraulic fluid; (ii) the actuator exhaut gae need not be collected, o fluid return line are unneceary and long term torage i not a roblem becaue neumatic ytem are virtually dry and no organic material need be ued. In addition, the neumatic actuator ha a lower ecific weight and a higher ower rate (torque-quared to inertia ratio) than an equivalent electromechanical actuator. In ome cae, a neumatic ytem may rovide a ignificant weight advantage. In hort duration miile alication, the weight of a elf-contained olid roellant neumatic ervo may be half that of an equivalent elf contained hydraulic ytem. Alo the neumatic actuator have many merit uch a eay maintenance and handling, relatively imle technology and low cot, clean, afe and eay to intalled (Tablin and Gregory, 1963). In thi aer a different methodologie for deriving the mathematical model of neumatic ervo actuator are reviewed. The model include cylinder dynamic, ayload motion, friction and valve characteritic. Correonding Author: Hazem I. Ali, Deartment of Electrical & Electronic Engineering, Faculty of Engineering, Univerity Putra Malayia, Malayia hazemcontrol2001@yahoo.com 3663

2 Fig. 1: Schematic diagram of double acting neumatic actuator ytem. Nonlinear Mathematical Modeling: Several aroache have been rooed for modeling the neumatic actuator. One of the widely ued method for finding the mathematical model of the neumatic actuator i a theoretical analyi. The analyi of neumatic actuator require a combination of thermodynamic, fluid dynamic and the dynamic of the motion. For contructing a mathematical model, three maor conideration mut be involved: 1) the determination of the ma flow rate through the valve. 2) the determination of the reure, volume and temerature of the air in cylinder. 3) the determination of the dynamic of the load (French and Cox, 1988). Valve Ma Flow Rate Model: The neumatic valve i a critical comonent of the actuator ytem. It i the command element, and hould be able to rovide fat and reciely controlled airflow in and out of the actuator chamber. There are many available deign for neumatic valve, which differ by geometry of the active orifice, tye of flow regulating element, number of ath and ort, and tye of actuating etc. One of the common ued valve i the neumatic roortional ool valve. The modeling of thi valve involve two aect: the dynamic of the valve ool, and the ma flow through the valve orifice. The equation of motion for the valve ool can be written a (Edmond and Yildirim, 2001). (1) where M i the ool and coil aembly ma, x i the ool dilacement, c i the vicou friction coefficient, Ff i the friction force and can be neglected, K i the ool ring contant, K fc i the coil force coefficient, and ic i the coil current. Figure 2 how the valve ool dynamic equilibrium (Edmod and Yildirim, 2001). The reure dro acro the valve orifice i uually large, and the flow ha to be treated a comreible and turbulent. If the utream to downtream reure ratio i larger than a critical value ( P cr ), the flow will attain onic velocity (choked flow) and will deend linearly on the utream reure. If the reure ratio i maller than Pcr the ma flow deend nonlinearly on both reure. The tandard equation for the ma flow through an orifice of area Av i (Edmod and Yildirim, 2001). (2) 3664

3 Fig. 2: Valve ool dynamic equilibrium. where i the ma flow through valve orifice, C f i a nondimenional dicharge coefficient, k i the ecific heat ratio, R i the univeral ga contant, T i the temerature, P u, P d are the utream and downtream reure, and (3) are contant for a given fluid. For air (k=1.4), o C 1 = ,C 2 = , and P cr =0.528 (Han et al., 2002;Xueong and Guangzheng, 2003; Shu and Gary, 2005; Xue et al., 2007; Zhihong and Gary, 2008). A new modeling aroach to develo the nonlinear ytem model uing a combination of mechanitic and emirical method i reented in (Zhihong and Gary, 2008). To model the valve flow rate, the ue of biolynomial function wa ued to roduce a more accurate olution than rior aroache. The dynamic model of the ytem wa given a follow (Zhihong and Gary, 2008). where and are ma flow rate through the four valve hown in Figure 3; and (4) are the ma flow rate of the two chamber; u 1, u, 2 u, 3 and u 4 are valve inut voltage; ë 1 to ë 4 are nonlinear function modeling the ma flow rate; P 1, P 2 are the reure of two chamber. The ma flow rate model of the roortional valve i a key art of the ytem model. The roortional valve that wa ued in (Zhihong and Gary, 2008). conit of a ring-loaded flat late covering a ray nozzle, and an electro-magnet for moving the late. It tructure i different from the traditional ool ervo valve o it wa neceary to derive a new ma flow rate model. The ma flow rate at different reure and valve inut voltage were firt etimated from the reure veru time reone obtained from te inut in valve voltage and a fixed iton oition. The fit of everal function wa evaluated in (Zhihong and Gary, 2008). and found that a 2nd order biolynomial function rovided the bet fit. The equation i: (5) where and i 1 are the coefficient. The coefficient for all equation were obtained uing nonlinear leat quare (Zhihong and Gary, 2008). Actuator Cylinder Model: The dynamic of the neumatic actuator i decribed uing equilibrium, continuity and energy equation, according to the lumed arameter model of the ytem. In the firt cae the equilibrium and the air ma continuity equation are choen to decribe the actuator dynamic. The fluid behavior i evaluated earately in the two chamber, with a variable volume, by mean of the continuity equation, auming that the air tranformation follow a generic olytroic. The chamber reure and the ytem kinetic variable are related by the equilibrium equation of the mobile element. The feedback on the continuity equation are the iton eed and dilacement, which determine the chamber volume. The continuity equation of air i (Sorli et al., 1999). 3665

4 (6) where ñ i the air denity, V i the volume of the chamber and rereent the chamber (where =1,2). Conidering a olytroic tranformation with index n which denote the olytroic index of exanion or comreion and auming that the air can be conidered a erfect ga, one ha (Sorli et al., 1999). (7) where the ubcrit i identifie initial condition. The volume V i a function of the troke, of the dead volume and of the iton dilacement. V = A (x 0 ±x )+V m = A (x 0 ±x +x m) (8) where A i the iton area, x 0 i the half troke dilacement, x i the iton dilacement, xm i the dead dilacement, V m i the dead volume. The oitive ign hould be conidered when =1 and the negative one when =2. By ubtituting equation (7) and (8) in (6), the following exreion i obtained (Sorli et al., 1999). and the reure rate i: (9) (10) The etimation of the value of n i quite comlex and motly it could not be contant with the troke and it could deend on the working condition. Generally iothermal tranformation can be aumed for ufficiently low cycle, while with high working frequencie the heat flow between the ytem and the extern can be neglected, and then the tranformation follow an adiabatic roce (Sorli et al., 1999). By auming that there i no heat tranfer during the running time, i.e. the roce i adiabatic and alying the firt law of thermodynamic and the ideal ga law to the two chamber, the dynamic equation of the cylinder chamber will be: where A 1 and A 2 are the cro-ectional area of the two chamber, V 1o and V 2o are the dead volume (including tube and connector) of the two chamber; K i the ratio of ecific heat of air; L i the troke length (Sorli, et al., 1999). Similar equation of cylinder chamber are reented in (Xueong and Guangzheng, 2003;, Zhihong and Gary, 2008; Nieto, et al., 2008 and Chen et al., 2003). The mot general model for a volume of ga conit of three equation: an equation of ideal ga law, the conervation of ma (continuity) equation, and the energy equation. Auming that (i) the ga i erfect, (ii) the reure and temerature within the chamber are homogeneou, and (iii) kinetic and otential energy term are negligible, thee equation can be written for each chamber (Edmod and Yildirim, 2001; Dorde and Novak, 2008). The energy equation i: (11) (12) (13) 3666

5 where q in and qout are the heat tranfer term, C v i the ecific heat at contant volume, are the ma flow entering and leaving the chamber, T in i the temerature of the incoming ga flow, i the rate of change in the work, and i the change of internal energy. The total change in internal energy i: in which the ideal ga relation wa ued, C v = R/ (k -1). (14) By ubtituting, equation (14) into (13), auming the incoming flow i already at the temerature of the ga in the chamber conidered for analyi, the roce i conidered to be adiabatic (q in - q = 0) and alying the ideal ga law, the time derivative of the chamber reure i: out If the roce i conidered to be iothermal (T=contant), then the change in internal energy, energy and the rate of change in reure equation are: (15) (16) (17) A comarion of equation (15) and (18) how that the only difference i the ecific heat ratio term k. Thu, both equation can be written a: (18) (19) with á, á in and á out taking value between 1 and k, deending on the actual heat tranfer during the roce. Chooing the origin of iton dilacement at the middle of the troke, the volume of each chamber can be exreed a: Subtituting equation (20) into (19), the time derivative for the reure in the neumatic cylinder chamber become: (20) (21) In thi above form the reure dynamic equation account for the different heat tranfer characteritic of the charging and dicharging roce, air comreion or exanion due to iton movement, the difference in effective area on the ooite ide of the iton, and the inactive volume at the end of troke and the admiion ort (Edmod and Yildirim, 2001). Motion Dynamic Model and Friction: The dynamic equilibrium of motion for the iton-rod-load aembly can be exreed a: (22) where M i the external load ma, M i the iton and rod aembly ma, i the iton oition, F f i the friction force, F d i the external force, P a i the abolute ambient reure, and A r i the rod cro ectional area (Edmod and Yildirim, 2001; Gary and Shu 2007). 3667

6 In order to comenate friction and achieve recie oition control, an accurate and feaible friction model need to be choen firt. Although friction occur in almot all mechanical ytem, there i no univeral friction model that can be ued for any ytem. For different ytem and control obective, different friction model are adoted to eae the tak. A imle Gauian exonential tatic friction model can be rereented in equation (23) a a function of intantaneou liding velocity, which cature three baic friction: coulomb, vicou and Stribeck friction (Armtrong and Wit, 1996). (23) where F c i the coulomb friction, F i the magnitude of the Stribeck friction, which i the exce of tatic friction over coulomb friction, F v i the vicou friction and v i the characteritic velocity of the Stribeck friction. By chooing different arameter, different friction model can be realized. Linear Mathematical Modeling: The nonlinear equation (2) that govern the ma flow rate of air through the control valve orifice can be linearized if the valve oerate within it mechanical oerating range (Mark and Nariman, 2006). Equation (2) can be linearized uing a Taylor erie exanion about oerating oint. Neglecting the econd and higher order term a well a any control valve leakage, the ma flow into each actuator chamber are written a: (24) where C fi and C i are known a the valve flow gain and flow reure coefficient, reectively. Their ecific value deend uon oerating oint reure, P 1o and P 2o a well a the oerating oint value of valve ool dilacement, x o (Mark and Nariman, 2006). By ubtituting equation (24) in equation (11) and (12) or (21) and then uing equation (22) with taking a vicou friction only, the linear neumatic actuator model i (Mark and Nariman, 2006). (25) (26) (27) where Kv i the valve gain, ô v i the valve time contant, V 1o, V 2o are volume oerating oint. Intelligent Modeling: It i known that the neural network ha learning ability and i a good choice for modeling dynamic and comlex roce. Levenberg-Marquardt method a the training algorithm of multilayered feedforward neural network, and build a neural network model for the neumatic ytem wa adoted in (Marumo and Tokhi, 2004). The conventional digital controller wa deigned baed on a linear model. So an ARX (auto-regreive with external inut) model wa derived from the weight of the trained neural network. A third order ARX model wa derived from the fixed weight of the trained neural network. The tructure of a two-layered feed forward neural network i hown in Figure (4). For the eae of the controller deign, linear activation function are choen for the hidden layer of the neural network. The goal of a neural network model i to determine it weight to minimize the following obective function: (28) (29) N where è contain all adutable neural network weight {W,w,i},Z i the dataet coniting of the lant outut y(t) and the neural network outut. To train the neural network the Levenberg-Marquardt method wa ued for training by taking the following form: 3668

7 (30) (i) i (i) (i) where è ecifie the ith iterate; f( ) f i the earch direction; ë l i an adutable arameter; G(è ) and (i) (i) N (i) R(è ) are the aroximate gradient matrix and heian matrix of V N(è Z ) with reect to è reectively. Alo an invetigation into the modeling and control of the low eed of an air motor incororating a neumatic equivalent of the electric H-bridge wa reented by (Marumo and Tokhi, 2004). The neumatic H-bridge had been devied for eed and direction control of the motor. The ytem wa divided into three main region of low, medium and high eed. The ytem wa highly nonlinear in the low eed region and hence a controller with an ability of intelligence uch a a neuro model and controller wa rooed. Thi i the mot common form of uervied learning, giving the neural network off-line data of the lant and maing the correlation between inut and outut, forming a black-box model of the lant hown in Figure (5). In the air motor neural model tructure, the module, which comute neumatic eed, i a nonlinear network, with ten igmoidal hidden neuron and one linear outut neuron. The entire control of the air motor ytem i achieved by the ue of modular control aroach cacade modeling. The modular aroach oen the way to modeling of more comlex behavior of the air motor. Each module can be conidered a a earate roce in the inut-outut form: Y (k) = f(y (k -1),...,Y (k -n), U(k -1),...,U(k -m) (30) where Y (k) i the outut vector and U(k) i the inut vector at time k. The goal of the modeling rocedure i to build a neuro-model (NM) which redict the roce outut Y (k) a accurately a oible, given the at inut and outut. Fig. 3: Four-valve neumatic ytem tructure. Fig. 4: A multilayered feedforward neural network. 3669

8 Fig. 5: Schematic of black box MISO model for neumatic actuator ytem. Concluion: The neumatic actuator rereent the main force control oerator in many indutrial alication and offer numerou advantage uch a cleanline, low cot, high ratio of ower to weight, eay to maintain, afe, long anti exloion, working life and working overload. But on the other hand, the control accuracy i affected badly by it nonlinear characteritic. The nonlinear characteritic, eecially the nonlinear friction and the thermodynamic of the air reure in the chamber of the cylinder have a bad influence on control accuracy of the dilacement controlling of the cylinder. In addition, there are a erie of nonlinear and time varying factor uch a load force, temerature, oition of the iton, taying time at till and wearing out during working rocedure. Alo the neumatic actuator are uncertain ytem. So, becaue of all the reaon above, it ha became neceary to find a model that can rereent the actual overall dynamic behavior of the neumatic ytem. In thi aer a different methodologie for deriving and imulating the model of the neumatic actuator controlled with roortional valve are reented. The model include cylinder dynamic, ayload motion and valve characteritic. Alo the intelligent method that were ued for modeling the neumatic actuator are dicued. REFERENCES Armtrong, B. and C. Canuda de Wit, Friction modeling and comenation, The Control Handbook (by William S. Levine), CRC Pre, Clement and Len., Electro-neumatic oitioner get electronic, Journal of control and Intrumentation, 17: Chen, Y.Y., J. Wang. and Q.H. Wu, A oftware tool develoment for neumatic actuator ytem imulation and deign, Comuter in indutry, 51: Dorde, D. and N. Novak, Simulation, animation and rogram uot for a high erformance neumatic force actuator ytem, Mathematical and Comuter Modelling. Edmod, R. and H. Yildirim, A high erformance neumatic force actuator ytem, ASME, Journal of Dynamic Sytem, Meaurement and Control, 122(3): French, L.G. and C.S. Cox, The robut control of a modern electroneumatic actuator, IFAC, Automatic Control In Sace. Gary, M.B. and N. Shu, Exerimental comarion of oition tracking control algorithm for neumatic cylinder actuator, IEEE/ASME Tranaction on Mechatronic, 12(5): Han, K.L., S.C. Gi. and H.C. Gi, A tudy on tracking oition control of neumatic actuator, Mechatronic, 12: Jihong, W., K. Ulle and K. Jia, Tracking control of nonlinear neumatic actuator ytem uing tatic tate feedback linearization of the inut-outut ma, Proceeding Etonian Academic Science of Phyic and Mathematic, 56: Mark, K. and S. Nariman, QFT ynthei of a oition controller for a neumatic actuator in the reence of wort-cae eritent diturbance, Proceeding of the American Control Conference, Minneaoli, Minneota, USA., Marumo, R. and O.M. Tokhi, Intelligent modeling and control of a neumatic motor, CCECE, IEEE. Nieto, A.J., A.L. Morale, A. Gonzalez, J.M. Chicharro. and P. Pintado, An analytical model of neumatic uenion baed on an exerimental characterization, Journal of Sound and Vibration, 313:

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