BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi Tomul LV (LIX), Fasc. 2, 2009 SecŃia AUTOMATICĂ şi CALCULATOARE MODELLING OF AN ELECTROMAGNETIC VALVE ACTUATOR BY CONSTANTIN FLORIN CĂRUNTU, MIHAELA HANAKO MATCOVSKI, ANDREEA ELENA BĂLĂU, DANIEL IONUł PĂTRAŞCU, CORNELIU LAZĂR and OCTAVIAN PĂSTRĂVANU Abstract. During the last few years automotive actuators have become mechatronic systems in which mechanical components coexist with electronics and computing devices and because pressure control valves are used as actuators in many control applications for automotive systems, a proper dynamic model is necessary. Starting from the modelling of a single stage pressure reducing valve found in literature, in this paper, the concept of modelling a real three land three way solenoid valve actuator for the clutch system in the automatic transmission is presented. Two simulators for an input-output model and a state-space model were developed and these were validated with data provided from experiments with the real valve actuator on a test bench. Key words: control valve, pressure reducing valve modelling. 2000 Mathematics Subject Classification: 93A30, 00A72. 1. Introduction During the last few years the major advances in automotive applications have been enabled by smart electronic devices that monitor and control the mechanical components. Cars have become complex systems in which electronic and mechanical subsystems are tightly connected and interact to achieve optimal performance. Automotive actuators, in particular, have become mechatronic systems in which mechanical components coexist with electronics and computing devices [1]. In recent years, the use of control systems for automated clutch and transmission actuation has been constantly increasing [2], in the powertrain sector, the trend towards higher levels of comfort and driving dynamics while at 10 Constantin Florin Căruntu et al. the same time minimizing fuel consumption representing a major challenge. Electromagnetic actuation provides reliable means and a popular alternative to hydraulic or pneumatic actuation for implementing control systems. It is a natural connection between electrical circuits and mechanical systems. All electromagnetic actuators work upon the same principle. By inducing current in a coil of wire, the actuator gives rise to a magnetic force, which is then used to affect the movement of a physical component. Such systems have two distinct advantages: the applied force is non-contacting, and often the response of the electromagnet is significantly faster then the dynamics of the system being controlled [3]. Electronic devices have improved the accuracy of control using closed loop control techniques in many applications that have traditionally been served by hydro-mechanical open loop systems [4]. Electromagnetic valves vary in arrangement and complexity, depending upon their function. Three broad functional types can be distinguished: directional control valves, pressure control valves and flow control valves. Pressure control valves act to regulate pressure in a circuit and may be subdivided into pressure relief valves and pressure reducing valves. Pressure relief valves, which are normally closed, open up to establish a maximum pressure and bypass excess flow to maintain the set pressure. Pressure reducing valves, which are normally open, close to maintain a minimum pressure by restricting flow in the line. There are many practical applications where this type of electromagnetic actuator is used: electromagnetic valve actuators of combustion engines, artificial heart actuators, electromagnetic brakes, electromagnetic actuator for the clutch system in automatic transmissions etc. [5]. The increasing amount of power available to man that required control and the stringent demands of modern control systems had focused attention on modeling different types of valves even since 1967, when in [6], pressure control valves and electro-hydraulic servovalves were analyzed. Recent attention has focused on modeling and developing advanced control methods for different valve types used as actuators in automotive control systems: physics-based nonlinear model for an exhausting valve [7], nonlinear state-space model description of the actuator that is derived based on physical principles and parameter identification [8], [9], [10], [11] nonlinear physical model for programmable valves [12], nonlinear model of an electromagnetic actuator used in brake system, based on system identification [5], mathematical model obtained using identification methods for a valve actuation system of an electro-hydraulic engine [13], linear model constructed based on gray-box approach which combines mathematical modelling and system identification for an electro-magnetic control valve [14]. Starting from the equations in [6], where a single stage pressure reducing valve is modeled, in this paper, the concept of modelling a three land three way pressure reducing valve used as actuator for the clutch system in the automatic transmission of a VW vehicle is presented. Two models were Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 11 developed: a linearized input-output model and a state-space model then implemented in Matlab/Simulink and validated by comparing the results with data obtained on a real test-bench provided by Continental Automotive Romania. The rest of the paper is organized as follows. In Sec. 2, the structure and the functional operation of the valve are presented and in Sec. 3 the mathematical models of the solenoid three way valve are developed. Sec. 4 presents the simulators for the electromagnetic valve and the concluding remarks are given in Sec. 5. 2. Valve Structure and Operation Pressure control valves employ feedback and may be properly regarded as servo control loops. Therefore proper dynamic design is necessary to achieve stability. In Fig. 1a a section through a real three stage pressure reducing valve is represented. Schematics of the three land three way pressure reducing valve are shown in Fig. 1b. The pressure to be controlled is sensed on the spool end areas C and D and compared with a magnetic force Fmag which actuates on the plunger. The feedback force Ffeed = FC – FD is the difference between the force applied on the left sensed pressure chamber FC, and the force applied on the right sensed pressure chamber FD. The difference in force is used to actuate the spool valve which controls the flow to maintain the pressure at the set value. In the charging phase, illustrated in Fig. 1c, the magnetic force is greater than the feedback force (Ffeed < Fmag) and moves the plunger to the left (x > 0), connecting the source with the hydraulic load. In the discharging phase, illustrated in Fig. 1d, the feedback force becomes grater that the magnetic force (Ffeed > Fmag) and the plunger is moved to the right (x < 0); the connection between the source and the hydraulic load is closed, the hydraulic load being connected to the tank. Using the magnetic force and the feedback force it results a force balance which describes the spool motion and the output pressure. This equation of force balance is the same for both positive and negative displacement of the spool: (1) Fmag − AC PC + AD PD = M v s 2 X + K e X , where PC represents the pressure in the left sensed chamber (C) that acts on the AC area, PD represents the pressure in the right sensed chamber (D) that acts on the AD area, Mv is the spool mass, Ke = 0.43w(PS0 – PR0) represents the flow force spring rate, PS is the supply pressure, PR is the reduced pressure, w represents the area gradient of the main orifice, X = X(s) is the Laplace transform of the spool displacement and s represents the Laplace operator. 12 Constantin Florin Căruntu et al. output pressure line pressure PS PR C Ffeed D +1mm x -1mm Fmag T hydraulic damper plunger tank pressure PT b a c d Fig. 1 − Section through a real three stage pressure reducing valve (a); Three stage valve schematic representation (b); Charging phase of the pressure reducing valve (c); Discharging phase of the pressure reducing valve (d). In Fig. 1a a hydraulic damper that acts to reduce the input pressure spike, which has negative effects on the output pressure, is also represented. 3. Valve Modelling The models designed in this Section are based on physical principles for flow and fluid dynamics and parameter identification. Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 13 3.1. Input-Output Mathematical Model for the Charging Phase The charging phase of the pressure reducing valve has been illustrated in Fig. 1c. A positive displacement of the spool allows connection between the source and the hydraulic load, while the channel that connects the hydraulic load with the tank is kept closed. The linearized continuity equation from [6] was used to describe the dynamics from the sensed pressure chambers: VC (2) QC = K1 ( PR − PC ) = (3) QD = K 2 ( PR − PD ) = βe VD βe sPC − AC sX , sPD + AD sX , where K1, K2 are the flow-pressure coefficients of restrictors, VC, VD are the sensing chamber volumes and βe represents the effective bulk modulus. Using the flow through the left and right sensed chambers, the flow through the main orifice (from the source to the hydraulic load) and the load flow, the linearized continuity equation at the chamber of the pressure being controlled is [15]: (4) K C ( PS − PR ) − QL − kl PR − K1 ( PR − PC ) − K 2 ( PR − PD ) + K q X = Vt βe sPR , where QL is the load flow, KC is the flow-pressure coefficient of main orifice, Kq is the flow gain of main orifice, kl is the leakage coefficient and Vt represents the total volume of the chamber where the pressure is being controlled. These equations define the valve dynamics and combining them into a more useful form, solving Eq. (2) and Eq. (3) w.r.t. PC and PD and substituting into (4) yields after some manipulation: 1 s 1 AC AD + 1 + 1 + K q X 1 + + + − s + ω1 ω2 ω1 ω2 K q K q s ( KC PS − QL ) (5) 1 V ω s A A + + C − D s 2 = PR K ce C 1 + 1 + ω ω Vt ω3 ω2 1 2 K qω2 K qω1 + where ω1 = s VD ω2 s s VC ω1 VD ω2 s + + + 1 , + 1 + 1 + 1 + Vt ω3 ω1 ω3 Vt ω3 Vt ω3 ω1 ω2 β e K1 VC , ω2 = βe K 2 VD are the break frequency of the left and right 14 Constantin Florin Căruntu et al. sensed chambers, ω3 = β e K ce is the break frequency of the main volume and Vt Kce = KC + kl represents the equivalent flow-pressure coefficient. Considering that VC « Vt and VD « Vt, the right side can be factored to give the final form for the reducing valve model in the charging phase: 1 s 1 AC AD + 1 + 1 + K q X 1 + + + − s + ω1 ω2 ω1 ω2 K q K q s ( KC PS − QL ) (6) 1 s s s A A + + C − D s 2 = PR Kce + 1 + 1 + 1 . ω1 ω2 ω3 ω1ω2 K qω2 K qω1 3.2. Input-Output Mathematical Model for the Discharging Phase A negative displacement of the pressure reducing valve spool allows connection between the hydraulic load and the tank, while the channel that connects the source with the hydraulic load is kept closed. The linearized continuity equations at the sensed pressure chambers for the discharging phase of the valve, illustrated in Fig. 1d, are: VC (7) −QC = K1 ( PC − PR ) = − (8) −QD = K 2 ( PD − PR ) = − βe VD βe sPC + AC sX , sPD − AD sX . Using the flow through the left and right sensed chambers, the flow through the main orifice (from the hydraulic load to the tank) and the load flow, the linearized continuity equation obtained for the chamber of the pressure being controlled is [15]: (9) QL + K1 ( PC − PR ) + K 2 ( PD − PR ) − K D ( PR − PT ) − kl PR + K q X = Vt βe sPR , where KD is the flow-pressure coefficient of main orifice and PT represents the tank pressure. Combining these equations into a more useful form, solving Eq. (7) and Eq. (8) for PC and PD and substituting into (9) yields after some manipulation: Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 15 1 s A A 1 + 1 + 1 + K q X 1 + + + C − D s + ω1 ω2 ω1 ω 2 K q K q s ( K D PT + Q L ) (10) 1 AC AD + + − ω ω 1 2 K q ω 2 K q ω1 + 2 s = PR K ce VC ω1 s + 1 + Vt ω 3 ω 2 . V ω s VD ω 2 s V ω s s + 1 + 1 + + C 1 + D 2 + 1 + 1 ω 3 Vt ω 3 Vt ω 3 ω1 Vt ω 3 ω1 ω2 In an entire analogue manner, again making the assumption that VC « Vt and VD « Vt like for the charging phase model and considering KD = KC the final form for the reducing valve in the discharging phase was obtained: 1 s 1 AC AD s+ + 1 + 1 + K q X 1 + + + − ω1 ω2 ω1 ω2 K q K q s ( K D PT + QL ) (11) 1 s s s A A + + C − D s 2 = PR K ce + 1 + 1 + 1 ω ω ω1 ω2 ω3 1 2 K qω2 K qω1 . 3.3. Transfer Function Block Diagram Equations (1), (2), (3), (6), for the charging phase of the valve, and Eqs (1), (7), (8), (11) for the discharging phase of the valve, define the pressure reducing valve dynamics and can be used to construct the transfer function block diagram represented in Fig. 2. Fig. 2 − Transfer function block diagram of valve. 16 Constantin Florin Căruntu et al. Considering the resulting force between the magnetic and the feedback force: F1 = Fmag − AC PC + AD PD , (12) solving PC and PD from the linearized continuity Eqs. (2) and (3) and substituting in the Eq. (1) of force balance, the following equation was obtained [15]: (13) Fmag − AC K1PR + AC sX + AD s + 1 K1 ω1 K 2 PR − AD sX = M v s2 X + Ke X . s + 1 K 2 ω2 Ke , Mv representing the mechanical natural frequency, and substituting (12) into (1) yields: After some manipulations, where it was considered that ωm = (14) AC2 AD2 s2 K1 K2 + F1 − sX = K e 2 + 1 X , ω s s m + 1 + 1 ω1 ω2 where (15) AD A F1 = Fmag − C + PR , s s +1 +1 ω2 ω1 illustrating the closed loop model from Fig. 2 for the displacement x. A switch is used in order to commutate between the three phases of the pressure reducing valve. Like seen in Fig. 2, switching between the charging and the discharging phase can be realized by selecting different disturbances for positive and negative displacement of the spool. 3.4. State-Space Model Starting from Eqs. (1), (2), (3) and (4), respectively Eqs. (1), (7), (8) and (9) a state-space model is designed [16]: (16) xɺ ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) + Du ( t ) Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 where: x (t ) = [ v (t ) u (t ) = PS ( t ) x (t ) PT ( t ) PC (t ) QL (t ) PD (t ) Fmag (t ) PR (t ) ] T T is 17 is the the state vector, input vector and y (t ) = [ x(t ) PR (t ) ] is the output vector. The A, C and D matrices are: T (17) 0 1 β e AC A = βe VC A − D VD 0 − Ke M v βe − AC M v βe AD M v βe 0 0 K1 VC 0 0 − 0 0 0 Kq K1 Vt Vt − K2 VD K2 Vt 0 K1 , VC K2 VD ( Kce + K 1+ K 2 ) − VD 0 0 1 0 0 0 C= , D = O2×4 , 0 0 0 0 1 and the matrix B has the Bc expression in the charging phase (for x > 0) and the B d expression in the discharging phase (for x < 0), where: (18) 0 0 Bc = βe 0 0 KC V t 0 0 0 0 0 0 0 0 0 − 1 Vt 0 0 M v βe 0 0 0 0 0 , B d = β e 0 0 0 0 0 − KC 0 Vt 1 0 0 0 0 1 Vt M v βe 0 0 . 0 0 1 This model is more precise because no approximations were used, as for the input-output model. 18 Constantin Florin Căruntu et al. 4. Simulators for Electromagnetic Valve In this section two simulators that were designed starting from the models described in Sec. 2 and developed in Matlab/Simulink program are presented. Parameter values used for testing in Simulink are presented in Table 1. Dimensional parameters were measured directly on the sectioned valve and the flow coefficients were determined through experiments with the real valve on a test bench at Continental Automotive Romania. Table 1 Coefficients Values Symbol Value Unit Mv 2.5 e-3 kg PS 1 e+6 N/m2 PT 1 e+5 N/m2 βe 1.6 e+9 N/m2 Ke 2 e+3 N/m KC 7.5772 e-11 m3/s·bar KD 7.5772 e-11 m3/s·bar K1 1.2634 e-10 m3/s·bar K2 1.3873 e-9 m3/s·bar Kq 8.0127 m3/s·bar kl VC 2 e-9 3 e-3 7.53 e-7 m3/s·bar m m3 VD 1.04 e-6 m3 Vt 3 e-4 m3 AC 3.664 e-5 m2 AD 2.94 e-5 m2 ω1 2.6845 e+6 rad/s ω2 2.1342 e+7 rad/s ω3 1.6606 e+4 rad/s α 2 e-5 m w The models were validated by comparing the results with data obtained on a real test-bench provided by Continental Automotive Romania. Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 19 4.1. Model Testing For testing purposes a Simulink model was created, using as input a step signal, like shown in Fig. 3. The commutation between the charging and the discharging phase was simulated by two switches that connect different perturbation depending on the value of the displacement. These switches were used in order to avoid the rapidly switching between the two flow perturbations caused by the oscillations of the plunger, using α as the threshold. [Ps] -K- Kc Fmag , Ql -K Fmag x 0 Pr In QL Model Load Flow Pt Kd -K - -K - Pr, x Fig. 3 − Simulink model with step signal input. In Fig. 3 two subsystems were used: one noted Model and representing the transfer functions of the reducing valve model (Fig. 4), that were represented as a block diagram in Fig. 2, and one noted as Load Flow representing the load flow. 1 Fmag MSD 1_num (s) MSD 1_den (s) P_num (s) 1 x 2 In P_den (s) MSD 2_num (s) Flow _num (s) MSD 2_den (s) Flow _den (s) FC_num (s) FC_den (s) FD_num (s) FD_den (s) Fig. 4 − Transfer functions represented in Simulink. 2 Pr 20 Constantin Florin Căruntu et al. For a step signal as the magnetic force and a sequence of pulses as the load flow represented in Fig. 5, the results obtained for spool displacement and reduced pressure are presented in Fig. 6. For modeling the load flow needed to actuate the clutch, two impulse signals, a positive one and a negative one, for 20 ms with a value of 10-4 m3/s were considered, value determined from measurements on the test bench. 5 QL Fmag Fmg[N], QL [104*m3/s] 4 3 2 1 0 -1 0 0.5 1 1.5 2 2.5 time [s] 3 3.5 4 4.5 Fig. 5 − Magnetic force and load flow. The displacement follows the step input behavior while the reducing pressure has almost the same value like the reference signal. 7 displacement pressure displacement [m], pressure [bar] 6 5 4 3 2 1 0 -1 0 1 2 3 4 5 time [s] Fig. 6 − Spool displacement and reduced pressure. The model shows good performance, being stable for input signals variations. Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 21 4.2. Input-Output Model Simulation In order to validate the results obtained for the solenoid valve actuator, a Simulink model (represented in Fig. 7) was created, using a magnetic force as input (Magnet block). The magnetic force block implements the connection between electric current trough solenoid and magnetic force generated by the magnetic flux. A force sensor was utilized to measure the magnetic force and the results were used in a form of a two dimensional look-up table, designed at Continental Automotive Romania for this type of valve. The blocks in the upper part of the model (Druck_A, Druck_P, Strommesszange and Weg_Magnet) represent the real data obtained from experiments made on a test bench at Continental Automotive Romania with this type of valve. The gains in the model are used to transform the values of the parameters that are in international units in other units used for display (meter to millimeter for the spool displacement and Pascal to bar for the reduced pressure). [time Druck _A] [Pr] Pr [xm] [time Druck _P] [time Strommesszange ] [time Weg _Magnet ] 1 [Ps] Ps [i] 0.003 s+1 Magnet [i] i -u+2.745 x [x] [xm] [x] -Kx_s vs. x_m [Ps] Fmag -K - Kc x 0 Pr In QL Model Load Flow Pt [Pr] Kd u-0.35 -KPr_s vs. Pr_m Fig. 7 − Input-output Simulink model. switching _filt Fmag 22 Constantin Florin Căruntu et al. In Fig. 7 the saturation block was used to allow only positive values for the magnetic force and the filter (switching_filt) to eliminate the high frequencies caused by the look-up table. In Fig. 8 a real input signal is illustrated, represented either by the magnetic force or by the current used to obtain the magnetic force through the look-up table. 8 Fmag i - current 7 Fmag [N], current [A] 6 5 4 3 2 1 0 -1 0 0.2 0.4 0.6 0.8 1 time [s] 1.2 1.4 1.6 Fig. 8 − Current and magnetic force used as input signals. The results of the simulations are presented in Figs. 9 and 10, where the spool displacement and the reduced pressure were compared with real data obtained from experiments made on a test bench with the input signal from Fig. 8. 1 measured simulated 0.8 0.3 0.25 0.2 displacement [mm] 0.6 0.2 0.15 0.1 0.1 0.4 0.05 0 0.8 0 0.8 0.2 0.82 0.84 0.85 0.86 0.88 0.9 0.9 0 -0.2 -0.4 0 0.2 0.4 0.6 0.8 1 time [s] 1.2 1.4 1.6 Fig. 9 − Compared spool displacements for input-output model. It can be seen that the simulated displacement of the spool has even smaller variations than the measured displacement while the behaviour is the same. Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 23 14 measured simulated 12 6 10 5 8 pressure [bar] 6 5.5 5.5 5 4.5 4.5 6 4 4 3.5 3.5 0.80.8 4 0.82 0.84 0.85 0.86 0.88 0.9 0.9 2 0 -2 0 0.2 0.4 0.6 0.8 1 time [s] 1.2 1.4 1.6 Fig. 10 − Compared reducing pressures for input-output model. Concerning the reduced pressure, the experimental results reveal that the simulated pressure follows the measured pressure behaviour, having in the steady state an irrelevant offset. The amplitude of the simulated pressures variations in steady state is lower than the amplitude of the measured pressures variations. 4.3. State-Space Model Simulation The state-space model was represented in Simulink as shown in Fig. 11, where a similar switch as in the input-output model was used to commutate the reduced pressure between the charging and the discharging phases. [time Druck _A ] Pr [time Druck _P ] Ps [Pr] [Ps] Load Flow [Ps] [time Strommesszange ] i [time Weg _Magnet ] x [Ql ] QL [x] -K - [Ps1] [x3] [i] -u+2.745 [x] -K - [Pr] x_s vs. x_m u-0.35 [xm] [Pr3] [xm] -K - 1 Pr_s vs. Pr_m [Fmag ] [i] 0.003 s+1 switching _filt Magnet [Ps1] [Ps1] Pt [Ql] x' = Ax+Bu y = Cx+Du Pt [x1] [Ql] [Pr1] x' = Ax+Bu y = Cx+Du [x2] [Pr2] State -Space _d State -Space _c [Fmag ] [Fmag ] [Pr1] [x1] [Pr2] [x1] 0 [Pr3] [x1] 0 [x2] Fig. 11 − State-space Simulink model. [x3] 24 Constantin Florin Căruntu et al. The results obtained for the spool displacement using the state-space model are similar to those obtained using the input-output model and are represented in Fig. 12. 1 measured simulated 0.8 0.3 0.3 displacement [mm] 0.25 0.6 0.2 0.2 0.4 0.1 0.1 0.15 0.05 00 0.8 0.8 0.2 0.82 0.850.86 0.84 0.88 0.9 0.9 0 -0.2 -0.4 0 0.2 0.4 0.6 0.8 1 time [s] 1.2 1.4 1.6 Fig. 12 − Compared spool displacements for state-space model. Fig. 13 illustrates the difference between the simulated and the measured reduced pressures. It can be seen that the amplitude of the simulated pressures variations in steady state is lower than the amplitude of the measured pressures variations. Also, the simulated pressure has in steady state a slight offset. 14 measured simulated 12 66 pressure [bar] 10 5.5 5.5 55 8 4.5 4.5 6 44 3.5 3.5 0.8 0.8 4 0.82 0.85 0.86 0.84 0.88 0.9 0.9 2 0 -2 0 0.2 0.4 0.6 0.8 1 time [s] 1.2 1.4 1.6 Fig. 13 − Compared reducing pressures for state-space model. 4.4. State-Space Model Simulation Using an S-Function After testing and validating the state-space model presented in the previous section, the state-space model was implemented as an S-function. Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009 25 Fig. 14 represents the Simulink model utilized for testing this block by comparing the results obtained through simulations with the measured values on the real test bench. [time Druck _A] [Pr] Pr [time Druck _P] Load Flow [Ps] Ps [Ps] [time Strommesszange ] -K- [Ps1] [i] i [time Weg _Magnet ] [Ql ] QL -u+2.745 x [x] [xm] [xm] 1 [Fmag ] [i] 0.003 s+1 Saturation Magnet switching _filt [Ps1] Pt [xs] sfun _valve _state [Ql] [Prs] S-Function [Fmag ] [x] [xs] -K - [Pr] u-0.35 [Prs] -K - x_s vs. x_m Pr_s vs. Pr_m Fig. 14 − State-space with S-function Simulink model. Fig. 15 illustrates the results and the measured values for the spool displacement and in Fig. 16 the compared reducing pressures, obtained using the input signals illustrated in Fig. 8, are represented. 1 measured simulated 0.8 0.3 0.25 displacement [mm] 0.6 0.2 0.2 0.15 0.4 0.1 0.1 0.05 0.2 00 0.8 0.8 0.82 0.84 0.85 0.86 0.88 0.9 0.9 0 -0.2 -0.4 0 0.2 0.4 0.6 0.8 1 time [s] 1.2 1.4 1.6 Fig. 15 − Compared spool displacements for S-function state-space model. 26 Constantin Florin Căruntu et al. 14 measured simulated 12 66 pressure [bar] 10 5.55.5 55 8 4.54.5 6 44 3.53.5 0.8 0.8 4 0.82 0.850.86 0.84 0.88 0.9 0.9 2 0 -2 0 0.2 0.4 0.6 0.8 1 time [s] 1.2 1.4 1.6 Fig. 16 − Compared reducing pressures for S-function state-space model. 5. Conclusions In this paper, two different models for a solenoid valve actuator used in the automotive control systems were developed: a liniarized input-output model, where simplifications were made in order to obtain a suitable transfer function to be implemented in Simulink and to obtain an appropriate behavior for the outputs, and a state-space model with no simplifications, that was implemented as a state-space model and as an S-function in Simulink. The results of the experiments illustrate a similar behavior of these three simulators. The models were validated by comparing the results with data obtained on a real test-bench provided by Continental Automotive Romania. It can be concluded that the simulators have good results illustrated by the similar behavior obtained for the spool displacement and the reduced pressure compared with the measured values. A c k n o w l e d g m e n t s. This work was partially supported by CNMP-SICONA project and Continental Automotive Romania. 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Bălău A.E., Căruntu C.F., Pătraşcu D.I., Lazăr C., Matcovschi M.H., Păstrăvanu O., Modelling of a Pressure Reducing Valve Actuator for Automotive Applications. accepted for the 3rd IEEE Multi-conference on Systems and Control MSC 2009, Saint Petersburg, Russia, July 8-10, 2009. MODELAREA UNEI VALVE ELECTROMAGNETICE UTILIZATE CA ELEMENT DE EXECUłIE (Rezumat) În ultimii ani elementele de execuŃie din autovehicule au devenit sisteme mecatronice în care componentele mecanice coexistă cu cele electronice. Deoarece valvele de control al presiunii sunt folosite ca elemente de execuŃie în multe aplicaŃii de 28 Constantin Florin Căruntu et al. control în autovehicule, a fost acordată o atenŃie deosebită modelării acestor tipuri de valve. Valvele hidraulice utilizează mişcarea mecanică pentru a controla presiunea unei surse de fluid, fiind folosite ca elemente de execuŃie în componenŃa structurilor de control ale autovehiculelor. Numeroasele tipuri de valve pot fi clasificate în funcŃie de rolul lor în: valve de control al direcŃiei fluidului, valve de control al presiunii şi valve de control al debitului. Obiectul studiului efectuat l-a constituit o valvă de reducere a presiunii cu 3 căi cu comanda electromagnetică ce intră în componenŃa transmisiei automate a unui autovehicul Volkswagen. Pornind de la ecuaŃiile generale prezentate în [6] au fost dezvoltate două modele neliniare (de tip liniar cu comutaŃie sau liniar pe porŃiuni) ale acestei valve: un model liniarizat intrare-ieşire şi un model de stare. Cele două modele au fost implementate în MATLAB-Simulink, pentru simulare fiind utilizate valorile parametrilor furnizate de firma Continental Automotive Romania. Pentru validarea modelelor, rezultatele obŃinute prin simulare au fost comparate cu rezultatele experimentale obŃinute de Continental Automotive Romania pe un stand de probă utilizând o valvă electromagnetică de tipul investigat. ConcordanŃa celor două tipuri de rezultate confirmă validitatea modelelor elaborate. Pentru modelul de stare a fost implementat în final un bloc Simulink de tip S-function.