Sensorless Plunger Position Control for a Switching Solenoid

Transcription

1 637 Sensorless Plunger Position Control for a Switching Solenoid Jyh-Chyang RENN and Yen-Sheng CHOU In this study, the sensorless control algorithm is applied to improve the position control accuracy of the output plunger of a switching solenoid. The main purpose of this study is the attempt to develop a fluid-technical proportional valve with simplest construction and lowest cost. Since the switching solenoid is not equipped with a position sensor and consequently no closed-loop control theory can be applied, the accuracy of the position control of the output plunger becomes a most critical question. Therefore, a modified open-loop controller using the sensorless control algorithm is proposed. The basic idea is the utilization of the sensorless control algorithm to estimate the steady-state position of the plunger. Then, this estimated position is fedback to the controller and the input current during the steady-state response is real-time adjusted according to the error signal. Finally, a series of experiments are carried out and the results show that the proposed modified open-loop controller improves significantly the steady-state accuracy of the plunger position and provides an alternative approach to design unconventional proportional valves. Key Words: Sensorless Control, Proportional Valve, Switching Solenoid, Electromagnetic 1. Introduction Nowadays, the electro-mechanical transducers are widely used in the design of electro-hydraulic or electropneumatic valves. Figure 1 shows the structures of two typical electro-mechanical transducers (1), (8). The first one is the switching solenoid, which is perhaps the most commonly used one and has been widely applied to the design of traditional fluid-technical directional control valves. The second one is the proportional solenoid, which is the standard component used in the design of conventional electro-hydraulic or electro-pneumatic proportional valves. As shown in Fig. 1, the switching solenoid and proportional solenoid have quite different force/stroke characteristics, though they possess almost the same structure (1). In details, the former possesses a highly nonlinear behavior regarding the force/stroke characteristics and the static force/stroke characteristic of the latter, however, is quite linear. Such a linear force/stroke relation of the proportional solenoid is the key requirement for the design of fluid-technical proportional valves. The valve spool, which is subjected to a constant force in the linear working range, reaches a definite position in the valve body ac- cording to Hooke s law. This definite position of the spool signifies a definite opening area of the valve orifice. Furthermore, it is also observed from Fig. 1 that the relation between the output force and the input current is quite linear. Consequently, the opening area of the valve orifice is continuously variable and is proportional to the input current. This is exactly the basic function of fluid-technical proportional valves. Surveying some previous reports (3), (8) (10), it is found that the research topic of introducing the switching solenoid instead of the proportional solenoid for the con- Received 7th July, 2003 (No ) Department of Mechanical Engineering, National Yunlin University of Science and Technology, 123 University Rd., Sec. 3, Yunlin, Taiwan 640, ROC. rennjc@yuntech.edu.tw Fig. 1 Structures of switching solenoid and proportional solenoid (1)

2 638 struction of the proportional valve has always been attracting the attentions of many researchers and engineers. One major reason for such attempts is the cost consideration. The cost of a switching solenoid is much lower than that of a proportional solenoid. According to the available literature, the most interesting method is perhaps the introduction of pseudo-force feedback, which linearizes the nonlinear force/stroke relation successfully (8) (10). Recently, the sensorless control algorithm has been widely discussed. Most of the reports, however, dealt with the control of rotatory electric machines, such (2), (7) as the reluctance motor, etc. One previous report described the attempt of applying the sensorless control algorithm to a closed-loop controlled proportional solenoid (6). Though the experimental results were not fully satisfying, the contribution of the paper was to reveal the feasibility of the new attempt. In addition, one interesting paper discussed the methods of estimating the plunger position of a solenoid actuator (4). However, the position control of the plunger was not reported. There is no further available literature concerning the application of sensorless control algorithm to the plunger position control of solenoid actuator. For the application of sensorless control algorithm to the plunger position control of a switching solenoid, the most difficult challenge is perhaps the unknown effect of the motional back emf during the transient response (6). Since the motional back emf during the transient period is basically a nonlinear function, it is very difficult to precisely acquire the effect of the back emf. However, it is reasonable to ignore the effect of motional back emf during the steady-state peroid because the plunger is almost stationary. Besides, the limited time-interval of the transient response is also a big problem because the implementation of the complex sensorless control algorithm demands quite large calculating effort. Thus, it is proposed in this study that the sensorless control algorithm may be applied to improve the steady-state performance of the plunger position control of a switching solenoid. Moreover, the closed-loop control of the plunger position during the transient response is excluded because of the obvious difficulties. In this study, therefore, a modified openloop controller using the sensorless control algorithm is proposed. The basic idea is the utilization of the sensorless control algorithm to estimate the steady-state plunger position. Then, this estimated position is fedback to the controller and the input current during the steady-state response is real-time adjusted according to the error signal. In the following, the experimental test devices are firstly outlined. i c : corrected new input current i : correction current K : gain of the controller L eq (x,i) : total equivalent inductance R : resistance of the coil V : applied voltage x : actual position of the plunger x : estimated plunger position λ(x,i) : flux linkage in the inductor 2. Experimental Test Devices In this study, two experimental test devices are developed for evaluating the performances of the switching solenoid with modified open-loop controller using sensorless control algorithm. The first one is the static test rig as shown in Fig. 2. An open-loop controlled micro-stepping motor (American Precision Industries, CMD-260) is utilized to control the plunger position of the tested switching solenoid. The angular displacement and angular speed of the micro-stepping motor are derived directly from the number and the frequency of the generated pulse signal sent to the driver respectively. In addition, the direction of rotation can be easily controlled by sending a Hi- (5 Volt) or Lo- (0 Volt) signal to one input port of the driver. Besides, the test device provides a position sensor (RDP- LVDT-D2/200) as well as a load cell (BAB-10 M) for the measurement of the stroke and the output force of the plunger. To measure the coil current as well, this test device is equipped with a current transformer. Finally, the control of the unit as well as the acquisition and processing of measured data are all integrated in a Pentium-IIIbased software controller. Figure 3 shows the measured force/stroke characteristics for the plunger extension of the tested switching solenoid. The test solenoid is type R02, which is a domestic product of Taiwan Seven Ocean hydraulic company and is originally developed for NG02 hydraulic directional solenoid valves. The second one is the dynamic test device as shown in Fig. 4. The output plunger of the switching solenoid e ss Nomenclature : steady-state error i : input current Fig. 2 The static test device Series C, Vol. 47, No. 2, 2004 JSME International Journal

3 639 Fig. 3 The measured force/stroke characteristics of the tested switching solenoid (for plunger extension) Fig. 5 The equivalent R L circuit model of the switching solenoid where V : applied voltage, R : resistance of the coil, λ(x,i) : flux linkage in the inductor, x : position of the plunger, i : input current. Fig. 4 The dynamic test device drives a mass-spring system, which is used to simulate the movement of the spool in a proportional valve body. The displacement of the mass is measured by an inductive displacement sensor (SENTEC, HA-222S). To produce the linearly controllable input current, a voltage-tocurrent transducer (i.e. amplifier) is employed. In addition, two separate AD/DA interface cards (ADLINK-PCI- 9112) are installed in the Pentium-III computer. One is employed to send the voltage control signal to the amplifier and to acquire the signal of the plunger position. The other is specially used to acquire the input current signal for the further derivation of the current slope, which is an important and necessary parameter for the implementation of sensorless control algorithm. 3. Modeling the Switching Solenoid The switching solenoid is modeled as an equivalent R L circuit as shown in Fig. 5, in which the coilinductance, L, is not a constant and is assumed to be a nonlinear function of the plunger position, x, and the input current, i. From the Kirchhoff s and Faraday slaw,we have V = ir+ dλ(x,i), (1) Surveying some previous reports, two different governing equations derived from the same Eq. (1) can be found, the first one (5) is the Eq. (2) and the second one (4) is the Eq. (3). V = ir+ L di +idl. (2) V = ir+ λ(x,i) i di + λ(x,i) x dx. (3) It is proved that these two equations are basically equivalent as follows. Since the inductance, L, is a function of the plunger position, x, and the input current, i. Eq. (2)can be rewritten as V = ir+l(x,i) di [ L(x,i) +i dx x + L(x,i) ] di, (4) i or [ V = ir+ L(x,i)+i L(x,i) ] di i +i L(x,i) x dx. (5) Using the definition λ = Li, Eq. (5) can further be rewritten as [ ] L(x,i) i di V = ir+ i + λ dx x. (6) Comparing the Eq. (6) with Eq. (3), it is obvious that these two equations possess the same form. Hence they are equivalent. In this study, therefore, the Eq. (3) will be utilized to model the switching solenoid. Moreover, the term λ(x, i) dx, which represents the motional back emf, is ignored because only the performance of the steady-state re- x sponse is investigated. Thus, Eq. (3) is simplified as V = ir+ L eq (x,i) di, (7) where L eq (x,i): total equivalent inductance.

4 640 Fig. 6 Examples of measured steady-state current signals at different plunger positions (i = 0.4 A) Fig. 7 The schematic illustration to calculate the slope of the current ripple 4. Sensorless Control Algorithm for Switching Solenoid The Eq. (7) can be rearranged as ( ) 1 di L eq (x,i) = (V ir). (8) Obviously, Eq. (8) is nontrivial if and only if di 0. This condition is generally troublesome in a step position control of the plunger at the steady-state because the current signal is nearly a constant at the steady-state and hence di = 0. In our experimental device, however, a digital amplifier with a PWM-controlled MOSFET is used. Though the plunger has reached its steady-state position, the current signal is not a constant and its derivative di is nonzero. Figure 6 shows some examples of measured steady-state current signals using oscilloscope at different plunger positions, in which the input current is set to be Series C, Vol. 47, No. 2, A. The schematic illustration to calculate the slope of the rising edge of the current ripple is shown in Fig. 7. After a series of measurements and observations, it is found that the slope of the rising edge of the steady-state current ripple is again a function of the plunger position and the filtered input current by a low-pass filter. The purpose of utilizing the low-pass filter is to acquire a constant input current by filtering out the current ripple. Figure 8 shows the corresponding graphic results. On the other hand, from the Eq. (7), it is clear that the value of the current slope, di, is related to the inverse of the total equivalent inductance, L eq. Therefore, the current slope may be regarded as a measure of the equivalent inductance and hence is suitable for the estimation of the plunger position. In the following, the unknown function relating the current slope, the plunger position and the filtered input current is derived by curve-fitting technique. First, the range of 0 mm x 4 mm for the plunger position and the range of 0.3 A i 0.6 A for the filtered input current are chosen for study, in which the curves approximate straight lines. For filtered input current over 0.7 A, however, the curves are almost horizontal and are not suitable for the implementation of sensorless control because of the difficulty to derive the current slope. After curve-fitting, the plot of the curves in the chosen ranges is shown in Fig. 9. For convenience, the value of the current slope shown in Figs. 8 and 9 is magnified by a factor of 50 to reduce the computational burden. In this study, it is proposed that the current slope is a function of the filtered JSME International Journal

5 641 Table 2 Values of the coefficients A(i)and B(i) as a function of the filtered input current Fig. 8 Experimental results relating the current slope, the plunger position and the filtered input current Fig. 10 Actual and estimated plunger position of tracking a triangular-wave motion (i = 0.3 A) Fig. 9 The plot of the curves after curve-fitting Table 1 The numerical relationship between the current slope, filtered input current and the plunger position Fig. 11 Actual and estimated plunger position of tracking a triangular-wave motion (i = 0.6 A) input current and the plunger position as follows di = A(i) x + B(i). (9) Table 1 shows the numerical relationship between the current slope, filtered input current and the plunger position. The corresponding values of the coefficients A(i) and B(i) as a function of the filtered input current are summarized in Table 2. By curve-fitting technique, the best-fit values of the coefficients are found to be A(i) = i i i i , (10) B(i) = i i i i (11) Obviously, from Eqs. (9) (11), the estimated plunger position, x, can be derived if the filtered input current and the current slope are both available. Because of the nonlinear hysteresis of the magnetic materials, it is worth mentioning that the best-fit values of the coefficients (Eqs. (10) and (11)) are only valid for the plunger extension. For the retraction of plunger, the same procedures have to be repeated to find a new set of the coefficients, which are found to be A(i) = i i i i , (12) B(i) = i i i i (13) 5. Experimental Results and Discussion As a first attempt, the plunger of the switching solenoid is forced to track a triangular-wave motion generated by the micro-stepping motor in the static test rig as shown in Fig. 2. Since the traverse speed of the plunger

6 642 is very low (0.36 mm/s), it is reasonable to assume that the motion of the plunger is basically the steady-state response. Figures 10 and 11 show two typical results. The filtered input currents are 0.3 A and 0.6 A respectively. Obviously, the estimated plunger positions agree very well with the actual ones. However, around dx = 0, there is inevitably a discontinuous change of the coefficient sets de- Fig. 12 (a) (b) Illustrations of setting the initial and desired plunger position as well as the initial guess of the input current pending on the sign of the plunger velocity dx because the set of coefficients, A(i) andb(i), for the plunger extension is different from that for the plunger retraction. Comparing the set of Eqs. (10) and (11) with the set of Eqs. (12) and (13), furthermore, it is found that the difference between these two sets of coefficients is quite small. Thus, a soft change of the coefficient sets around the peak is expected. On the other hand, such a discontinuous change indeed results in the larger deviation and oscillation of the estimated plunger position around the peak of the triangular wave as shown in Fig. 11. Nevertheless, these successful results show the feasibility of utilizing the estimated plunger position as the feedback signal to improve the steady-state performance. For a given initial plunger position and a desired plunger stroke in the range of 0 mm x 4 mm, an arbitrary initial input current in the range of 0.3 A i 0.6 A is applied to the solenoid. The initial plunger position is defined as the travel distance of the plunger measured from zero to the position, where the plunger contacts the movable mass as shown in Fig. 12 (a). This initial plunger position may be arbitrarily set in the range of 0 mm x 4 mm. After the setting of the initial plunger position, the excitation current is given to produce the magnetic driving force. The initial guess of the input current and the setting of the desired plunger position are obtained by the intersection of the experimental magnetic-force/stroke and spring-force/stroke curves as shown in Fig. 12 (b). The (a) Measured step responses (b) Estimated position and input current Fig. 13 Experimental results using the proposed controller with sensorless control algorithm (initial position: 2.0 mm, initial current: 0.3 A) Series C, Vol. 47, No. 2, 2004 JSME International Journal

7 643 (a) Measured step responses (b) Estimated position and input current Fig. 14 Experimental results using the proposed controller with sensorless control algorithm (initial position: 2.5 mm, initial current: 0.4 A) springconstantis 6.7N/mm. For example, the intersection of the two curves shown in Fig. 12 (b) implies that the initial input current is 0.3 A and the desired plunger position is 2.26 mm. As a result of the possible friction and other nonlinearities, however, the mass driven by the switching solenoid cannot reach the desired position. Thus, it is expected that the beginning steady-state position error of the plunger is quite large because the initial input current is not enough to overcome the friction and other nonlinearities. Figure 13 shows one experimental result, in which the step response is measured using the apparatus illustrated in Fig. 3. During the phases of transient response, it is observed that the estimated plunger position, x, is fully un- reliable. However, the estimated steady-state plunger position after 0.5 second reaches a stable and reliable range and may be used as the feedback signal to correct the plunger position to the desired one. In this study, it is proposed that the quantity of correction of the input current is proportional to the steady-state error between the desired and the estimated steady-state plunger position. Thus i c = i+ i, (14) where i = K e ss, (15) e ss = x x, (16) i c : corrected new input current, i : correction current, K : gain of the controller (K = 0.4 Amp/mm), e ss : steady-state error, x : actual desired plunger position, x :estimated plunger position. In this study, the modified open-loop controller described by Eqs. (14) (16) is carried out only once at 0.5 second. As shown in Fig. 13, the initial position of the plunger is 2.0 mm and the desired plunger position is set to be 2.26 mm. It is observed that the actual plunger position using the proposed modified open loop controller reaches gradually the desired position after 0.5 second. Obviously, the initial input current of 0.3 A is not enough to drive the mass to the desired position. According to Eqs.(14) (16), the corrected new input current is found to be 0.36 A. Similarly, Fig. 14 shows the other experimental result. The initial and desired plunger positions are defined as 2.5 mm and 2.98 mm respectively. The initial input current is set to be 0.4 A and the corrected new input current after 0.5 second is 0.43 A. For the both experiments, the motion of the plunger is always extension. Thus, the set of coefficients given by Eqs. (10) and (11) is used. However, for the case that the plunger has to retract, the other set of coefficients described by Eqs. (12) and (13) may be utilized. In this study, a dynamic test for the case of plunger retraction is executed. The basic idea of the test is that the experiment shown in Fig. 14 is continued and the input excitation current is decreased. It is expected that the plunger will retract as a result of the decreased excitation current. Figure 15 shows the experimental result. The initial in-

8 644 (a) Measured step responses (b) Estimated position and input current Fig. 15 Experimental results for the plunger retraction using the proposed controller with sensorless control algorithm Series C, Vol. 47, No. 2, 2004 put current is set to be 0.43 A and the plunger is initially kept at the position of 2.98 mm. Similarly, the desired steady-state plunger position is now set to be 2.84 mm corresponding to a input current of 0.35 A. However, the experimental result shows that the current 0.35 A is not small enough to bring the plunger back to the desired steadystate position. Applying the proposed method, therefore, the corrected new input current after 0.5 sec is found to be 0.33 A. Furthermore, from Figs. 13, 14 and 15, it is noticeable that there are still small steady-state position errors. This is chiefly because the constant gain, K, of the controller may not be globally optimal for the whole range of operation. In addition, such steady-state position errors may also arise from the inevitable deviation between the actual plunger position and the estimated one. Theoretically, continuous utilization of the proposed P-controller or the introduction of PI or PID controller may increase the control accuracy. One prerequisite, however, is that the estimated plunger position must be accurate enough. Otherwise, the deviation between the actual and estimated plunger position may even deteriorate the control accuracy. As shown in Figs. 10 and 11, it is obvious that such deviations are inevitable. Therefore, it is suggested in this study that single application of a simple P-controller is adequate for improving the control accuracy. Nevertheless, using the proposed modified open-loop controller with sensorless control algorithm indeed reveals the possibility of controlling the plunger position accurately without the need to install a position sensor or to linearize the force/stroke characteristics. 6. Conclusions In this study, we have successfully developed a modified open-loop controller with sensorless control algorithm for the plunger position control of a switching solenoid. Three conclusions may be drawn from this research. ( 1 ) Instead of the well-known method of linearizing the force/stroke characteristics, the proposed controller with sensorless control algorithm provides an alternative approach to design unconventional proportional valves based on switching solenoid. ( 2 ) Utilizing the estimated plunger position as the feedback signal effectively compensates the steady-state error of the plunger position of a switching solenoid for an arbitrarily given input current. ( 3 ) Since the proposed controller is focused on the steady-state response, it has no influence on the transient response. Consequently, the application field of the developed proportional valve based on the switching solenoid and sensorless control algorithm is limited to the cases, in which the response speed is not the critical performance. Possible examples of application may be the openloop speed control of a proportional-valve-controlled motor in a conveyer system, the open-loop velocity control JSME International Journal

9 645 of a proportional-valve-controlled cylinder in a hydraulic press, etc. Acknowledgements The financial supports of the National Science Council under grant number NSC E is greatly appreciated. References ( 1 ) Backe, W., Steuerung- und Schaltungstechnik II, Umdruck zur Vorlesung, Vol.4, (1993), Auflage, RWTH Aachen, Germany. ( 2 ) Lagerquist, R., Boldea, I. and Miller, T.J.E., Sensorless Control of Synchronous Reluctance Motor, IEEE Trans. on Industrial Applications, Vol.30, No.3 (1994), pp ( 3 ) Rahman, M.F., Cheung, N.C. and Lim, K.M., Proportional Control of a Solenoid Actuator, IEEE Proc. on IECON, Vol.3 (1994), pp ( 4 ) Rahman, M.F., Cheung, N.C. and Lim, K.M., Position Estimation in Solenoid Actuators, IEEE Trans. on Industrial Applications, Vol.1 (1995), pp ( 5 ) Vaughan, N.D. and Gamble, J.B., The Modeling and Simulation of a Proportional Solenoid Valve, Trans. of the ASME, J. of Dynamic System, Measurement and Control, Vol.118 (1996), pp ( 6 ) Renn, J.-C. and Liu, J.-S., Sensorless Position Control of a Proportional Solenoid, Proc. of the 2nd International Fluid Power Conference (2. IFK), Dresden, Germany, Vol.1, (2000), pp ( 7 ) Park, H.W., Lee, S.H. and Won, T.H., Position Sensorless Speed Control Scheme for Permanent Magnet Synchronous Motor Drives, IEEE Proc. of Int. Symposium on Industrial Electronics, Vol.1, (2001), pp ( 8 ) Renn, J.-C. and Tsai, C., Linearization of the Force/Stroke Characteristic of Switching Solenoid Using Fuzzy-Logic-Controller, Proc. of Fifth JFPS International Symposium on Fluid Power, Nara, Japan, Vol.1, (2002), pp ( 9 ) Tappe, P. and Kleinert, D., Stetige Ventilschieberlageregelung durch Magnete mit Schaltmagnetcharakteristik, O+P, Vol.42, No.5 (1998), pp (10) Kleinert, D. and Tappe, P., Betaetigungsmagnete mit Schaltmagnetcharakteristik fuer den Einsatz in Proportionalventilen, O+P, Vol.44, No.7 (2000), pp