Self-Sensing Actuators in Electrohydraulic Valves - CiteSeerX

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Proceedings of IMECE04 2004 ASME International Mechanical Engineering Congress and Exposition November 13-20, 2004, Anaheim, California USA

IMECE2004-62104

SELF-SENSING ACTUATORS IN ELECTROHYDRAULIC VALVES ∗

QingHui Yuan Dept. of Mechanical Engineering University of Minnesota Minneapolis, Minnesota 55455 Email: [email protected]

Perry Y. Li Dept. of Mechanical Engineering University of Minnesota Minneapolis, Minnesota 55455 Email: [email protected]

ABSTRACT Self-sensing refers to extracting the position information from the electromagnetic signals instead of from a physical position sensor. Self-sensing actuator has the benefit of significantly reducing hardware effort. In this paper, we investigate selfsensing actuators applied in direct acing proportional valves. The study is based on the configuration of a dual-solenoid actuator where each solenoid provides the independent single direction force. By considering the electrical dynamics of the solenoids and the nonlinear constraint between the two solenoids , an observer for the flux linkages is obtained. The position estimate is then calculated based on the flux linkage estimates. Both simulation and experiments verify the validity of the proposed algorithm.

position feedback, usually furnished by a linear variable differential transformer (LVDT), is needed for higher performance control. In our own research of high performance single stage electro-hydraulic valves that use open-loop fluid flow induced instability to increase spool agility [4], [5], [6], the spool position is critical for stablizing the closed loop system. Incorporating LVDTs into electrohydraulic valves raises the cost of the product. For example, consider the cost of the LVDTs themselves, the electric circuits for exciting and decoding the LVDTs, and the overhead of assembling the LVDTs. Therefore, the concept of self-sensing actuators can be very useful to reduce hardware effort of electrohydraulic valves. Of course it will raise the complexity of software though. However, software is expensive in development, but actually inexpensive in mass-production. Therefore, the overall cost can be significantly reduced.

1

Signal processing technique and observer approach are used to implement self-sensing actuators. Examples of signal processing technique include Noh et al [1] who propose a forward path filter that includes a high pass filter, a rectifier, and a low pass filter. The current switching waveform from a PWM amplifier is fed into the forward path filter. Since the switching waveform is an amplitude-modulated signal that carries position information in its amplitude, the signal out of the forward path filter, or demodulation of the switching waveform, will be dependent on the position. This demodulated signal, however, is a nonlinear function of the duty cycle, thus making the performance of the filter less reliable. Examples of observer approach include Vischer et al [7] who propose to treat an active magnetic bearing as two-port system with bidirectional information flow. The locally linearized system can be proved to be both controllable and ob-

INTRODUCTION Self-sensing refers to extracting the position information from the electromagnetic signals of an actuator, like currents and voltages, instead of from a physical position sensor. This technique has attracted the attention of researchers in some fields. One example is self-sensing magnetic bearing, which is stimulated by the requirement of minimizing the number of wires in some applications, like heart pump [1]. Another example is selfsensing switched reluctance motor, which is the cost effective and robust solution for electric motors [2] [3]. Similarly, self-sensing actuators can be applied in flow power control engineering. In the electrohydralic valves, spool

∗ THIS RESEARCH IS SUPPORTED BY THE NATIONAL SCIENCE FOUNDATION ENG/CMS-0088964.

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c 2004 by ASME Copyright °

servable. However, this method has several drawbacks. First of all, it is only valid in the local region of the equilibrium where the linearization is applied. Secondly, it cannot provide the position estimates when the signal frequency is either too high or too low. Therefore, the behavior of so called negative stiffness occurs in self-sensing magnetic bearing [7]. In this paper, we investigate self-sensing actuators applied to direct acing proportional valves. The study is based on the configuration of a dual-solenoid actuator where each solenoid provides an independent single direction force. An observer approach is proposed to estimate the position information. Unlike in [7] where both the electric and the mechanical ports of the actuator system are used to construct the unknown state vector, our solution only relies on the inductance characteristic of the two solenoids in a push-pull configuration. This results in a lower order and more robust observer, since it circumvents the uncertainty due to mechanical load on the system. The observer design is based on the electrical dynamics of the solenoids, and an nonlinear output equation which is furnished by the geometric constraint between the two solenoids. The rest of paper is organized as follows. In section 2, we formulate the model of a dual-solenoid actuator. Section 3 presents the principle and the analysis of the observer. In section 4, simulation is presented to verify the observer design. Some experimental results are presented in Section 5. Section 6 contains concluding remarks.

dual-solenoid actuator is modeled as [2] ˙ 1 = −R d + x λ1 + u1 λ β ˙ 2 = −R d − x λ2 + u2 λ β x˙ = v 1 v˙ = (F2 − F1 + F∆ ) m

where λ1 , λ2 are the flux linkages of the two solenoids, v is the velocity of the spool, m is the lumped mass of the spool and the armatures, u1 and u2 are the voltages across the coils, and R is the resistances for solenoid 1 and 2 since two solenoids are identical. The currents through two coils are related to the flux linkages by d +x 1 λ1 = λ1 β L1 (x) d −x 1 i2 = λ2 = λ2 β L2 (x) i1 =

MODELING OF A DUAL SOLENOID ACTUATOR The nonlinear model is developed based on the differential configuration of a dual-solenoid actuator, as depicted in Fig. 1. Two solenoids produce contraction forces F1 and F2 acting on the spool. For simplicity, assume that the two solenoids are identical. Then the electromagnetic forces are given by [8] [9] i21 β 2 (d + x)2

(1)

F2 =

i22 β 2 (d − x)2

(2)

(4)

Note that the inductances of coils are given by L1 (x) = β/(d + x) and L2 (x) = β/(d − x). For simplicity, we assume that the currents are biased and unidirectional, i.e., i1 > 0, i2 > 0. We assume that both the currents i1 , i2 and the voltages u1 , u2 are measurements.

2

F1 =

(3)

3 FLUX LINKAGE OBSERVER We will construct an observer for the flux linkages in a dualsolenoid actuator. Once the flux linkages are obtained, the spool position can be easily estimated. The required signals are the voltages u1 , u2 and the currents i1 , i2 . As we have mentioned, a full order observer can be constructed. Since v˙ equation in Eq. (3) is subject to unmodeled or poorly modeled forces, the estimated states obtained by an observer that estimate both the electrical (λ1 , λ2 ) and the the mechanical (v, x) states would intuitively have poor precision and robustness performance. However, notice that if we are able to correctly estimate the flux linkages, Eq. (4) shows that position information can be obtained from the estimated flux linkages and the measured currents. Therefore, our task is to find the reduced-order observer ˙ 1, λ ˙ 2 in Eq. (3). The that only involves the electrical dynamics λ mechanical dynamics will not be important any more. The principle of the observer is presented as follows. Eq. (4) provides a constraint between the inductances in the form of

where β and d are two parameters that are related to the number of turns, the geometries, magnetic permeability of the air and the iron core of the solenoids, i1 ,i2 are the currents through the coils, and x is the displacement of the spool. Define x = 0 as the spool is at the central position. Under the operating condition, the spool also experiences the flow induced forces that are well modeled in [6]. The flow induced forces, friction, spring forces, and other uncertainties, are denoted as F∆ . If we ignore the leakage flux linkages in the actuator and hysteresis effect, then the system dynamics of the

1 1 2d + = L1 (x) L2 (x) β 2

(5)


+u2-

+u1i1

d-x

d+x

i2

Spool

F1

F2 x Solenoid 2

Solenoid 1

Figure 1. Configuration of a dual-solenoid actuator in a directional valve. The spool also functions as a rigid connection of the armatures of two solenoids.

or i1 i2 2d + = λ1 λ2 β

I1 (t 0 , t k )

(6)

∆I1,0 (k )

I1 (t 0 , t k − N )

If we can get both the currents i1 , i2 and the flux linkages λ1 , λ2 , then the position information can be captured via the following equation x(i1 , i2 , λ1 , λ2 ) =

βi1 βi2 −d = d − λ1 λ2

∆I1,1 (k )

I1 (t 0 , t k −1 )

(7) t0

which is again derived from Eq. (4). According to our proposal, the currents i1 , i2 are obtained by measurement. Then the key issue is how to obtain λ1 , λ2 correctly without measurement. From Eqs. (3) (4), we can represent the flux linkages at ∀t > t0 with respect to the initial conditions λ1 (t0 ), λ2 (t0 ) as λ1 (t) = λ1 (t0 ) + I1 (t0 ,t) λ2 (t) = λ2 (t0 ) + I2 (t0 ,t)

Z b

Z

I2 (a, b) :=

a b

a

t k −1

tk ∆T

Figure 2. Least squares method is used to calculate the local initial condition λ1,0 (k) = λ1 (tk−N ),λ2,0 (k) = λ2 (tk−N ), for the horizon with the fixed width N∆T . See text for details.

(8)

In our algorithm, we attempt to update the initial conditions in a uniform interval. In Fig. 2, tk for k = 0, 1, 2, . . . are the update points. The update interval ∆T := tk − tk−1 is identical. In our algorithm, we define a horizon which consists of multiple update intervals, as can be seen in Fig. 2. For simplicity, a time horizon [tk−N ,tk ] is so chosen to possess the fixed width N∆T where N is a positive integer. In addition, it is worth mentioning that the initial conditions are defined to be the staring point of a time horizon. Given a horizon associated with tk , we can define the local variables for solenoid 1: the differential integral ∆I1, j (k) = I1 (t0 ,tk− j )−I1 (t0 ,tk−N ), the local initial condition λ1,0 (k) = λ1 (tk−N ), and the local current i1, j (k) = i1 (tk− j ) for j = 0, 1, . . . , N −1. Similarly, we define the variables for solenoid 2: ∆I2, j (k) = I2 (t0 ,tk− j ) − I2 (t0 ,tk−N ) ,λ2,0 (k) = λ2 (tk−N ), and i2, j (k) = i2 (tk− j ) for j = 0, 1, . . . , N − 1. Substituting the above variables at j = 0, 1, . . . , N − 1 (or at t = tk−N+1 , . . . ,tk−1 ,tk ) into Eq. (6), we have N nonlinear equa-

in which I1 (a, b) :=

N∆ T

tk−N

[u1 (τ) − i1 (τ)R] dτ

[u2 (τ) − i2 (τ)R] dτ.

Since u1 , u2 , i1 , i2 are known, so are I1 (t0 ,t), I2 (t0 ,t). The problem of finding λ1 , λ2 can be transformed into one in which we are required to solve the initial conditions λ1 (t0 ), λ2 (t0 ) . Theoretically, we can calculate λ1 (t0 ), λ2 (t0 ) by substituting Eq. (8) into Eq. (6) for two different time t = ta , and t = tb respectively, where ta 6= tb . However, this method is not robust since any disturbance in the open loop dynamic would cause the significant errors of I1 (t0 ,t), I2 (t0 ,t), hence the errors of the flux linkages. 3


0.1

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λ1 (H A)

tions

0.05

for j = 0, 1, . . . , N − 1

λ1 error (H A)

i1, j (k) i2, j (k) 2d + − =0 λ1,0 (k) + ∆I1, j (k) λ2,0 (k) + ∆I2, j (k) β (9)

0 0 0.05

λ2 (H A)

0.02

λ2 error (H A)

f ( j) = i1, j (k)[λ2,0 (k) + ∆I2, j (k)] + i2, j (k)[λ1,0 (k) + ∆I1, j (k)] 2d − [λ1,0 (k) + ∆I1, j (k)][λ2,0 (k) + ∆I2, j (k)] = 0 β for j = 0, 1, . . . , N − 1 (10)

0 0 0.02

−0.02 0

b j (k) := i1, j (k) − i1, j−1 (k) −

∆I1, j−1 (k)],

¸ λ1,0 (k) = [M(k)T M(k)]−1 M(k)T C(k) λ2,0 (k) 





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Simulation results for the flux linkages estimates. From top to

the following formulas to get the flux linkages. bλ1 (t) = λ1,0 (k) + ∆I1,0 (k) +

and c j (k) := 2d β [∆I1, j (k)∆I2, j (k) − ∆I1, j−1 (k)∆I2, j−1 (k)] + i1, j−1 (k)∆I2, j−1 (k) + i2, j−1 (k)∆I1, j−1 (k) − i1, j (k)∆I2, j (k) − i2, j (k)∆I1, j (k), for j = 1, . . . , N − 1. Notice that although Eq. (9) represent N nonlinear equations in the initial flux linkages λ1,0 (k), λ2,0 (k), Eq. (11) represent N − 1 equations that are linear. The least squares algorithm is then utilized to calculate the local initial conditions λ1,0 (k) and λ2,0 (k). In specific, we have ·

1

λ1 and the observer estimate bλ1 in Eq. (13) for solenoid 1; estimate error for solenoid 1 bλ1 − λ1 ; the actual flux linkage λ2 and the observer estimate b λ2 in Eq. (13) of the solenoid 2; estimate error for solenoid 2 b λ2 − λ2 .

a j (k)λ1,0 (k) + b j (k)λ2,0 (k) = c j (k) for j = 1, . . . , N − 1 (11) ∆I2, j−1 (k)],

0.8

bottom: The actual flux linkage

Subtracting f ( j) from f ( j − 1) in Eq. (10), for j = 1, . . . , N − 1, can eliminate the nonlinear terms, and give N − 1 equations

2d β [∆I2, j (k) − 2d β [∆I1, j (k) −

0.6

0

Figure 3.

where a j (k) := i2, j (k) − i2, j−1 (k) −

0.4

0

−0.05 0 0.04

Multiplying the above equation by [λ1,0 (k) + ∆I1, j (k)][λ2,0 (k) + ∆I2, j (k)] gives

0.2

bλ2 (t) = λ2,0 (k) + ∆I2,0 (k) +

Z t tk

Z t tk

(u1 − i1 R) dτ (u2 − i2 R) dτ

(13)

Notice that although the initial flux linkages are updated discretely, the flux linkage estimates are available at all times. The position estimate can then be obtained based on the average of the two formulas in Eq. (7) 1 xˆ = β

(12)

µ

i1 i2 − bλ1 bλ2

¶ (14)

Before we conclude our algorithm in this section, it is worth mentioning the initial behavior of the observer starting from t0 . Because we cannot compute the local initial conditions before a complete horizon is formed, our algorithm doesn’t provide the correct flux linkage information in the period [t0 ,t0 + N∆T ).



a1 (k) b1 (k) c1 (k)  a2 (k) b2 (k)   c2 (k)      where M(k) =  , and C(k) =  . .. .. ..     . . . aN−1 (k) bN−1 (k) cN−1 (k) In short, we can summarize our observer as follows. For t = tk , this is so called the update time point. All the local variables i1, j (k), i2, j (k), ∆I1, j−1 (k), ∆I2, j−1 (k) for j = 1, 2, . . . , N − 1, are refreshed. Eq. (12) are then utilized to obtain the local initial conditions λ1,0 (k), λ2,0 (k). Next, for ∀t ∈ (tk ,tk+1 ), we will use

4 SIMULATION The observer is tested by simulation in Matlab. First, we test the observer design when the parameters d, β are well known. The parameters used in the simulation are β = 2.64 × 4


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x 10

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0 0 0.05

λ2 (H A)

0.2 −3

1

Actual Estimated

λ1 (H A)

4

2

Figure 5. Simulation results for the flux linkages estimates in the case where the parameter perturbation is presented. From top to bottom: The

Figure 4. Simulation results for the position estimates. Upper: actual position x of the dual-solenoid actuator and the estimate xˆ by the observer in Eq. (14). Lower: the position error x − xˆ.

λ1 and the observer estimate bλ1 in Eq. (13) for solenoid 1; estimate error for solenoid 1 b λ1 − λ1 ; the actual flux linkage λ2 and the observer estimate bλ2 in Eq. (13) of the solenoid 2; estimate error for solenoid 2 b λ2 − λ2 . actual flux linkage

10−4 NA−2 m2 , d = 7.76 × 10−3 m. The spool is controlled to move in a sinusoidal manner with 3 × 10−3 m amplitude and 4Hz frequency. The observer is described in Section 3. We choose N = 4, and ∆T = 0.05s. The flux linkages of the solenoids and the estimates of the observer are shown in Fig. 3. Note that in the time period [0, 0.2], the errors between λ1 , λ2 and bλ1 , bλ2 are significant. As explained in Section 3, the errors are due to the fact that the horizon with the width of N × ∆T (= 0.2s) is yet formed. As t > 0.2s, the local variables and the local initial conditions in Section 3 can be computed, as well as the observer in Eq. (13). The flux linkage errors approach zero very quickly. In Fig 4, the actual displacement and the estimate from the observer are presented. The observer provides a very good prediction of position. In simulation, we will investigate the case where the solenoid parameters are not precisely identified. In the simulation, the observer is designed using the parameter βˆ = 2.4 × 10−4 NA−2 m2 , dˆ = 6.9 × 10−3 m, which are 10% less than the real value. As can be seen in Fig. 5, the flux linkage errors are apparently more significant than the case in Fig. 3, so does the position error in Fig. 6. Therefore, this algorithm is sensitive to the parameter estimate errors.

which the control input Vc and the sensing resistor R determine the current value over the solenoid (coil). In order to precisely measure the voltage and the current of the solenoid, two differential amplifiers are connected across the solenoid and a small sensing resistor Rs . The output signals Vv and Vi are then proportional to the voltage and current, respectively. The control and measurement are implemented in Simulink/XPC Target(Mathworks Inc., USA), a real time environment. The sample rate is set to be 1K Hz. In Fig. 7, we have R = 1.2Ω, Rs = 50mΩ, Rcoil = 3Ω, Vdd = 10V . The solenoid parameter values are identified to be β = 1.94×10−3 NA−2 m2 , d = 2.01×10−3 m. It is worth mentioning that the signal of Vv is very noisy due to the high frequency switching of the MOSFET M1. Therefore, A voltage-controlled voltage-source (VCVS) Butterworth filter is placed afterwards with fc = 1.34K Hz. In our experiment, two biased current commands, 0.6 + 0.4 sin(2πt) A and 0.6 + 0.4 sin(2πt + π) A, are used to excite each of the solenoid actuators. The motion of the spool is recorded via a linear variable displacement transducer (LVDT), as shown in Fig. 9. In addition, we set N = 10, ∆T = 0.004s. The observer in Eq. (13) is utilized to estimate the flux linkages as shown in Fig. 8, and accordingly the position of the spool. In Fig. 9, it can be seen that the prediction using self-sensing technique agrees well with the LVDT measurement, and the selfsensing concept is valid. We also investigate the case where the higher frequency ex-

5

EXPERIMENTAL RESULTS The experimental results are presented in this section. We use the standard solenoids in a commercial valve (KDFV5, Eaton Corporation) as the study object. The driving and measuring circuits are shown in Fig. 7. This consists of a current driver, in 5


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Figure 8. Figure 6. Simulation results for the position estimates in the case where the parameter perturbation is presented. Upper: actual position x of the dual-solenoid actuator and the estimate xˆ by the observer in Eq. (14).

1 Time (s)

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The estimated flux linkages for 1 Hz excitation using the ob-

server in Eq.

(13).

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Figure 9. The measured displacement using a LVDT, and the prediction from the observer for the 1Hz excitation frequency.

Figure 7. The schematic of electric circuit for driving the solenoids and measuring the voltage and current signals. The output signals Vv and Vi reflect the voltage across the solenoid and the current, respectively. See text for details.

6 CONCLUSION Self-sensing refers to extracting the position information from the electromagnetic signals instead of from a physical position sensor. Self-sensing actuator has the benefit of significantly reducing hardware effort. Considering the fact that LVDTs are widely used in electrohydraulic industry, applying self-sensing technology into electrohydraulics could save the product cost significantly. In this paper, we investigate self-sensing actuators applied in direct acing proportional valves, in which a dual-

citation is applied. In specific, 0.6 + 0.4 sin(8πt) A and 0.6 + 0.4 sin(8πt + π) A, are used to excite each of the solenoid actuators. As shown in Figs. 10, 11, the self-sensing technique again provide the satisfactory position signal. 6


1

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Figure 10. The estimated flux linkages for 4 Hz excitation using the observer in Eq. (13).

[6]

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actions on Instrumentation and Measurement, 46 (1) Feb. , pp. 45–50. Zhan, Y. J., Chan, C. C., and Chau, K. T., 1999. “A novel sliding-mode observer for indirect position sensing of switched reluctance motor drives”. IEEE Transactions on Industrial Electronics, 46 (2) April. , pp. 390–397. McCann, R. A., Islam, M. S., and Husain, I., 2001. “Application of a sliding mode observer for position and speed estimation in switched reluctance motor drives”. IEEE Transactions on Industrial Electronics, 37 (1) Jan/Feb , pp. 51–58. Krishnaswamy, K., and Li, P. Y., 2002. “On using unstable electrolydraulic valves for control”. Transactions of ASME. Journal of Dynamic Systems, Measurement and Control, 124 (1) March , pp. 182–190. Yuan, Q., and Li, P. Y., 2002. “Using viscosity to improve spool agility”. In Proceedings of the 2002 Power Transmission and Motion Control (PTMC), Bath, UK. Editors: C.R. Burrows, K.A. Edge, pp 263-286, Professional Engineering Publishing LTD., London, UK, 2002. Yuan, Q., and Li, P. Y., 2003. “Modeling and experimental study of flow forces for unstable valve design”. In Proceedings of the 2003 ASME international mechanical engineering congress, Washington DC., USA, no. IMECE200342924. Vischer, D., and bleuler, H., 1993. “Self-sensing active magnetic levitation”. IEEE Transactions on Magnetics, 29 (2) march . Xu, Y., and Jones, B., 1997. “A simple means of predicting the dynamic response of electromagnetic actuators”. Mechatronics, 7 (7) , pp. 589–598. Koch, C. R., Lynch, A. F., and Chladny, B. R., 2002. “Modeling and control of solenoid valves for internal combustion engines”. International Federation of Automatic Control, Berkeley, California, USA Dec 9-11 , pp. 213–218.

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Figure 11. The measured displacement using a LVDT, and the prediction from the observer ( for the 4Hz excitation frequency.

solenoid actuator is used to provide bidirectional forces. An observer is proposed to estimate the flux linkages in the solenoids and from which the spool position is obtained. The self-sensing concept has been verified both in simulation and in experiment.

REFERENCES [1] Noh, M. D., and Maslen, E. H., 1997. “Self-sensing magnetic bearings using parameter estimation”. IEEE Trans7
