August 21, 2005 3:10

Proceedings of IMECE’052005 ASME International Mechanical Engineering Congress and RD&D Expo

November 5-11, 2005, Orlando, Florida USA

IMECE2005-80284

ON SELF-SENSING ACTUATORS FOR ELECTROHYDRAULIC VALVES:COMPARISONS BETWEEN BOXCAR WINDOW OBSERVER AND KALMAN FILTER

Qinghui YuanEaton Corp. Innovation Center

Eden Prairie, MN 55344Email: QinghuiYuan@eaton.com

Perry Y. LiDepartment of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455Email: pli@me.umn.edu

ABSTRACTSelf-sensing refers to extracting the position information

from the electromagnetic signals instead of from a physical posi-tion sensor. Self-sensing actuator has the benefit of significantlyreducing hardware effort. In this paper, we investigate two typesof observers: Boxcar Window Observer and Kalman Filter. Theperformance of both observers has been compared in simulationand experimentally.

1 IntroductionSelf-sensing refers to extracting the position information

from the electromagnetic signals of an actuator, like currentsand voltages, instead of from a physical position sensor such asLVDT. This technique has attracted the attention of researchersin some fields. One example is self-sensing magnetic bearing,which is useful for minimizing the number of wires in some ap-plications, like heart pump [1]. Another example is self-sensingswitched reluctance motor, which is the cost effective and robustsolution for electric motors [2] [3].

Similarly, self-sensing actuators can be applied in fluidpower control engineering. In the electrohydralic valves, spoolposition feedback, usually furnished by a linear variable differen-tial transformer (LVDT), is needed for higher performance con-trol. Incorporating LVDTs into electrohydraulic valves incursthe cost of the LVDTs themselves, the electric circuits for excit-ing and decoding the LVDTs, and the overhead of assembling theLVDTs. Therefore, the concept of self-sensing actuators can bevery useful to reduce hardware requirement of electrohydraulic

valves at the expense of the increased software complexity. Al-though software development is expensive, the overall cost canbe significantly reduced when the product is produced in largequantities.

In this paper, we investigate self-sensing actuators appliedto direct acting proportional valves. The study is based on theconfiguration of a dual-solenoid actuator where each solenoidprovides an independent single direction force. An observer ap-proach is proposed to estimate the position information. Unlikein [4] where both the electric and the mechanical ports of theactuator system are used to construct the unknown state vector,our solution only relies on the inductance characteristic of thetwo solenoids in a push-pull configuration. This results in lowerorder and more robust observers, since they circumvent the un-certainty due to mechanical load on the system. The main diffi-culty in the observer design is that the output equation of the ob-server is highly nonlinear. An exact linearizing approach basedon multiple time samples and a Boxcar filter [5] and a Kalmanfilter [6] were proposed. However, experimental evaluation andcomparisons have not been made. The objective of this paper isto compare the result of two different methods. It is shown thatthe Kalman filter offers the smoother estimate.

A simple solenoid model is used to model how the solenoidinductance varies with the actuator displacement, which is foun-dational to the self-sensing principle. The model neglects hys-teresis, saturation, and a number of details of model which maybe very significant for the solenoids in electrohydraulics. A self-calibration procedure has been proposed [7]. This approach isused in identifying the parameters experimentally in this paper.

1 Copyright c© 2005 by ASME

Our approach that relies on simple solenoid models differ fromother self-sensing approach [8] that uses much more complicatedsolenoid models.

The rest of the paper is organized as follows. In section 2,we formulate the model of a dual-solenoid actuator. Section 3reviews the principle and the analysis of two types of observers:the Boxcar Window observer and the Kalman Filter in [5] and[6], respectively. In section 4, simulation is presented to verifythe observer design. Some experimental results are presented inSection 5. Section 6 contains concluding remarks.

2 Modeling of a dual solenoid actuatorConsider two identical solenoid (with subscriptsi = 1,2) are

set up in the push-pull configuration as in Fig. 1. Let the inputvoltages across the coils beu1 andu2, and the resulting currentsthrough the coils bei1 andi2. The dynamics of the flux linkagesλ1,λ2 are given by

λ1 = −Ri1 +u1

λ2 = −Ri2 +u2 (1)

whereR is the resistance of solenoid 1 and 2. By definition, theflux linkages are related to the currents via the inductances:

λ1 = L1(x)i1λ2 = L2(x)i2 (2)

whereL1(x),L2(x) are the inductances of solenoid 1 and 2 asso-ciated with the spool displacementx by

L1(x) =β

d+x

L2(x) =β

d−x(3)

where β,d are the parameters of the model, in which bothsolenoids are assumed to be identical. This meets most of casein ElectroHydraulics. Note that the pulling forcesF1 andF2 (seeFig. 1) acting on the spool share the identical parameters toL1,L2

F1 =β2

i21(d+x)2

F2 =β2

i22(d−x)2 (4)

Figure 1. Configuration of a dual-solenoid actuator in a directional valve.

In the above analysis, saturation and hysteresis of thesolenoids are not taken into account. Therefore, the inductanceof the coil can be simplified as a function of the spool displace-ment. If saturation is considered, then the inductance would be afunction of both the spool displacement and the current. If hys-teresis is, then the inductance has to be modeled according to theB−H curve, which is very complicated. In most of the high pro-file commercial valves, the solenoid is designed to work in thelinear region with little hysteresis. In this case, the models arevalid.

3 Flux linkage observerWe will construct an observer for the flux linkages in a dual-

solenoid actuator. Once the flux linkages are obtained, the spoolposition can be easily estimated. The required signals are thevoltagesu1,u2 and the currentsi1, i2. As we have mentioned,a full order observer with the states(λ1,λ2,x,v) can be con-structed, where the flux linkagesλ1,λ2 are the electrical sig-nals, and the displacement and velocity(x,v) are the mechan-ical signals. Sincev is subject to unmodeled or poorly mod-eled forces, the estimated states obtained by an observer that es-timate both the electrical(λ1,λ2) and the the mechanical(v,x)states would intuitively have poor precision and robustness per-formance. However, notice that if we are able to correctly es-timate the flux linkages, position information can be obtainedfrom the estimated flux linkages and the measured currents, asdiscussed next. Therefore, our task is to find the reduced-orderobserver that only involves the electrical dynamicsλ1, λ2 in Eq.(1). The mechanical dynamics will not be important any more.

3.1 PrincipleThe principle of the observer is presented as follows. If the

inductancesL1,L2 are known, then from Eq. (3), the displace-mentx is given by

x(L1,L2) =β2

(1L1− 1

L2

)

2 Copyright c© 2005 by ASME

Unfortunately, the inductance values cannot be directly mea-sured. We would like to find an alternative way to estimate in-ductances, and accordingly the displacement. Note that the in-ductance is associated with the corresponding flux linkage andthe current. Therefore, our objective becomes to estimate theflux linkages, provided that the currents are measured. Then theposition information can be captured via the following equation

x(λ1,λ2) =β2

(i1λ1− i2

λ2

)(5)

wherei1, i2 are the currents, andλ1,λ2 are the flux linkages.In the section to follow, we will derive an observer to re-

construct the information of the flux linkages. In the observerdesign procedure, we use the following constraint between theinductances (derived from Eq. (3))

1L1(x)

+1

L2(x)=

2dβ

(6)

or

i1λ1

+i2λ2

=2dβ

(7)

Eq. (7) serves as an output equation to the dynamics for the ob-server design. Notice thatλ1 andλ2, which are to be estimated,appear nonlinearly.

3.2 Boxcar Window ObserverDue to nonlinearity of Eq. (7), we cannot easily design the

observer. To overcome the difficulty, we define a moving Boxcarwindow in time domain, and solve the initial conditions of fluxlinkages in the wondow. Reader are referred to [5] for details ofthe Boxcar window observer.

From Eqs. (1), we can represent the flux linkages at∀t > t0with respect to the initial conditionsλ1(t0),λ2(t0) as

λ1(t) = λ1(t0)+ I1(t0, t)λ2(t) = λ2(t0)+ I2(t0, t) (8)

in which

I1(a,b) :=∫ b

a[u1(τ)− i1(τ)R] dτ (9)

I2(a,b) :=∫ b

a[u2(τ)− i2(τ)R] dτ.

kt1−ktNkt −0t

),( 01 NkttI −

),( 01 ttI

1,1I∆

0,1I∆

2−kt

2,1I∆

NI ,1∆

),( 01 kttI

),( 101 +−NkttI

),( 101 −kttI

),( 201 −kttI

1+−Nkt

Figure 2. Least squares method is used to calculate the local initial con-

dition 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.

We assume thatR is insensitive to the temperature variation, andcan be measured precisely. Sincet, u1(t),u2(t), i1(t), i2(t) areknown (via measurement) at allt, so areI1(t0, t), I2(t0, t). Theproblem of findingλ1,λ2 can be transformed into one in whichthe initial conditionsλ1(t0),λ2(t0) are required to be solved.

Theoretically, we can evaluate Eq. (8) att = ta, andt = tb(ta 6= tb) respectively, and then combine them with (7)

i1λ1(t0)+ I1(t0, ta)

+i2

λ2(t0)+ I2(t0, ta)=

2dβ

i1λ1(t0)+ I1(t0, tb)

+i2

λ2(t0)+ I2(t0, tb)=

2dβ

two unknown variablesλ1(t0),λ2(t0) can be obtained from theabove equations. However, this method is not robust since theerrors of I1(t0, t), I2(t0, t), and henceλ1,λ2, would accumulatequickly due to the disturbance in the open loop dynamics.

In order to avoid the error accumulation, a Boxcar windowis defined in the time history, and the initial conditions of theflux linkages are solved for the particular window. The windowmoves uniformly, therefore we will have the uniformly updatedinitial conditions. The details are explained as follows.

Define a update interval∆T > 0, and tk = k∆T for k =0,1,2,3, . . ., as shown in Fig. 2. In our algorithm, a boxcar win-dow is defined to cover a time horizon[tk−N, tk] whereN is apositive integer. In other words, the horizon has the widthN∆T .Given a horizon associated withtk, we can define the local vari-ables

∆I1, j(k) = I1(tk−N, tk− j) = I1(t0, tk− j)− I1(t0, tk−N)λ1,0(k) = λ1(tk−N)

i1, j(k) = i1(tk− j)∆I2, j(k) = I2(tk−N, tk− j) = I2(t0, tk− j)− I2(t0, tk−N)

λ2,0(k) = λ2(tk−N)i2, j(k) = i2(tk− j)

3 Copyright c© 2005 by ASME

for j = 0,1, . . . ,N−1, whereI1(·, ·), I2(·, ·) refer to Eq. (9).Combining the above variables atj = 0,1, . . . ,N−1 (or at

t = tk, tk−1, . . . , tk−N+1) with Eq. (7) yieldsN nonlinear equations

i1, j(k)λ1,0(k)+∆I1, j(k)

+i2, j(k)

λ2,0(k)+∆I2, j(k)− 2d

β= 0

for j = 0,1, . . . ,N−1 (10)

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

f ( j) = i1, j(k)[λ2,0(k)+∆I2, j(k)]+ i2, j(k)[λ1,0(k)+∆I1, j(k)]

−2dβ

[λ1,0(k)+∆I1, j(k)][λ2,0(k)+∆I2, j(k)] = 0

for j = 0,1, . . . ,N−1(11)

Subtracting f ( j − 1) from f ( j) in Eq. (11), for j =1, . . . ,N− 1, can eliminate the nonlinear terms, and giveN− 1equations

a j(k)λ1,0(k)+b j(k)λ2,0(k) = c j(k) for j = 1, . . . ,N−1 (12)

where

a j(k) :=i2, j(k)− i2, j−1(k)− 2dβ

[∆I2, j(k)−∆I2, j−1(k)]

b j(k) :=i1, j(k)− i1, j−1(k)− 2dβ

[∆I1, j(k)−∆I1, j−1(k)]

c j(k) :=2dβ

[∆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)(13)

for j = 1, . . . ,N−1.Notice that although Eq. (10) represent N nonlinear equa-

tions in the initial flux linkagesλ1,0(k),λ2,0(k), Eq. (12) repre-sentN−1 equations that are linear. The least squares algorithmis then utilized to calculate the local initial conditionsλ1,0(k) andλ2,0(k) associated with the boxcar window[tk−N, tk]

[λ1,0(k)λ2,0(k)

]= [M(k)TM(k)]−1M(k)TC(k) (14)

whereM(k) =

a1(k) b1(k)a2(k) b2(k)

......

aN−1(k) bN−1(k)

, andC(k) =

c1(k)c2(k)

...cN−1(k)

.

In short, we can summarize our observer as follows. Fort = tk, this is so called the update time point. All the local vari-ablesi1, j(k), i2, j(k),∆I1, j−1(k),∆I2, j−1(k) for j = 1,2, . . . ,N−1,are refreshed. Eq. (14) are then utilized to obtain the local initialconditionsλ1,0(k),λ2,0(k). Next, for∀t ∈ (tk, tk+1), we will usethe following formulas to get the flux linkages.

λ1(t) = λ1,0(k)+∆I1,0(k)+∫ t

tk(u1− i1R) dτ

λ2(t) = λ2,0(k)+∆I2,0(k)+∫ t

tk(u2− i2R) dτ (15)

Notice that although the initial flux linkages are updated dis-cretely, the flux linkage estimates are available at all times. Theposition estimate can then be obtained via

[i1/λ1

i2/λ2

]=

[d/βd/β

]+

[1/β−1/β

]x (16)

from Eq. (3) using the least squares method.Before we conclude our algorithm in this section, it is worth

mentioning the initial behavior of the observer starting fromt0.Because we cannot compute the local initial conditions before acomplete horizon is formed, the Boxcar window observer cannotprovide the correct flux linkage information in the period[t0, t0+N∆T).

3.3 Kalman FilterIn the Boxcar Window observer design (section 3.2), we

transform the flux linkage estimate problem into one of solvingthe initial conditions. However, from Eq. (1), it is also clear thatwe may use the Kalman Filter method if a term both measuredand estimated can be found so that we may construct an outputinjection for the Kalman filter.

First, an output equation is developed. Again using Eqs. (8)(9), the inductance constraint in Eq. (7) can be rewritten as:

[βi2(t)−2dI2(t, t0)]λ1(t)+ [βi1(t)−2dI1(t, t0)]λ2(t)+2dI1(t, t0)I2(t, t0) = 2dλ1(t0)λ2(t0) (17)

where Eq. (8) has been used repeatedly. The left side of Eq. (17)is linear in(λ1,λ2). The right hand side of Eq. (17), althoughnonlinear in the unknown initial states(λ1(t0),λ2(t0)), is not afunction of timet. The unknown initial states can be eliminatedvia taking the difference of Eq. (17) evaluated att and att0:

[βi2(t)−2dI2(t, t0)]λ1(t)+ [βi1(t)−2dI1(t, t0)]λ2(t)+2dI1(t, t0)I2(t, t0)−βi2(t0)λ1(t0)−βi1(t0)λ2(t0) = 0

4 Copyright c© 2005 by ASME

Then substitute(λ1(t),λ2(t)) for (λ1(t0),λ2(t0)) using Eq. (8):

β[i2(t)− i2(t0)]−2dI2(t, t0)λ1(t)+β[i1(t)− i1(t0)]−2dI1(t, t0)λ2(t)

=−2dI1(t, t0)I2(t, t0)−βi2(t0)I1(t, t0)−βi1(t0)I2(t, t0)(18)

Eq. (18) can be interpreted as the output equation for the fluxlinkages(λ1,λ2) in Eq. (1). The left side of Eq. (18) is the linearcombination of the flux linkages, with the right side furnishingthe ”measurement”.

Similar to the case in Section 3.2, the error may accumulatein I1(t, t0), I2(t, t0). We may restrict the computation in a (finite)boxcar horizon right before the current timet, thus having aneffect of forgetting the the past currents and voltages. We cantaket0 = t−T whereT is a fixed time interval. We denote theoutput ”measurement” by

y(t) =−2dI1(t, t−T)I2(t, t−T)−βi2(t−T)I1(t, t−T) (19)

−βi1(t−T)I2(t, t−T)

The dynamics of the flux linkages Eq. (1), the linear outputequation Eq. (18), and the measurement Eq. (19) are in a stan-dard time varying linear state space form. Thus we can design aKalman Filter [9] to reconstruct the unknown statesλ1(t),λ2(t):

ddt

[λ1

λ2

]=

[−Ri1 +u1

−Ri2 +u2

]+G(t)[y(t)−y(t)] (20)

y(t) = C(t)

[λ1

λ2

]

where

C(t) =[

β[i2(t)− i2(t−T)]−2dI2(t, t−T)β[i1(t)− i1(t−T)]−2dI1(t, t−T)

]

And the Kalman Filter gainG(t) is obtained by solving the Ric-cati Differential Equation (RDE):

P(t) =−P(t)CT(t)V−1C(t)P(t)+W

G(t) = P(t)C(t)TV−1

whereP(t) ∈R2×2 is covariance matrix of the states,V ∈R,W ∈R2×2 are the spectrum density of the measurement noise and pro-cess noises.

The stability of the Kalman Filter ensures that in the absenceof model uncertainty, process noises or measurement noise, wehaveλ1(t)→ λ1(t), λ2(t)→ λ2(t).

0 1 2 3 4 5 6

0.8

1

1.2

1.4

Vol

tage

s (V

olt)

Voltage 1Voltage 2

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

Time (sec)

Cur

rent

s (A

mp)

Current 1Current 2

Figure 3. Simulation results: input voltages (with process noise) and

output currents (with measurement noise) of solenoids .

4 Simulation study of self-sensing actuatorBoth observers are tested by simulation in Matlab/Simulink.

The parameters used in the simulation areβ = 2.64×10−4H−m,d = 7.76×10−3m, andR= 0.5Ω. The spool dynamics are givenby a mass-spring-damper (0.01Kg,2029N/m,6N− s/m) systemwith sufficiently large spring constant so that the valve is openloop stable. System nonlinearity is neglected in simulation. Thespool is commanded to move according to a0.1Hz to 3Hz chirpsignal in a range of[0,6] sec. The input voltages are corrupted bythe processing noise, while the currents are disturbed by the mea-surement noise (Fig. 4). Both types of noises are implementedusing ”Band Limited White Noise” block in simulink. For theBoxcar Window observer, we chooseN = 4, and∆T = 0.05sec.For the Kalman Filter, we chooseT = 0.02sec, V = 2e5, andW = diag(2e5,2e5).

First, we assume the parametersd,β are well known. Thesimulation results for the Boxcar Window Observer can be seenin Figs. 4-5. Note that in the beginning, the estimation errorsare significant. This is due to the Boxcar window can be onlyformed after a certain time. Then the estimated flux linkagesapproach the actual one very quickly. A property of this typeof observer is that the estimates may be discontinuous becausethe local initial conditions are designed to be updating discretely.

Similarly, as can be seen in Figs 6- 7, the Kalman Filter iseffective in estimating the unknown flux linkages and the spooldisplacement.

Next, we assume that the solenoid parameters are not pre-cisely identified. In the simulation, both of the observers areimplemented using the parameterβ = 2.64× 10−4H −m, d =6.9×10−3m. In other words,d is 10% less than the real valued. As can be seen in Fig. 8, the displacement estimation errors,

5 Copyright c© 2005 by ASME

0 1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

Time (sec)

λ 1 and

est

imat

e (W

b−se

c)

actualestimated

0 1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

Time (sec)

λ 2 and

est

imat

e (W

b−se

c)

actualestimated

Figure 4. Simulation results for the Boxcar Window Observer: the actual

and estimated flux linkages .

0 1 2 3 4 5 6−5

0

5x 10

−3

Time (sec)

Dis

plac

emen

t (m

)

ActualEstimate

0 1 2 3 4 5 6−2

−1

0

1

2x 10

−3

Time (sec)

x es

timat

e er

ror

(m)

Figure 5. Simulation results for the Boxcar Window Observer. Top: the

actual and estimated displacement; Bottom: the displacement estimation

error.

0 1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

Time (sec)

λ 1 and

est

imat

e (W

b−se

c)

actualestimated

0 1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

Time (sec)

λ 2 and

est

imat

e (W

b−se

c)

actualestimated

Figure 6. Simulation results for the Kalman Filter: the actual and esti-

mated flux linkages .

0 1 2 3 4 5 6−5

0

5x 10

−3

Time (sec)

Dis

plac

emen

t (m

)

ActualEstimate

0 1 2 3 4 5 6−2

−1

0

1

2x 10

−3

Time (sec)

x es

timat

e er

ror

(m)

Figure 7. Simulation results for the Kalman Filter. Top: the actual and

estimated displacement; Bottom: the displacement estimation error.

0 1 2 3 4 5 6−5

0

5x 10

−3

Time (sec)

Dis

plac

emen

t (m

)ActualBoxcarKalman

0 1 2 3 4 5 6−2

−1

0

1

2x 10

−3

Time (sec)

x es

timat

e er

ror

(m) Boxcar

Kalman

Figure 8. Simulation results for the case where the solenoid parameters

are not well known (d = 90%d). Top: the actual and estimated displace-

ment by both observers; Bottom: the estimation error for both observers.

either by the Boxcar Window Observer or by the Kalman Filter,are more significant than those in Figs. 5 and 7. However, theresults are still good.

5 Experimental resultsThe experimental study has been conducted as well. We use

xPC Target prototyping system that enables us to execute Mat-lab/Simulink models with physical systems in realtime. The di-agram of the experimental setup can be seen in Fig. 9. Thehost computer is a Laptop (T40, IBM), while the target com-puter is a desktop (Dimension 2400, Dell). The TCP/IP com-munication is chosen between the host and the target system forthe fast program-uploading and data-downloading. On the targetcomputer are installed two I/O boards: CIO-DDA06 (Comput-

6 Copyright c© 2005 by ASME

!

"

Figure 9. The diagram of the experimental apparatus.

Figure 10. 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.

erboard, US) and AD512 (Humusoft, Czech). They total have 6channels of DA and 8 channels of AD. We also build up a elec-tric circuit on the breadboard to drive the solenoids and to mea-sure the voltages and currents. The details of the electric circuitwill be presented later. A commercial proportional valve (D1F,Parker Hannifin Corp.), with two solenoids at each end, is uti-lized in our experimental study. The LVDT in the valve providesus with the actual spool displacement that can be used to verifythe observer estimates.

Next we will introduce the electric circuit, whose function-ality consists of actuation and measurement. The left side ofFig. 10 (including an Op amp, a MOSFET M1, a diode D1, acoil/solenoid), represents a current driver circuit that is used tofeed the desired current value over the solenoid (coil) accordingto control inputVc and the sensing resistorRs. The actual currentthough the coil can be measured by the sensing resistorRs (Vi).The right side of Fig. 10 is for measuring the voltage across the

coil. First, the voltage across the coil is fed into a differentialamplifier (upper right). The signal out of the differential am-plifier is very noisy due to the high frequency switching of theMOSFET M1. Therefore, a voltage-controlled voltage-source(VCVS) Butterworth filter is placed afterwards (lower right). InFig. 10, we setVdd = 10V, Rs = 1Ω, Rcoil = 6.2Ω, R1 = 54kΩ,R2 = 28kΩ, R3 = 54kΩ, C = 2.2nF, so the cutoff frequency ofthe low pass active filter isfc = 1

2πR3C = 1.34K Hz. The sampletime of the xPC Target system is set to beTs = 1m sec.

In Fig. 10, the output(Vv,Vi) are the measurements, i.e.,Vi

is for the current, whileVv is for the voltage. Since we have twosolenoids, two independent circuits are built accordingly, withthe outputs(Vv1,Vi1) and(Vv2,Vi2). In the experiment, we assumethat the current measurements are precise. The output signals canbe so calibrated that the actual voltages are

u1 = 5.70Vv1−4.44, i1 = Vi1

u2 = 6.17Vv2−4.91, i2 = Vv2

for solenoid 1 and 2, respectively.The calibration of the solenoid parameters will be presented

as follows. A self-calibration method in [7] is used to identifythe model parameters. The basic idea is that the parameters ofthe solenoid could be retrieved from the limited knowledge ofthe displacement. For example, using two inexpensive switchsensors to detect the end stop displacement (please see [7] formore detail). Since our valve is equipped with a LVDT, the end-stroke information is obtained from the LVDT. Fig. 11 illustratesthe iteration procedure of the Newton method. We arbitrarilychoose the initial guess of the parameters to beβ0 = 1×10−3H−m, d0 = 0.03m. The solution converges toβ = 1.2× 10−6H −m, d = 1.3×10−3m at the163rd iteration step. These values areused to to design the observers. Note that in reality,d,β do nothave to be the same for both solenoids.

In our experiment, we design a PI controller to implementthe spool displacement tracking. The controller output is con-nected to theVc port in Fig. 10. The proportional gain andintegral gain are tuned so that the system behaves mildly. Wewill compare two estimation methods, the Boxcar Window Ob-server and the Kalman Filter, with the actual spool displacementis measured by the LVDT. We setN = 4,∆T = 0.01secfor theBoxcar Window Observer, whileT = 0.02sec for the KalmanFilter. In Fig. 12, it can be seen that the prediction using self-sensing technique agrees well with the LVDT measurement, andthe self-sensing concept is valid. It is also observed that thereare some spikes in the the Boxcar window Observer estimatesdue to the discrete correction of the initial states. In addition,there is less steady state error (see the time period when the dis-placement is at a standstill in Fig. 12) for the Kalman filter thanfor the Boxcar Window observer. Larger errors in the transientprocedure could be probably due to the simple solenoid model

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0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1x 10

−3

β^ (

H−

m)

Iteration steps

0 20 40 60 80 100 120 140 160 1800

0.01

0.02

0.03

0.04

d^ (m

)

Iteration steps

Figure 11. Self-calibration using experimental results: Estimated param-

eters β, d.

0 1 2 3 4 5 6−5

−4

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−2

−1

0

1

2

3

4

5x 10

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0 1 2 3 4 5 6−5

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−2

−1

0

1

2

3

4

5x 10

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Time (sec)

Disp

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m)

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Figure 12. Experimental results for the Kalman Filter (upper) and the

Boxcar Filter (lower).

we utilized, which does not match the actual solenoids installedin the Electrohydraulic valves. Therefore, the actual inductance-displacement curve passing through two points may be expressedother than β

d±x.

The self-sensing algorithm does not consider thermal effect.The experimental process lasts in the order of seconds, therefore,the temperature, and accordingly the resistance, can be assumedto be constant.

6 ConclusionSelf-sensing refers to extracting the position information

from the electromagnetic signals instead of from a physical posi-tion sensor. Self-sensing actuator has the benefit of significantlyreducing hardware effort. In this paper, two types of observers,the Boxcar window observer and the Kalman Filter, have beencompared both in simulation and experimentally. It is shown thatthe Kalman Filter has smoother estimate and smaller steady error.Finally, it is also worth mentioning that due to lack of details ofthe simple solenoid model we used in algorithm, both observersmay be sensitive to the model structure or the model parameters.In that case, more care needs to be taken to identify the modelparameters precisely, or the observers based on the augmentedmodel could be redesigned based on the current observers.

Acknowledgement:The paper is based on work supportedby the National Science Foundation under grant ENG/CMS-0088964.

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[5] Yuan, Q., and Li, P. Y., 2004. “Self-sensing actuators in elec-trohydraulic valves”. In ASME Fluid System TechnologyDivision Publication - 2004 International Mechanical Engi-neering Congress, Anaheim, CA, no. IMECE2004-62104.

[6] Li, P. Y., and Yuan, Q., 2005. “Flux observer for spool dis-placement sensing in self-sensing push-pull solenoids”. Inthe Sixth International Conference on Fluid Power Transmis-sion and Control (ICFP’ 2005), Hangzhou, China,.

[7] Yuan, Q., and Li, P. Y., 2004. “Self-calibration of push pullsolenoid actuators in electrohydraulic valves”. In ASMEFluid System Technology Division Publication - 2004 Inter-national Mechanical Engineering Congress, Anaheim, CA,no. IMECE2004-62109.

[8] Eyabi, P., 2003. Modeling and Sensorless Control ofSolenoidal Actuators. Phd thesis, The Ohio State University.

[9] Bryson, A. E., 2002. Applied Linear Optimal Control:Exaples and Algorithms. Cambridge university press.

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