Self-Sensing Dual Push-Pull Solenoids using a Finite Dimension Flux-observer

Perry Y. Li

Abstract— Position feedback in a solenoid actuated systemtypically requires a position sensing device such as a LVDT.The goal of self-sensing is to obtain position information directlyfrom the electrical signals to the solenoid actuators, thus obvi-ating the additional cost and footprint of a LVDT or anotherdisplacement sensing device. Such measurement is possibledue to the position dependence of electrical inductance in thesolenoids. This paper proposes a finite-dimensional nonlinearobserver for the magnetic flux linkage for the solenoids. Oncethe flux linkage has been identified, the solenoid position canbe determined via the position-inductance relationship. Thealgorithm has been adapted for actual solenoids modeled asa third-order system that includes two eddy current modesaccurate up to 1024 Hz. Implementation on commerciallylow-cost solenoids (with 5mm stroke) has demonstrated RMSposition accuracy up to 0.061mm. The ability to self-senseaccurately can enable solenoids to be deployed at low-cost formany motion control applications besides hydraulic valves.

I. INTRODUCTION

Solenoids are low-cost actuators but position feedback isrequired to utilize them for position control. The cost forposition sensors such as LVDT, however, can be prohibitive.Self-sensing refers to extracting position information fromthe inherent signals like currents and voltages. This techniquehas attracted the attention of researchers in fields such asmagnetic bearing for heart pumps in order to minimize thenumber of wires [1], or self-sensing switched reluctance mo-tor, which is the cost effective and robust solution for electricmotors[2], [4]. Self-sensing solenoids will also be useful toreduce the cost of high performance electro-hydraulic valveswhich are typically solenoid actuated but currently use LDVTfor spool position feedback.

In this paper, we focus on dual solenoid actuators in thepush-pull configuration in which the solenoids are mechani-cally connected together (see Fig. 1). As one solenoid retractsthe other extends. Since solenoids produce only force in onedirection, the alternative is a single solenoid with a springreturn. This requires only one solenoid but has a lower forcecapability and needs a bias current overcome the spring forceso as to stay in place.

A finite dimensional nonlinear flux linkage observer isproposed to estimate the position information of a pairof push-pull solenoids. This is based upon the positiondependent inductance characteristic of the solenoids. Theobserver design combines the inductance information fromboth solenoids to form a nonlinear time varying outputequation. Once the flux is estimated, the position can beobtained from the position-inductance relationships.

P. Y. Li is with the Department of Mechanical Engineering, Universityof Minnesota, MN 55455 USA. E-mail: lixxx099@umn.edu.

Fig. 1. Two low-cost solenoids in push-pull configuration. The stroke ofthis experimental setup is 5 mm. The LVDT at the top of the setup is usedfor measuring the actual displacement.

Our earlier approaches to self-sensing in [6], [7] alsofocused on the dual solenoids in push-pull configuration. Theapproach in [6] explicitly solves for the initial flux linkageover a finite moving time horizon and utilizes several timedelayed (hence infinite dimensional) filters. Consequently,it is rather complicated and is susceptible to singularity.The approach in [7] advances the self-sensing problem asan observer problem but still relies on an infinite dimen-sional boxcar filter. In the present paper, a nonlinear time-varying output equation is formulated which allows us tocast the problem as a finite dimensional observer problem.The Kresselmeier estimator [11] originally developed forestimating constant unknown parameters is adapted for useas an observer. The observer has some unique properties suchan forgetting factor and direct access to the observer errorthat can be useful.

Other self-sensing approaches for solenoids include [8]which uses the position dependent phase information at aresonant frequency signal to detect position. In [9], theposition dependence of the rate of current rise with PWMinput is used to estimate position. In both cases, the specificinput wave-forms are assumed and the detection algorithmsdo not account for the motion induced variation in thecurrent. In [10], a nonlinear observer is designed for asolenoid system that includes eddy current dynamics. Bothmechanical and electrical aspects are used in the design. Incontrast to these works, our proposed flux observer allows forarbitrary voltage waveforms and fully takes into account themotion induced current variation. In addition, the flux linkageobserver primarily utilizes the electrical characteristics anddoes not require a model of the mechanical system.

The proposed flux observer is experimentally tested on

Fig. 2. Idealize first order L-R circuit model for a solenoid.

a pair of low cost solenoids. System identification revealsthe solenoids to be 3rd order systems which incorporatetwo modes of eddy current dynamics. To apply the fluxobserver, the 3rd order solenoid system is first converted intoa first order inductance model by applying a pre-feedbackto compensate for the eddy current effect. RMS positionestimation accuracy of 0.061 mm (over a stroke of 5mm)has been achieved.

The rest of paper is organized as follows. In section II, weformulate the model of a dual-solenoid actuator. Section IIIpresents the principle and the analysis of the observer. Someexperimental results are presented in Section ??. Section VIcontains concluding remarks.

II. SYSTEM MODELS

A. Ideal dynamics

We consider two ideal solenoids connected in an opposingpush-pull manner (Fig. 1). Suppose x = 0 corresponds tothe actuator being at one extreme position. The positiondependent inductances of the two solenoids are modeled as:

L1(x) =�1

d1 + x� L1, L2(x) =

�2

d2 � x� L2 (1)

Let V1, V2 be the voltage inputs and i1, i2 be the currentsgoing through the two inductances. The flux linkages for thetwo solenoids are given by:

�1 = L1(x)i1, �2 = L2(x)i2 (2)

An idealized model of a solenoid is a L-R circuit (Fig. 2).Therefore, the dynamics for the ideal push-pull solenoids aretherefore:

�1 = �Rs1 i1 + V1

�2 = �Rs2 i2 + V2

(3)

Assuming that V1, V2 and i1 and i2 can be measured, ourgoal is to estimate the flux linkages �1(t) and �2(t), and thenfrom using the measured currents i1(t), i2(t) and inductancemodels L1(x), L2(x) in (1) to determine position x(t). Notethat this enterprise does not rely on models of the mechanicalaspect of the solenoids.

Fig. 3. Third order circuit model for a solenoid with two modes of eddycurrent.

B. Eddy current dynamics

The model in (2)-(3) would suggest that the frequencyresponse (with voltage input and current output) at fixedpositions will be that of a first order system

I(s) =1

L(x)s+Rs

V (s)

Detailed frequency domain identification (see Section V)however suggests that a third order model is needed toaccurately model the dynamics up to 1 khz. To explain thethird order dynamics, a 2 mode eddy current model (for eachsolenoid) is proposed:

d

dt

2

666664L(x)

0

@1 k1 k2k1 1 0k2 0 1

1

A

| {z }K

0

@iie1

ie2

1

A

3

777775=

0

@�R

s

0 00 �R

e1 00 0 �R

e2

1

A

0

@iie1

ie2

1

A+

0

@V00

1

A

(4)

Here, ie1 and i

e2 represents the eddy currents induced bythe primary current i, k1 and k2 are respectively the mutualinductance coefficients, and R

e1 and Re2 are the resistances

of the eddy current loops. A useful way to visualize theeddy current effect is that of an AC transformer in which theprimary coil and the two secondary coils are wound arounda common iron core (Fig. 3). In this way, changes in the airgap (via x) would change the inductances of both the primaryand secondary coils. This hypothesis is supported by the factthat frequency responses at different locations collapse to acommon frequency response with appropriate 1/L(x) scalingof the frequencies. The mutual inductance coefficients k1 andk2 as well as L(x) were experimentally identified from thefrequency response data. Although it would have been morecomplete to include the mutual inductance term between thethe 2 eddy current loops, it is found not to improve the fitwith the experimental data.

DenoteK�1 =

✓1 gT

g G

◆/det(K)

Multiplying K�1 on both sides of (4), we can see that theeddy current is equivalent to the simple first order model in(3):

� =d

dt[L(x)i] = F [V �R

s

i] (5)

where F [u] is the output of the second order system:

�E

= �GRE

L(x)�E

+ g · u

y =

✓� gTR

E

L(x(t))�E

+ u

◆/ det(K) (6)

Here, �E

= (L(x)ie1, L(x)ie2)T are in fact the flux linkages

of the eddy currents. It can be shown that F [V � Rs

i] isa low pass filtered version of (V � R

s

i). Although (6) istime-varying and requires knowledge of x, the bandwidthsof filters are quite high so that using an estimate of x seemsadequate as long as V (t)�R

s

i(t) does not have significanthigh frequency content.

C. Nonlinear output equation

For the state equations (3), i1(t) = �1(t)/L1(x) andi2(t) = �2(t)/L2(x) can be considered output equations.If L1(x) and L2(x) are known, then the observer designis quite trivial. The difficulty lies in the dependence on theactuator displacement, x, being unknown.

With the push-pull solenoid configuration, the two fluxlinkage equations in (2) can be combined to form a newnonlinear output equation that does not depend on x. Byeliminating x from the inductances in (1), an implicit rela-tionship between the two inductances can be obtained:

�1

L1(x)� L1| {z }d1+x

+�2

L2(x)� L2| {z }d2�x

= d1 + d2 =: d (7)

Re-arranging, we have:

�L1(x)L2(x) + aL1(x) + bL2(x) + c = 0.

where

a := �(L2 + �2/d)

b := �(L1 + �1/d)

c := (L2�1 + L1�2)/d+ L1L2

Substituting

�1(t) = L1(x)i1(t); �2(t) = L2(x)i2(t)

into (7) and rearranging, we have

0 = h(t,�1,�2)

:= a · i2(t)�1 + b · i1(t)�2 + �1�2 + c · i1(t)i2(t) (8)

so that the dependence on x has now been eliminated. Eq.(8) will be treated as a time varying output equation of thesystem (with output identically 0). Assuming observabilitycondition holds, our approach is to design an observer toestimate �1(t) and �2(t) based on output h(t,�1(t),�2(t)).In particular, if the flux estimates �1 and �2 are incorrect,h(t, �1, �2) will be non-zero and will provide error foradjusting the estimates.

III. FLUX OBSERVER DESIGNS

The general approach is to utilize the nonlinear outputequation (8) as feedback for an observer for the flux (�1,�2).With the estimate of (�1,�2), the actuator displacement x isobtained from the inductance relationships (2) and (1). Thissection presents the flux linkage observer. Recovery of theactuator displacement x will be presented in section IV.

A. General structure

Since the right hand side of (3) (1st order) or (5) (3rdorder) is known (either as filtered version of the input andmeasurement), we consider a flux observer structure of theform:

˙�1 = F []V1 �R

s1 i1] + w1(t)

˙�2 = F [V2 �R

s2 i2] + w2(t)(9)

where w(t) = [w1(t), w2(t)]T is the output injection to bedesigned based on the output prediction error:

e := h(t, �1, �2)

= a · i2(t)�1 + b · i1(t)�2 � �1�2 + c · i1(t)i2(t)(10)

Using the notation,

� := [�1, �2]T

and subtracting h(t,�1,�2) = 0 in (8), (9) can be writtenexactly as:

e = �(t) �� �1�2

where �(t) is a measurable regressor given by:

�(t) := [a · i2(t) + �2(t), b · i1(t) + �1(t)] (11)

The flux estimation error dynamics are:˙� = w(t); �(0) = �0 (12)

e = �(t)�(t)� �1(t)�2(t) (13)

B. Kresselmeier Estimator

Here, the exponential observer first proposed by Kres-selmeier in [11] is adapted for the flux observer.

The goal is to construct the output injection to be similarto:

w(t) = ��R(t)�

where R(t) is a positive definite matrix. If this is possible,we would have the stable error dynamics:

˙� = ��R(t)�

The obvious challenge is that � is unknown.The key to the Kresselmeier method is a set of low pass

filters of the regressor and of the output error:

R(t) = �q ·�R(t)� �T (t)�(t)

�(14)

p(t) = �q ·�p(t)� �T (t)e(t)

�+R(t)w(t) (15)

where q > 0 is the filter’s bandwidth. For simplicity, assumethat these filters have zero initial conditions. The outputinjection is then defined as:

w = ��p (16)

Lemma 1 With the filters in (16),

p(t) = R(t)�+ pe

(t) (17)

where pe

(t) is the output of the filter equation:

pe

= �q(pe

� �T (t)�1�2) (18)

Therefore without pe

, we can use p to achieve of desiredinjection. The presence of p

e

in (17) is due to the �1�2

term in the output equation (13).

Proof: Both sides of (17) have the same zero initial con-ditions. Differentiating both sides, we see that both sidessatisfy the same linear differential equation:

X = �q ·⇣X � �T (t)(e(t) + �1�2)

⌘+R(t)w(t)

By the existence and uniqueness property of linear differen-tial equations, both sides of (17) are the same. ⇧

Remarks:1) R(t) is the filtered version of �(t)T�(t). While �(t) is

seldom full rank, the collection of �(t)T�(t) over timecan be. The filter gain q controls the time constant andcan be thought of as a forgetting factor.

2) If R(t) becomes full rank at some time t0 it remainsfull rank for all t > t0 although its co-norm maybecome smaller and smaller.

3) In the absence of the nonlinear term in (13), whenR(t) is full rank, the observer error � can be obtainedfrom � = R�1(t)p(t). This is unlike other estimatoror observer.

Let r(t) = �(R(t)) be the minimum eigenvalue of R(t)such that for all x 2 <n

xTR(t)x � r(t)kxk22We assume that the regressor �(·) is sufficiently exciting suchthat 0 < r r(t). In practice, this can be achieved bysuperimposing on any input to the solenoid a persistent highfrequency excitation signal.

Since p(t) = R(t)�(t) + pe

, we have the dynamics:

˙� = ��R(t)�� �p

e

pe

= �q · (pe

� �T (t)�1�2)

Using a Lyapunov function

W =1

2�T �+

1

2pTe

pe

W = ���(R(t)�+ pe

) + pTe

(�qpe

� �(t)�1�2)

= ���T pT

e

�✓ �R(t) �I2⇥2

�s(t, �) qI2⇥2

◆✓�pe

◆(19)

wheres(t, �) =

q

2�T (t)[�2, �1]

so that ks(t, �)k ! 0 ans � ! 0. Therefore, in theneighborhood of � = 0, the matrix in (19) is positive definiteif � < 4r(t)q so that (�, p

e

) = (0, 0) is asymptotically stable.

IV. ACTUATOR DISPLACEMENT ESTIMATION

With the flux linkage estimates �1(t) and �2(t), we nowestimate the solenoid position x. Expanding (1) and assumingthe estimated fluxes are the actual fluxes in (2):

(d1 + x)(�1 + L1i1(t)) = �1i1(t) (20)

(d2 � x)(�2 + L2i2(t)) = �2i2(t) (21)

Rearranging, we have✓�1i1(t)� d1(�1(t) + L1i1(t))�2i2(t)� d2(�2(t) + L2i2(t))

◆

| {z }y

x

(t)

=

✓1�1

◆x(t) (22)

While x(t) can be obtained using a least square algorithmusing the instantaneous estimate �, we design an observer toprovide some temporal averaging.

By assuming the displacement dynamics,

x = v(t)

where velocity v(t) is a zero mean random process, theobserver of x is:

˙x = ��x

�1 �1

�✓✓ 1�1

◆x� y

x

(t)

◆

such that 2�x

is the bandwidth of the observer. This observercan be designed using standard Kalman methodology.

V. SYSTEM IDENTIFICATION AND EXPERIMENTS

1) System Identification: The system models are obtainedfrom the frequency response data taken at different solenoidpositions. To identify the solenoid model, at each posi-tion (measured using a LVDT), we apply biased sinusoidalvoltages at various frequencies (from 0.5Hz to 1024Hz) tothe solenoids, and measure the corresponding total currentoutput. Sampling times between 0.05ms to 0.1ms are used.Figure 4 shows the frequency response plots at variouspositions.

According to the eddy current model (4), the positiondependence only resides on the LHS via L(x) which affectsthe derivative term. In the frequency domain, L(x) willmultiply each s term. Therefore, by applying 1/L(x) in Fig.5 to the frequencies, the frequency response plots shouldcollapse when plotted with this frequency scaling. This isillustrated in Fig. 6. The collapsed frequency response isthen fitted with a third order model (4) that incorporates 2eddy current modes. This shows that the 2nd order modelis accurate up to 100Hz (referenced to the middle stroke)whereas the 3rd order model is accurate up to 1024Hz.

A. Testing the self-sensing algorithm

To test the self-sensing algorithm experimentally, openloop tests were first performed. This involves sending in-put signals to the solenoids and physically perturbing andmoving the dual solenoids. A LVDT is used to record themovement while the input signals and physical perturbationsare applied. Figure 10 shows some typical results. The inputvoltage signals consists in D.C. signal superimposed by high

10 -1 10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

10 -3

10 -2

Ga

in

-1

Solenoid 1: Collapsed Frequency Responses

10 -1 10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

-60

-40

-20

0

Ph

ase

[d

eg

]

Increasing position

Increasing positionIncreasing position

10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

10 -3

10 -2

Gain

-1

Solenoid 2: Frequency Responses

10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

-60

-40

-20

0

Phase

[deg] Increasing position

Increasing position

Increasing position

Fig. 4. Frequency response plots at different positions. Top: Solenoid 1.Bottom: Solenoid 2.

0 1 2 3 4 5 6

Position - mm

0.6

0.8

1

1.2

1.4

1.6

Re

lative

in

du

cta

nce

Solenoid 1

Solenoid 2

Fig. 5. Inductance values used to collapse the frequency response.

10 -1 10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

10 -3

10 -2

Ga

in

-1

Solenoid 1: Collapsed Frequency Responses

10 -1 10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

-60

-40

-20

0

Ph

ase

[d

eg

]

10 -1 10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

10 -3

10 -2

Ga

in

-1

Solenoid 2: Collapsed Frequency Responses

10 -1 10 0 10 1 10 2 10 3

Scaled Frequency [Hz]

-60

-40

-20

0

Ph

ase

[d

eg

]

Fig. 6. Inductance scaled frequency responses at 11 locations and the fitwith the eddy current models. Top: Solenoid 1. Bottom: Solenoid 2. Solidblue = 3rd order model with 2 eddy current modes. Dashed pink = 2ndorder model with 1 eddy current mode.

-1

-0.75

-0.5

-0.25 0

0.25

0.5

0.75

1

1.25

1.5

Contours of h(L1, L

2)

L1(x)

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

L2(x

)

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5x=4.56mm

x=0.75mm

x=2.86mm

x=3.90mm

x=1.45mm

x=1.91mm

x=3.44mm

Fig. 7. Inductances L2 verus L1 at various locations superimposed on thecontour lines of h(L1, L2).

5 5.1 5.2 5.3 5.4Time - s

0

2

4

6

8S

ol.

1 v

olta

ge -

V

5 5.1 5.2 5.3 5.4Time - s

0

2

4

6

8

Sol.

2 v

olta

ge -

V

Fig. 8. Sample input voltage signals.

0 5 10 150.08

0.1

0.12

0.14

Sol. 1

curr

ent -

A

0 5 10 15

Time - s

0.11

0.12

0.13

0.14

Sol. 2

curr

ent -

A

Fig. 9. Sample output current signals.

frequency A.C. are as shown in Fig. 8. The correspondingcurrent outputs are shown in Fig. 9. The flux observerand position estimation algorithms are then applied to thecollected offline. The bandwidth of the low pass filter inthe flux observer is 16Hz. The RMS error is found to be0.061mm.

VI. CONCLUSIONS

A finite dimension flux observer has been developed todetermine the position of a pair of solenoids in push-pullconfiguration. The key to the design is the formulationof a nonlinear output equation based upon the geometricrelationship between the inductances of the two solenoids.Another key ingredient is the accurate system identificationof the solenoids. Although the flux observer was developedbased upon a first order solenoid model, system identificationrevealed that the system is in fact a 3rd order model. Thisis reconciled by using a pre-feedback to undo the effects of

0 5 10 15

Time - s

1

1.5

2

2.5

3

3.5

4

Positio

n -

mm

Estimate

Measured

Fig. 10. Estimated and actual position. Input signals are biased sinusoids.Solenoid 1’s input is a biased 128 Hz sinusoid. Solenoid 2’s inout is a biased16 Hz sinusoid. The RMS error is 0.061mm.

the additional eddy current dynamics. Experimental resultsindicate that the RMS accuracy of the algorithm is less than0.065mm over the stroke length of 5mm. The next step is toincorporate the self-sensing function and utilize it in closedloop position control.

ACKNOWLEDGEMENT

Assistance by a number of former graduate and under-graduate students at the University of Minnesota who con-structed the experimental setup and conducted various systemidentification experiments are gratefully acknowledged. Thestudents include: Abirim Vankatesh, Benjamin Sohn, WeiXian Lee, Jim Crist, Jin Ahn and Jonathan Lee,

REFERENCES

[1] Noh, M.D. and Maslen, E.H., “Self-sensing magnetic bearings usingparameter estimation,” IEEE Transactions on Instrumentation andMeasurement, Vol. 45, No. 1, pp. 45–50, 1997.

[2] Zhan Z. J., Chan C. C. and Chau, K. T., “A novel sliding modeobserver for indirect position sensing of switched reluctance motordrives,” IEEE Transactions on Industrial Electronics, Vol. 46, No. 2,pp. 390-397, April, 1999.

[3] Hussain, I.; and Ehsani, M., “Error analysis in indirect rotor positionsensing of switched reluctance.” IEEE Transactions on IndustrialElectronics, Vol. 41:3, pp. 301-307, 1994.

[4] McCann, R. A., Islam, M.S., and Hussain, I., “Application of a slidingmode observer for position and speed estimation in switched reluc-tance motor drives,” IEEE Transactions on Industrial Applications,Vol. 37:1, p. 51-58, 2001.

[5] Vischer, D. and Bleuler, H., “Self-sensing active magnetic levitation,”IEEE Transactions on Magnetics, Vol. 29:2, p. 1276-1281, March1993.

[6] Yuan, Q.-H., Li, P. Y. Self-sensing actuators in electrohydraulic valves.Proceedings of the 2004 ASME-IMECE, Anaheim, CA, USA, Paper#62104, 2004.

[7] Li, P.Y. and Yuan, Qinghui, “Flux observer for spool displacementsensing in self-sensing push-pull solenoids”, Proceedings of the SixthInternational Conference on Fluid Power Transmission and Control(ICFP’2005), Hangzhou, China, 2005.

[8] Wu, Shang-Teh, and Wei-Nian Chen. ”Self-sensing of a solenoid valvevia phase detection.” In 2009 IEEE/ASME International Conferenceon Advanced Intelligent Mechatronics, pp. 1165-1170. IEEE, 2009.

[9] Rahman, M. F., Cheung, N. C., and Lim, K. W. “Position Estimationin Solenoid Actuators”, IEEE Transactions on Industry Applications,Vol. 32, No. 3., pp. 552-559, May/June 1996.

[10] Braun T., Reuter T., and Rudolph J.. “Observer Design for Self-Sensing of Solenoid Actuators with Application to Soft Landing”,IEEE Transactions on Control Systems Technology, Vol. 27, No. 4,pp. 1720-1727, 2019.

[11] Kreisselmeier, Gerhard, “Adaptive observers with exponential rate ofconvergence.” IEEE transactions on automatic control 22.1 (1977):2-8.