Survey of Controller Designs and Implementations on a MagLev-BasedFloating Light Bulb

Fernando Zigunov

Abstract— In this short paper, a working floating light bulbprototype will be presented along with simulations on itscontroller design. The light bulb contains a permanent magnetthat is used in conjunction with an electromagnet solenoid toperform the levitation task. The energy is transferred to thelight bulb with near-field wireless power transmission, whereefficiencies of the order of 25% were achieved.

Details on the failed implementations will be provided alongwith simulations that provide insight on the reasons why some ofthe controller topologies applied failed. Three controller topolo-gies will be presented: An LQR-Observer based controller, adigital hysteresis controller and an analog hysteresis controller.

I. INTRODUCTION

Magnetic levitation is a very interesting control problemthat has several potential applications in the industry. AsEarnshaw’s theorem demonstrates, there is no configurationof permanent magnets in space that is statically stable. There-fore, in order to achieve magnetic levitation for a specificapplication involving permanent magnets, it is necessary toemploy some level of active control that changes the systempoles to the left-half of the Laplace plane.

Although applications of magnetic levitation are usuallyrelated to high-technology industrial applications such aslevitating trains and heavy-duty shaft bearings, devices acces-sible to the general population are already becoming ubiq-uitous and being mass-produced by several manufacturersas decorative household items. This work will focus onthe engineering design of a device that targets a similardecorative application, where a functional light bulb is keptfloating through a closed-loop magnetic levitation system.

II. HARDWARE DESIGN AND IMPLEMENTATION

A. Wireless Energy Transfer

In order to provide the floating LED with the energyrequired for light emission, a wireless transmitter/receivercircuit was designed and implemented. A simple resonantair-cored transformer was employed for near-field energytransfer. This form factor was chosen primarily due to itsimplementation simplicity and the ability to package intoa lightweight form factor, which is required for levitationwithin the electromagnet force capacity.

The driver consists of a low-side MOSFET driver, asshown in Figure 1, driven by a 600kHz square wave gen-erated by a dedicated Arduino Nano (ATMega 328P). Thefrequency corresponds to the LC tank fundamental resonantfrequency. Since the coupling coefficient of the TX/RX coilsis low, the effect of the RX circuit on the resonant frequencyis negligible. The microprocessed waveform generation was

particularly chosen due to its temperature stability, crucialfor high Q-factor transformers.

The capacitors C1 and C2 are polyester with very lowESR, which is also crucial for the formation of a high Qtank circuit, necessary for the strong coupling of the two coils[1]. The coils L1 and L2 were four manually wound, 20 mmdiameter loops of AWG24 wire. The inductance indicated inFigure 1 was measured experimentally by finding the tankcircuit resonant frequency with a function generator and anoscilloscope. Although a design study on the coil parameterswas not performed, the guidelines used involved maximizingthe Q factor of the tank circuits by minimizing the parasiticESR from the coil wire resistance. This was accomplishedby increasing the wire diameter and minimizing the numberof loops. On the other hand, these actions will reduce theself-inductance of the coil, increasing the resonant frequency.Although in the general case it is not an issue to resonateat higher frequencies, the efficiency of the amplifier circuitwould be affected by the parasitic capacitance of the MOS-FET, requiring faster-switching drivers and a more complexarchitecture to support the higher switching current.

Although the design was settled by experimentally build-ing a few different coil configurations with the target LEDload, simulations on the coupling coefficients that couldbe achieved with the coils used were performed to under-stand the performance levels achieved and whether a strongresonant coupling regime was achieved with the describeddesign. Figure 2 shows the resulting coupling coefficientsobtained through FEMM [2]. The desired distance betweenthe the TX and RX coil was about 13 millimeters, whichcorresponds to a coupling coefficient of κ = 0.065, how-ever simulations were performed across a larger range tounderstand the behavior as the RX coil is moved fartherfrom the TX coil. Both tank circuits have a damping ratio ofζTX = ζRX = 0.01, estimated from the ESR values of thecomponents. As defined by [1], the strong coupling regimewhere efficient wireless energy transfer happens when:

κ2

ζTXζRX> 1 (1)

The strong coupling regime happens throughout the rangeplotted as long as the coil axes are aligned, however thevoltage generated in the RX is only sufficient to light upthe LED for x < 15 mm. Unfortunately, the test equipmentrequired to characterize the true power transfer efficiency ofthis implementation was not available, however it is believedthat the overall efficiency was less than 25% based on the

IRF630Q2

220R1

700nL1

700nL2

100nC1

100nC2

LEDLuminusSST-10-G

+7.5V

Square Wave(from Arduino)

TX RX

600kHz

Fig. 1. Air-cored transformer driver circuit and receiver circuit.

0 5 10 15 20 25 30

0

20

40

60

80

100

[%

]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

[-]

Fig. 2. Coupling coefficients obtained through FEMM simulations andtheir corresponding power transfer efficiencies for a 30Ω load, consideringthe driver resonates the transmitter coil.

power supply power consumption and the LED specifica-tions.

B. Magnetic Levitation

The magnetic levitation hardware implementation usedmaterials available to an electronics hobbyist. During thecontroller implementation phase it was found that this de-cision introduced a large amount of uncertainty in the modeland led to the failure of the theoretical approaches used.However, since this is a very realistic scenario in engineeringdesign, it serves as a benchmark to understand the require-ments for control robustness.

The hardware, depicted in Figure 3, consisted of a hand-wound electromagnetic coil attached to a structure in theform of a lamp shade. The lamp shade, however, has the solepurpose of making space to house the electronics for boththe magnetic levitation driver and the wireless transmitter. AHall effect sensor (Honeywell SS41) was used to measurethe position of the light bulb by sensing the magnetic fieldof the neodymium magnet that was embedded on the lightbulb construction. An undesired side effect, however, is thatthe hall effect sensor also senses the magnetic field of theelectromagnet, producing an output for the controller thatcontains two states encoded (electromagnet current and lightbulb position).

Electromagnet coil

TX coil

RX coil

Neodymium magnet

LED

Space for electronics

Structure

Hall effect sensor

Fig. 3. Mechanical arrangement of the hardware in the lamp shade formfactor.

The electromagnet consisted of 1 Kg of AWG24 copperwire wound over a spool of approximately 25 ID x 75 ODx 55 L mm. The coil resistance was measured to be 20 Ω atambient temperature and its inductance L = 33.6 mH. Theinductance was easily measured from the step response ofthe coil in series with a 1 kΩ resistor.

A low-side driver of similar architecture shown in Figure 1was used to drive the electromagnet through high frequencyPWM switching. The full circuit implementation will bediscussed in Section IV, since the circuit topology wasdifferent for the analog and digital implementations.

III. CONTROLLER DESIGN AND SIMULATIONS

A. Traditional Equations of Motion

Although the floating magnet plant is a true six-degree-of-freedom (6-DOF) system with all spatial and rotationalfreedom, it is generally sufficient to restrict the model to theunstable degree of freedom for control purposes. The onlyunstable mode is associated with the motion of the permanentmagnet in the axis of the solenoid, which is going to be calledz in this work. The positive direction of z is aligned withthe gravity vector. In the Appendix, a model based on aninfinitesimal magnetic dipole pair is shown to indicate theunstable eigenvalues of the linearized 6-DOF plant.

It is widespread in the literature ([3], [4]) the use ofthe same simplified non-linear model for a solenoid-basedfloating magnet, and it is customary to write the equationsof motion in the following form:

z = g − k I2

z2

I =u

L− IR

L

(2)

Where k is an arbitrary constant proportional to theelectromagnet strength, g is the gravitational acceleration, I

is the solenoid current and R,L are the solenoid inductanceand equivalent series resistance. The control input u is thevoltage given to the solenoid. The equations simply modela force balance between the gravity and the electromagnetforce, as well as the linear dynamics of the electromagnetcircuit.

B. Identification of System Non-linearities

Although the model presented in the last section is basedon theoretical results, the simplifying assumptions requiredto build the model are too restrictive and the real devicedid not follow the non-linearity F ∝ z−2, according tomeasurements made.

In order to build a model that more accurately describesthe real device, measurements of the relevant parameterswere performed and converted into polynomial fits that de-scribed the non-linear functions observed. Then the equationsof motion for the controllable modes can be modified, giventhat only the solenoid intensity (ms in the Appendix) isaccessible through the control input.

1) Electromagnet Force: The first non-linearity analyzedwas the electromagnetic pulling force exerted on the per-manent magnet, both as a function of the magnet distanceand the current input. A measurement of the force versusdistance was performed with a scale (resolution 0.1 g) andan micrometric traverse. The magnet to be levitated wasplaced in the scale with an extra plate that added 120 gramsof mass to prevent the magnet from being lifted off thetraverse during the test. With the scale tared, data pointswere then taken at discrete locations with the magnet runningthe maximum current available and a polynomial curve fitwas performed, as displayed in Figure 4. Since the weightof the permanent magnet was 24.2 grams, data points atgreater distances were not taken as the system would justdrop the magnet. The polynomial fit chosen was a quadratic,mostly due to its low VC dimension (to prevent overfit). Acomparison with the theoretical F ∝ z−2 is shown in a reddashed line, showing how inappropriate the theoretical modelis to describe the real device behavior. It is believed that themain source of discrepancy is the permanent magnet beingon the near field of the electromagnet (z−2 decay assumesfar field of two infinitesimal magnetic dipoles).

Since the measurements were performed manually, it wasnoticed that temperature effects due to the electromagnetwarming up when mounted in the enclosure were signif-icantly affecting the curves obtained as detailed measure-ments took around 10 minutes to perform. Because of thisadded variable, the number of data points was limited tominimize the effect of warming up on the results. After thislimitation, the measurements took approximately 1 minuteto perform, which significantly improved the repeatability ofthe measurements. All measurements shown in this sectionwere started with a cold coil (Tcoil ≈ 25C).

The non-linearity model for the magnetic attraction forceas a function of current, which is usually regarded as afunction F ∝ I2 (or, if following the Biot-Savart law, F ∝I), was also found to be inadequate. A measurement with the

0 5 10 15 200

50

100

150MeasurementsQuadratic fitC/(z2) fit

Fig. 4. Measured electromagnet force and comparison of a polynomial fitversus a C/z4 fit, showing the traditional model inadequacy. The maximumcontrol input was given to the electromagnet.

0 0.2 0.4 0.6 0.8 1

I/Imax

0

0.2

0.4

0.6

0.8

1

F/F

max

Fig. 5. Measured electromagnet force for z = 10 mm and comparisonwith a square-root fit showing excellent agreement. Force is normalized bythe maximum pulling force Fmax measured at full current Imax.

permanent magnet fixed at z = 10 mm was performed and,as shown in Figure 5, a square root fit F ∝

√I was found

to show excellent agreement. The traditional F ∝ I2 is ananalytical solution that does not correspond to the physicaldesign, as it assumes a high permeability material in thesolenoid core with only a small air gap. The Biot-Savartcurrent loop (F ∝ I) does not consider the effect of finitelength of the solenoid. Limited discussion on the physicswill be provided here to explain the measured F ∝

√I

dependence, as a more comprehensive analytical solutionwould be necessary.

Although the plot shown in Figure 4 uses for simplicitymillimeters and grams-force as length and force units, re-spectively, the curve fits were performed in SI units to ensureconsistency during modeling and controller implementation.The polynomial coefficients for force as a function of currentF1(I) and as a function of magnet distance F2(z) in SI unitswere, then:

F1(I) = 1.14√I

F2(z) = 1367z2 − 59.2z + 0.87(3)

It was verified that the magnet force Fm = F1(I)F2(z)predicted the force correctly for other combinations of (I, z).

2) Hall Effect Sensor Reading: Due to the placement ofthe Hall effect sensor and its principle of operation, thesensor will respond to both the intensity of the permanent

0 0.2 0.4 0.6 0.8 1

I/Imax

520

530

540

550

Fig. 6. Raw signal from the hall effect sensor (from the Arduino 10-bitADC) as a function of electromagnet current.

magnet field (the desired measurement, z) and the intensityof the electromagnet field (which is a function of the controlinput and the system dynamics). This is inevitable as theHall effect sensor is required to be placed between the twocomponents in order to have a meaningful signal-to-noiseratio. Placing the Hall sensor between the permanent magnetand the structure base was an option that was ruled out due toaesthetical constraints. Another alternative would be to use asensor that does not rely on magnetic effects to measure thecoil position, where an example would be a time-of-flightoptical sensor. These sensors, however, are prohibitively ex-pensive for the small distances that the electromagnet couldoperate in this implementation, especially at the samplingrates necessary for control.

For these reasons, it is imperative to measure the in-fluence of the electromagnet on the sensor reading. Suchmeasurement is shown in Figure 6 as a raw signal from theArduino ADC, which was fit to a quadratic polynomial. Forreference, the reading at I/Imax = 1 when the permanentmagnet is at z = 10 mm is 681 counts, which shows theinterference generated by the electromagnet is not negligible.Other measurements not shown in this report showed thatthe readings from the Hall effect sensor when both theelectromagnet and the permanent magnet were present couldaccurately be described as a superposition of the independentreadings from each component. Therefore, a model for theoutput as measured by the microcontroller was defined as:

y = A1(I) +A2(z) (4)

Where both A1(I) and A2(z) are non-linear functions.A1(I) is the polynomial fit shown in Figure 6. For brevity,the chart for A2(z) was omitted as a similar procedure wasused. The functions A1(I) and A2(z) were measured to be:

A1(I) = −29.7I2 + 54.14I

A2(z) = 8.94× 105z2 − 3.73× 104z + 954.4(5)

3) Generalized Non-linear Model: Since most of themeasurements made resulted in non-linear curve fits that arenot of the form shown in Section III-A, the equations ofmotion need to be generalized to account for the arbitrarynon-linear force and input functions. These functions will

then be replaced with the polynomial coefficients fitted.Furthermore, the linear system solenoid-resistor is also goingto be introduced in the model:

z = g − 1M F1(I)F2(z)

I = uL −

RIL

y = A1(I) +A2(z)

(6)

Where R is the solenoid wire parasitic resistance, L isthe solenoid inductance and u is the voltage applied tothe solenoid (which is also the control input given by thecontroller).

C. Linearization Around an Equilibrium Point

A linear controller can be built based on the non-linearequations of motion (6) if they are linearized around anequilibrium point (x0, u0). First, we define the system statevector x as:

x =

zzI

(7)

The equations of motion (6) can then be linearized bydefining the Jacobian matrix of the system dynamics withoutinput:

J =

∂z/∂z ∂z/∂z ∂z/∂I∂z/∂z ∂z/∂z ∂z/∂I

∂I/∂z ∂I/∂z ∂I/∂I

(8)

Which, evaluated for Equation (6), results in:

J =

0 −F1(I) 1M

dF2

dz −F2(z) 1M

dF1

dI1 0 00 0 −R

L

(9)

Now, assuming there is an equilibrium point for (x0, u0),it is possible to linearize the equations by assuming theequilibrium point is disturbed by ∆x and ∆u:

∆x = A∆x+B∆u

∆y = C∆x(10)

Where A = J is the uncontrolled system Jacobian. Sincethere is no non-linearity in the input, B is simply:

B =

00

1/L

(11)

As discussed in Equation 4, the output also is non-linearin relation to the system state. A similar linearization processis used to find C:

C =[0 dA2

dzdA1

dI

](12)

Note all derivatives in A and C are evaluated at theequilibrium point x0.

0 0.1 0.2

13

13.2

13.4

13.6

z [

mm

]

0 0.01 0.02 0.03

0.6

0.65

0.7

0.75

0.8

I [A

]

Fig. 7. Effect of the input cost matrix R on the system states for an initialdisturbance ∆z(t = 0) = 0.5mm and ∆z(t = 0) = 10mm/s.

D. LQR Controller Design

With a linearized system equation in hands, the designof a controller can be done to place the system poles atstable frequencies. Defining the system variables and theequilibrium point for the analysis:

R = 20Ω

L = 33.6× 10−3H

M = 30× 10−3Kg

z0 = 13× 10−3m

z0 = 0m/s

I0 = 0.605A

(13)

Calculating the eigenvalues of A for the equilibrium pointgiven, one unstable eigenvalue (λ1 = 26.4) and two stableeigenvalues (λ2 = −26.4 and λ3 = −595.2) are found.Therefore, the uncontrolled system is unstable, as expectedfrom the physics. It is worth noting that, for the valuesprovided, the system is controllable and observable.

Since the desired equilibrium point is somewhat close tosaturation (Imax = 0.81 A), the control input during thetransient cannot exceed ∆I = 0.205 A, corresponding to amaximum control input of ∆u = 4.05 V. The LQR algorithmprovides a means to tune the aggressiveness of a control lawby varying the input cost matrix R. In Figure 7 the linearsystem response for differentR values is shown for the linearcase, for an initial disturbance of ∆z(t = 0) = 0.5mmand ∆z(t = 0) = 10mm/s. The initial disturbance wasarbitrarily chosen, but it reflects the inaccuracy of the lightbulb placement during the initialization of the system. The Qmatrix shown in Equation 14 was used for the optimization.Its form was chosen to bring all states (which have differentunits) to similar orders of magnitude such that the LQRalgorithm would weight deviations in all states equally.

Q =

1× 102 0 00 1× 103 00 0 1

(14)

To understand the impact of the system non-linearities inthe controller performance, the same controllers obtained forFigure 7 were simulated again for the full non-linear systemusing the Matlab ODE45 integrator, and their responses are

0 0.1 0.2 0.3

13

13.2

13.4

13.6

z [

mm

]

0 0.1 0.2

0.4

0.6

0.8

1

I [A

]

Fig. 8. Same control laws used for Figure 7 applied in the non-linearsystem modeled in Section III-B.

plotted in Figure 8. It is very interesting how the controllerresponses are vastly different due to the non-linearity. Thisbehavior, however, is expected because the non-linearity isvery strong considering the displacements analyzed. Eventhough the system stability was not compromised, the con-trollers withR = 1 andR = 1×10−2 now display saturationconsidering the system capacity, which means they wouldfail in the physical device. Surprisingly, the only controllerthat does not display saturation is the most aggressive (R =1×10−4), where ∆I peaked at 0.2 A. On the other hand, thecontrol input is very close to saturation, leaving little marginfor inaccuracies. Other controllers were designed with poleplacement at different locations in the s-plane, with similarperformance degradation when simulated in the non-linearsystem.

The controller gain matrix K obtained forR = 1×10−4 isshown below. It will be further analyzed in the next sections,as it was one of the few candidates that did not saturate thecontrol input when applied to the non-linear system.

K =[−1214 −18467 85.2

](15)

With this controller matrix, the eigenvalues of the linearsystem were placed at λ1 = −3034s−1, λ2 = −87.3s−1

and λ3 = −8.5s−1. However, the first eigenvalue placesa very strong requirement when the system is digitizedfor microcontroller implementation, as it sets the order ofmagnitude of the sampling rates necessary to avoid aliasing.

Reducing the magnitude of λ1 is possible at the expenseof saturating the controller and even making the systemunstable. For example, for λ1 = 510s−1 and maintainingthe other two pole locations, the non-linear system becomesunstable for the initial disturbance studied, as shown inFigure 9. As will further be discussed later, this is believedto be one of the reasons why the digital implementations ofthis controller (as well as many other attempts not reportedhere) failed.

E. Observer Design

The controller designed in the former subsection requiresfull state feedback. However, as discussed in Section III-B,the Hall effect sensor will only provide indirect informationabout the system state, requiring the design of an observer to

0 0.2 0.4 0.612.5

13

13.5

z [m

m]

0 0.2 0.4 0.6-2

0

2

4

I [A

]

Non-linear system

Linear system

Fig. 9. Failed pole placement in non-linear system for linearized systempoles at λ1 = −510s−1, λ2 = −87.3s−1 and λ3 = −8.5s−1.

enable the use of the LQR controller in the practical device.Only the case without noisy measurements will be examinedhere, as the linear observer was found to lack the robustnessnecessary to deal with the system non-linearities.

A linear system observer can be simply designed by usingAckermann’s method on the linearized system matrix AT andthe linearized output matrix CT . Pole placement can then beperformed arbitrarily, but in general as a rule of thumb itis desired to have the observer settle around 5 to 10 timesfaster than the controller poles.

A Luenberger observer L was designed with pole place-ment at locations 10 times faster than the controller poles(λo,i = 10λc,i). This observer was applied together with thecontrol law K from Equation 15 for the linear system. Ascan be observed from Figure 10 (compare against Figure 7),the observer degrades the performance but is still stable.

If this observer is now applied to the fully non-linearsystem of Equation 6, a very fast instability develops (notshown). In order to investigate the effect of the non-linearitythat makes the system unstable, the output non-linearity wasremoved by replacing the y equation in Equation 6 with itslinear version at the equilibrium point. This change makes thesystem stable again, and the results are ploted in Figure 11.The system settles in a different equilibrium point because itslinear observer applied to the non-linear system makes thecontroller believe it is elsewhere in the state space, whichis clearly observed by the steady-state error between thetrue and observed state lines. When the input non-linearityis added to the simulation, the error in the observed statebecomes so much larger that it destabilizes the controller.

Because the simplified, noiseless version of the linearizedcontroller/observer already saturates the control input (notshown in Figure 11, but can be inferred from I > Imax),the linearization technique is unlikely to produce any mean-ingful results for the physical system. Therefore, effects ofnoise, time discretization, output digitization, control inputdiscretization and PWM will not be further analyzed asincreased performance degradation from these effects isexpected.

F. Hysteresis Controller

Perhaps on the opposite side of the spectrum of controllercomplexity lies the hysteresis controller, sometimes also

0 0.1 0.2

13

13.2

13.4

13.6

13.8

z [

mm

]

0 0.01 0.02 0.03

0.6

0.8

1

1.2

1.4

I [A

]

True State

Observed State

Fig. 10. Performance degradation of Luenberger observer implementationwith poles 10 times faster than the LQR controller poles. Degradationobserved against Figure 7 for the fully linearized system. Blue traces indicateobserved state, which is used for control input computation.

0 0.5 1

14

16

18

20

22

24

z [

mm

]0 0.5 1

0

0.5

1

1.5

2

I [A

]

True State

Observed State

Fig. 11. Further degradation of performance for the same observerimplemented in Fig. 10 when simulated in the non-linear system. Outputequation is still simulated as fully linear.

called the Bang-Bang controller. It is very common to usea hysteresis controller in magnetic levitation apparatuses,especially in hobby level projects. Apart from its obvioussimplicity, it seems to be popular belief that this kind of con-troller is unconditionally stable under high enough samplingfrequencies. However, no theoretical evidence supportingsuch claim was found in the literature review performed inthis work.

The hyteresis controller has, in its simplest form, onlyone parameter. A signal s is generated as a function of thesensor inputs and compared with a threshold value t. Then,the controller is simply defined as:

u =

0, for s ≤ tKu, for s > t

(16)

In the system studied, the signal s is a function of theonly input (s = −y) and the threshold is a function of theequilibrium point input value (t = −y0 = −642). The signsare reversed because y decreases as the permanent magnetfalls.

Simulation results of this control rule applied on the fullynon-linear controller for the same initial conditions presentedin Section III-D are shown in Figure 12. Apart from theeffect state and output non-linearity, this simulation was alsoperformed with digitization effects (sensor bit rounding).

In this simulation, the physics were resolved with a timestep of 1 × 10−6 s and the control input was updated

0 1 212

12.5

13

13.5

14

14.5

z [m

m]

0 1 20

0.2

0.4

0.6

0.8

I [A

]

Sampling 10kHz

Sampling 1kHz

Fig. 12. Failed hysteresis controller simulated at initial conditions ∆z(t =0) = 10 mm/s, ∆z(t = 0) = 0.5 mm and ∆I(t = 0) = 0 A at twodifferent sampling rates.

0 2 4

13

13.02

13.04

13.06

13.08

13.1

z [

mm

]

0 2 4

0.56

0.58

0.6

0.62

0.64

0.66

I [A

]

Fig. 13. Successful hysteresis controller simulated at initial conditions∆z(t = 0) = 0 mm/s, ∆z(t = 0) = 0.2 mm and ∆I(t = 0) = 0 A. Inthe limit cycle, the control input waveform is a 2470 Hz square wave.

every 1 × 10−4 s or 1 × 10−3 s (the two sampling ratesshown in Figure 12)). Lower sampling rates for the controllermake the system more unstable, but increasing the samplingfrequency does not improve the system stability for theinitial conditions used. The results show that the hysteresiscontroller is also unstable for this initial condition. However,it is worth noting the time scale of the simulation, which istwo seconds long. The LQR-observer based controller, forcomparison, exceeded z = 20 mm in 1 millisecond for thesame conditions with the full non-linear system.

Since the hysteresis-based controller was more promisingon its stability and robustness characteristics, another simu-lation was performed with a more lenient initial conditionof ∆z(t = 0) = 0 mm/s, ∆z(t = 0) = 0.2 mm and∆I(t = 0) = 0, for which the LQR controller applied to thelinear system still produces an unstable output. The resultsfor the hysteresis controller, shown in Figure 13, display thatthe controller is now stable to these initial conditions. Thelonger simulation time of 5 seconds show that the controllerreaches a stable point, which is at z = 13.05 mm due tothe rounding of the sensor reading. A somewhat narrow-band limit cycle switching at 2470 Hz is reached at thisequilibrium point. However, the control input is not a puresquare wave, having noticeable frequency jitter.

Finally, the effect of bit noise was assessed for the sameinitial conditions of ∆z(t = 0) = 0.2 mm and the other twostates being the equilibrium conditions. The bit noise wasobserved to be an issue during the practical implementation,

0 2 4

12.95

13

13.05

13.1

13.15

z [

mm

]

0 2 4

0.5

0.55

0.6

0.65

0.7

I [A

]

Fig. 14. Effect of random bit noise in the hysteresis controller stability forthree bit change probabilities Pbit. Initial conditions are the same as Fig.13.

and for this study it was defined as a uniformly distributedrandom process that would add or subtract one unit to thesensor reading y with a probability Pbit. The adding orsubtracting would have equal probabilities. Other models,like introducing white noise in the analog signal prior todigitization had similar effects, however not reported here.The results of this study are summarized in Figure 14 andshow that the system is robust to a bit noise probability ofapproximately Pbit = 2× 10−4. Above this noise level, thesystem becomes unstable. Unfortunately, it was observed em-pirically from reading the Hall effect sensor to the ArduinoADC that the probability of last bit flipping with the coil andthe wireless transmitter off was of Pbit ≈ 0.1 when reading1× 105 samples. Similar tests at arbitrary magnet positions(still with the electromagnet off) were performed to removethe bias that an analog level between the two bit values couldbe the cause of bit flipping, with similar probability levels.

IV. PHYSICAL IMPLEMENTATION

Since the observer-based controller was found to lack therobustness necessary to be implemented in the real hardware,experimental data on its instability will not be presentedherein for brevity. Many experiments were performed withthe digital version of the observer-based controller, but allresulted in unstable systems. The deeper analysis shown inSection III-E reveals the causes of the controller failure,which in summary are related to the high non-linearity ofthe physical system and the non-linear coupling of twostate variables in the input. The remaining non-idealities(digitization, sampling, noise) only exacerbated the effect ofimplementing an inadequate controller algorithm.

In the following subsections the results from the imple-mentation of the hysteresis controller will be shown andanalyzed.

A. Digital Hysteresis Controller

The digital implementation using an Arduino Nano boardwas implemented for both the LQR and the hysteresiscontroller. Its electrical circuit is shown in Figure 15 (a).The only input is the Hall effect sensor, and it drives theoutput MOSFET with a digital output. For the failed LQRimplementations, the output signal to the MOSFET wasan 16kHz, 8-bit PWM signal generated from the ATmega

IRF630Q2

220R1

LD1

+16.2V

Hall Sensor+5V

Vref LM741

+5V

IRF630Q2

220R1

LD1

+16.2VH

all S

enso

r +5V

ArduinoNano

A0 D3

(a) (b)

Fig. 15. Schematics of the driver circuits for the electromagnet. (a) Digitalcontroller using an Arduino Nano. (b) Analog controller using an operationalamplifier operating as a comparator.

Timer0 clock. Every iteration of the main loop would updatethe register that controls the duty cycle of the PWM signal,which had 8 bits of resolution, for the LQR controller case.

For the digital hysteresis controller, however, the im-plementation was much simpler. The signal read directlyfrom the Hall effect sensor was directly compared with thethreshold value and updated the digital output according toEquation 16. Since the sampling rate of the microcontrollerdepends heavily on the main loop overhead, the fastest con-troller obtained (operating at approximately 3.2kS/s) couldnot send signals to the computer for recording and analysis.However, by using the USB to Serial converter embeddedin the Arduino, the signals could be sent with increasedoverhead, which reduced the sampling rate to 1.95 kS/s.

Both hysteresis controllers, however, were unstable atthe sampling rates achieved. An example signal from thecontroller that used the USB to Serial interface (slower) isshown in Figure 16. It displays the raw Hall effect sensorsignal, as read directly by the Arduino from its 10-bit ADC.This signal can be directly compared with the function fory given in Equation (4). As the figure displays, the digitalhysteresis controller is unstable. The simulations performedin Section III-F point to two main causes for the instability:The ADC noise and the very low sampling rate. The problemwith the sampling rate, specifically, is that the controllercannot capture the effect of the electromagnet dynamics,which is very fast (about 1.7 ms time constant). Since afaster sampling rate is not viable with the Arduino hardware,a decision was made to change the hardware and implementan analog controller.

B. Analog Hysteresis Controller

The analog controller implemented is a simple comparatoroperational amplifier and replaces the Arduino, as shown inFigure 15 (b). The unit gain bandwidth of the LM741 is verylarge, typically 1.5 MHz, but for the input changes expected(order 50mV) the output is expected to have a bandwidth inthe order of 15 kHz. Therefore, a performance improvementin relation to the digital controller is expected, obviouslyat the cost of controller flexibility. The voltage Vref to thecomparator was supplied by the Arduino board, as there was

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

600

620

640

660

680

700

y

threshold

Placement

Inaccuracy

Magnet

fell off

Fig. 16. Input seen by the Arduino controller operating a hysteresiscontroller at 1950 Hz. Input plateau at final time is the signal when theelectromagnet is on but the permanent magnet is far.

Fig. 17. Snapshot of the tracking video representing the tracking performedin the physical device. Blue box represents the feature window. Blue staris an arbitrarily placed point for visual confirmation of tracking success.Video of the tracking point available here. A more complete video of theworking device is provided in this link

0 5 10 15 20

17

17.2

17.4

0 5 10 15 20

-0.2

0

0.2

x [

mm

]

Fig. 18. Tracked image data for the magnet location as a function oftime for the analog hysteresis controller based on the LM741 operationalamplifier. ∆x represents the sideways motion. Note the very long time scale.

no reason to remove it from the hardware. It was generatedby supplying a higher PWM rate (64kHz) through an RClow-pass filter. The Arduino Nano hardware does not haveany digital-to-analog converters.

In order to assess and report the efficacy of the analog con-troller, a digital image correlation algorithm was developed totrack images of a video of the floating light bulb in operation,giving quantitative evidence of the results achieved. A simpledot pattern was used to re-scale the images to physicalcoordinates.

A frame of the tracking video is shown in Figure 17 withthe light bulb successfully floating and lighting up. The fullvideo is available online in this link. The figure also showsthe feature window used, which provided good trackingaccuracy due to its contrasting colors. It also provided anaxisymmetric surface, which was necessary as the lamprotated during the capture due to the expected misalignmentduring initial placement. The tracking was performed ingrayscale images, and the z and x displacements are shownfor 20 seconds of a video in Figure 18. They showcasethe effectiveness of the analog hysteresis controller, whichin practice holds the lamp for very long times (about 2-10minutes). Some discussion on the reasons why the controllerbecomes unstable will be provided in the next section.

V. CONCLUSIONS AND PRACTICALCONSIDERATIONS

In summary, this practical controller design exerciseshowed several engineering challenges that must be over-taken in order to successfully build a magnetic levitator. Inthis last section, some practical considerations and lessonslearned will be briefly discussed to summarize the difficul-ties encountered during the physical implementation of thecontroller.

1) The Danger of Simplifications: It is very attractive toattempt to simplify the real-world applications to tractablemathematical constructs in order to gain insight about a prob-lem. However, it is more often than not that the practicingengineer will find the models insufficient to provide usefulsolutions in real applications.

In the case studied, the linearizations performed provedinadequate even in simulated models. Thus, one must be ex-tremely careful in new applications when carrying analyticalmodels to real applications to ensure they describe the realsystem with sufficient accuracy. However, it could have beenpossible, with further modeling and the application of non-linear control techniques, to control the fully non-linear plant.

2) Model Complexity / Robustness Trade-off: In thisstudy, opposite ends of the spectrum in model complexitywere examined. On one hand, the LQR-Observer imple-mentation requires the model of the system and its state-space equation. On the other hand, the hysteresis controlleris model-free and requires a single parameter, its set-pointthreshold.

The increased model complexity of the LQR controllerprovides more confidence on the placement of the systempoles (if the system was linear), at the expense of narrowing

down the range of its applicability. Uncertainty in the curvefits performed to get the non-linear functions, as well asin the plant characteristics due to external factors will leadthe controller to unknown regions of the Laplace plane. Forboth cases, robustness needs to be assessed if the applicationis critical. However, due to the lack of extra parameters inthe model-free approach, it is very likely that the set ofparameters for which the model is stable constitutes a largerfraction of the parameter space.

3) Effect of Coil Temperature: Although very briefly dis-cussed in this work, it was found the coil power dissipationwas found to be the major reason why the analog hysteresiscontroller eventually becomes unstable. The warming up ofthe coil causes its wire resistance to increase, changing notonly its current but also its dynamics. The ratio of resistancesof the cold coil versus its equilibrium temperature (which isvery specific to this construction geometry) was found to beapproximately Rhot/Rcold = 1.2.

One can tune the voltage Vref in the analog controllerto be stable when the coil is cold and change it as thecoil heats up, which would require a temperature sensorinput and fine-tuning to get the temperature curve stableacross all temperatures. In this work, however, the referencevoltage was constant, meaning that the stable operation timeis limited.

4) Presence of the Wireless Transmitter: Another interest-ing practical observation specific to levitating objects wherepower transfer takes place is that the wireless transmittersignal will be captured by the Hall effect sensor. If the Halleffect sensor is sufficiently slow, the high frequency signalwill be low-passed and not be a problem. However, the smallsignal can still contribute to noise at the digitization levels,systematically biasing the rounding of the digital readingsand potentially causing the destabilization of the system.

Specifically to the ATmega328P microprocessor, the ab-sence of analog filters in its analog inputs will cause thehigh frequency signal to be aliased, characterizing a non-white sensor noise reading. A signal of 2 mV amplitude at600 kHz was measured at the analog input port where theHall effect sensor was connected when the TX coil was on.This was also a reason why the analog hysteresis controllerpotentially performed better than its digital counterpart, asthe comparator amplifier provides another layer of lowfrequency filtering.

REFERENCES

[1] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, andM. Soljacic, “Wireless power transfer via strongly coupled magneticresonances,” Science, vol. 317, no. 5834, pp. 83–86, 2007.

[2] D. C. Meeker, “Finite element method magnetics, version 4.2(28feb2018 build).”

[3] P. Suster and A. Jadlovska, “Modeling and control design of magneticlevitation system,” pp. 295–299, 01 2012.

[4] F. Gomez-Salas, Y. Wang, and Q. Zhu, “Design of a discrete trackingcontroller for a magnetic levitation system: A nonlinear rational modelapproach,” Mathematical Problems in Engineering, vol. 2015, pp. 1–8,03 2015.

[5] P. B. L. Kar W. Yung and D. D. Villani, “An analytic solution forthe force between two magnetic dipoles,” Magnetic and ElectricalSeparation, vol. 9, no. 1, pp. 39–52, 1998.

-6 -4 -2 0 2 4 6-8

-6

-4

-2

0

2

ms

mp

Fig. 19. Slice along the axis of the solenoid magnetic dipole ms displayingthe magnetic field ~B of two aligned magnetic dipoles according to the Biot-Savart law.

[6] D. D. V. Peter B. Landecker and K. W. Yung, “An analytic solutionfor the torque between two magnetic dipoles,” Magnetic and ElectricalSeparation, vol. 10, no. 1, pp. 29–33, 1999.

APPENDIX

A. Three-dimensional system stability

As briefly discussed in Section III-A, the floating magnetproblem is inherently a three-dimensional control problem,where the floating element has all 6 degrees of freedom.In order to facilitate an analytical understanding of thecontrol problem of the real system, both the solenoid andthe permanent magnet will be simplified as magnetic dipolesof dipole moment ~ms and ~mp, respectively. A depiction ofthe ~B field generated by these magnetic dipoles is shown inFigure 19 for reference.

An analytical solution for the force between two magneticdipoles based on the Biot-Savart law is provided by [5] andshown in Equation 17, where ~r is the distance vector betweenthe two dipoles, r is its unit vector and r is its magnitude.Furthermore, µ0 is the permeability of free space.

~F =3µ0

4πr4

((r × ~ms)× ~mp + (r × ~mp)× ~ms

− 2r( ~ms · ~mp) + 5r[(r × ~ms) · (r × ~mp)]

) (17)

The same authors later [6] provided a calculation for thetorque between dipoles, where ms and mp are the magnitudeof the dipole moments:

~τ =µ0msmp

4πr3[3(ms · r)(mp × r) + (mp × ms)] (18)

Equations 17 and 18 are approximate under the assumptionthat the distance r between the dipoles is large. Let’s furtherconsider the configuration of Figure 3, where the solenoidis above the permanent magnet in a Cartesian coordinatesystem where z is downwards. In the desired state, the two

dipoles are aligned in the vertical direction (ms = mp = z)and positioned directly under another (~r = rz).

In order to perform a linearized disturbance analysis,let’s first add a small rotation to the dipole moment of thepermanent magnet ~mp and a small perturbation to its positionvector ~r. The disturbed vectors ~m∗p and ~r∗ would then beexpressed as:

~m∗p = Rx(θ)Ry(φ) ~mp = mp

ζxζy1

(19)

~r∗ =

εxεyz

(20)

Where ζx = θ and ζy = φ are small because θ and φ inthe rotation matrices Ri are small angles. No rotation aroundthe z axis was considered. Similarly, εx and εy are also smallcompared to z. It is straightforward to replace the disturbedvectors in the force and torque Equations 17 and 18 and dropthe higher order terms to obtain:

~F =

3µ0msmp(zζx − 4εx)

4πz4|z|3µ0msmp(zζy − 4εy)

4πz4|z|−3µ0msmp

2πz3|z|

(21)

~τ =

−msmpµ0

4πz4|z|(3εyz − 2ζyz

2)

msmpµ0

4πz4|z|(3εxz − 2ζxz

2)

0

(22)

Where the z component of torque is zero because it onlycontains higher order terms. To assess the stability of thesystem, it is more convenient to lump the positive constantC = mpµ0/4π in the force and torque vectors. It is alsofurther assumed for now that ms, the solenoid dipole, isconstant:

~F =

ms

z4|z|(3Czζx − 12Cεx)

ms

z4|z|(3Czζy − 12Cεy)

−2Cms

z3|z|

(23)

~τ =

− ms

z4|z|(3Cεyz − Cζyz2)

ms

z4|z|(3Cεxz − Cζxz2)

0

(24)

All dependences are now linearized, except for the zdependence. In order to keep the equations of motion of thefloating magnet analytically tractable, the permanent magnetwill be assumed to be spherical, such that its products ofinertia are zero and its tensor of inertia I is a diagonal matrixwhere all non-zero elements are equal (Ixx = Iyy = Izz =I). The classical equations of motion:

M ~r∗ = ~F

I~ω = ~τ(25)

Can then be used for assessing the system stability. Notethe angular acceleration ~ω is the second time derivative of~m∗p/mp, which means the z component equation is already

identically satisfied by the zero torque τz .In order to perform a linearization of the z displacement

around an equilibrium point z0, let’s define a position vectorx as the combination of the disturbance displacement androtation, where εz is a z disturbance around the equilibriumpoint z0.

x =

εxεyεzζxζy

(26)

The linearization of the force and torque equations givesthe following Jacobian matrix J :

J =

−12Cms

Mz40 |z0|

0 0 3Cms

Mz30 |z0|

0

0 −−12Cms

Mz40 |z0|

0 0 3Cms

Mz30 |z0|

0 0 −8Cms

Mz40 |z0|

0 0

0 −3Cms

Iz30 |z0|

0 0 Cms

Iz20 |z0|

3Cms

Iz30 |z0|

0 0 −Cms

Iz20 |z0|

0

(27)

The equations of motion (25) then can be written in state-space form: [

xx

]=

[0 JI 0

] [xx

](28)

Where[0 JI 0

]= A is the system matrix.

Finding the eigenvalues of A then allows one to assess thestability of this 3D system for small disturbances. The ex-pression forms for the eigenvalues are non-trivial and will notbe reproduced herein. Numerical analysis of the expressionslead to unstable eigenmodes that are related only to the εz

-3 -2 -1 0 1 2 3

r

-4

-2

0

2

4

i

z modes

lateral modes

Fig. 20. Eigenvalues of A obtained by numerically sweeping 0.1 < C < 1for mp = ms = µ0 = M = I = z0 = 1 plotted in the complex plane.

motion, as well as lateral modes that combine the ζx,y andεx,y displacement. Figure 20 displays how the eigenvaluesrelated to these modes distribute in the Laplace plane, for arange of C values between 0.1 and 1 when the remainingconstants are maintained equal to unity. Similar numericalstudies have been performed when the other constants arevaried, resulting in very similar eigenvalue plots. For eachcase, 4 eigenvalues related to the lateral modes are alwaysunstable, apart from the unstable εz eigenmode.

These results lead to believe that, apart from the expectedunstable z mode, there are also unstable lateral and torsionalmodes that even though are fully decopuled from the zmode from analysis of the eigenvectors of A, can alsobecome unstable. Further analysis needs to be performed tounderstand if the instability of the lateral modes is relatedto the z mode instability or if the nature of the instabilityis related to the precession motion arising from the angulardisturbances introduced.