Model based, nonlinear control of near-field acoustically ...past.isma-isaac.be/downloads/isma2016/papers/isma2016_0498.pdf · 1579. As shown below, Reynolds equation is a nonlinear

Model based, nonlinear control of near-field acoustically levitated objects

D. Ilssar1, I. Bucher1, H. Flashner2 1 Technion – Israel Institute of Technology

Dynamics laboratory, Mechanical engineering

Technion 3200003, Haifa, Israel

e-mail: [email protected]

2 University of Southern California

The department of Aerospace and Mechanical Engineering

Los Angeles, CA 90089, USA

Abstract The near-field acoustic levitation phenomenon uses ultrasonic vibrations of a driving surface, to elevate

the average pressure in the thin gas layer trapped between this surface, and a freely suspended planar

object. This is done, exploiting the compressibility and the viscosity of the entrapped gas. Due to the

pressure elevation, the planar object can be manipulated vertically between dozens and hundreds of

microns, without any mechanical contact. This paper introduces a novel, simplified model, describing the

governing, slow dynamics of near-field acoustically levitated objects. This model replaces the commonly

used Reynolds equation based model, with a second order, ordinary differential equation, where the

stiffness and damping terms are given explicitly. Due to the simplicity of the model presented here, it sets

a convenient foundation for model based control algorithms. Indeed, based on this model, a continuous

gain scheduling controller is designed and implemented experimentally, showing good performance.

1 Introduction

When two closely placed surfaces, surrounded by gas, experience relative normal oscillations, a force,

consisting both oscillatory and constant terms, is acting on the two surfaces, by a thin layer of gas trapped

between them, commonly known as squeeze-film. This phenomenon occurs due to the viscosity and the

compressibility of the gas. Namely, due to its viscosity, the gas trapped between the surfaces cannot be

immediately squeezed out, causing the amount of gas leaking out of the film during compression, to be

equal to the amount of gas enters it during decompression [1]. Furthermore, since the gas is compressible,

the pressure distribution between the surfaces fluctuates around a nominal profile whose average value is

higher than the ambient pressure (e.g. [2–4]). Accordingly, the average force induced by the entrapped gas

is repulsive, thus by these means, a freely suspended planar object, can be levitated above a rapidly

oscillating driving surface, without any mechanical contact.

It was formerly shown that the abovementioned levitation mechanism, commonly known as near-field

acoustic levitation, is governed by the amplitude of the driving surface’s oscillations (e.g. [2,5,6]). Thus it

can be used for manipulating planar objects vertically, without contact. A possible application for near-

field acoustic levitation is handling silicon wafers throughout inspection and manufacturing processes (e.g.

[7]), replacing traditional chucks, that make mechanical contact with the substrates, and thus,

contaminating the highly controlled work environment.

The two prevailing approaches for modelling the behavior of the squeeze-film, are the radiation pressure

theory (e.g. [8–10]), and Reynolds equation (e.g. [3,4]). However, when dealing with the dynamic

response of near field acoustically levitated objects, only the latter can be used, since the radiation

pressure approach refer to the entrapped gas as inviscid, and thus cannot describe the effect of damping.

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As shown below, Reynolds equation is a nonlinear partial differential equation, thus it is too complex for

practical uses, such as serving as a basis for model based control algorithms. Therefore, Ilssar and Bucher

[2], formulated a simplified model, describing the governing, slow dynamics of near-field acoustically

levitated objects, using a single, second order ordinary differential equation. This model is based on

Reynolds equation, so it comprises both the conservative levitation force originating from the

compressibility of the gas, and the damping force acting due to its viscosity, in an explicit manner.

However, since the model derived by Ilssar and Bucher, is restricted to the case where the driving surface

oscillates uniformly, as a rigid piston, Ilssar et al. [6] Generalized it so it represent the dynamics of a

system, where the driving surface vibrates as a general standing wave. The abovementioned general model

is introduced in the present paper, alongside two empirical methods, adjusting this formulation to a

specific configuration. Next, after numerical and experimental verification, this model is exploited for

sake of formulating a height dependent, gain-scheduled PID controller, enabling rapid and accurate

vertical positioning of a levitated object. Finally, the necessity of a resonance tracking loop, operating

along the levitation height control loop, is demonstrated.

2 The theoretical model

The drawing appears in Figure 1, describes schematically the system discussed in the present paper. This

system consists a driving surface, oscillating as a pure standing, flexural wave, in a constant frequency

denoted , and according to a constant spatial profile a . The investigated system also includes a freely

suspended planar object, levitated above the driving surface, due to the pressure elevation produced by the

rapid oscillations of the driving surface. The system considered throughout this paper is axisymmetric,

implying that the levitated object moves only vertically, and the spatial profile of the driving surface’s

oscillations depends only on the radial coordinate, denoted r .

Figure 1: Schematic layout of the investigated system, consisting a driving surface, a squeezed gas layer,

and a levitated object

It was shown in the past (e.g. [2,11–13]), that the dynamic response of the levitated object consists of two

time scales; a fast time scale, associated with the oscillations of the driving surface and the resulting low

amplitude oscillations exhibited by the levitated object, and a slow time scale, describing the transient

dynamics of the object. As seen in Figure 4, presenting four typical dynamic responses of levitated

objects, the slow evolution of the levitated object, is 2-3 orders of magnitude slower than the excitation,

implying that the dynamic response of the levitated object can be decomposed to a slow process, denoted

h , and a fast process.

2.1 Formulating the governing equations

Before formulating the system’s governing equations, the following non-dimensional measures are

defined

2 2

0 0

2 2 2

0 0 0 0 0

12 12, , , , , , .

a a a

r rp h h r a aP H H R T t

p h h r p h p h H h h H

(1)

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Here, , ap p are the pressure distribution inside the squeeze-film, and the ambient pressure respectively, 0h

denotes a reference, typical clearance between the driving surface and the levitated object, and stands

for the dynamic viscosity of the entrapped fluid. Moreover, denotes the non-dimensional excitation

function, whose magnitude is much smaller than unity, and is the squeeze number, governing the

instantaneous behavior of the squeeze-film. It was formerly shown that , , prevail the instantaneous

behavior of the system (see [2]).

To formulate the coupled dynamics of the system illustrated in Figure 1, the behavior of the squeeze-film

should be coupled with the dynamics of the levitated object. The first equation considered here is

Reynolds equation, representing the behavior of the fluid residing inside the squeeze-film. Assuming the

film is axisymmetric and isothermal, this equation takes the following non-dimensional form [2]:

3 2 .PHP

RH P H RR R T

(2)

Reynolds equation originates in the Navier-Stokes momentum and continuity equations, referring to the

entrapped fluid as an ideal gas. An essential assumption made in the derivation of Reynolds equation is

that the clearance between the levitated object and the driving surface is much smaller than the system’s

lateral dimensions. This assumption leads to omission of all inertia terms [3,4], leaving the equation only

with viscous and pressure terms.

In order to fully define the pressure distribution inside the squeeze-film, assuming the clearance H and its

slow component H are known, boundary conditions need to be imposed. The first boundary condition

needs to be taken, demands pressure continuity at the center of the film. Moreover, it is customary to

assume pressure release boundary conditions, dictating the pressure at the boundaries of the film to be

equal to the ambient pressure. I.e., the boundary conditions are taken as following

0,

,0, 1, 1.

R T

P R TP R T

R

(3)

It was shown by Minikes and Bucher [14,15], and by Nomura et al. [16] that when the driving surface and

the levitated object do not fully overlap, they induce pressure radiation. The latter results in energy

leakage, reducing the accuracy of the pressure release conditions. However, since in the system discussed

in this paper, the levitated object and the driving surface overlap, the boundary conditions given in (3)

seem valid.

As mentioned above, in order to describe the overall dynamics of the system, Reynolds equation needs to

be coupled with the equation of motion, describing the vertical translation of the levitated object. This

equation considers the pressure acting on the levitated object by the surrounding fluid, and the gravity

force. For sake of consistency with Reynolds equation, the equation of motion is formulated in terms of

the pressure distribution and the thickness of the squeeze-film, thus it takes the following form [2]:

122

0

2 2 2

0 00

21 d sin ,ap rH g

R P R H TT mh h

(4)

where g denote the gravitational acceleration.

3 Simplified modelling

As seen in Figure 4, the slow time scale, related to the transient response of the system, is much more

significant than the fast time scale, associated with the rapid oscillations of both the levitated object and

the driving surface. Thus, for practical applications, such as parametric studies or controlling the dynamics

of near-field acoustic levitation based system, it is sufficient to find a model, describing merely the slow

dynamics. Ilssar and Bucher [2], formulated a second order, ordinary differential equation, representing

only the slow time scale, assuming the driving surface oscillates uniformly, and Ilssar et al. [6],

HIGH FREQUENCY ACTUATORS AND PROCESSES 1581

generalized it so it suits the more general case considered here, where the driving surface oscillates as a

pure standing wave. This model is indeed simple enough to set the foundation for a model based

controller. Therefore, since it is used in the upcoming section for control purposes, the generalized model

derived by Ilssar et al is introduced here. Next, in order to make the model applicable, a numerical method

and an experiment based method, used to adjust it to a specific system, are introduced. Finally, an

experimental setup is introduced and modelled using both methods, in order to compare them, and to set

the ground for a model based control.

3.1 Introducing the general simplified model

It was shown by Ilssar and Bucher [2] that first term on the right hand side of the equation of motion (4),

representing the influence of the surrounding pressure on the levitated object, can be decomposed into two

distinct forces. The first force is the conservative levitation force emanating from the compression of the

entrapped gas, and the second force is the dissipative damping force, originating from its viscosity.

Moreover, in order to describe the damping associated with slow evolution of the system, the terms

connected with the fast time scale can be ignored. The latter implies that since the oscillations of the

driving surface are related to the fast time scale, the excitation form (i.e. the mode shape of the driving

surface being excited) does not affect the dissipative force related to the slow dynamics of the system.

Consequently, the non-dimensional form of this force, is taken as formulated by Ilssar and Bucher, who

derived it for the case of uniform excitation (see [2], equations 35-36)

2

0

2 3

0

3 d,

2 ddamping

a

r HF

p h H T

(5)

where dampingF is normalized by 2

0ap r .

Contrary to the damping force, the levitation force does depend on the excitation form, so it needs to be

modified according to a specific form of excitation. In order to calculate this force, Ilssar et al. [6]

considered a system where the driving surface oscillates arbitrarily, in proximity of a fixed wall whose

nominal distance from the driving surface is H . Moreover, in order to take nonlinear effects into account

they developed the pressure distribution into a second order asymptotic series, so they described the

degenerated system according to

2 3ˆ1 sin , , 1 , , .A BH T H H R T P R T R T R T (6)

where since the magnitude of is much smaller than unity, the small parameter is defined by

ˆ ˆ, , .H R H R R H R (7)

Obviously, since the parameter takes the small magnitude of the non-dimensional excitation function,

the norm of ̂ equals to unity, where for sake of consistency with the model derived in [2], the weight

function of the norm is chosen as 1.

The average force acting on the wall in the degenerated system, yields a good approximation for the

averaged levitation force acting on a levitated object whose nominal height is H . The latter is since the

levitation force is associated with the compression of the entrapped fluid which is hardly affected by the

levitated object’s motion. Namely, the evolution of the system is much slower than the rapid oscillations,

thus it hardly affects the compression of the gas. Moreover, the amplitude of the levitated object’s rapid

oscillations of the are 2-3 orders of magnitude smaller than the excitation magnitude (e.g. [2]), so the

levitated object’s fast motion can also be ignored.

Ilssar et al. [6] substituted (6) into Reynolds equation (2), and decomposed the obtained expression into

two equations, related to the terms of order and 2 . By harmonic balancing these equations, they

achieved approximated solutions for ,A B , depending on the excitation form ̂ . Once an approximated

expression for the pressure is found, the time averaged force acting on the wall, corresponding to the


slowly varying levitation force acting on a freely suspended object, levitated at the same average height,

can be found. This is done by integrating the pressure acting on the wall from both its sides, over its area,

and averaging over a single excitation cycle, in order to eliminate terms associated with the fast time scale.

Thus, the non-dimensional, slowly varying levitation force is taken as following

2 2 1

0 0 0

211 d d d ,

2levitationF R P R T K

(8)

where levitationF is normalized by 2

0ap r .

The function K is an excitation form dependent calibration function, denoting the dependency of the

averaged levitation force on the squeeze number. This function captures the relatively complex features of

the levitation process, considering only the slow dynamics of the system. It should be noted that it is

usually impractical to find a closed form solution for the second order approximation of the pressure

distribution, implying that a closed form for the function K can also not be found. Therefore, in the

following sub-section, two empirical methods for finding this function, are presented.

The slowly varying damping and levitation forces found above, provide a second order approximation for

the slowly varying influence of the surrounding gas, meaning that they describe the first term in the

equation of motion (4), averaged over one period of excitation. Therefore, (4) is averaged over a single

excitation period, followed by substitution of the forces given in (5) and (8), providing the following

simplified equation, accounting for the slow dynamics of the system illustrated in Figure 1

* 2

2

2 2 3

d d.

d d

s dC u K H CH H

GT H H T

(9)

Here, u representing the excitation magnitude, *

sC denoting a coefficient of the stiffness term, dC

standing for the coefficient of the damping term and G symbolizing the gravity, are given as followings:

* 2 2 4 3 2

0 0 0 0 0 0, , 3 2 , .s a du a r h C r p m h C r m h G g h (10)

Obviously, since the levitated object is no longer fixed, the squeeze number is a functional of the slow

evolution H .

3.2 Approximating the calibration function, empirically

As stated above, in order to use (9)-(10) for modelling of the slow dynamics of an acoustically levitated

object, a closed form for the function K must be found. In the upcoming sub-sections, two empirical

methods, enabling to approximate this function, using either numerical simulations or experiments, are

presented.

3.2.1 The numerical approach

Since the motion of the levitated object was neglected when formulating the levitation force, it should also

be ignored when calculating the function K . Therefore, when calculating K based on numerical

simulations, the levitated object can be assumed fixed, so the numerical scheme consists of numerous

ordinary differential equations, achieved by discretizing Reynolds equation (2) as following:


2

2

2 22 2

2 2

13

1

n nn n

n n n nn

nn n n

n nnn

H PP P H PH P

T H R R R R

PP P HH H P

H R R H T

(11)

where 1,n N denotes the index of the radial node residing at 1 1R n N .

In the analyses made in this study, the spatial derivatives in (11) were calculated using central differences

formulas, while the needed values at the external nodes 1,n N were determined according to the

boundary conditions (3). Moreover, assuming the excitation form and magnitude are known, the radial

dependent, instantaneous clearance H is known, leaving the pressure as the only unknown function. In

order to find the instantaneous values of the pressure distribution, the coupled system (11) is solved using

numerical integration.

It should be noted that the derivation of the levitation force presented above, ignores the transient behavior

of the pressure, which is a valid assumption since the time constants of the pressure are much lower than

those of the levitated object's evolution. Hence, according to (8), the levitation force of a specific

configuration, is obtained by integration of the pressure distribution obtained from a numerical simulation,

over the system’s area, and averaging over an integer number of excitation periods, after convergence to

steady state. Calculating the levitation force for various squeeze numbers in the desired work range, and

division by 2 , results in the desired values of K , where should be kept small in the simulations,

in order to comply with the asymptotic assumptions, stating that this parameter is much smaller than unity.

Finally, in order to approximate a closed form expression for K , curve fitting on the obtained discrete

values should be done.

3.2.2 The experiment-based approach

As in the numerical approach, the dynamics of the levitated object can also be disregarded when

approximating the function K , based on experimental data. Thus, approximating K

experimentally can be done by measuring the time averaged levitation heights at steady state eqH ,

obtained under various constant excitation magnitudes equ . The steady state measures ,eq eqH u are related

by the following transcendental relation, obtained from (9), when nulling the time derivatives of H

* 2 2 .s eq eq eqC u K H GH (12)

This equation, obtained for every pair ,eq eqH u , provides a single value of K . Curve fitting on these

discrete values, yields a closed form approximation that can be used in (9).

Obviously, since it is based on experimental data, the experiment-based approach helps correcting

inaccuracies in the model, such as pressure release boundary conditions, thus it is more reliable than the

numerical approach. Moreover, the experiment-based approach can take additional static forces, such as

magnetic forces acting on the system, into account. These forces are taken as supplementary terms on the

right hand side of (12), thus when using the experiment-based approach, K is calculated considering

the additional static forces (see sub-section 3.3).

3.3 Modelling the experimental setup

Figure 2 shows the setup on which all the experiments discussed in this paper were performed. This setup

consists a piezoelectric actuator, composed of a Langevin transducer, and a structure used to magnify the


displacements at its base, in order to produce significant excitation amplitudes. This actuator was used to

supply the required excitation, for levitating a planar object whose weight and radius are 134.7 g and 50

mm respectively. For sake of high efficiency, all components comprising the actuator are designated to

operate at approximately 28.5 kHz, where all axial elements resonate at their first elastic longitudinal

mode, and the upper plate whose top surface functions as the driving surface, resonates at its second

axisymmetric flexural mode. Obviously, the system cannot be not perfectly leveled, thus undesired

degrees of freedom such as lateral translation and tilting can arise (e.g. [17,18]). Therefore, as seen in

Figure 2, a magnetic centering array, using magnetic repulsion forces between three centering elements

and a series of magnets placed on the perimeter of the levitated object, was implemented.

In order to preserve high performance, and to meet the model's assumption, referring to the excitation as a

pure standing wave, in all of the experiments discussed throughout this paper, the excitation frequency

was determined according to a Phase-Locked loop, resonance tracking algorithm (e.g. [19,20]).

Nevertheless, in all of the experiments the frequency changed by less than 0.1%. Hence, these variations

are not taken into account in the analyses.

Figure 2: Right: A photograph of the experimental rig, showing the piezoelectric actuator in the middle

and the levitated object with centering the passive magnets (set aside for visibility). Left: Schematic

layout of the piezoelectric actuator, and a sample levitated object.

In order to model the experimental setup, using both approaches suggested above (see sub-sections 3.2),

three sets of experiments were performed. In the first experiment set, the excitation form ̂ , denoting the

mode-shape of the driving surface being excited, was measured using a laser Doppler sensor (PolytecTM,

OFV-303), under different excitation magnitudes, when the system was loaded with the mass mentioned

above. According to these experiments, the excitation form does not change with the levitation height,

thus ̂ is taken as the measured, levitation-height independent function, complying with (7). Next, the

relation between the amplitude of the voltage supplied to the system, denoted as V , and the amplitude at

the center of the loaded driving surface, was measured using a laser Doppler sensor (PolytecTM, OFV-

551), under excitation with several intensities in the allowable range. Using this experiment set, and the

measured excitation form, relating the amplitude at the center of the driving surface to a , the relation

between V and the excitation magnitude u was found. Finally, the steady-state height of the levitated

object was measured several times under numerous constant excitation magnitudes, using a laser

triangulation sensor (KeyenceTM, LK-H008). These experiments, combined with the data from the former

experiments, relate the steady-state levitation height to the excitation magnitude, enabling the calculation

of K , according to the experiment-based approach. It should be noted that although all three

experiment sets are essential in order to apply the experiment-based approach, the first experiment set is

sufficient for the numerical approach.


Figure 3 compares the numerical and the experimental forms of K , obtained according to the

experiments detailed above. This figure shows that the experimental values of K , achieved over 20

experiments, in which more than 20 steady state excitation magnitudes and their corresponding steady

state heights were measured, are quite repetitive. However, there are significant deviations between the

values obtained from the different approaches, indicating an error in the numerical approximation of

K . These deviations can strongly be attributed to the fact that the numerical approach does not take

the force induced by the magnetic array presented in Figure 2, into account, where the experiment based

approach does consider it. Namely, as mentioned above, when using the experiment based approach, the

magnetic force is taken as an additional term in the right hand side of (12), thus it is included in K .

Figure 3: Comparison between the values of K(σ) obtained using the different methods. (a) The curve

fitted to the measurements, (b) The area enclosing all of the measurements, (c) The curve fitted to the

numerically obtained values (d) The values calculated according to the numerical approach, (e) The

difference between fitted curves.

Since the disagreement between the experimental and the numerical values of K , originate vastly

from the magnetic forces, it seems appropriate to validate the simplified model (9) with the numerically

obtained K , ignoring the effect of these forces. For this sake, in Figure 4, a few typical dynamic

responses, obtained using this model are compared to their corresponding responses achieved using the

original, Reynolds equation based model (2)-(4). The latter was calculated numerically by solving the

equation of motion (4) together with the ordinary differential equation system (11), using finite differences

in space and numerical integration in time.

Curve Mass [g] Excitation

frequency [kHz] ‖a‖ [μm]

δ at steady

state

Steady-state

error

a 134.7 28.5 7 0.0696 0.1659%

b 134.7 28.5 2.5 0.0360 0.0779%

c 1000 4 2 0.0870 0.3976%

d 20000 2 4 0.3244 2.3425%

Table 1: Significant characteristics of the dynamic responses presented in Figure 4

From Figure 4 and from Table 1, presenting important properties of the compared responses, it can be

seen that the simplified model calculated utilizing the numerical approach, accurately describes the slow

dynamics of the system formulated by the Reynolds equation based model, when is small. However,

since the simplified model assumes that 1 , as this condition wakens, the correlation between the

simplified model and the original model deteriorates. Nevertheless, in most practical uses the small


parameter does not reach such high values. The good correlation between both models, supports the

hypothesis that the major cause of the deviations between the numerical approach and the experiment-

based approach is the force applied by the magnetic array.

Figure 4: Comparison between typical dynamic responses whose properties are given in Table 1. In solid

black curves: the results obtained from the original model. In dashed orange curves: the results obtained

from the simplified model utilizing the numerically obtained values of K(σ).

Figure 5: Solid black curve: a measured dynamic response of the levitated object under excitation with

several constant magnitudes. Dashed orange curve: The corresponding responses obtained using the

simplified model (9) utilizing the experimentally obtained K(σ).

Figure 5 compares between a measured and its corresponding theoretical dynamic response, obtained

under excitation with several constant magnitudes. Here, the theoretical results were obtained using the

simplified model (9), utilizing the experimentally obtained K . It can be seen from Figure 5, that the

deviations between the theoretical and the measured steady-state heights do not exceed 2%, over the entire

work range. Moreover, at low levitation heights, the theoretical model accurately predicts the dynamic


behavior of the system as well. The latter implies that beside the stiffness term found empirically, the

damping term calculated theoretically almost perfectly predicts the actual damping. However, as the

levitation height increases, the theoretical prediction of the transient response deteriorates. For instance,

when the levitation height is around 190 µm, the predicted rising time is about 9 times shorter than its

experimental value. These deviations can be explained by the fact that the simplified model (9) is based on

Reynolds equation, derived assuming the clearance between the levitated object and the driving surface is

very small. Thus, as this clearance grows, the validity of (9) deteriorates.

Obviously, the theoretical damping term can be adjusted empirically, as shown by Ilssar and Bucher [2].

However, the simplified model (9) seems adequate for purposes of designing a controller for the system

presented in Figure 2, assuming the controller is sufficiently robust. Thus in the next section, the model (9)

is utilized as a basis for a model based controller, governing the slow dynamics of the system.

4 Closed loop control

Both the conservative and the dissipative terms, comprising the simplified model (9), were derived under

the assumption that the excitation magnitude and frequency are constant, considering terms up to order 2 . The latter implicitly suggests that (9) is valid if the slow dynamics of the system is sufficiently slow

such that

2d d .T (13)

From the definition of , that includes both the slow evolution of the system, and the excitation

magnitude, it is implied that if the excitation magnitude varies slowly so the latter is met, the simplified

model still holds. Thus, (9) can function as a basis for a model based controller, governing the slow

dynamics of the system, by changing the excitation magnitude sufficiently slow. Indeed, in the

forthcoming sub-sections, a gain-scheduled controller, governing the slow dynamics of the experimental

setup is designed based on (9). Next, after numerical simulations used to assess the performance of the

controller, the latter is evaluated experimentally.

4.1 Controller design

A gain-scheduled controller consists of a family of linear controllers of the same form, designed for small

deviations around different operating points. The latter leads to a nonlinear controller whose coefficients

vary with a scheduling variable, indicating on the appropriate operating point and thus also on the suitable

coefficients. Thus, in order to devise a gain-scheduled controller, relying on the simplified model (9), the

latter is linearized around a family of equilibrium points, using first order Taylor expansion, providing the

following LPV (linear parameter-varying) model

*

2* 2

2 2 3 3 2

2 2dd 1 d.

d d deq

eq s eq eqd

s eq

eq eq eq eqH

K H C u K HK H CH HC u H u

T H H H H T H

(14)

Here, , 1u H are small deviations around the equilibrium point ,eq equ H denoting a constant

excitation magnitude and its resulting steady state levitation height respectively, where the relation

between ,eq equ H is obtained from (12).

The controller chosen to be implemented on the system is of a PID form, based on the velocity algorithm

suggested by Kaminer et al. [21]. Here, the proportional and the integral actions act on the error between

the reference height r and the levitation height H , whereas the derivative action acts on H [22].

Moreover, the three controller parameters , ,p i dk k k , denoting the proportional, integral, and derivative

gains respectively, vary with the levitation height H , implying that H is the scheduling variable. For


causality, and in order to avoid amplification of high frequency noise, the differentiator is connected in

series to a low-pass filter whose cut-off frequency is denoted LPF , as illustrated in Figure 6. According to

the above, the gain-scheduled controller is given by

p du k H r H k H (15)

where

d d d

, .d d d

i LPF LPF

Hk H r H

T T T

(16)

As seen in Figure 6, the measured signal goes through an analog low-pass filter, that was not taken into

account in (15), (16) since its cut-off frequency is chosen to be much higher than those of the closed loop

and the filter connected to the differentiator. Another component that is disregarded in the controller

design is an anti-windup loop, also appear in Figure 6.

Figure 6: Schematic layout of the controlled system.

It should be noted that under the assumption of zero steady-state error, the controller given by (15), (16)

has no hidden coupling terms, meaning that around each equilibrium point, the gain-scheduled controller

is similar to the linear controller designed to achieve the desired performance at this point [23]. Therefore,

the controller coefficients at every operating point were chosen in a way that satisfies the required

performance locally, according to the LPV (14). Here, in order to achieve rapid convergence to steady

state with reduced oscillations, the controllers were designed to have two couples of identical and thus real

valued poles, implying on critical damping. These requirement reduces the controller designs into two

degrees of freedom, determined by setting the cut-off frequencies of the closed loop and the

differentiator's low-pass filters to desired values, chosen to attenuate high frequency noise without

affecting the system’s performance.

Ilssar et al. [6] stated that if the scheduling variable of the closed-loop system discussed above, is slowly

varying and the initial state is sufficiently close to the initial equilibrium point, than the response of the

system is uniformly bounded. Furthermore, if the scheduling variable H converges to a constant steady

state value, than the error vanishes as T . The abovementioned properties can be proven following

Khalil [24], who verified these properties for the velocity algorithm given in [21].

Cut-off frequency of the closed loop 10 Hz

Cut-off frequency of the differentiator's filter (ωLPF) 300 Hz

Cut-off frequency of the analog Low-pass filter 2000 Hz

Sampling rate 2000 Hz

Table 2: The parameters of the closed loop system.


In order to assess the behavior of the chosen gain-scheduled controller, its performance was examined

utilizing a numerical scheme, describing the closed-loop system discussed above. Here, the levitation

system was modelled based on the simplified model, where K was approximated experimentally using

the data, collected in the preliminary experiments described in the sub-section 3.3.

Figure 7 shows the step responses of the simulated closed loop system, to a reference signal, varying

between 60 µm and 180 µm, obtained utilizing the abovementioned numerical scheme, with the values

appear in Table 2. In order to describe a more realistic situation, measurement noise was added to the

abovementioned simulation, where, since the simplified model is valid only if the condition (13) is met,

the measurement noise was taken as colored noise with maximal magnitude of 2 µm, whose cut-off

frequency is 300 Hz. As seen in Figure 7 (orange curves), the gain-scheduled controller is expected to

provide a rapid, and non-oscillatory convergence to steady state, with no steady-state errors, in the entire

work range.

Figure 7: A theoretical (orange curves), and its corresponding measured (blue curves) dynamic response

of the closed loop, to an input reference, varying in steps between 60 µm and 180 µm.

4.2 Implementing the control algorithm

Following the satisfying theoretical performance of the controller, the latter was implemented

experimentally, using the system illustrated schematically in Figure 6. The system consists of the

experimental setup presented in Figure 2, whose excitation frequency was determined by a resonance-

tracking algorithm, an analog low-pass filter, reducing high frequency noise, and the control loop designed

above. As seen in Figure 6, the controller closes a feedback on the levitation height, measured using a

laser triangulation sensor (KeyenceTM, LK-H008), where since the controller’s input is non-dimensional,

the physical levitation height yielded by the sensor, was converted to a non-dimensional form using (1).

Moreover, the controller outputs the excitation magnitude u , but in practice it commands the amplitude of

the input voltage V . Thus, the former was converted into the desired values using the relation found in the

second preliminary experiment set, discussed in sub-section 3.3.

In addition to the theoretical results discussed above, Figure 7 also presents the corresponding measured

results, filtered such that only frequencies lower than 35 Hz are taken into account. It can be seen that

there is a good correlation between the theoretical and the measured results, however, at significant

levitation heights, the agreement between the theoretical and the measured transient responses, weaken.

These deviations are due to the fact that as shown above, the theoretical model overestimates the actual

damping at significant heights, resulting in an oscillatory measured response where the theoretical model

predicts a critically damped response. Moreover, it can be seen from both the theoretical and the


experimental responses that at low altitudes, the controller is more sensitive to noise. Thus around 60 µm,

the experimental gain-scheduled controller did not keep the desired steady-state height. Yet, around 70 µm

and beyond, it tracked the commended steady-state height, accurately.

Finally, although it is not the focus of this paper, the importance of the phase-locked loop, determining the

excitation frequency of the experimental system, should be emphasized. Figure 8 shows the measured

dynamic response of the levitated object, as presented in Figure 7, and the instantaneous excitation

frequency, determined by a phase-locked loop resonance tracking algorithm. From this figure one should

notice that the resonance frequency of the system depend on the levitation height. This phenomenon

occurs since the thickness of the squeeze-film affects the stiffness and damping of the system (e.g. [3]).

Moreover, additional causes such as temperature and moisture variations, change the resonance frequency

as well. Therefore, as seen in Figure 8, the frequency variation also has a time scale, much slower than the

slow dynamics of the system. Obviously, since the system discussed in this paper is a high-Q system,

slight deviations from resonance can reduce the excitation magnitude, significantly.

In order to enable the controller designed above to work properly, the excitation magnitude should depend

directly on the controlled amplitude of the input voltage. Thus, working with a constant excitation

frequency seems impractical, and so, a resonance-tracking algorithm must be an integral part of any

control loop, governing the dynamics of near-field acoustically levitated objects.

Figure 8: Blue curve: the measured levitation height of the system (as presented in Figure 7). Black

curve: the excitation frequency, determined by the phase-locked loop, resonance tracking algorithm.

5 Conclusions

It was shown in the past that the dynamic behavior of near-field acoustically levitated objects consists of

two time scales. A slow time scale, associated with the transient response of the levitated object and a fast

time scale, related to the excitation, and the resulting rapid oscillations of the levitated object. It was also

shown that the slow time scale is much more significant than the fast time scale. Thus, in order to control

the dynamics of near-field acoustically levitated objects, it is usually sufficient to consider merely their

slow dynamics, implying that a relatively slow control system is adequate. Nevertheless, due to the

nonlinear behavior of the system, originating in the gas trapped between the driving surface and the

levitated object, to achieve satisfactory performance, the control algorithm should be nonlinear as well.

In this paper, a novel semi-analytical model, capturing the slow dynamics of a near-field acoustically

levitated object, using a single, second order ordinary differential equation, was derived. Based on this

model, a continuous gain-scheduled PID controller was designed, and provided adequately good

performance, although the simplified model on which it is based underestimates the real damping of the

system at significant air-gaps. Thus, it is the authors' opinion that based on the performance presented

above, the simplified model developed in this paper is suitable to serve as the basis for more sophisticated


model based controllers. Moreover, it is safe to assume that an empirically corrected damping term would

have resulted in a superior controller.

Finally, it was demonstrated that since the thickness of the squeeze-film affects the stiffness and damping

of the system, and because temperature and moisture variations may change the properties of the

entrapped gas, the resonance frequency is constantly varying. Moreover, due to the low damping of the

piezoelectric actuator, small deviations from the resonance, can reduce the excitation magnitude

dramatically. Thus, in order to implement a control loop, governing the dynamics of near-field

acoustically levitated objects, the latter must include a resonance-tracking algorithm.

Acknowledgements

This research was funded by the Israeli Ministry of commerce under the Metro 450, Magnet program and

the Ministry of Science, Technology and Space.

The authors would like to express their gratitude to Dr. Nadav Cohen who designed the experimental

setup, and Mr. Ran Shaham who built the resonance-tracking loop and helped implementing the control

algorithm.

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