actuators
Article
Active Magnetic Bearing Online Levitation Recovery
through µ-Synthesis Robust Control
Alexander H. Pesch 1, * and Jerzy T. Sawicki 2
1
2
*
Department of Engineering, Hofstra University, Hempstead, NY 11549, USA
The Center for Rotating Machinery Dynamics and Control, Department of Mechanical Engineering,
Cleveland State University, Cleveland, OH 44115, USA; [email protected]
Correspondence: [email protected]; Tel.: +1-516-463-7175
Academic Editor: Eric H. Maslen
Received: 1 December 2016; Accepted: 5 January 2017; Published: 8 January 2017
Abstract: A rotor supported on active magnetic bearings (AMBs) is levitated inside an air gap by
electromagnets controlled in feedback. In the event of momentary loss of levitation due to an acute
exogenous disturbance or external fault, reestablishing levitation may be prevented by unbalanced
forces, contact forces, and the rotor’s dynamics. A novel robust control strategy is proposed for
ensuring levitation recovery. The proposed strategy utilizes model-based µ-synthesis to find the
requisite AMB control law with unique provisions to account for the contact forces and to prevent
control effort saturation at the large deflections that occur during levitation failure. The proposed
strategy is demonstrated experimentally with an AMB test rig. First, rotor drop tests are performed
to tune a simple touchdown-bearing model. That model is then used to identify a performance
weight, which bounds the contact forces during controller synthesis. Then, levitation recovery trials
are conducted at 1000 and 2000 RPM, in which current to the AMB coils is momentarily stopped,
representing an external fault. The motor is allowed to drive the rotor on the touchdown bearings until
coil current is restored. For both cases, the proposed control strategy shows a marked improvement
in relevitation transients.
Keywords: AMB; levitation; recovery; robust control; touchdown bearing
1. Introduction
Active magnetic bearings (AMBs) support a rotating shaft with a magnetic field generated by
electromagnets controlled in feedback. The shaft, consequently, is levitated in an air gap and has
no physical contacts. This has the advantages of no friction losses, less maintenance, longer life, no
need for lubrication, etc. Practical AMB technology has been commercially available for many years.
However, the application of AMBs in industry remains limited. This is due, in part, to the lack of
familiarity with AMBs among the body of practicing engineers. As a result, end users are reluctant
to purchase AMB systems for fear of the consequences of levitation failure, such as damage to the
shaft, housing, or process. This work proposes an AMB control method that provides robust levitation
recovery in the event of the acute loss of levitation.
In general, there are two types of AMB failures; internal and external faults [1]. Internal faults are
associated with the components of the AMB. An example of an internal fault is a position sensor failure,
which would result in a severing of the feedback control loop and loss of levitation. The prevention of
internal faults has been addressed by the robustness of individual components and, in certain cases,
component redundancy, such as having more magnetic poles than is necessary. There has also been
research on proper control of redundant AMB systems that are experiencing component failure [2].
Conversely, external faults are those that arise from factors outside of the AMB system. For example, a
shaft supported on AMBs that processes natural material may encounter an aberration in the material’s
Actuators 2017, 6, 2; doi:10.3390/act6010002
www.mdpi.com/journal/actuators
Actuators 2017, 6, 2
2 of 14
consistency, which applies a onetime unsupportable load. Another example of an external fault is a
momentary loss of power. In both cases, the components of the AMB system remain intact.
The control solution proposed in the current work addresses the problem of external faults. In the
event of an external fault, there is an acute loss of levitation and, although there are no sustained
component failures, recovery of levitation may be prevented by two phenomena. First, the fact that the
rotor is at a large deflection can drive the control current into the saturation region [3]. Saturation of
control actuation is widely known to have adverse effects on stability. The situation is exacerbated by
the coincidence of unbalance forces. Second, shaft contact with the touchdown (TD) bearings while
rotating will impart other forces on the rotor and may result in a so-called contact mode of vibration [4].
TD bearings are usually traditional bearings that are oversized for the shaft and do not touch the
shaft during levitation but rather catch the shaft in the event of levitation failure. A contact mode is
a non-linear but stable mode of vibration of the rotor in intermittent contact with the TD bearings.
Contact modes are sustained by the motor feeding energy into the system. Different types of contact
modes are possible, including forward whirl (continuous sliding), backward whirl (continuous rolling),
and repeated impact (bouncing in a pattern). Whether and which type of contact mode occurs depends
on many factors such as unbalance magnitude and angle, running speed, TD bearing parameters
such as stiffness and friction, free rotor natural frequencies, etc. The levitation recovery control
method proposed and demonstrated in this work uses the robust control strategy µ-synthesis, which is
model-based. These two phenomena, which may prevent levitation recovery while rotating, are taken
into account with the system model such that the synthesized control is able to achieve relevitation.
This development is novel compared to typical robust controllers for AMBs such as µ-synthesis [5]
and robust H ∞ [6], which can guarantee stability and performance inside the specified operating region
(including deflection, load, rotor speed, etc.), utilizing parametric uncertainties and performance
weights cast as physical limits on input and output signals, but have no consideration for stability or
performance if that operating region is violated in any way. Once the linear control law is designed,
that law is enforced regardless of whether the situation it was synthesized for is actually the case. For a
simple example, consider an AMB controller that is designed to support a certain load with deflection
limited to 10 µm and current limited to 10 A. Assume that, after robust controller synthesis, the resulting
controller has a stiffness of 1 A/µm. If, during an acute levitation failure, the rotor is temporarily at
100 µm deflection, the robust controller will attempt to send 100 A to the AMB coil. This may saturate
the current limit of the hardware and the formerly robust system is actually uncontrollable.
Much of the existing body of literature in the area of control for relevitation addresses the
synchronous disturbance due to TD bearing contact, e.g., [7–9]. Therefore, algorithms similar to those
for synchronous unbalance compensation have been developed. Good experimental results have been
obtained using these approaches; however, a period of automatic disturbance learning is often required,
which diminishes their practicality. There have also been efforts in the area of control of actuated TD
bearings towards levitation recovery such as [10]. More similar to the method proposed in this work
is [11], which implements a switching algorithm for an AMB controller manually tuned for levitation
recovery of an industrial system. Levitation fault recovery with differential flatness control using
noise filtering and derivative estimation through B-Splines has also been studied [12], although no TD
bearing interaction was considered. That approach naturally lends itself to non-linear systems due to
the parameterization afforded by the concept of differential flatness. However, the method proposed
in the current study is not directly compared to non-linear control methods. Preliminary results for the
current study were presented in [13].
The paper is structured as follows: Section 2 covers the materials and methods used to develop
and demonstrate the proposed levitation recovery control, including an explanation of µ-synthesis and
its application for levitation recovery, the experimental test rig, crafting of the contact force performance
weight, and details of controller synthesis for the test rig. Then, Section 3 presents results of controller
synthesis and experimental implementation of the proposed method on the test rig. Section 4 gives
discussion of the significance of the proposed technique and findings.
Actuators 2017, 6, 2
3 of 14
of controller synthesis and experimental implementation of the proposed method on the test rig.
Actuators 4
2017,
6, 2 discussion of the significance of the proposed technique and findings.
3 of 14
Section
gives
2. Materials and Methods
2. Materials and Methods
This section covers the materials and methods used in development and experimental
This section
covers
the materials
and methods
used in development
demonstration
of the
proposed
AMB levitation
recovery technique,
starting with aand
briefexperimental
explanation
demonstration
of
the
proposed
AMB
levitation
recovery
technique,
starting
with
a
brief
of μ-synthesis robust controller design. The application of μ-synthesis to controllerexplanation
design for of
a
µ-synthesis
robust
controller
design.
The
application
of
µ-synthesis
to
controller
design
for
a
typical
typical AMB system under normal operating conditions is discussed. Then, the way that procedure
AMB
systemto
under
normal
operatinginconditions
discussed.
Then,
the way
procedure
is modified
is
modified
ensure
relevitation
the eventisof
momentary
failure
is that
explained.
Details
of the
to
ensure
relevitation
in
the
event
of
momentary
failure
is
explained.
Details
of
the
experimental
AMB
experimental AMB test rig on which the proposed technique is demonstrated are then presented.
TD
test rig on
which
the proposed
technique
is the
demonstrated
then presented.
TD
bearing
contact
bearing
contact
forces
are estimated
through
tuning of a are
numerical
simulation
with
a simple
TD
forces are
estimated
through
tuning
a numerical
with
a simple
TD bearing
model
bearing
model
to drop
test the
data.
The of
contact
forces simulation
are used to
craft
a frequency
dependent
to
drop
test
data.
The
contact
forces
are
used
to
craft
a
frequency
dependent
performance
weight
for
performance weight for synthesis of the recovery controller. Finally, the remaining details of the
synthesis of
the recovery
controller.
Finally,
the remaining details of the controller synthesis for the test
controller
synthesis
for the
test rig are
reported.
rig are reported.
2.1. μ-Synthesis Robust Controller Design for Typical AMB Operation and Levitation Recovery
2.1. µ-Synthesis Robust Controller Design for Typical AMB Operation and Levitation Recovery
The controller design strategy μ-synthesis was developed to accommodate a system with
The controller design strategy µ-synthesis was developed to accommodate a system with
structured uncertainties. The controller synthesis is carried out with an uncertain plant model using
structured uncertainties. The controller synthesis is carried out with an uncertain plant model using
the concepts of liner fractional transformation (LFT) and the structured singular value μ. These tools
the concepts of liner fractional transformation (LFT) and the structured singular value µ. These tools
allow for consideration of multiple parametric uncertainties at the points where they appear in the
allow for consideration of multiple parametric uncertainties at the points where they appear in the
model, including the magnitude and type of each uncertainty and therefore any interactions between
model, including the magnitude and type of each uncertainty and therefore any interactions between
them. Consequently, a robust closed-loop can be designed which is not overly conservative at the
them. Consequently, a robust closed-loop can be designed which is not overly conservative at the cost
cost of performance.
of performance.
The μ-synthesis problem formulation for a generalized system is shown in Figure 1a where P is
The µ-synthesis problem formulation for a generalized system is shown in Figure 1a where P is
the plant, K is the controller, and Δ is the uncertain matrix. The structure of matrix Δ is a result of
the plant, K is the controller, and ∆ is the uncertain matrix. The structure of matrix ∆ is a result of
where each parametric uncertainty occurs in the system model.
where each parametric uncertainty occurs in the system model.
(a)
(b)
(c)
Figure 1. (a) Generalized μ-synthesis framework; (b) μ-synthesis framework for basic AMB controller
Figure 1. (a) Generalized µ-synthesis framework; (b) µ-synthesis framework for basic AMB controller
design,
AMB controller
controller design.
design.
design, and;
and; (c)
(c) μ-synthesis
µ-synthesis framework
framework for
for robust
robust levitation
levitation recovery
recovery AMB
The weights WI and WO are transfer function matrices that are selected to stipulate the frequency
The weights WI and WO are transfer function matrices that are selected to stipulate the frequency
dependent performance
from the closed-loop system. The matrix M is found as the lower LFT, Fl, of
dependent performance from the closed-loop system. The matrix M is found as the lower LFT, Fl , of
the weighted plant P′0 and the controller.
the weighted plant P and the controller.
(1)
M = Fl (P′, K )
M = Fl (P0 , K)
(1)
The μ-synthesis framework is then cast with the upper LFT of the weighted closed-loop with the
uncertainty
perturbation,
which maps
input
w to
performance
response
z.
The µ-synthesis
framework
is thendisturbance
cast with the
upper
LFT
of the weighted
closed-loop
with the
(2)
uncertainty perturbation, which maps disturbance
input
w
to
performance
response
z.
z = Fu (M , Δ)w
The smallest perturbation matrix Δ, aszevaluated
= Fu ( M, ∆with
)w the maximum singular value, which has
(2)
the defined structure Δ and which destabilized the system M, yields the structured singular value μ.
The smallest perturbation matrix ∆, as evaluated with the maximum singular value, which has
the defined structure ∆ and which destabilized the system M, yields the structured singular value µ.
Actuators 2017, 6, 2
4 of 14
µ( M) =
1
inf{σ (∆) : det( I − M∆) = 0, ∆ ∈ ∆}
(3)
It is the structured singular value from which µ-synthesis is named.
The controller synthesis procedure seeks to find a controller by iterating to minimize µ(M). If the
resulting µ-value is less than unity, it indicates that a greater than allowed uncertainty perturbation is
required to destabilize the system. Therefore, the system closed-loop with the synthesized controller
is stable and robust to the bounded uncertainties. For a derivation and thorough discussion of
µ-synthesis, the reader is referred to the excellent tomes [14,15].
There are many specialized applications of µ-synthesis for control of AMBs, which take advantage
of the model-based nature of µ-synthesis, such as for machining chatter attenuation [16], machining
tooltip tracking [17], and hydrodynamic bearing oil-whip stabilization [18]. However, for a basic
AMB system, control is mainly concerned with stabilizing the inherently unstable open-loop and
creating practical bearing stiffness in the presence of the flexible modes of the rotor and gyroscopic
effect. Therefore, µ-synthesis for a basic AMB system is as illustrated in Figure 1b. The nominal
plant to be controlled is the AMB-Rotor System. This consists of a finite element (FE) model of the
rotor, four radial AMB forces that are linearized and expressed as current and position stiffnesses, and
an amplifier model, which quantifies the AMB slew-rate and delays due to digital implementation.
The parametric uncertainty perturbations account for AMB linearization errors and a varying running
speed. An exogenous input and performance output are defined to achieve bearing stiffness and
are physical load and allowable deflection, respectively. The load is external load on the rotor
(often taken at the AMB force center location for convenient plant assembly) and position is the
rotor lateral deflection (often taken at the AMB sensor location for convenient posteriori evaluation).
The performance weight W L on load is crafted to account for rotor weight at low frequency and
unbalance load across the operational speed range and then roll off at high frequencies outside of the
range of interest. The weight on position W P1 is crafted to require small deflection at low frequency and
a practical orbit size across the operational speed range before rolling up at high frequency. The noise
exogenous input is disturbance on the four sensor signals and is weighted to reflect the quality of the
actual signals. The current output is the control current. It is weighted with W C , which requires the
controller response to stay below the AMB bias current level, and then rolls off as to not saturate the
amplifier-coil slew-rate. Earlier roll-off may be specified to prevent spillover effect where the controller
excites rotor flexible modes that were neglected in the model.
The proposed µ-synthesis scheme for fault recovery, shown in Figure 1c, is similar to that for
basic AMB operation with two modifications. First, the weight on position performance is relaxed
to allow for deflections up to the TD bearing clearance. Therefore, the resulting controller will not
violate the AMB current limits in the case that the rotor should momentarily lose levitation. Second,
an additional exogenous input is defined, which accounts for any forces on the rotor due to contact
with the TD bearings. The input is applied at the FE nodes corresponding to the TD bearing locations.
The weight W TD is selected to bound the magnitude of the TD bearing force at all frequencies of
interest. This weighted input requires that the closed-loop system be stable in the presence of contact
with the TD bearing. This is true for any phase angle of TD bearing force due to the nature of the
µ criterion.
For the remainder of this paper, the µ-controller designed for acceptable performance under
normal operating conditions is to be called the performance controller and µ-controller designed for
robust fault recovery is called the recovery controller. The recovery controller, although stabilizing
under extreme deflections, is not expected to yield closed-loop bearing stiffness acceptable for healthy
rotation. Therefore, a recovery scheme is utilized in which the performance controller is used until
a delevitating event is detected, at which point the AMB is automatically switched to the recovery
controller. When the acute fault has passed, the AMB is automatically switched back to the performance
controller after a prescribed time within an acceptable deflection limit.
Actuators
Actuators 2017,
2017, 6,
6, 22
55of
of14
14
2.2. Experimental Test Rig
2.2. Experimental Test Rig
The proposed fault recovery scheme is demonstrated on the experimental AMB test rig shown
The 2.
proposed
recovery scheme
is demonstrated
on the (Calgary,
experimental
AMB testconsists
rig shown
in Figure
The testfault
rig, manufactured
by SKF
Magnetic Bearings
AB, Canada),
of
in
Figure
2.
The
test
rig,
manufactured
by
SKF
Magnetic
Bearings
(Calgary,
AB,
Canada),
consists
a rotor supported on two radial AMBs and one thrust AMB. The rotor is driven by a DC electric motor
of aarotor
supported
two radial
and onethe
thrust
AMB.
The
rotor
is driven
by a DC
electric
via
flexible
coupling.onThere
is one AMBs
disk between
bearings
and
each
AMB
has a target
rotor.
The
motorrotors
via a flexible
coupling.
There
is one disksized
between
the bearings
and each
AMB
targetstator
rotor.
target
are of material
and
appropriately
optimize
the magnetic
force
fromhas
theaAMB
The target
rotors
are ofare
material
appropriately
optimize
forceoutside
from the
coils.
The TD
bearings
rollingand
element
type and sized
are situated
on the
the magnetic
shaft directly
of AMB
each
stator coils.
TDrotor.
bearings
element
and areissituated
theshaft
shaft is
directly
outside
radial
AMB The
target
Theare
TDrolling
bearing
radialtype
clearance
190 µm.onThe
steel and
has of
a
each
radial
AMB
target
rotor.
The
TD
bearing
radial
clearance
is
190
µm.
The
shaft
is
steel
and
has
a
diameter of 16 mm.
diameter of 16 mm.
(a)
(b)
Figure
Figure 2.
2. AMB
AMBtest
testrig
rig(a)
(a)image,
image,and
and(b)
(b)basic
basic dimensions.
dimensions.
The basic parameters of the AMB system are listed in Table 1. The values of current stiffness and
The basic parameters of the AMB system are listed in Table 1. The values of current stiffness and
position stiffness are resulting from the linearization of the AMB magnetic force at the operating
position stiffness are resulting from the linearization of the AMB magnetic force at the operating point,
point, which is at the center of the AMB with 1.25 A bias current. Both inboard and outboard AMBs
which is at the center of the AMB with 1.25 A bias current. Both inboard and outboard AMBs are of
are of the same type and have the same parameters. The rotordynamic values of the components on
the same type and have the same parameters. The rotordynamic values of the components on the shaft
the shaft are listed in Table 2. These components are affixed to the shaft with collets.
are listed in Table 2. These components are affixed to the shaft with collets.
Table 1. AMB system parameter values.
Table 1. AMB system parameter values.
Parameter
Value Units
Parameter
Value
Units
Current Stiffness
32
N/A
Current Stiffness
32
N/A
Position
Stiffness
0.1
N/µm
Position
Stiffness
0.1
N/µm
1
Amp Bandwidth
2.5
kHz
2.5
kHz
Amp Bandwidth 1
Shaft Diameter
16
mm
1
Shaft Diameter
16
mm
Amplifier
bandwidth includes load of AMB coil.
1
Amplifier bandwidth includes load of AMB coil.
Component
Balance
Disk
Component
Thrust AMB
Balance Disk
Radial AMB
Thrust AMB
Radial AMB
Table 2. Rotor component parameter values.
Table 2. Rotor component parameter values.
2
Mass (kg)
6.5 × 101
Mass (kg)
2.1 × 10−41
6.5 × 10
2.5 × 10−1 4
2.1 × 10
2.5 × 101
Polar Moment (kg·m ) Trans. Moment (kg·m2)
4.4 × 10−4
2.5 × 10−4
Polar Moment (kg·m2 )
Trans. Moment (kg·m2 )
−5
9.9 × 10
9.7 × 10−5
4.4 × 10−4−5
2.5 × 10−4 −5
4.1 × 10
5.8 ×−10
−5
5
9.9 × 10
4.1 × 10−5
9.7 × 10
5.8 × 10−5
The rotor is modeled with the FE method using 32 Timoshenko beam elements. Each collet
affixed rotor component is accounted for with a lumped mass occurring at a FE node placed at its
The rotor is modeled with the FE method using 32 Timoshenko beam elements. Each collet affixed
center of mass. FE nodes are also placed at the AMB force and sensor locations for typical assembly
rotor component is accounted for with a lumped mass occurring at a FE node placed at its center
of the open-loop plant model. FE nodes are also placed at the TD bearing locations for fault recovery
of mass. FE nodes are also placed at the AMB force and sensor locations for typical assembly of
controller synthesis. The FE model is reduced through modal truncation, retaining two rigid body
the open-loop plant model. FE nodes are also placed at the TD bearing locations for fault recovery
modes and two flexible modes and neglecting higher order modes.
controller synthesis. The FE model is reduced through modal truncation, retaining two rigid body
The open-loop plant is assembled including the FE rotor model, linear AMB model, and pulse
modes and two flexible modes and neglecting higher order modes.
width modulation amplifier model, which includes digital phase lag and AMB slew rate. The
The open-loop plant is assembled including the FE rotor model, linear AMB model, and pulse
open-loop model is confirmed against experimental system identification data. Figure 3 shows the
width modulation amplifier model, which includes digital phase lag and AMB slew rate. The open-loop
Bode plot of the open-loop model from control current input to position sensor output. The model is
model is confirmed against experimental system identification data. Figure 3 shows the Bode plot of the
4-input 4-output for each radial AMB axis and assumes the thrust axis is dynamically decoupled from
Actuators 2017, 6, 2
6 of 14
open-loop
model
Actuators 2017,
6, 2 from control current input to position sensor output. The model is 4-input 4-output
6 of 14
for each radial AMB axis and assumes the thrust axis is dynamically decoupled from the radial axes.
the radial
The
figurecharacteristic
shows a single
characteristic
radial
axis
of the roughly
symmetric
The
figure axes.
shows
a single
radial
control axis
of thecontrol
roughly
symmetric
system.
Figure 3
system.
Figure
also shows
the results sine-sweep
of experimental
from and
the same
input
output.
also
shows
the 3results
of experimental
fromsine-sweep
the same input
output.
Theand
open-loop
The open-loop
AMB system,
the bias
current,
whichand
effects
thestiffnesses,
position is
and
current
AMB
system, including
the bias including
current, which
effects
the position
current
inherently
stiffnesses,
is
inherently
unstable.
To
perform
the
experimental
sine
sweep,
a
stabilizing
AMB
unstable. To perform the experimental sine sweep, a stabilizing AMB control (PID) is used andcontrol
input
(PID) is used is
and
input perturbation
superimposed
on the
closed-loop
current.frequency
Then, the
perturbation
superimposed
on theisclosed-loop
control
current.
Then, control
the open-loop
open-loopisfrequency
extracted
total current
and
position
response data.
The
response
extracted response
from the istotal
currentfrom
and the
position
response
data.
The comparison
shows
comparison
showsthe
agreement
between
the model
experimental
system, which
is a critical
agreement
between
model and
experimental
system,and
which
is a critical prerequisite
for model-based
prerequisite
for model-based
design.
The
two rotor
flexible
modes
are indicated
by
controller
design.
The first twocontroller
rotor flexible
modes
arefirst
indicated
by peaks
in the
magnitude
at 164 Hz
peaks
in
the
magnitude
at
164
Hz
and
387
Hz,
ω
n1
and
ω
n2
,
respectively.
Higher
frequency
flexible
and 387 Hz, ω n1 and ω n2 , respectively. Higher frequency flexible modes, which are neglected in the
modes, can
which
are
in the
model,
can alsoof
bethese
seen in
the figure.
of theseismodes
by
model,
also
beneglected
seen in the
figure.
Excitation
modes
by theExcitation
AMB controller
avoided
thecontroller
AMB controller
is avoided
by controller
whichweighting,
is achieved
through inperformance
by
roll-off, which
is achieved
throughroll-off,
performance
as detailed
Section 2.4.
weighting,
as the
detailed
in Section
2.4. More
detail identification
on the mathematical
modeling
andbesystem
More
detail on
mathematical
modeling
and system
of the AMB
test rig can
found
identification
of the[19].
AMB test rig can be found in the earlier study [19].
in
the earlier study
(a)
(b)
Figure 3. Bode plot of open-loop plant system identification data and model at a characteristic radial
Figure 3. Bode plot of open-loop plant system identification data and model at a characteristic radial
input-output pair on the inboard AMB. (a) Magnitude; (b) Phase.
input-output pair on the inboard AMB. (a) Magnitude; (b) Phase.
2.3. Contact Force Estimation and Performance Weight
2.3. Contact Force Estimation and Performance Weight
The worst case TD bearing force on the rotor when running on the TD bearings must be bounded
The worst case TD bearing force on the rotor when running on the TD bearings must be bounded
in order to synthesize the recovery controller, which is required to be robust to that disturbance. A
in order to synthesize the recovery controller, which is required to be robust to that disturbance.
simulation is performed in order to estimate those forces. A simplistic TD bearing model is used in
A simulation is performed in order to estimate those forces. A simplistic TD bearing model is used in
the simulation, which is then tuned to match experimental time response data.
the simulation, which is then tuned to match experimental time response data.
The experimental rotor is levitated and run at 2000 RPM. The current to the AMB coils is abruptly
The experimental rotor is levitated and run at 2000 RPM. The current to the AMB coils is abruptly
turned off and the rotor is allowed to fall freely onto the TD bearings while the motor is still driving
turned off and the rotor is allowed to fall freely onto the TD bearings while the motor is still driving
rotation. Figure 4 shows the orbit response and time response from the outboard AMB sensor after
rotation. Figure 4 shows the orbit response and time response from the outboard AMB sensor after
the AMB support is removed. The rotor starts at 0 µm, the center of the AMB. The rotor then falls in
the AMB support is removed. The rotor starts at 0 µm, the center of the AMB. The rotor then falls
the negative vertical direction until it hits the TD bearing which is shown as the dashed line in the
in the negative vertical direction until it hits the TD bearing which is shown as the dashed line in
figure. The position signal goes outside of the TD bearing clearance due to a combination of TD
the figure. The position signal goes outside of the TD bearing clearance due to a combination of TD
bearing deflection and bending of the flexible rotor between the non-collocated TD bearing and
bearing deflection and bending of the flexible rotor between the non-collocated TD bearing and sensor.
sensor. The rotor bounces up and to the right due to the stiffness of the TD bearing and initial relative
The rotor bounces up and to the right due to the stiffness of the TD bearing and initial relative velocity
velocity between the TD bearing and rotor surface, which is rotating clockwise. After the initial
between the TD bearing and rotor surface, which is rotating clockwise. After the initial bounce, the
bounce, the rotor rolls back and forth on the bottom of the TD bearing, driven by the motor and
rotor rolls back and forth on the bottom of the TD bearing, driven by the motor and residual unbalance.
residual unbalance.
Actuators 2017, 6, 2
7 of 14
(a)
(b)
Figure 4. Response at outboard AMB sensor during 2000 RPM drop test, (a) orbit, and; (b) time response.
Figure 4. Response at outboard AMB sensor during 2000 RPM drop test, (a) orbit, and;
(b) time response.
The numerical simulation of the rotor drop test is performed in MATLAB™ via ode45. The same
FE rotor model constructed and experimentally confirmed for controller design is used for the
The numerical simulation of the rotor drop test is performed in MATLAB™ via ode45. The same FE
numerical drop simulation. The initial conditions are at rest (although subject to the gyroscopic effect)
rotor model constructed and experimentally confirmed for controller design is used for the numerical
on the AMB centerline. No AMB force or controller is needed for the simulation as these have been
drop simulation. The initial conditions are at rest (although subject to the gyroscopic effect) on
deactivated by the simulation start time. Rotor weight and an unbalance force at 2000 RPM are
the AMB centerline. No AMB force or controller is needed for the simulation as these have been
applied. A static unbalance is assumed to be distributed to the AMB force centers, the magnitude of
deactivated by the simulation start time. Rotor weight and an unbalance force at 2000 RPM are applied.
which was originally tuned to rotor orbits in [19]. Initially, the TD bearing is modeled as a linear
A static unbalance is assumed to be distributed to the AMB force centers, the magnitude of which
stiffness and damping on the FE rotor at the TD bearing nodes with a discontinuity to reflect if the
was originally tuned to rotor orbits in [19]. Initially, the TD bearing is modeled as a linear stiffness
rotor is in contact. It is found that such a simple TD bearing model is not able to explain the first rotor
and damping on the FE rotor at the TD bearing nodes with a discontinuity to reflect if the rotor is in
bounce
but can return a reasonable response on subsequent bounces. Therefore, it is further assumed
contact. It is found that such a simple TD bearing model is not able to explain the first rotor bounce
that the first bounce is affected by the initial relative velocity between the rotor surface and TD
but can return a reasonable response on subsequent bounces. Therefore, it is further assumed that
bearing
inner raceway but that, for subsequent contact, the TD bearing has accelerated to a similar
the first bounce
is affected
by the TD
initial
relative
velocity
between the
rotor
andforce
TD bearing
velocity
as the rotor.
The simple
bearing
model
is augmented
with
an surface
exogenous
on the
inner
raceway
but
that,
for
subsequent
contact,
the
TD
bearing
has
accelerated
to
a
similar
velocity
as
rotor, which occurs immediately prior to first contact. The magnitude and direction of that force are
the
rotor.
The
simple
TD
bearing
model
is
augmented
with
an
exogenous
force
on
the
rotor,
which
then tuned to match the first experimental bounce. Figure 4 also shows the responses for the tuned
occurs immediately
firstsame
contact.
The magnitude
and direction
of that
are then
tuned
simulated
drop test prior
undertothe
conditions
as the experiment.
There
areforce
obvious
disparities
to match the
the first
experimental
bounce. Figure
shows the
responses
for simplicity
the tuned of
simulated
between
experiment
and simulation,
which4 also
the authors
attribute
to the
the TD
drop
test
under
the
same
conditions
as
the
experiment.
There
are
obvious
disparities
between
the
bearing model. However, the similar magnitude of the overall response suggests that the simulated
experiment
and
simulation,
which
the
authors
attribute
to
the
simplicity
of
the
TD
bearing
model.
TD bearing force is of realistic magnitude. Therefore, the simple simulation is used as a practical
However,tothe
similarthe
magnitude
of the
overall response suggests that the simulated TD bearing force
solution
estimate
TD bearing
force.
is of The
realistic
magnitude.
Therefore,
the
simple
simulation
is used
as adrop
practical
solutionistoused
estimate
simple TD bearing model with
parameters
tuned
to the
experiment
in a
the
TD
bearing
force.
simulation to bound the expected TD bearing force. The simulation includes a rotor drop impact
The simple
TD bearing
model
withTD
parameters
tuned
the drop
experiment
used inby
a
followed
by continuous
running
on the
bearings for
10 s,toduring
which
the rotor is
is driven
simulation
to
bound
the
expected
TD
bearing
force.
The
simulation
includes
a
rotor
drop
impact
residual unbalance to bounce on the bottom of the TD bearings in a chaotic like fashion. For this time,
followed
by continuous
running
on the
TDare
bearings
forThe
10 s,
during which
the rotor
driven
by
the
TD bearing
forces in all
four radial
axes
collected.
frequency
spectrum
for theiseach
of the
residual
unbalance
to
bounce
on
the
bottom
of
the
TD
bearings
in
a
chaotic
like
fashion.
For
this
four forces is found using FFT of the sampled numerical data. Then, for each frequency, the largest
time, from
the TD
bearing
in allisfour
radial
collected.
The spectrum
frequencyshown
spectrum
for the5.each
force
each
of theforces
four axes
taken.
The axes
resultare
is the
frequency
in Figure
The
of the four forces
is found
using
of the sampled
numerical
for each frequency,
the
apparently
noisy result
is due
to FFT
the chaotic
like bouncing
of thedata.
rotorThen,
on discontinuous
stiffnesses
largest
force
from
each
of
the
four
axes
is
taken.
The
result
is
the
frequency
spectrum
shown
and the frequency-by-frequency combination of the four axes forces. The TD bearing force in
is
Figure
5.
The
apparently
noisy
result
is
due
to
the
chaotic
like
bouncing
of
the
rotor
on
discontinuous
dominated by harmonics of the running speed, 1X, 2X, and 3X. The 2X peak also has sidebands at
stiffnesses
±10.2
Hz. and the frequency-by-frequency combination of the four axes forces. The TD bearing force
is dominated by harmonics of the running speed, 1X, 2X, and 3X. The 2X peak also has sidebands at
±10.2 Hz.
Actuators 2017, 6, 2
Actuators 2017, 6, 2
8 of 14
8 of 14
Figure 5. Frequency spectrum of the simulated TD bearing force on the rotor during rolling/bouncing
Figure
Frequency
of the
simulatedweighting
TD bearing
force on
rotorforce
during
rolling/bouncing
at 5.
2000
RPM and spectrum
corresponding
performance
function.
Thethe
largest
magnitude
at each
at 2000
RPM
and
corresponding
performance
weighting
function.
The
largest
force
magnitude
at each
radial axis is taken at each frequency.
radial axis is taken at each frequency.
Figure 5 also shows the weighting function WTD. WTD is crafted manually to have greater
magnitude
thanshows
the simulated
worst case
TD bearing
force
all frequencies below the anticipated
Figure 5 also
the weighting
function
W TD
. Wat
TD is crafted manually to have greater
Nyquist frequency of the digital controller. Then, WTD is used to weight the exogenous perturbation
magnitude than the simulated worst case TD bearing
force at all frequencies below the anticipated
input to the FE nodes at the TD bearing locations during controller synthesis. The transfer function
Nyquist frequency of the digital controller. Then, W TD is used to weight the exogenous perturbation
of the final TD bearing force weight is presented in the next section on details of the experimental
inputcontroller
to the FEdesign.
nodes at the TD bearing locations during controller synthesis. The transfer function
of the final TD bearing force weight is presented in the next section on details of the experimental
controller
design. Synthesis for the Experimental System
2.4. Controller
The parametric uncertainties used for robust AMB controller synthesis are as follows and are
summarized in Table 3. Uncertainties are placed on the two retained natural frequencies of the rotor
model.
The uncertainty
bound isused
selected
capture
any controller
mismatching
between are
the as
nominal
model
The
parametric
uncertainties
for to
robust
AMB
synthesis
follows
and are
and experimental
system
identification
shown in
3. As
the FEnatural
rotor model
is expressed
summarized
in Table 3.
Uncertainties
are placed
onFigure
the two
retained
frequencies
of theinrotor
modal
coordinates, the
parametric
uncertainty
is conveniently
placedbetween
on the square
of the natural
model.
The uncertainty
bound
is selected
to capture
any mismatching
the nominal
model and
frequencies. The natural frequency uncertainty is of the complex type for the added benefit of
experimental system identification shown in Figure 3. As the FE rotor model is expressed in modal
robustness to modeling error of modal damping.
2.4. Controller Synthesis for the Experimental System
coordinates, the parametric uncertainty is conveniently placed on the square of the natural frequencies.
The natural frequency uncertainty is
of the
complex
type for the added benefit of robustness to
Table
3. Parametric
uncertainties.
modeling error of modal damping.
Parameter
Type
Range
Instances
1
1st Natural Frequency
Complex
±3%
2
Table 3. Parametric uncertainties.
2nd Natural Frequency 1
Complex
±5%
2
Current Stiffness
Real
±5%
4
Parameter
Type
Range
Instances
Position Stiffness
Real
±20%
4
1
Complex
±3%
26
1st Natural
Frequency
Running
Speed
Real
±100%
Complex
±5%
2
2nd Natural
Frequency 1
1 Uncertainty placed on square of the natural frequency.
Current Stiffness
Real
±5%
4
Position Stiffness
Real
±20%
4
Real uncertainties are placed on the AMB current stiffness and position stiffness. The
Running Speed
Real
±100%
6
uncertainties are intended to account for errors due to linearization. The uncertainty bounds are taken
1 Uncertainty placed on square of the natural frequency.
from [19] and found to yield
good effective control. The uncertainty on position stiffness is relatively
large because position stiffness is the destabilizing term in a linearized AMB system. Also, real
Real
uncertainties
areonplaced
on speed,
the AMB
current
stiffness
positioneffect
stiffness.
uncertainties
uncertainty
is placed
running
which
appears
in theand
gyroscopic
in theThe
rotor
model.
are intended
to
account
for
errors
due
to
linearization.
The
uncertainty
bounds
are
taken
from
[19] and
The nominal running speed is set at 1000 RPM and a ±100% uncertainty covers the speed range from
up togood
targeteffective
speed of control.
2000 RPM.
This
running speed
is relatively
low compared
to most
foundstart
to yield
The
uncertainty
on position
stiffness
is relatively
largeAMB
because
applications.
The
rig is modular
and
configured
without
special
making
position
stiffness is
thetest
destabilizing
term
in manually
a linearized
AMB system.
Also,
realbalancing,
uncertainty
is placed
higher speeds
convenient.
the low speed
demonstration
is foundThe
to benominal
sufficientrunning
to
on running
speed, not
which
appearsHowever,
in the gyroscopic
effect
in the rotor model.
show the efficacy of the proposed control method. Higher speeds are expected to make TD bearing
speed is set at 1000 RPM and a ±100% uncertainty covers the speed range from start up to target speed
forces, and therefore the need for a recovery controller, greater.
of 2000 RPM. This running speed is relatively low compared to most AMB applications. The test rig is
modular and manually configured without special balancing, making higher speeds not convenient.
However, the low speed demonstration is found to be sufficient to show the efficacy of the proposed
control method. Higher speeds are expected to make TD bearing forces, and therefore the need for a
recovery controller, greater.
Actuators 2017, 6, 2
9 of 14
Each natural frequency uncertainty perturbation appears twice, once for each perpendicular
rotor plane. The running speed uncertainty appears 6 times because the gyroscopic effect effects
the two retained bending modes and the second rigid body mode, all in two perpendicular planes.
The current and position stiffness uncertainties are assumed to be the same in all radial AMB axes.
A more conservative approach would be to allow all radial AMB axes to vary independently; however,
the present assumption provides a more tractable problem for µ-synthesis and is found to yield
good results.
The performance weights used for controller synthesis for the test rig are shown in Table 4.
These weighting transfer functions are equally applied to each of the four radial AMB axes. The weights
on sensor noise, external load, and control current limit are common to both performance and recovery
controllers. The allowable rotor position, however, differs in that the performance controller is required
to hold the rotor close to the center of the bearing whereas the recovery controller my allow deflection
up to the TD bearing clearance. Also, the recovery controller is required to tolerate the TD bearing
force as discussed in the previous subsection.
Table 4. Performance weights for µ-synthesis controller designs.
Signal
Weight
Units
Controller
WN = 0.9
µm
Both
N
Both
A−1
Both
µm−1
Performance
2.632 ×
s + 1.336
s + 253.9
µm−1
Recovery
1.382 × 104 s + 5.052 × 104
s2 + 3748 s + 1.01 × 106
N
Recovery
Noise
Load
WL =
4
1.6 × 10 s + 5.224 × 10
s + 6.53 × 107
0.03125 s + 1.834
WP1 =
s + 16.5
Current
WC =
Position
Position
TD Bearing Force
3 × 10−4 s2 + 1.524 × 103 s + 889.8
s2 + 254.1 s + 49.43
WP2 =
WTD =
7
10−7
3. Results
In this section, the results of controller synthesis for the test rig and parameters discussed in the
previous section are presented. Then, the results for a fault recovery experiment using the recovery
controller and the performance controller are presented.
3.1. Results of Controller Synthesis
µ-synthesis is performed on the two uncertain and weighted plant models using the MATLAB™
Robust Control Toolbox to perform the D-K iterations. The performance controller results from 2 D-K
iterations, achieving a µ-value of 0.89. The recovery controller results from 2 D-K iterations, achieving a
µ-value of 0.96. Therefore, each controller is expected to yield robust stability and robust performance
for their corresponding operation regiments.
Each controller is 4-input 4-output for each radial AMB axis. The Bode plots of two I/O pairs on
the main diagonal are shown in Figure 6. This figure is for one axis on the inboard AMB and one axis
on the outboard AMB in the same plane. The controllers are similar due to the approximate symmetry
of the rotor. Those transfer functions for the perpendicular plane are also similar due to the rotor being
axisymmetric. These are on the main diagonal of the 4 input 4 output controllers; the off diagonal
transfer functions follow similar trends but are at least one order of magnitude lower.
Actuators 2017, 6, 2
Actuators 2017, 6, 2
10 of 14
10 of 14
Figure 6. Bode plots of the AMB controllers designed for performance under normal operating
Figure 6. Bode plots of the AMB controllers designed for performance under normal operating
conditions and for robust levitation recovery. Left plot is for the input-output on the inboard AMB
conditions and for robust levitation recovery. Left plot is for the input-output on the inboard AMB and
and the right plot is for the pair on the outboard AMB in the same plane.
the right plot is for the pair on the outboard AMB in the same plane.
Both performance and recovery controllers exhibit negative feedback for stabilization of the
Both
performance
and recovery
controllers
exhibit
negative
feedback
for stabilization
of theoff
AMBs.
AMBs. Both
have features
for control
of the rotor
flexible
modes
at ωn1 and
ωn2 before rolling
at
Both
have
features
for
control
of
the
rotor
flexible
modes
at
ω
and
ω
before
rolling
off
at
high
n2
high frequency. The recovery controller has a significantly lowern1
low frequency
gain, which prevents
frequency.
The recovery
controller
has a is
significantly
lower lowsuch
frequency
gain,
which
prevents
high control
currents when
the rotor
at large deflections
as would
occur
during
loss high
of
control
currents
thecontroller
rotor is atalso
large
would
during
levitation.
levitation.
The when
recovery
hasdeflections
lower gainsuch
at theasfirst
rotoroccur
flexible
modeloss
but of
higher
for
The
recovery
also has
lowerisgain
the first
rotorthe
flexible
mode
but higher
thedue
second.
the
second. controller
The performance
control
40that
order,
whereas
recovery
control
is 48th for
order
to
additional states
in the
plant
model
for the the
TD recovery
bearing weight.
Thethe
performance
control
is 40th
order,
whereas
control is 48th order due to the additional
controllers
are fairly
high order. For digital implementation, they are both
states inThe
theμ-synthesized
plant model for
the TD bearing
weight.
reduced
to the 36th order
and discretized
at 20order.
kHz with
ZOH via
MATLAB reduce
and
The µ-synthesized
controllers
are fairly high
For digital
implementation,
they
arec2d,
both
respectively.
Then,
the
discrete
controllers
are
put
in
canonical
state-space
form
for
optimization
of
reduced to the 36th order and discretized at 20 kHz with ZOH via MATLAB reduce and c2d, respectively.
real
time
calculations
via
MATLAB
canon.
The
controller
implementation
is
executed
with
a
Then, the discrete controllers are put in canonical state-space form for optimization of real time
dSPACE1103
control
prototyping
board,implementation
which is laboratory
grade equipment.
For industrial
calculations
via rapid
MATLAB
canon.
The controller
is executed
with a dSPACE1103
rapid
implementation,
further
reduction
of
the
weighted
plant
model
or
the
synthesized
controller
may be
control prototyping board, which is laboratory grade equipment. For industrial implementation,
required for practical implementation.
further reduction of the weighted plant model or the synthesized controller may be required for
practical implementation.
3.2. Results of Controller Implementation
3.2. Results
Controllerthat
Implementation
It is of
anticipated
the performance controller may have difficulty relevitating a delevitated
rotor at running speed. It is also anticipated that the recovery controller will yield low bearing static
It is anticipated that the performance controller may have difficulty relevitating a delevitated rotor
stiffness and therefore poor steady-state support. Therefore, a switching scheme is utilized for
at running speed. It is also anticipated that the recovery controller will yield low bearing static stiffness
implementation of the recovery controller [4]. In AMB systems, for safety reasons, it is common to set
and therefore poor steady-state support. Therefore, a switching scheme is utilized for implementation
a deflection limit at which power is removed from the AMBs and motor, in order to prevent the loss
of the
recovery controller [4]. In AMB systems, for safety reasons, it is common to set a deflection limit
of stability. In the event that an acute external fault causes violation of this safety limit, an attempt is
at which
power
removed
from
AMBs
andrelevitation.
motor, in order
to preventscheme
the loss
stability. In the
made to
safelyispower
down
the the
system
before
The proposed
forofimplementing
event
that
an
acute
external
fault
causes
violation
of
this
safety
limit,
an
attempt
is
made
safely
the recovery controller utilizes a similar safety limit; however, if that limit is violated thetoAMB
power
down theswitches
systemto
before
relevitation.
Theinstead
proposed
scheme for
implementing
recovery
automatically
the recovery
controller
of powering
down.
The recoverythe
controller
controller
utilizes
a
similar
safety
limit;
however,
if
that
limit
is
violated
the
AMB
automatically
can return the rotor to within the safety limit after the external fault has passed. Then, after a
switches
to the
recovery
instead
of powering
down.
recovery controller
prescribed
amount
of controller
time within
the safety
limit, the
AMBThe
automatically
switches can
backreturn
to thethe
rotor
to within controller.
the safety limit after the external fault has passed. Then, after a prescribed amount of
performance
time within
the
limit,recovery
the AMB
automatically
switches
back toanthe
performance
controller.
To test
thesafety
proposed
controller
and switching
scheme,
experiment
is conducted.
The
test
is levitated
and rotated
at thecontroller
nominal controller
designscheme,
speed, 1000
When steady
state
Torigtest
the proposed
recovery
and switching
an RPM.
experiment
is conducted.
The test rig is levitated and rotated at the nominal controller design speed, 1000 RPM. When steady
Actuators 2017, 6, 2
Actuators 2017, 6, 2
11 of 14
11 of 14
state is reached, current to the AMB coils is abruptly stopped to mimic a power failure. The rotor is
is reached, current to the AMB coils is abruptly stopped to mimic a power failure. The rotor is allowed
allowed to fall onto the TD bearings. After approximately 2 s running on the TD bearings, the current
to fall onto the TD bearings. After approximately 2 s running on the TD bearings, the current is
is restored
and
thethe
rotor
rotating.Figure
Figure7 shows
7 shows
time
history
oflevitation
the levitation
restored
and
rotorrelevitates
relevitateswhile
while still
still rotating.
thethe
time
history
of the
failure
and
recovery
test
when
using
(Figure
7a,c)
just
the
performance
controller
and
(Figure
7b,d,e)
failure and recovery test when using (Figure 7a,c) just the performance controller and (Figure 7b,d,e)
the proposed
recovery
controller
and
switching
scheme.
the proposed recovery controller and switching scheme.
Figure
7. Outboard
AMBsensor
sensorsignals
signals during
during 1000
failure
andand
recovery
test. test.
Figure
7. Outboard
AMB
1000RPM
RPMlevitation
levitation
failure
recovery
(a)
Time
response
using
no
recovery
scheme;
(b)
Time
response
using
recovery
scheme;
(c)
orbit
when
(a) Time response using no recovery scheme; (b) Time response using recovery scheme; (c) orbit
with performance controller; (d) orbit when relevitating with recovery controller; and
whenrelevitating
relevitating
with performance controller; (d) orbit when relevitating with recovery controller;
(e) orbit when switching from recovery controller to performance controller.
and (e) orbit when switching from recovery controller to performance controller.
In each case, the rotor starts at 0 µm, corresponding to the center of the AMB. At 1 s, the current
In
each
case,
theisrotor
starts
atAMBs
0 µm,incorresponding
to thetrial
center
of the AMB. switched
At 1 s, the
current
to the
AMB
coils
stopped.
The
the recovery scheme
are automatically
from
performance
controllerThe
to the
recovery
controller
when
the sensor
surpasses the predefined
to thethe
AMB
coils is stopped.
AMBs
in the
recovery
scheme
trialsignal
are automatically
switched from
safety limit of
±100 µmto
(although
current
is still not
supplied
to thesignal
coils). surpasses
For both cases,
from
the performance
controller
the recovery
controller
when
the sensor
the predefined
approximately
1
s
to
3
s,
the
rotor
runs
supported
loosely
by
the
TD
bearings,
resulting
in
a
chaotic
safety limit of ±100 µm (although current is still not supplied to the coils). For both cases, from
bouncing/rolling motion. At approximately 3 s, power is restored. The system with no recovery
approximately 1 s to 3 s, the rotor runs supported loosely by the TD bearings, resulting in a chaotic
scheme responds violently with overshoot, causing impact with the top of the TD bearing, but it is
bouncing/rolling motion. At approximately 3 s, power is restored. The system with no recovery
ultimately able to return to stable levitation. The system with the recovery controller returns to stable
scheme
responds
violently
with
overshoot,
causing
impact
with the 2top
of thethe
TD
bearing,
but it is
levitation
quickly
with little
unwanted
dynamics.
After
approximately
s inside
±100
µm safety
ultimately
able
to
return
to
stable
levitation.
The
system
with
the
recovery
controller
returns
to
stable
limit, the AMBs automatically switch back to the performance controller.
levitationPlots
quickly
with 7c,d
littleshow
unwanted
dynamics.
After approximately
2 s insidecontroller
the ±100
µm
of Figure
the orbits
during relevitation
with the performance
and
thesafety
controller,
respectively.
The
TD bearing
radial clearance
is indicated with the dashed line.
limit,recovery
the AMBs
automatically
switch
back
to the performance
controller.
The orbit
plot including
horizontal
andduring
verticalrelevitation
response better
the violent overshoot
when
Plots
of Figure
7c,d show
the orbits
withshows
the performance
controller
and the
attempting
relevitation
with
the
performance
controller
and
the
possibility
of
developing
into
a line.
recovery controller, respectively. The TD bearing radial clearance is indicated with the dashed
contact
mode.
Plot
of
Figure
7e
shows
the
orbit
of
the
recovery
scheme
trial
when
switching
from
the
The orbit plot including horizontal and vertical response better shows the violent overshoot when
recovery controller back to the performance controller. Smaller orbit size and less overall deflection
attempting relevitation with the performance controller and the possibility of developing into a contact
from the AMB center indicates superior performance of the performance controller when inside the
mode. Plot of Figure 7e shows the orbit of the recovery scheme trial when switching from the recovery
proper operating region.
controllerThe
back
to the
controller.
Smaller
orbit sizecontroller
and lessdesign
overall
deflection
from the
power
lossperformance
and recovery test
is repeated
at the maximum
speed
of 2000 RPM.
AMBThe
center
indicates
superior
performance
offormat
the performance
controller
whenexacerbates
inside thethe
proper
results
are shown
in Figure
8 in the same
as Figure 7. The
higher speed
operating
region.
problems
with levitation recovery while rotating.
The power loss and recovery test is repeated at the maximum controller design speed of 2000 RPM.
The results are shown in Figure 8 in the same format as Figure 7. The higher speed exacerbates the
problems with levitation recovery while rotating.
Actuators 2017, 6, 2
Actuators 2017, 6, 2
12 of 14
12 of 14
Figure
8. 8.Outboard
2000 RPM
RPM levitation
levitationfailure
failureand
andrecovery
recovery
test.
Figure
OutboardAMB
AMB sensor
sensor signals
signals during
during 2000
test.
Time
responseusing
using no recovery
(b)(b)
Time
response
usingusing
recovery
scheme;
(c) orbit(c)
when
(a) (a)
Time
response
recoveryscheme;
scheme;
Time
response
recovery
scheme;
orbit
relevitating
withwith
performance
controller;
(d) orbit
whenwhen
relevitating
with recovery
controller;
and
when
relevitating
performance
controller;
(d) orbit
relevitating
with recovery
controller;
when
switching
from
recovery
controller
to performance
controller.
and(e)(e)orbit
orbit
when
switching
from
recovery
controller
to performance
controller.
The time response and orbit when relevitating with the performance controller show an even
The time response and orbit when relevitating with the performance controller show an even
more violent transient. However, the proposed control method is still able to cope with the higher
more violent transient. However, the proposed control method is still able to cope with the higher
speed and achieves levitation recovery without unwanted dynamics. These results demonstrate the
speed
and achieves levitation recovery without unwanted dynamics. These results demonstrate the
efficacy of the proposed method.
efficacy of the proposed method.
4. Discussion and Conclusion
4. Discussion and Conclusion
An AMB control method for recovering levitation of a rotor that has been temporarily
An AMB by
control
method
for has
recovering
levitation
of proposed
a rotor that
has been
delevitated
an external
fault
been proposed.
The
method
usestemporarily
μ-synthesis delevitated
to design
by an
anAMB
external
fault
has
been
proposed.
The
proposed
method
uses
µ-synthesis
to
design
controller, which is robust to very large rotor deflections and TD bearing forces on the
rotor.an
AMB
controller,
which
is
robust
to
very
large
rotor
deflections
and
TD
bearing
forces
on the
The resulting recovery controller exhibited lower static gains to prevent current saturation during
rotor.
The
resulting
recovery
controller
exhibited
lower
static
gains
to
prevent
current
saturation
relevitation while maintaining high frequency dynamics to control the rotor flexible modes. The
during
relevitation
maintaining
high frequency
dynamicsistodetected.
control The
the proposed
rotor flexible
modes.
recovery
controllerwhile
is automatically
activated
if rotor delevitation
levitation
Therecovery
recovery
controller
is automatically
activated
rotor
is1000
detected.
The
proposed
method
was demonstrated
on an
AMB testifrig.
Thedelevitation
rig was run at
and 2000
RPM,
and
levitation
recovery
method was
onstopped
an AMB
test rig.
The rig fault.
was run
1000
and
the current
to the magnetic
coilsdemonstrated
was temporarily
to cause
an external
Afteratthe
rotor
was
running
thecurrent
TD bearings,
power was
restored.
The trials using
a baseline
AMB an
controller
2000
RPM,
andonthe
to thethe
magnetic
coils
was temporarily
stopped
to cause
external
responded
violently,
hitting
the topon
of the
the TD
before
trial
using
the a
fault.
After the
rotor was
running
TD bearing
bearings,
the recovering
power waslevitation.
restored.The
The
trials
using
proposed
recovery
controller
was
able
to
recover
levitation
quickly
with
no
unwanted
dynamics.
baseline AMB controller responded violently, hitting the top of the TD bearing before recovering
These
therecovery
current controller
state-of-the-art
by making
the levitation
overall AMB-system
levitation.
Thedevelopments
trial using theadvance
proposed
was able
to recover
quickly with
reliable.
The assurance of levitation recovery will lead to increased acceptance of AMBs in
no more
unwanted
dynamics.
industry
at large. Therefore,
AMBs
will
become
more widespread
and thethe
already
proven
advantages
These developments
advance
the
current
state-of-the-art
by making
overall
AMB-system
more
of
AMBs,
e.g.,
higher
efficiency
and
longer
life,
will
more
benefit
society.
reliable. The assurance of levitation recovery will lead to increased acceptance of AMBs in industry at
A deficiency in this work is the simplistic model used to estimate a bound on the TD bearing
large. Therefore, AMBs will become more widespread and the already proven advantages of AMBs,
force used for controller design. Therefore, a direction of further research would be to apply this
e.g., higher efficiency and longer life, will more benefit society.
controller design method using a more sophisticated TD bearing model. Alternatively,
A deficiency in this work is the simplistic model used to estimate a bound on the TD bearing force
experimentally gathered TD bearing force data may be used to make the performance bound. Also,
used for controller design. Therefore, a direction of further research would be to apply this controller
Actuators 2017, 6, 2
13 of 14
design method using a more sophisticated TD bearing model. Alternatively, experimentally gathered
TD bearing force data may be used to make the performance bound. Also, the test of a typical PID
controller, which has been designed following the frequency response trends found effective with the
recovery controller, would be interesting and may prove more useful for industrial application.
Author Contributions: Alexander H. Pesch and Jerzy T. Sawicki conceived the proposed method and designed
the experiment. Then, Alexander H. Pesch conducted the experiment and analyzed the data.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Keogh, P.S.; Cole, M.O.T. Dynamics and Control Issues for Fault Tolerance in Magnetic Bearings: Theory, Design,
and Application to Rotating Machinery, 1st ed.; Schweitzer, G., Maslen, E.H., Eds.; Springer: Berlin, Germany,
2009; pp. 407–433.
Maslen, E.H.; Sortore, C.K.; Gillies, G.T.; Williams, R.D.; Fedigan, S.J.; Aimone, R.J. Fault Tolerant Magnetic
Bearings. ASME J. Eng. Gas Turbines Power 1999, 121, 504–508. [CrossRef]
Hawkins, L.; Wang, Z.; Wadhvani, V. Transient Simulation of Magnetic Bearing and Backup Bearing
Interaction in a High Speed Rotary Atomizer Subject to Impulsive Loads. In Proceedings of the 15th
International Symposium on Magnetic Bearings, Kitakyushu, Japan, 2–6 August 2016.
Cole, M.O.T.; Keogh, P.S.; Sahinkaya, M.N.; Burrows, C.R. Towards Fault-Tolerant Active Control of
Rotor-Magnetic Bearing Systems. Control Eng. Pract. 2004, 12, 491–501. [CrossRef]
Sawicki, J.T.; Maslen, E.H. Mu-Synthesis for Magnetic Bearings: Why Use Such a Complicated Tool?
In Proceedings of the ASME 2007 International Mechanical Engineering Congress and Exposition, Seattle,
WA, USA, 11–15 November 2007; pp. 1103–1112.
Smirnov, A. AMB System for High-Speed Motors Using Automatic Commissioning. Ph.D. Thesis,
Lappeenranta University of Technology, Lappeenranta, Finland, 18 December 2012.
Cole, M.O.T.; Keogh, P.S. Rotor Vibration with Auxiliary Bearing Contact in Magnetic Bearing Systems Part 2:
Robust Synchronous Control for Rotor Position Recovery. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
2003, 217, 393–409. [CrossRef]
Abulrub, A.G.; Sahinkaya, M.N.; Keogh, P.S.; Burrows, C.R. Experiments on ROLAC to Recover Rotor
Position Following Contact. In Proceeding of the 10th International Symposium on Magnetic Bearings,
Martigny, Switzerland, 21–23 August 2006.
Schlotter, M.; Keogh, P.S. Synchronous Position Recovery Control for Flexible Rotors in Contact with
Auxiliary Bearings. J. Vib. Acoust. 2007, 129, 550–558. [CrossRef]
Li, P.; Sahinkaya, M.N.; Keogh, P.S. Active Recovery of Contact-Free Levitation in Magnetic Bearing Systems.
In Proceedings of the ASME 2012 International Design Engineering Technical Conference & Computers and
Information in Engineering Conference, Chicago, IL, USA, 12–15 August 2012; pp. 673–680.
Khatri, R.K.; Hawkins, L.A.; Bazergui, C. Demonstrated Operability and Reliability Improvements for
A Prototype High-Speed Rotary-Disc Atomizer Supported on Active Magnetic Bearings. In Proceedings
of the ASME Turbo Expo 2015: Turbine Technical Conference and Exposition, Montreal, QC, Canada,
15–19 June 2015; p. V07AT30A015.
Suryawan, F.; Doná, J.; Seron, M. Fault Detection, Isolation, and Recovery Using Spline Tools and Differential
Flatness with Application to Magnetic Levitation System. In Proceedings of the 2010 Conference on Control
and Fault Tolerant Systems, Nice, France, 6–8 October 2010.
Pesch, A.H.; Sawicki, J.T. Robust Controller Design for AMB Levitation Recovery. In Proceedings of the 15th
International Symposium on Magnetic Bearings, Kitakyushu, Japan, 2–6 August 2016.
Zhou, K.; Doyle, J.C. Essentials of Robust Control; Prentice Hall: Upper Saddle River, NJ, USA, 1998.
Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control: Analysis and Design; Wiley: Chichester, UK, 2005.
Pesch, A.H.; Sawicki, J.T. Application of Robust Control to Chatter Attenuation for a High-Speed Machining
Spindle on Active Magnetic Bearings. In Proceedings of the 13th International Symposium on Magnetic
Bearings, Arlington, VA, USA, 6–9 August 2012.
Pesch, A.H.; Smirnov, A.; Pyrhonen, O.; Sawicki, J.T. Magnetic Bearing Spindle Tool Tracking through
µ-synthesis Robust Control. IEEE/ASME Trans. Mech. 2015, 20, 1448–1457. [CrossRef]
Actuators 2017, 6, 2
18.
19.
14 of 14
Pesch, A.H.; Sawicki, J.T. Stabilizing Hydrodynamic Bearing Oil Whip with µ-synthesis Control of an
Active Magnetic Bearing. In Proceedings of the ASME Turbo Expo 2015: Turbine Technical Conference and
Exposition, Montreal, QC, Canada, 15–19 June 2015; p. V07AT31A029.
Pesch, A.H.; Hanawalt, S.P.; Sawicki, J.T. A Case Study in Control Methods for Active Magnetic
Bearings. In Proceddings of the ASME Dynamic Systems and Control Conference, San Antonio, TX,
USA, 22–24 October 2014; p. V001T12A003.
© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC-BY) license (http://creativecommons.org/licenses/by/4.0/).