Albert F. Kascak

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

Gerald V. Brown

Glenn Research Center, Cleveland, Ohio

Ralph H. Jansen

University of Toledo, Toledo, Ohio

Timothy P. Dever

QSS Group, Inc., Cleveland, Ohio

Stability Limits of a PD Controller for a

Flywheel Supported on Rigid Rotor and

Magnetic Bearings

NASA/TM—2006-213979

January 2006

ARL–TR–3745

AIAA–2005–5956 U.S. ARMY

RESEARCH LABORATORY
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Albert F. Kascak

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

Gerald V. Brown

Glenn Research Center, Cleveland, Ohio

Ralph H. Jansen

University of Toledo, Toledo, Ohio

Timothy P. Dever

QSS Group, Inc., Cleveland, Ohio

Stability Limits of a PD Controller for a

Flywheel Supported on Rigid Rotor and

Magnetic Bearings

NASA/TM—2006-213979

January 2006

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

Guidance, Navigation, and Control Conference and Exhibit

sponsored by the American Institute of Aeronautics and Astronautics

San Francisco, California, August 15–18, 2005

ARL–TR–3745

AIAA–2005–5956 U.S. ARMY

RESEARCH LABORATORY
Available from

NASA Center for Aerospace Information

7121 Standard Drive

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National Technical Information Service

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Available electronically at http://gltrs.grc.nasa.gov

This work was sponsored by the Low Emissions Alternative

Power Project of the Vehicle Systems Program at the

NASA Glenn Research Center.
NASA/TM—2006-213979 1 

Stability Limits of a PD Controller for a Flywheel Supported 

on Rigid Rotor and Magnetic Bearings

Albert F. Kascak 

U.S. Army Research Laboratory 

Glenn Research Center 

Cleveland, Ohio 44135 

Gerald V. Brown 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Ralph H. Jansen 

University of Toledo 

Toledo, Ohio 43606 

Timothy P. Dever 

QSS Group, Inc. 

Cleveland, Ohio 44135 

Abstract 

Active magnetic bearings are used to provide a long-life, low-loss suspension of a high-speed 

flywheel rotor. This paper describes a modeling effort used to understand the stability boundaries of the 

PD controller used to control the active magnetic bearings on a high speed test rig. Limits of stability are 

described in terms of allowable stiffness and damping values which result in stable levitation of the 

nonrotating rig. Small signal stability limits for the system is defined as a nongrowth in vibration 

amplitude of a small disturbance. A simple mass-force model was analyzed. The force resulting from the 

magnetic bearing was linearized to include negative displacement stiffness and a current stiffness. The 

current stiffness was then used in a PD controller. The phase lag of the control loop was modeled by a 

simple time delay. The stability limits and the associated vibration frequencies were measured and 

compared to the theoretical values. The results show a region on stiffness versus damping plot that have 

the same qualitative tendencies as experimental measurements. 

The resulting stability model was then extended to a flywheel system. The rotor dynamics of the 

flywheel was modeled using a rigid rotor supported on magnetic bearings. The equations of motion were 

written for the center of mass and a small angle linearization of the rotations about the center of mass. The 

stability limits and the associated vibration frequencies were found as a function of nondimensional 

magnetic bearing stiffness and damping and nondimensional parameters of flywheel speed and time 

delay.

Nomenclature 

A amplitude of oscillation 

c damping coefficient 

C damping coefficient of combined bearings 

F force 

IP polar moment of inertia 

IT transverse moment of inertia 

k stiffness coefficient 
NASA/TM—2006-213979 2 

K stiffness coefficient of combined bearings 

m modal mass 

x lateral displacement 

R complex lateral displacement 

R’ complex rotation about -x and +y-axis 

α real part nondimensional eigenvalue or rotation about y-axis 

β imaginary part nondimensional eigenvalue or rotation about x-axis 

η ratio of polar to transverse moment of inertia 

Δt time delay 

Δφ phase change of transfer function 

Δω change of frequency 

ω circular frequency 

λ eigenvalue 

μ nondimensional eigenvalue 

τ nondimensional time 

ζ damping ratio 

Subscripts 

n negative stiffness 

p proportional 

d derivative 

0 un-damp natural frequency

Introduction 

High speed flywheel systems are being developed at the NASA Glenn Research Center (NASA 

Glenn) in Cleveland, Ohio. Flywheels show promise as an alternative to batteries and reaction wheels for 

space systems. Strengths of this technology include high energy density, long life, capability for up to 

90 percent depth of discharge, and peaking or pulse power capability. Flywheels can also be deployed in 

an array which provides both energy storage and attitude control. A system level flywheel test bed called 

the High Energy Flywheel Facility (HEFF) is operational at NASA Glenn, which includes two high-speed 

flywheel modules. 

 The system level flywheel test bed consists of a rotating carbon fiber ring supported on a hub with a 

shaft. The flywheel modules use a motor/generator coaxial with the rotor to facilitate energy storage and 

retrieval. The shaft is supported on two noncontacting magnetic bearings. One of the bearings gives both 

a radial and axial support. The system utilizes a feedback loop in which the position of the rotor is 

measured with eddy current sensors and used as the input to the magnetic bearing control algorithm. 

The magnetic bearing system uses magnetic forces to levitate the shaft between opposing magnetic 

poles (Fig. 1). The rotor is attracted to one pole or the other pole and is inherently unstable. The magnetic 

bearing system is stabilized with an active control system. The magnetic bearing control system for a 

flywheel measures the location of the rotor with eddy current proximity sensors. The signal is low pass 

filtered with an anti-alias filter. The signal is digitized and sent to a computer which applies a control 

algorithm, (digital controller). The digital signal is then converted to an analog signal and sent to a low 

pass re-construction filter. This signal commands the power amplifiers, which produce current in the 

magnetic bearing coils. This magnetic bearing actuator produces forces which stably suspend the rotor 

between the opposite magnetic poles. 
NASA/TM—2006-213979 3 

Figure 1.—Shows the magnetic bearing system: proximity 

sensor, anti-alias filter, digital controller, re-construction 

filter, power amplifier, magnetic bearing actuator. 

Several groups have developed magnetic bearing controllers for high speed flywheel systems. Many 

different approaches have been used, varying from PID to modal or adaptive methods. The controller 

complexity depends on the details of the flywheel mechanical design and the performance requirements. 

Pichot (Ref. 1) discusses the benefits of a notch filter based controller in comparison to a PID control for 

a large flywheel. Tsiotras has used an H infinity and variable bias controllers for small flywheel and 

gyroscope systems (Refs. 2 and 3). Dr. Alan Palazollo at Texas A&M developed a modal control system 

which was applied to a 60,000 rpm flywheel (Ref. 4). At NASA Glenn this approach has been refined 

and is discussed in a paper by Dever (Ref. 5). 

Each of the components involved in the magnetic bearing and control system has a time delay 

associated with the components. For the analog devices, they are approximated by a linear time delay low 

pass filter. The amplitude of the transfer function is assumed to be a constant and the phase is linear. The 

time delay is the change in phase of the transfer function divided by the change in frequency. For the 

digital devices this time delay is the loop time for each measurement. The total time delay is the sum of 

the individual time delays. 

ana device ana device log log

prox alias control recon amp mag

t

tt t t t t t

Δ = Δφ Δω − −

Δ =Δ +Δ +Δ +Δ +Δ +Δ (1) 

The magnetic bearing and controller have other requirements beside stability. Another requirement is 

its response to transients. To improve the magnetic bearing’s transient response a bias magnetic field is 

applied. The source of this field is a permanent magnet placed in the magnetic circuit of the bearing. The 

result of this bias field causes the rotor to behave as though it was supported on a spring with a negative 

stiffness. There is no time delay associated with the bias field since it is caused by a permanent magnet. 

This paper describes a modeling effort which was used to theoretically map the stability boundaries of the 

magnetic bearing controller which levitates the high speed flywheel rotor. The controller analyzed is a 

P-D controller which causes the magnetic bearing to produce two forces. The first force is proportional to 

the displacement and the second force is proportional to the derivative of the displacement, the velocity. 
NASA/TM—2006-213979 4 

Discussion of Test Rig Stability Data 

Tests were performed on a high speed shaft rig to determine the region of stable operation. A high 

speed shaft rig was used in place of the actual flywheel rig. This rig was simpler, easer to operate, and 

analyze than the actual flywheel rig. The high speed shaft rig was simply a shaft supported on two 

magnetic bearings (Fig. 2). A simple plant model was used as an estimator for the velocity and 

displacement, and the difference between the measured and estimated displacement was used for control. 

The shaft was not rotating. Similar test were performed on each bearing, (one at a time), and in a given 

lateral direction. For a given proportional gain, the derivative gain was varied noting the region of stable 

operation and the frequencies of vibration at the boundaries. 

The measured results for the lower and upper magnetic bearing are shown in Figures 3 and 4. Both 

bearings have similar shaped regions of stability, but the numerical values are different because the 

bearings are different. The region of stable operation is limited at small proportional gains by the gain 

necessary to cancel the negative stiffness. If the negative stiffness is not exceeded by the proportion 

stiffness, the rotor will just go to one pole or the other, (static instability). At high proportional gains there 

is no stable region of operation. At intermediate proportional gains there is a stable region between two 

derivative gains. Intuitively one might expect that a minimum amount of damping is necessary for 

stability, but why would one expect that a large amount of damping would cause instability? 

Intuitively one might expect the proportional gain to be parabolic with respect to the frequency, 

(ω2

 = K/M). This seems to be true at low frequencies, (the lower curve on the stability plot), but is not 

true for the higher frequencies, (the upper curve on the stability plot). The derivative gain seems to be 

parabolic with respect to the frequency. For a mass-stiffness-damped system any damping would result in 

a damped oscillation, not a stable oscillation. 

Figure 2.—Shows the High Speed Shaft Rig use to test magnetic bearing 

control systems to be used on the Energy Storage Flywheel.
NASA/TM—2006-213979 5 

Figure 3.—Shows the stability data for the bottom magnetic bearing. The 

top magnet bearing’s parameters were held constant. 

Figure 4.—Shows the stability data for the top magnetic bearing. 

The bottom magnetic bearing’s parameters were held constant. 

Test Rig Stability Analysis 

A simple model of the high speed shaft rig is a modal mass of the shaft being accelerated by a radial 

force produced by the magnetic bearing. The equation of motion is then: 

mx t F k x t cx t t kx t t   ( ) = = − −Δ − −Δ mag n ( ) ( ) ( ) (2) 

The force produced by the magnetic bearing has a negative stiffness term associated with the 

permanent magnet bias field; and, stiffness and damping terms produced by the magnetic bearing 

controller. The controller terms have a time delay associated with the various components in the control 

loop. The classical small signal stability analysis assumes an eigenvalue solution of the form: 
NASA/TM—2006-213979 6 

() ( ) () t t x t Ae x t t x t e λ −λΔ = ⇒ −Δ = (3) 

Substituting this solution into the equation of motion yields: 

 

2 0 t t m k c e ke n −λΔ −λΔ = λ − +λ + (4) 

If the following nondimensional parameters are defined: 

2 2

0

0

2 2

2

n n

t

kt m

c tm

ktm

μ = λΔ

τ=Δ

ςτ = Δ

τ= Δ

 (5) 

Then the eigenvalue equation becomes: 

22 2

0 0 0 2 n e e −μ −μ = μ − τ + ςτ μ + τ (6) 

The eigenvalue equation has two solutions depending on weather the eigenvalue μ is real or complex. If 

the eigenvalue is real, the nondimensional stiffness becomes: 

( ) ( ) 2 22 Re 2 0 0 n al eα μ = α ⇒ τ = − α − τ − ςτ α (7) 

If the eigenvalue is complex, then the imaginary portion of the eigenvalue is the circular frequency of the 

vibration times the time delay: 

c μ =α+ β⇒β=ω Δ i t (8) 

Solving the real and imaginary portions of the eigenvalue equation, the nondimensional damping 

becomes: 

(( ) ) 2 2

0 2 Im n eμ ςτ =− μ −τ β (9) 

And the nondimensional stiffness becomes: 

(( ) ) ( ) (( ) ) 2 22 22

0 Re Im n n e e

μ μ τ =− μ −τ + α β μ −τ (10) 

If the real part of the eigenvalue α is positive, the vibrations grow in time and the system is unstable. 

If α is negative, the vibrations decay in time and the system is stable. α equal to zero defines the stability 

boundary. 
NASA/TM—2006-213979 7 

( ) ( )

( ) ( )

2 2

0

2 22

0

0

2 sin

cos

n

n

α = ⇒μ= βi

ςτ = β +τ β β

τ = β +τ β

 (11) 

The stiffness and damping are defined in term of the proportional and derivative gains of the controller: 

p p

d d

k kG

c cG

=

= (12) 

The nondimensional stiffness and damping are related to the proportional and derivative gains by 

2 2

0

0 2

p p

d d

G

G

τ =τ

ςτ = τ

 (13) 

Correlation of Test Rig Data 

A simple model of the high speed shaft rig is a modal mass of the shaft being accelerated by a radial 

force produced by the magnetic bearing. The force produced by the magnetic bearing has a negative 

stiffness term associated with the permanent magnet bias field; and, stiffness and damping terms 

produced by the magnetic bearing controller. The controller terms have a time delay associated with the 

various components in the control loop. The classical small signal stability analysis assumes an 

eigenvalue solution of the equation of motion. This solution can be nondimensional by defining time 

constants which are the product of various frequencies associated with the negative stiffness, proportional 

stiffness, and the eigenvalue multiplied by the time delay. 

Figure 5 shows the stable region for a magnetic bearing with time delay. It also shows the growth 

rate, α (real part of the eigenvalue), for both the static, and dynamic stability, and the frequency of 

oscillations, β (imaginary part of the eigenvalue), for dynamic stability. The nondimensional stiffness for 

the static stability analysis is given by Equation (7), and for the dynamic stability analysis is given by 

Equation (10). The nondimensional damping for the static stability analysis is an independent variable, 

and for the dynamic stability analysis is given by Equation (9). For both the static and dynamic stability 

analysis, if the real part of the eigenvalue α is defined positive, the vibrations grow in time and the system 

is unstable. If α is defined negative, the vibrations decay in time and the system is stable. α equal to zero 

defines the stability boundary. The absolute value of the imaginary part of the eigenvalue, β, is a 

parameter which varies from 0 to π/2. A nondimensional negative stiffness of 0.2 was assumed. 

Figure 5 shows a plot of the nondimensional damping versus nondimensional stiffness for various 

growth rates. The nondimensional stiffness and nondimensional damping are related to the proportional 

and derivative gains (Eq. (13)). Figure 5 shows a stable region similar to that shown in Figures 3 and 4. 

Figure 6 shows the dynamic stability map for a magnetic bearing with time delay for various negative 

stiffness, Kn. The solid lines represent dynamic stability limits and the dotted lines represent the static 

stability limits. For both cases, α (real part of the eigenvalue) equal to zero. There is a stable region for 

nondimensional negative stiffness, nondimensional stiffness, and nondimensional damping between 0 and 

2. The nondimensional stiffness and nondimensional damping have to be greater then the nondimensional 

negative stiffness. 
NASA/TM—2006-213979 8 

k (Δt)2/m

cΔt/m

kn(Δt)2/m=.2

ωcΔt=π/2

3π/8

π/4

0

π/8

α=+.1 +.05 0 -.05 -.1 (Real μ)

α=+.1

α=+.2

α=+.3

Stable 

Region

Growth Rate of Disturbance of PD Controlled 

(Magnetic Bearing with Time Delay)

α=0

α=-.1

α=-.2

k (Δt)2/m

cΔt/m

kn(Δt)2/m=.2

ωcΔt=π/2

3π/8

π/4

0

π/8

α=+.1 +.05 0 -.05 -.1 (Real μ)

α=+.1

α=+.2

α=+.3

Stable 

Region

Growth Rate of Disturbance of PD Controlled 

(Magnetic Bearing with Time Delay)

α=0

α=-.1

α=-.2

Figure 5.—Shows the stable region for a magnetic bearing with 

time delay. It also shows the growth rate, α (real part of the 

eigenvalue), for both the static and dynamic stability, and the 

frequency of oscillations, β (imaginary part of the eigenvalue) 

for dynamic stability. 

cΔt/m

kn(Δt)2/m

k (Δt)2/m

0 .4 .8

1.2

1.6

2

ωcΔt

π/4

3π/8

0

STABLE

Stability Map of PD Controlled 

(Magnetic Bearing with Time Delay)

π/2

cΔt/m

kn(Δt)2/m

k (Δt)2/m

0 .4 .8

1.2

1.6

2

ωcΔt

π/4

3π/8

0

STABLE

Stability Map of PD Controlled 

(Magnetic Bearing with Time Delay)

π/2

Figure 6.—Shows the dynamic stability map for a magnetic 

bearing with time delay for various negative stiffness, Kn. 

The dotted line represents the static stability limit. 
NASA/TM—2006-213979 9 

Figure 7 shows the correlation of stability data for the bottom magnetic bearing shown on Figure 3. 

Figure 8 shows the correlation of stability data for the top magnetic bearing shown on Figure 4. The data 

shown on Figures 3 and 4 were correlated with different proportionality coefficients (Eq. (13)), for the 

proportional and derivative gain for each magnetic bearing. (The top magnetic bearing actuator was 

different than the bottom. The top magnetic bearing actuator could only apply a radial force and the 

bottom magnetic bearing actuator could apply both a radial and axial force.) There was only one digital 

controller for both bearings and the rest of the control system components were identical so the same time 

delay was used for both magnetic bearings. The time delay was calculated from the zero crossing 

frequency of the proportional gain, (2πfcΔt = π/2). The nondimensional negative stiffness for the top 

magnetic bearing was 0.99X1.2 = 1.19, and for the bottom magnetic bearing was 0.155X0.4 = 0.062. The 

correlation showed a stable region on derivative gain versus proportional gain plot that have the same 

qualitative tendencies as experimental measurements. In general the correlation of the proportional gain 

and derivative gain with the oscillation frequency at the stability boundary was very good. 

Figure 7.—Shows the correlation of stability data for the bottom magnetic bearing. 

Figure 8.—Shows the correlation of stability data for the top magnetic bearing. 
NASA/TM—2006-213979 10 

Flywheel Stability Analysis 

Figure 9 shows the rigid body rotordynamic model of a flywheel. The rotordynamics of the flywheel 

can be described in terms of the motion of the center of mass and rotations about the center of mass 

(Ref. 6). For small displacements the lateral motion is uncoupled from the axial motion. The lateral 

equations of motion of the center of mass are: 

1 2

1 2

cm x x

cm y y

mX F F

mY F F

= +

= +



 (14) 

For small rotations, the equations of angular motion about the center of mass are: 

11 2 2

2 2 11

TP y y

TP x x

I I LF L F

I I L F LF

β+ω α= −

α−ω β= −

 

  (15) 

If the x-direction is real and the y-direction is imaginary, then the complex displacement, rotation, and 

force can be defined as: 

cm cm cm

x y

R X iY

R i

F F iF

= +

′ =α− β

= +

 (16) 

The equations of motion become: 

2 1

22 1

cm

T P

FF F

mR F

T L F LF

IR iIR T

= +

=

= −

′ ′ −ω =

∑

∑

∑

∑



 

 (17) 

Figure 9.—Shows the rigid body rotordynamic model of a 

flywheel. The rotordynamics of the flywheel can be described 

in terms of the motion of the center of mass and rotations 

about the center of mass. 
NASA/TM—2006-213979 11 

For a centralized controller the summation of the forces of the magnetic bearings can be expressed as 

a function of the position and velocity of the center of mass. The summation of the torques about the 

center of mass can be expressed as a function of the angular position and angular velocity about the center 

of mass. For a centralized PD controller, (including the negative stiffness), the summation of the force 

and torque are: 

n cm cm cm

n cm cm cm

F K R KR CR

T K R K R CR

= −−

= −− ′ ′ ′′ ′

∑

∑



 (18) 

The centralized controller decoupled the motion of the center of mass and the rotation about the 

center of mass. If a time delay is included in the controller the equations of motion become: 

( ) ( ) ( ) ( )

() () () ( ) ( )

cm n cm cm cm

T Pn

mR t K R t KR t t CR t t

I R t i I R t K R t KR t t CR t t

= − −Δ − −Δ

′ ′ ′ ′ ′′ ′′ − ω = − −Δ − −Δ

 

   (19) 

The solution for the motion of the center of mass and that for the rotation about the center of mass is 

of the same form, if the shaft speed ω is set equal to zero. The solution for the motion of the center of 

mass is similar to the previous solution presented in the rig analysis. Therefore only motion of the rotation 

about the center of mass will be solved. If the eigenvalue solution is assumed to be: 

( ) t R t Aeλ ′ = (20) 

Then the equation of motion becomes: 

2 0 t t

T Pn I i I K C e Ke −λΔ −λΔ λ − ω λ− + λ + = ′′ ′ (21) 

If the following nondimensional parameters are defined: 

2 2

0

0

2 2

2

T

T

nn T

P T

t

kt I

c tI

ktI

tI I

μ = λΔ

τ=Δ

ςτ = Δ

τ= Δ

τη = ωΔ

 (22) 

Then the eigenvalue equation becomes: 

( ) 22 2

0 0 0 2 n i eμ = μ − τημ − τ + ςτ μ + τ (23) 

The eigenvalue equation does not have a real solution unless ω is zero. If the eigenvalue is complex, 

then the imaginary portion of the eigenvalue is the circular frequency of the vibration times the time 

delay: 

c μ =α+ β⇒β=ω Δ i t (24) 
NASA/TM—2006-213979 12 

Solving the real and imaginary portions of the eigenvalue equation, the nondimensional damping 

becomes: 

(( ) ) 2 2

0 2 Im n i eμ ςτ = − μ − τημ − τ β (25) 

And the nondimensional stiffness becomes: 

((( )) ) ( ) ((( )) ) 2 22 22

0 Re Im n n ie ie μ μ τ = − μ − τημ − τ + α β μ − τημ − τ (26) 

If the real part of the eigenvalue α is positive, the vibrations grow in time and the system is unstable. 

If α is negative, the vibrations decay in time and the system is stable. α equal to zero defines the stability 

boundary. Equation (27) defines the nondimensional damping and stiffness at the transition between 

stable and unstable operation of the flywheel. 

( ) ( )

( ) ( )

2 2

0

22 2

0

0

2 sin

cos

n

n

α = ⇒μ= βi

ςτ = β − τηβ + τ β β

τ = β − τηβ + τ β

 (27) 

Discussion of Results 

A simple model of the flywheel is shown in Figure 9. The rotordynamics of the flywheel can be 

described in terms of the motion of the center of mass and rotations about the center of mass (Ref. 6). For 

small displacements the lateral motion is uncoupled from the axial motion. The rotor center of mass is 

being accelerated by radial forces produced by the magnetic bearings. The rotation of the rotor about the 

center of mass is being accelerated by the torques produced by the radial forces of the magnetic bearings. 

The forces produced by the magnetic bearings have a negative stiffness term associated with the 

permanent magnet bias fields; and a stiffness and damping terms produced by the magnetic bearing 

controller. A centralized controller decouples the motion of the center of mass and the rotation about the 

center of mass. The controller terms have a time delay associated with the various components in the 

control loop. The classical small signal stability analysis assumes an eigenvalue solution of the equation 

of motion. This solution can be nondimensional by defining time constants which are the product of 

various frequencies associated with the negative stiffness, proportional stiffness, rotational speed, and the 

eigenvalue multiplied by the time delay. The nondimensional equations for the motion of the center of 

mass are of the same form as the nondimensional equations for the motion of rotation about the center of 

mass at zero rotational speed. These equations were solved in the previous section, and only the rotation 

equations about the center of mass will be solved. 

Figure 10 shows the stable region for a rigid flywheel supported on magnetic bearings using a 

centralized controller with time delay. Gyroscopic effects result in either a forward or backward whirl. It 

also shows the growth rate, α (real part of the eigenvalue), and the frequency of oscillations, β (imaginary 

part of the eigenvalue). The nondimensional stiffness is given by Equation (26) and the nondimensional 

damping is given by Equation (24). If the real part of the eigenvalue α is defined positive, the vibrations 

grow in time and the system is unstable. If α is defined negative, the vibrations decay in time and the 

system is stable. α equal to zero defines the stability boundary. The imaginary part of the eigenvalue, β, 

is a parameter which varies from 0 to +π/2. The nondimensional speed of the flywheel was +0.1 for 

forward whirl, 0 for nonrotation, and –0.1 for backward whirl. A nondimensional negative stiffness of 0.2 

was assumed. 
NASA/TM—2006-213979 13 

Figure 10.—Shows the stable region for a rigid flywheel supported 

on magnetic bearings using a centralized controller with time 

delay. Gyroscopic effects result in either a forward or backward 

whirl. If the growth rate, α (real part of the eigenvalue), is less 

then zero the system is stable.

Figure 11.—Shows the dynamic stability map for a flywheel 

supported on magnetic bearings with time delay for various 

values of speed, ω. The stiffness must be greater then the 

negative stiffness for the system to be stable.

Figure 10 shows a plot of the nondimensional damping versus nondimensional stiffness for various 

growth rates. Figure 10 shows a stable region similar to that shown in Figure 5 except it is now limited by 

the forward whirl. The stable region is smaller then that for the nonrotating rotor. 

Figure 11 shows the dynamic stability map for a flywheel supported on magnetic bearings with time 

delay for various values of speed, ω. The stiffness must be greater then the negative stiffness for the 

system to be stable. To the left of the solid lines represent stability limits resulting from the forward whirl. 

(The real part of the eigenvalue, α, equals zero.) The stable region becomes smaller for higher 

nondimensional rotor speeds. There is no stable region for nondimensional rotor speeds higher 0.8. 
NASA/TM—2006-213979 14 

Figure 12.—Shows the dynamic stability map for a flywheel 

supported on magnetic bearings with time delay for various 

values of negative stiffness, Kn. The stiffness must be greater 

then the negative stiffness for the system to be stable. 

Figure 12 shows the dynamic stability map for a flywheel supported on magnetic bearings with time 

delay for various values of negative stiffness, Kn. It shows plots of nondimensional damping versus 

nondimensional flywheel speed. It shows both forward and backward whirl, and the stable region is 

between the forward and backward whirl for the appropriate stiffness. The stiffness must be greater than 

the negative stiffness for the system to be stable. As the system negative stiffness increases the stable 

region decreases and the maximum flywheel speed decreases. 

Conclusion 

Active magnetic bearings are used to provide a long-life, low-loss suspension of a high-speed 

flywheel rotor. This paper describes a modeling effort used to understand the stability boundaries of the 

PD controller used to control the active magnetic bearings. Limits of stable levitation and associated 

vibration frequencies were measured for a nonrotating rotor in terms of allowable stiffness and damping 

values. The following conclusions were made: 

1. These results could be correlated by a simple time delay model used for the active magnetic 

bearing controller. 

2. The time delay could be calculated from the zero crossing frequency of the proportional gain. 

3. The product of the frequency of vibration times the delay time varied between 0 and π/2. 

The analysis was extended to a rotating rigid flywheel supported on active magnetic bearings. The 

following conclusions were made: 

1. The larger the negative stiffness, the smaller the region of stability. 

2. The larger the speed, the smaller the region of stability. 

3. The forward whirl limited the region of stability 
NASA/TM—2006-213979 15 

The results of this modeling effort showed a region (on damping versus stiffness), plot that has the 

same qualitative tendencies as experimental measurements and the analysis could be extended to more 

complicated cases including gyroscopic effects. 

References 

1. M.A. Pichot, et al., “Active Magnetic Bearings for Energy Storage Systems for Combat Vehicles,” 

IEEE Transactions on Magnetics, Vol. 37, No. 1, January 2001. 

2. P. Tsiotras, S. Mason, “Self Scheduled H∞ Controllers for Magnetic Bearings,” University of Virginia, 

Charlottesville, VA. 

3. P. Tsiotras, M. Arcak, “Low-Bias Control of AMB Subject to Voltage Saturation: State-Feedback and 

Observer Designs,” IEEE, 2005. 

4. “System and Method for Controlling Suspension Using a Magnetic Field,” U.S. Patent: (US 

6,323,614B1); Date of Patent: Nov. 27, 2001; Inventors: Palazzolo, A., Mu Li, Uhn Joo Na, Erwin 

Thomas. 

5. T. Dever, et al., “Modeling and Development of a Magnetic Bearing Controller for a High Speed 

Flywheel System,” IECEC, 2004. 

6. “Rotordynamics of Turbomachinery,” Dr. John M. Vance, Texas A&M, Wiley-Interscience 

Publication, pp. 121–122.
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E–15301

21

Stability Limits of a PD Controller for a Flywheel Supported on Rigid Rotor and

Magnetic Bearings

Albert F. Kascak, Gerald V. Brown, Ralph H. Jansen, and Timothy P. Dever

Flywheels; Magnetic bearings; Gyroscopic; Energy storage

Unclassified -Unlimited

Subject Category: 37

WBS–22–066–30–02

1L161102AF20

Prepared for the Guidance, Navigation, and Control Conference and Exhibit sponsored by the American Institute of

Aeronautics and Astronautics, San Francisco, California, August 15–18, 2005. Albert F. Kascak, e-mail:

Albert.F.Kascak@grc.nasa.gov, U.S. Army Research Laboratory, NASA Glenn Research Center; Gerald V. Brown,

e-mail: Gerald.V.Brown@grc.nasa.gov, NASA Glenn Research Center; Ralph H. Jansen, e-mail:

Ralph.H.Jansen@grc.nasa.gov, University of Toledo, 2801 W. Bancroft Street, Toledo, Ohio 43606–3390; and

Timothy P. Dever, e-mail: Timothy.P.Dever@grc.nasa.gov, QSS Group, Inc., 21000 Brookpark Road, Cleveland, Ohio

44135. Responsible person, Albert F. Kascak, organization code RSS, 216–433–6024.

Active magnetic bearings are used to provide a long-life, low-loss suspension of a high-speed flywheel rotor. This paper

describes a modeling effort used to understand the stability boundaries of the PD controller used to control the active

magnetic bearings on a high speed test rig. Limits of stability are described in terms of allowable stiffness and damping

values which result in stable levitation of the nonrotating rig. Small signal stability limits for the system is defined as a

nongrowth in vibration amplitude of a small disturbance. A simple mass-force model was analyzed. The force resulting

from the magnetic bearing was linearized to include negative displacement stiffness and a current stiffness. The current

stiffness was then used in a PD controller. The phase lag of the control loop was modeled by a simple time delay. The

stability limits and the associated vibration frequencies were measured and compared to the theoretical values. The results

show a region on stiffness versus damping plot that have the same qualitative tendencies as experimental measurements.

The resulting stability model was then extended to a flywheel system. The rotor dynamics of the flywheel was modeled

using a rigid rotor supported on magnetic bearings. The equations of motion were written for the center of mass and a

small angle linearization of the rotations about the center of mass. The stability limits and the associated vibration

frequencies were found as a function of nondimensional magnetic bearing stiffness and damping and nondimensional

parameters of flywheel speed and time delay.