concerning performance as wellas costs are high. stabilized [5]. Thus, this motor design can only be applied, if. The proposed bearingless brushless DC motor ...
IEEE PEDS 2005
Nonlinear Feedback Control of a Bearingless Brushless DC Motor Herbert Grabner, Wolfgang Amrhein, Siegfried Silber, Klaus Nenninger LCM - Linz Center of Competence in Mechatronics Johannes Kepler University Linz, Dep. Electrial Drives and Power Electronics A-4040 Linz, Austria herbert.grabnerglcm.at
axial position of the rotor as well as tilting is passively Abstract-The demands on bearingless drive configurations concerning performance as well as costs are high. stabilized [5]. Thus, this motor design can only be applied, if The proposed bearingless brushless DC motor consists of five the requirements on stiffness in axial and tilting direction are concentrated coils in a symmetrical arrangement, which generate radial forces and motor torque simultaneously in interaction with a permanent magnet excited disc shaped rotor. Additionally, as a centrifugal pump in semiconductor, chemical or medical tilting deflection and the axial position of the rotor are stabilized industry. Owing to the rare earth rotor permanent magnets, air passively by means of magnetic reluctance forces. Thus, system gaps of approximately 4mm can be realized that allow much costs can be reduced significantly compared to a conventional freedom in constructive design. bearingless motor - . setup, . . which stabilizes all six degrees of further mechanical simplification can be achieved by em~~~~~~~~~~~~~A freedom actively. the the of linear ploying concentrated instead of distributed coils [6]. Thereby use Owing to the nonlinearity of plant, control design methods alone is not suitable for achieving a the simplest motor design, which fulfills a bearingless operahigh operation performance. This paper introduces a new radial tion, features four concentrated coils in the stator and has a position and motor torque control algorithm based on the four pole permanent magnet rotor-disk [7]. However, this drive theory of feedback linearization for a bearingless brushless DC . . t motor. Thereby, the combined model of translatory and rotatory configuration leads to single-phase motor characteristics with dynamics can be split into independent linear systems by means a relatively high cogging torque. To overcome these problems, a bearingless motor with five concentrated coils and a two pole of a nonlinear change of system coordinates and a static state feedback. rotor has been developed [8]. Experimental results demonstrate the effectiveness of the Because of the concentrated coils, force and torque genproposed approach. eration are coupled and highly nonlinear. Therefore, they Keywords-bearingless motors; brushless DC-motors; nonlin- cannot be considered separately. In the following sections a performance optimization of a bearingless slice motor with ear control system five concentrated coils using nonlinear control techniques is presented. I. INTRODUCTION 11. MATHEMATICAL MODEL During the past decade there have been some significant advances in the areas of bearingless motor technology. Over A Description of the Model t of the Me l Fir the years, first serial products entered the market and it tumed is fal h ehnclpato h oo ilb ae out that, compared to usual motor bearing concepts, this kind out that, compared to usual motor bearing concepts, this kind into account. According to [9], any radial displacement of the of technology has a lot of advantages. Bearingless drives needestalng d esabi orce depending on the no lubrication, they have an almost unlimited lifetime and they rotor ause s nonlinear can be used in several applications, where high demands on rtorfane mathis cleanliness, chemical resistance and tightness are important stifiess matrix 1 + c2 cos(2-y) c sin(2-y) 0 [1], [2], [3]. Additionally, they can perform movements of the rotating or levitating part during operation. However, Kx(y) = kx cx sin(2-y) 1 - cx cos(2-y) 0 , (1) 0 0 mechanical and electrical complexity is high and therefore in o0 L the range of low cost applications, they hardly gain ground. where k denotes the natural negative stiffness of the rotor. stfns reuc system coplxiy To reduce mechanical Theoretically, the coupling parameter cx is 0.5, but it basically complexity, a very simple mechanica design can be achieved by means of a bearingless brushless depends on the motor arrangement. K (7) contains a row and DC motor with a permanent magnet excited rotor-disk [4]. An a column of zeros, because there is no coupling to rotational advantage of this drive configuration is that only the radial movements. Moreover, the stiffness matrix position and the rotor angle need to be actively controlled. Due to the high magnetic flux density in the air gap, the Kz(-y) = K1+ Kn(y)
To~~~~~~
aeoe
0-7803-9296-5/os/$20.00 ©2005 IEEE
simplel
gftertr
366
\:~ ~ ~ ~ ~ v
damping is not considered, because it is application specific
and has no influence on the proposed approach. The bearingless motor is fed from a PWM voltage source inverter. Hence, the control inputs are the terminal voltages
[Ul
i 2-li!0' 5 J 5 u~~~~U=
U2u2u3 U3
5]T.
U4~U4 5
Then, current dynamics is determined by the stator voltage
equation
u = Ri + Li + TT
(y)vr.
(4)
Here, the velocities of the rotor in radial direction and the angular velocity are combined in
A complete symmetrical motor design is assumed, which involves equal resistances R=R1 =R2 = =Rs ,
Fig. 1. Bearingless brushless DC motor with five concentraited coils
equal self inductances
is split into a linear and a nonlinear part. The exact electromagnetically generated force and torque values are quadratic functions of the phase currents [10]. However, a large air gap and a rotor featured with permanent magnets lead to an almost linear dependency. In addition, the influence of the radial rotor position can be largely neglected and eddy current and hysteresis losses are not considered. The current to force and torque relationship
FrF,
[1 =
[T11(7) T12(yr) T13(>y) T14(7r) T15(-r)1 I2 T21(Y)XY) T22(-y) T22(7) T23(Y) T23X-Y) T24(Y) T24(Y) T25(Y) T25X-YI I33
[TJ [T31(y) T32(-y) T33(y) T34(y) T35(y)j v 4
Tm(y)
5
(2)
is a nonlinear function of , where F. and Fv are the forces acting on the rotor and T specifies the motor torque [10]. Furthermore, the vector of the phase currents, arranged as shown in Fig. 1, can be written as .3 .1T
.= .2
X~ ~ ~ ~ ~ ~ ~ ~
Setting up the equation of motion leads to MXr = K.(-y)xr + Tm(-y)i where Xr = [2r Yr ^1 indicates the radial coordinates Xr, Yr and the angle generally referred to as the mass matrix [mr ° 0]
(3)
y. M is
with mr the mass of the rotor disc and Ip is the polar moment of inertia. If additional damping is applied by a medium (e.g. in pump applications), the rotor stability will increase. Here,
L1i = L22 =.-=L55
and equal couplig iductances L12 = L15 = L21 = L23 = L13 = L14 = L24 = L25 = in all five phases. B. Clarke Transformation One of the main properties of the proposed bearingless motor is that each coil generates radial forces and motor torque simultaneously. The permanent magnets in the rotor form one pole pair. Thus, force generation needs a four pole and torque a two pole magnet stator field generation needs [7]. A convenient way of mapping multi-phase currents into a two-phase orthogonal stator axis is a Clarke transformation 1
V
i2kkir
cos( kI
CO(8k7r)
...
c ^ 5 0 sin( 5) ... sin Therefore, the two-phase orthogonal auxiliary winding generates a two-pole stator field with k = 1 or a four-pole stator field with k = 2. Using a transformation in the form of
= [Vc, V.J 13 (4 ~~~~~~~ = [v,2 Vc,i] i
5
(5)
vc
where
i = [if,D if,Q it,D it,Q] T denotes the currents in the orthogonal systems, one can find a decoupled current to force and torque relationship'
TF
T(-y) i
T12(7)
[-Th2(-y) T11({y)
0
0
1
_.
IA complete decoupling of Tm(y) only takes place if the rotor magnethis is the practical case to keep the
tization is almost sinusoidal. However, motor losses to a minimum.
367
Furthermore, the five concentrated coils are star connected to decrease the complexity of the power converter. Due to i =0, there is no homopolar component and only four of the five phase currents occur independently. Then using the transformation according to (5), the new phase currents become independent. The transformation (5) of the phase currents is invariant with respect to the electrical power, which leads to the phase voltage transformation U =VTU.
(6)
Applying (5) and (6) the stator voltage equation (4) can be rewritten in the form u = R7i+ L i + TM(-OX'r
.(7)
The inductance matrix holds the relation L = VTLV _ C LV - FLfI
Lt
with z as the new state
x = -'(z) .
+
[L
O
1]
=
Then, the following coordinates Fl 117i+b,()r X r Z IZ2| = |
1
= [z1 Z2 Zl are defined, relating to 2 2 Tm(7)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -T34() T33 00
-
1
Lz3J
[
V
T
Lf=Li,-Li,-+L13 2an
0
)
LKn°1 rank 4~1,1(t 4
1,2(7) =
for any value of
easily.
with the state vector x = Xr
LVr'j
(12)
Z
(8)
L0 J
(10)
ax dt x=$ (z)
Z
~~~0
M1Tm(y)
(9)
Differentiate (9) with respect to time to yield ()d
Combining the stator voltage equation (7) and the mechanical and part (3), the system of a bearingless brushless DC motor can be directly expressed in the nonlinear state space representation F-RL Detto -Lr~~ ---1 --1-T 1 TM(-Y)~~~Due x = x O I
LMR Tm(L M-'K,(7) M1K~(y)
z= (x), vector, satisfies the relation
with
Li,_+Liz_2 -L13
=
x=A(x)+B(x)U can only be done, if the state transformation
0] LtIJ
with I as the identity matrix and +1
Changing the coordinates of a general plant described by a system of nonlinear differential equations
Clearly, force and torque generation as well as translatory and rotatory dynamics are not yet coupled, but there is still the nonlinearity Tm(-y). Additionally, the non-constant stiffness matrix Kx(-y) can lead to resonance phenomena, which were discussed in [11]. III. FEEDBACK LINEARIZATION A. Nonlinear Change of Coordinates Since explaining the derivation of the nonlinear state transformation of bearingless motors is beyond the scope of this paper, only the basic idea will be presented. For more details regarding feedback linearization of nonlinear systems, please
the inversion (10) of (12) ay,
can be found
Changing the coordinates is reserved to the phase currents of the auxiliary winding systems, which are mapped concerning a physical background. In particular, z1 and Z2 express the radial levitation forces, corrected to compensate for any nonlinearity in stifffess. Z3 allows direct access to the motor torque. In order to get a fully invertible transformation, a fourth coordinate Z4 is added, which allows field weakening. Thus, an operation above base speed can be achieved when accepting higher copper losses and a lower efficiency. B. Static Feedback Control Law The nonlinear change of states applying (11) leads to a system that is still not linear, but nonlinearity disappears in the lower six state equations. Choosing a static state feedback law of the forn
=
1,1(y)v - RE7, (-y)1,2Qy)z2 + Tm(Y)z3
+QL8 L
-'
( z)
-1
-A(d (. I2(AY)Z2)
368
linear
linear system
r-----------------
eedbacklinearization
tLes
plant
Fig. 2. z= Block~ diagramm of the proposed feedbackOBv.u linearization > I xz+ ~~~~~ OAO ~x
and the
new
system
respectively.
Thus, the resulting
closed
loop system including the transformation (12) and the state feedback (13) becomes linear and can be written as
*~ ~~
-RLz+[° °
M
K
;i ''.
-L-1
Fig. 3.
. Bearingless brushless DC motor with five concentrated coils
Additionally, the system of tenth order may be partitioned into using the states of the original system. four independent subsystems C. Controller Implementation loop system including the transforaccording to2(6)nthehrelatio = -L 1+k j-v1 The proposed control scheme is outlined in Fig. 2. Z1Yr= t (6) nonlinear acoring. bruhessDl Beriges mtrwihfvicnenrtdnol z 14 Thanks to the feedback linearization, the main position and 7 = 0n Z2 m, zYr speed controllers can be defined separately by means of any Vr - 1 2+ B k=2 d t fe r Lsingle input single output (SISO) design method for linear time m, pro invariant (LTI) systems. According to (18), all state variables = + have to be known. To keep system costs low, state observers (15) based on a mathematical model can also be employed. Finally, k
u= VCU z3 =-o z3 + mV3 fou = Q = Z,
holds, (16)
to transform the
stator voltages
back into the actual
physical winding configuration.
IV. EXPERIMENTAL RESULTS The setup of the bearingless brushless DC motor is shown and Z4 = -L-Z4 + .1V (17) in Fig. 3. With an outer rotor diameter of46 mm and a rotor t t ~~~~~~~heightof 15 mm, this motor can drive a rated power of 300 W. The characteristics of (14)and (15) are equivalent to a voltage Th air egap has a thickness of 4 mm andthe radial operating controlled active magnetic bearing system. Additionally, (16) ...prange isc0.6 mm. Its synchronous speed is 10000 rpm and corresponds to the model of a DC machine with EMF comth force e capacity is about 30 N at rated current. Another pensation. ..... important i tem is the passive stiffness in axial direction which be is about 6500 m ' § Lf dc.2 (5)) Asfar as the realizationand implementation in hardware is 1,2 f r concerned, a TMS320F28 11 controller by Texas Instruments 1,1
