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Description  |
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BACKGROUND OF THE INVENTION
The present invention relates to an electromagnetic bearing apparatus
having five degrees of freedom for use in a turbo molecular pump, a
spindle of a working machine or the like.
FIG. 1 is a view showing the general structure of a five degrees of freedom
control type magnetic bearing of conventional construction. In FIG. 1, 1
is an axis position sensor for detecting the position of a rotating body 2
in the axis direction, 3 is a target corresponding to the axis position
sensor 1 mounted on the rotating body 2, 4 is a motor for rotating the
rotating body 2, 5 is an armature disc mounted on the rotating body 2, 6
is an axis direction electromagnet for providing an axis direction
controlling force to the armature disc 5, 7 and 8 are radial direction
magnetic bearings, and 9 and 10 are radial direction position sensors.
FIG. 2 is an embodiment of a prior art control system for the five degrees
of freedom control type magnetic bearing arranged as shown in FIG. 1. The
arrangement of the control system is disclosed in French Pat. No. 2149644
in which the translating movement parallel with the rotating axis of the
rotating body and the rotating movement with respect to the center of mass
of the rotating body are controlled individually. In the drawing, 11 is an
adder for a pair of radial direction position sensors X.sub.1 and X.sub.1
' or X.sub.2 and X.sub.2 ' and 12 is an adder for carrying out the adding
operation for the output from the adders 11. The output from the adder 12
is indicative of the translating movement in the X axis direction and is
applied to a phase advance compensating circuit 13. The output from the
phase advance compensating circuit 13 is applied to adders 14 and 19 whose
outputs control a power amplifier 29 by which electromagnet coils A.sub.1,
A.sub.1 ', A.sub.2 and A.sub.2 ' are driven. In a similar way, a control
device for restricting the translating movement in the Y direction
comprises adders 20 and 21, a phase advance compensating circuit 22,
adders 23 and 28 and the power amplifier 29 to control the power supplied
to electromagnet coils B.sub.1, B.sub.1 ', B.sub.2 and B.sub.2 '.
A signal component of the movement around the center of mass of the
rotating body is obtained by adding the output of an inverter 15 to the
output of the adder 11 for the radial direction detectors X.sub.2 and
X.sub.2 ' by the use of an adder 16. The output of the adder 16 is applied
to a wide band phase advance compensating circuit 17 whose output drives
the electromagnet coil A.sub.1 or A.sub.1 ' and the electromagnet coil
A.sub.2 or A.sub.2 ' is driven by the output of an inverter 18. The
movement rotating about the Y axis at the center of the mass of the
rotating body is restricted by the control device mentioned above. In a
similar way, the movement control around the X axis is carried out by the
control device composed of the adder 20, an inverter 24, a wide band phase
advance compensating circuit 26, an inverter 27 and the power amplifier
29. The electromagnetic coil B.sub.1 or B.sub.1 ' is driven by the output
of the power amplifier 29 and the electromagnetic coil B.sub.2 or B.sub.2
' is driven by the output of the inverter 27 whereby to attain the desired
control.
In order to carry out the restriction control in the Z axis direction, that
is, the thrust direction of the rotating body, the signals of the axis
direction position detectors Z.sub.1 and Z.sub.2 are applied to an adder
30 and the control signal corresponding to the signal is produced by a
phase advance compensating circuit 31. An electromagnet coil C.sub.2 is
driven by controlling a power amplifier 29' in accordance with the control
signal mentioned above, and the output of the phase advance compensating
circuit 31 is applied to the inverter 32. The power amplifier 29' is
controlled by the output signal to drive the electromagnet coil C.sub.1.
As a result, the restriction control along the Z axis is attained.
The meanings of the symbols X.sub.1, X.sub.1 ', . . . A.sub.1, A.sub.1 ' .
. . used for explaining the control block diagram shown in FIG. 2 are
illustrated in FIG. 3. In FIG. 3, 33 is a rotating body, P.sub.1, P.sub.2
are radial direction magnetic bearings, and P.sub.3 is an axis direction
magnetic bearing. A.sub.1, A.sub.1 ' designate the mounting positions of
vertical direction electromagnet coils for the radial direction magnetic
bearing and B.sub.1 and B.sub.1 ' designate the mounting positions of
horizontal direction electromagnet coils for the radial direction magnetic
bearing. C.sub.1 and C.sub.2 designate the mounting positions of
electromagnet coils for the axis direction magnetic bearing P.sub.3. In
FIG. 3, the directions of the arrows indicate the direction of
electromagnetic force. X.sub.1 and X.sub.1 ' form a pair of position
detectors which are structural members of the bearing P.sub.1 and placed
in the vertical direction. Y.sub.1 and Y.sub.1 ' form a pair of position
detectors which are placed in the horizontal direction. Similarly, X.sub.2
and X.sub.2 ', and Y.sub.2 and Y.sub.2 ' are pairs of position detectors
by which the bearing P.sub.2 is arranged and Z.sub.1 and Z.sub.2 is a pair
of position detectors by which the bearing P.sub.3 is arranged.
With the arrangement of the control block shown in FIG. 2, it is possible
to control three translating movements other than the movements around the
rotating axis of the rotating body and two rotating movements around the
center of the mass. However, when the precession and the nutation occurs
in the rotating body due to a gyro effect, the aforedescribed arrangement
is not so effective as to control it. The reasons for this are as follows.
For example, when the rotating motion around the X axis occurs due to the
influence of the gyro effect during the high speed rotation of the
rotating body 33, the rotating body 33 starts to rotate around the Y axis.
However, in the control system shown in FIG. 2, no control for suppressing
this effect is taken into consideration.
SUMMARY OF THE INVENTION
An object of the present invention to provide a control system which is
able to quickly suppress the precession and the nutation due to the gyro
effect. The control system of the present invention has been achieved by
applying thereto the optimum regulator problem taught by modern control
theory, and excellent results have been obtained by noting the interior
structure of the object to be controlled in the analysis.
BRIEF EXPLANATION OF THE DRAWINGS
FIG. 1 is a structural view showing a common prior art structure of the
magnetic bearing for controlling five degrees of freedom,
FIG. 2 is a circuit diagram showing a prior art control system,
FIG. 3 is an explanatory view of a part of the prior art structure shown in
FIG. 2 showing the mounting position of an electromagnetic coil or a
position sensor,
FIG. 4 is a circuit diagram showing the optimum feedback mechanism of the
present invention,
FIG. 5 is an explanatory view illustrating a coordinate system which is
used in the analysis,
FIG. 6 is a view illustrating the optimum regulator of a one degree of
freedom system,
FIG. 7 is a circuit diagram showing the structure of the inside of an
object to be controlled,
FIG. 8 is a circuit diagram showing one example of the circuit arrangement
of the present invention, and
FIG. 9 is a view showing the response waveform.
DETAILED DESCRIPTION OF THE INVENTION
In FIG. 4, there is shown a block diagram of the control system of the
present invention which enables attenuation of the precession and the
nutation of a controlled object 37.
The analysis used to obtain the block diagram shown in FIG. 4 will be
described hereinafter.
In FIG. 5, a rotating body 34 comprises an axial symmetric rigid body which
is symmetric with respect to the center of mass G and the body 34 rotates
around the rotating axis at a constant angular speed .omega..sub.z by
means of a motor. The coordinate system O-XYZ is determined in such a way
that the position of the center of mass G of the rotating body 34 when it
is balanced is selected as an origin and the rotating axis is coincident
with the Z axis. When the attracting force of an electromagnet is
represented by F.sub.k (k=1, . . . , 10), the equation of motion for the
rotating body 34 will be written as follows if the high-order terms more
than .theta..sub.x.sup.2 and .theta..sub.y.sup.2.
mx.sub.G =F.sub.1 -F.sub.3 +F.sub.5 -F.sub.7 (1)
my.sub.G =F.sub.2 -F.sub.4 +F.sub.6 -F.sub.8 (2)
mz.sub.G =F.sub.9 -F.sub.10 (3)
I.sub.r .theta..sub.y -I.sub.a .omega..sub.z .theta..sub.x =l(F.sub.1
-F.sub.3 -F.sub.5 -F.sub.7) (4)
I.sub.r .theta..sub.x +I.sub.a .omega..sub.z .theta..sub.y =l(-F.sub.2
+F.sub.4 +F.sub.6 -F.sub.8) (5)
Where,
m: mass of the rotating body,
I.sub.a : moment of inertia around the rotating axis
I.sub.r : moment of inertia around the diameter through the center of mass
G
(x.sub.G, y.sub.G,z.sub.G): coordinates of the center of mass G of the
rotating body
l: distance between the center of mass G and the point of application of
the electromagnetic force
(.theta..sub.x,.theta..sub.y): magnitude of the angular displacement around
the X and Y axes of the rotating body
F: electromagnetic force (F.sub.1, F.sub.2 . . . F.sub.10 as shown in FIG.
5)
The attracting force F of the electromagnet is expressed as follows:
F=Ki.sup..rho. /d.sup..sigma. (K>0, >1, .sigma.>1) (6)
where,
d: gap between the electromagnet and the rotating body
i: exciting current for the electromagnet
Expanding the foregoing equation in the vicinity of the balanced condition,
the following equation will be given.
F=F+K.sub.i .DELTA..sub.i -K.sub.d .DELTA..sub.d (7)
wherein,
F=Ki.sup..rho. /d.sup..sigma.
K.sub.i =.rho.Ki.sup..rho.-1 /d.sup..sigma.
K.sub.d =.sigma.Ki.sup..rho. /d.sup..sigma.+1
.DELTA..sub.i =the amount of infinitesimal change in i
.DELTA..sub.d : the amount of infinitesimal change in d
K,.rho.,.sigma.: constants
If the equation (7) is applicable to any electromagnet and the amount of
change in the exciting current of each electromagnet is represented by
i.sub.k (k=1, 2, . . . , 10), the equations (1) to (5) will be written as:
x=Ax+Bu (8)
where,
x=[x.sub.x.sup.T, x.sub..theta..sup.T, x.sub.y.sup.T ].sup.T
x.sub.x =[x.sub.G, x.sub.G ].sup.T
x.sub..theta. =[.theta..sub.y, .theta..sub.y, .theta..sub.x, .theta..sub.x
].sup.T
x.sub.y =[y.sub.G, y.sub.G ].sup.T
u=[i.sub.1 -i.sub.3, i.sub.5 -i.sub.7, i.sub.2 -i.sub.4, i.sub.6 -i.sub.8
].sup.T
##EQU1##
Now, consider the problem in which the input variable u(t) to minimize the
following quadratic relationship of an estimating function when the object
to be controlled is in any initial state x(0):
##EQU2##
Where, Q: non-negative definite matrix
R: positive definite matrix
If the symmetricity of the system is considered, it is reasonable to select
the following form:
##EQU3##
If the input variable is rewritten as follows:
u=[(i.sub.1 -i.sub.3)+(i.sub.5 -i.sub.7), (i.sub.1 -i.sub.3)-(i.sub.5
-i.sub.7), -(i.sub.2 -i.sub.4)+(i.sub.6 -i.sub.8), (i.sub.2
-i.sub.4)+(i.sub.6 -i.sub.8)].sup.T (11)
and the equations (8) and (9) are also rewritten, the problem of the
optimum regulator of the system described by the equation (8) may be two
optimum regulator problems of a one degree of freedom system and an
optimum regulator problem of a two degrees of freedom system.
In this case, if A.sub.p, b.sub.p, u.sub.p, Q.sub.p and r are properly set
with respect to the axis direction of the rotating body, it is easy to
derive that these problems become the optimum regulator problem of a one
degree of freedom system.
When the object to be controlled is described by the following equation,
x.sub.p =A.sub.p x.sub.p +b.sub.p u.sub.p (12)
the problem of the optimum regulator of a one degree of freedom system is
equal to the problem of finding the u.sub.p to minimize the following
estimating function:
##EQU4##
wherein, when x.sub.p =x.sub.x, U.sub.p =i.sub.1 -i.sub.3 +i.sub.5
-i.sub.7
when
x.sub.p =x.sub.y, u.sub.p =i.sub.z -i.sub.4 +i.sub.6 -i.sub.8
b.sub.p =[0, Ki/m].sup.T, Q.sub.p =diag (q.sub.d, q.sub..nu.)
when 4Kd/m=.alpha., Ki/m=.beta., 2q/r=.gamma.d, 2q/r=.gamma..nu.,
the input u.sub.p for minimizing the value of J.sub.p will be given as the
following equation.
where,
##EQU5##
Therefore, the optimum regulator of one degree of freedom will be the
arrangement shown in FIG. 6. In the figure, 35 is an object to be
controlled, and 36 is a feedback compensator for producing a feedback
control signal corresponding to the radial displacement or deviation of
the controlled object 35. In this optimum regulator system, the feedback
for the displacement and the speed is applied to the object 35 which has
an unstable pole S=+.sqroot..alpha.. As a result, the system comes to the
stable state having a proper attenuation characteristics. The magnitude of
the attenuation is adjustable by the selection of the weight matrix. The
optimum regulator of the translating movement of Z axis direction about
Z.sub.G is same as in the aforemention.
On the other hand, when the object to be controlled is described as
x.sub.p =A.sub..theta. x.sub..theta. +B.sub..theta. u.sub..theta.(15)
the problem of the optimum regulator of a two degrees of freedom is equal
to the problem of finding the u.sub..theta. for minimizing the following
evaluating function:
##EQU6##
wherein,
##EQU7##
The problem of the optimum regulator of two degrees of freedom system is
related to a gyro effect which occurs at the rotating body. The
arrangement of the optimum regulator will now be described.
For providing a common solution, the following reference relations,
##EQU8##
are used, and the orderless variables t(.DELTA.t/t.sub.0), .theta..sub.x
(.DELTA..theta..sub.x /.theta..sub.0), .theta..sub.y (.DELTA..theta..sub.y
/.theta..sub.0), u(.DELTA.u.sub..theta. /i.sub.0).
When the dynamic characteristics of the object to be controlled is
described as
X.sub..theta. =A.sub..theta. X.sub..theta. +B.sub..theta. u.sub..theta.(18)
the control input u.sub..theta. (t) can be found by minimizing the
following evaluating function:
##EQU9##
where:
##EQU10##
Since i.sub.0.sup.2 r/2>0, the following relationship will be established
without losing the generality:
R.sub..theta. =diag (1, 1)
When the positive solution for a constant value of the following
PA.sub..theta. +A.sub..theta..sup.T P-PB.sub..theta. R.sub..theta..sup.-1
B.sub..theta..sup.T P+Q.sub..theta. =0 (20)
the optimum input u.sub..theta. * for minimizing J.sub..theta. is expressed
as the following expression:
u.sub..theta. *(t)=-R.sub..theta..sup.-1 B.sub..theta..sup.T PX.sub..theta.
(t) (21)
Therefore, for closed loop systems, it will be written as:
X.sub..theta. =(A.sub..theta. -B.sub..theta. R.sub..theta..sup.-1
B.sub..theta. P)X.sub..theta. (22)
In general, although the solution of the equation (22) is found by
numerical calculation, the resulting solution is not so useful for finding
the physical meaning. In this analysis, the solution is found by noting
the structure of the inside of the system following which an excellent
solution is obtained. If the equation (21) is expressed by the use of each
component of the matrix, it will be as follows:
.theta.y-k.theta..sub.x -.theta..sub.y =-P.sub.12 .theta..sub.y -P.sub.22
.theta..sub.y -P.sub.23 .theta..sub.x -P.sub.24 .theta..sub.x (23)
.theta..sub.x +k.theta..sub.y -.theta..sub.x =-P.sub.34 .theta..sub.x
-P.sub.44 .theta..sub.x -P.sub.14 .theta..sub.y -P.sub.24 .theta..sub.y (
24)
However, as P is a symmetrical matrix, each component of P is shown only by
the upper triangular elements. The equations (23) and (24) show a system
having an inverse symmetrical cross-linking in which the systems each of
which has the same transfer function 1/(s.sup.2 -1) are linked through
transfer elements having an opposite sign with respect to each other. This
state is shown in FIG. 7. In the relation for minimizing the input energy
required for the control in response to the structure of the inside of
such a controlled object 37, the optimum state feedback compensating
mechanism has also a similar structure. That is,
##EQU11##
Substituting the equation (25) for the equation (20) and clearing it,
P.sub.12, P.sub.22 and P.sub.14 can be obtained. If these solutions are
expressed as P.sub.12 *, P.sub.22 * and P.sub.14 *, P.sub.12 * is the
roots of the following equation
2P.sub.12.sup.3 +(k.sup.2 +q.sub..omega. -4)P.sub.12.sup.2 -2(q.sub..theta.
+q.sub..omega.)P.sub.12 -q.sub..theta. q.sub..omega. =0 (26)
and should satisfy the following relationship:
##EQU12##
Also,
##EQU13##
As a result, the optimum regulator will be a structure as shown in FIG. 4.
In the light of the mutual interference produced by the gyro effect
between the movements of .theta..sub.y and .theta..sub.x, in order to
arrange the optimum state feedback compensating mechanism 38, the inverse
symmetrical cross-linking feedback, that is P.sub.14 *.theta. and
-P.sub.14 *.theta. are required.
The function of the inverse symmetrical cross-linking feedback is as
follows: When the rotating body is rotating at relatively high speed, if
any control operation is not provided, the rotating axis carries out the
movement combining the precession and the nutation. The nutation will be
decreased by providing a relatively small damping. In the optimum state
feedback compensating mechanism 38, the portion of P.sub.22 *S effects the
damping motion. The precession is a motion of rotating with respect to the
Z axis while keeping the angle between the rotating axis and the Z axis.
If the rotating axis is inclined to a certain direction by a disturbance,
the direction of the incline of the rotating axis rotates in a
predetermined direction as time is passed. Therefore, for the attenuation
of the precession, it is effective to apply the torque around the X axis
(Y axis) to the rotating body in accordance with the magnitude of
.theta..sub.y (.theta..sub.x) so as to disturb the rotation of the
rotating axis. In this manner, the optimum state feedback compensating
mechanism 38 produces a feedback control signal corresponding to the
angular displacement or deviation of the controlled object to compensate
for the gyro effect and such is achieved by the inverse symmetrical
cross-linking beedback, that is, the portions of P.sub.14 *.theta..sub.x
and -P.sub.14 *.theta..sub.y.
In FIG. 8, a more detailed embodiment of the circuit is shown and, in the
figure, 39 and 45 are proportion-differentiation compensators which
correspond to the feedback compensator 36. 40 is a
proportion-differentiation compensator which corresponds to P.sub.12
*+P.sub.22 *S of the optimum feedback compensating mechanism 38. 41 is a
proportion compensator which corresponds to P.sub.14 * of the optimum
feedback compensating mechanism 38, and 42 is an inverter.
As described above, according to the present invention, it is possible to
quickly suppress the precession or the nutation caused by the gyro effect.
One example of the result of numerical simulation is shown in FIG. 9. In
the figure, 46 is the response waveform in the system which does not have
the compensating mechanism and 47 is the response waveform in the system
having it. It is obvious that the suppression of the precession and the
nutation caused by the gyro effect is carried out quickly by using the
optimum condition feedback mechanism 38. Furthermore, according to the
design technique of the present invention, when the parameters of an
object to be controlled have various values, the parameter of the
compensating mechanism is gained by carrying out the calculation in
accordance with the established calculating procedure, so it is possible
to quickly design the control device.
* * * * *
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