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BACKGROUND OF THE INVENTION
This invention relates to a magnetic bearing used, for example, in a
magnetically levitating apparatus and, in particular, a non-linear control
type magnetic bearing.
A variety of magnetic bearings utilizing the attractive force of
electromagnets have recently been developed, but this type of magnetic
bearing is, in general, very expensive, since a position displacement
sensor and power amplifier necessary for magnetic bearings are much higher
in cost and greater in dissipation power. If these drawbacks are overcome,
then it may be considered that such magnetic bearings having a
contactless, non-friction and oilless feature will find a wider range of
practical application.
In the field of a linear control method, a magnetic bearing has already
been implemented and commercialized which utilizes a velocity sensor using
an inexpensive, easy-to-manufacture search coil in place of the expensive
position displacement sensor. It is disclosed in, for example, "Magnetic
Suspension with Passive Radial Centering for Satellite Flywheel and
Kinetic Energy Storage Systems", the Third International Workshop on Rare
Earth-Cobalt Permanent Magnets and Their Applications, June 27 to 30, 1978
by P. C. Poubeau. This system also has advantages as a "zero-power" system
and has already been established as such. However, this system has a
drawback that it cannot start to levitate an object from an at-rest
position. For this reason, an extra lift-off circuit is provided so as to
levitate the object. It is therefore necessary to provide another sensor
corresponding to the position displacement sensor. This is a substantial
bar to a wide acceptance of this system.
A general non-linear type control theory is disclosed in "Theoretical Basis
of Technical Cybernetics" and "Optimal Quick-Action Line System", 1966.
However, this theory has not been applied to magnetic bearings so far.
SUMMARY OF THE INVENTION
An object of this invention is to provide a non-linear control type
magnetic bearing which obviates the necessity of providing any expensive
position displacement sensor and power amplifier.
Another object of this invention is to provide a non-linear control type
magnetic bearing which can start to levitate an object from an at-rest
position.
Another object of this invention is to provide a non-linear control type
magnetic bearing requiring a less power dissipation.
The essence of this invention resides in that, in the non-linear control
type magnetic bearing, the displacement of an object to be levitated is
found based on the output signal of a velocity sensor and excitation
current of electromagnets without the necessity of providing any expensive
position displacement sensor and the need of using any expensive power
amplifier is eliminated by a pair of electromagnets excited alternately.
A non-linear control type magnetic bearing according to this invention
comprises (a1) a pair of mutually opposite electromagnets, sandwiching an
object to be levitated therebetween with a gap left between the
electromagnet and the object, for attracting the object under their
magnetic force; (a2) current supply means for alternately supplying
excitation current to the respective electromagnets; (a3) a means for
detecting the level of current flowing through the respective
electromagnet and for generating a pair of current level signals; (a4) a
velocity sensor for detecting a velocity with which the object to be
levitated is moved in a direction in which the pair of electromagnets face
each other and for generating a velocity signal; (a5) difference means for
finding a difference between the pair of current level signals and for
generating a difference signal; (a6) differentiating means for
differentiating the velocity signal and for determining an acceleration
rate of the object to generate an acceleration signal; (a7) analog
operation means for performing an analog operation on an equation of
motion of the object with the use of the difference signal and
acceleration signal to find a position displacement of the object and for
generating a position displacement signal; and (a8) switching
discriminator means for identifying a positive or a negative sign of a
linear equation of the position displacement signal and velocity signal
and for supplying to the current supply means a switching signal for
alternately switching a supply of current to the pair of electromagnets.
The above-mentioned non-linear control type magnetic bearing essentially
possesses a levitation start function and thus implements an energy-saving
effect analogous to a zero-power system. Furthermore, the magnetic bearing
also performs a lift-off function in the gravitational field. Even if as a
switching element use is made of a transistor, a switching amplifier can
be compactized as it utilizes the switching amplifying principle in which
the transistor is operated constantly at a minimum heat-generating region.
This arrangement obviates the need of providing any expensive position
displacement sensor and power amplifier. Therefore, an inexpensive
magnetic bearing can be readily implemented having a contactless,
non-friction and oilless feature.
BRIEF DESCRIPTION OF THE DRAWINGS
The other objects and advantages will be apparent from the following
description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a schematic diagram showing a non-linear control type magnetic
bearing according to a first embodiment of this invention;
FIG. 2 is a block diagram showing an operating circuit of the first
embodiment of this invention;
FIGS. 3A and 3B are graphs showing a relation of a velocity dx/dt to an
object displacement x and a relation of excitation current through an
electromagnet coil to a time t when the non-linear control type magnetic
bearing is operated with the use of a special switching discriminant;
FIGS. 4A and 4B are graphs showing a relation between the object
displacement x and the velocity (dx/dt) and relation of excitation current
through the electromagnet coil to a time t when the non-linear control
type magnetic bearing is operated with the use of another special
switching discriminant;
FIGS. 5A to 5D show graphs a relation of the object displacement x to the
velocity dx/dt, a relation of the velocity dx/dt to the acceleration
d.sup.2 x/dt.sup.2, a relation of the displacement x and estimated
displacement y to the time t, and a relation of excitation current through
the electromagnet coil to the time t, when the non-linear control type
magnetic bearing, according to this invention, is operated under no
disturbance with the use of the switching discriminant;
FIGS. 6A to 6E are graphs showing a relation of the object displacement x
to the velocity dx/dt, a relation of the velocity dx/dt to the
acceleration d.sup.2 x/dt.sup.2, a relation of the estimated displacement
y to the time t, a relation of the displacement x to the time t, and a
relation of excitation current through the electromagnet coil to the time
t, when the non-linear control type magnetic bearing is operated under the
gravitational field as a disturbance using the switching discriminant
according to this invention;
FIG. 7 is a schematic diagram showing a non-linear control type magnetic
bearing according to a second embodiment of this invention; and
FIG. 8 is a block diagram showing an operating circuit of the second
embodiment of this invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The non-linear control type magnetic bearing according to the first
embodiment of this invention will be explained below by referring to FIGS.
1 and 2.
First and second electromagnets 12 and 14 are arranged opposite to each
other in a manner to sandwich an object 10 therebetween with a gap left
there. Coils of the first and second electromagnets 12 and 14 are
connected at one end to one terminal of a power supply 16. The coil of the
first electromagnet 12 is connected at the other end to an S1 terminal of
a changeover switch 20 through a first resistor 18 for current detection
and the coil of the second electromagnet 14 is connected at the other end
to an S2 terminal of the changeover switch 20 through a second resistor 22
for current detection. An S0 terminal of the changeover switch 20 is
connected to the other terminal of the power supply 16. The first
electromagnet 12 is excited with the S1 terminal of the changeover switch
20 in the ON state and the second electromagnet 14 is excited with the S2
terminal of the changeover switch 20 in the ON state. That is, the first
and second electromagnets 12 and 14 are excited alternately. In this
connection it is to be noted that the changeover switch 20 is comprised
of, for example, a transistor as a switching element.
Voltages across the first and second resistors 18 and 22 for current
detection are supplied, as current level signals iA and iB, to an
operating circuit 24. A velocity sensor 26 comprised of, for example, a
search coil is disposed opposite to the lower surface of the object 10.
The velocity with which the object 10 is moved up and down is detected by
the velocity sensor 26 and a velocity signal dx/dt is supplied to the
operating circuit 24.
The operating circuit 24 is comprised of a subtracter 28, differentiator
30, analog operating circuit 32 and switching discriminator 34, as shown
in FIG. 2. The current level signals iA and iB detected are delivered to
the subtracter 28 where a calculation of i=iA-iB is carried out. A
difference signal i as an output of the subtracter 28 is supplied to the
analog operating circuit 32. A velocity signal dx/dt which is detected at
the velocity sensor 26 is supplied to the differentiator 30 where it is
differentiated. The differentiated signal is supplied as the acceleration
signal d.sup.2 x/dt.sup.2, together with the current level signal, to the
analog operating circuit 32. In the analog operating circuit 32, the
current level signal i and acceleration signal d.sup.2 x/dt.sup.2 are
substituted into the equation of motion of the object 10 to effect an
analog operation and thus to produce a position displacement signal x of
the object 10. The position displacement signal x is supplied, together
with the velocity signal dx/dt, to the switching discriminator 34. The
switching discriminator 34 identifies the positive or negative sign of a
linear equation for the position displacement signal and velocity signal,
and supplies to the changeover switch 20 a switching signal which
alternately switches a supply of current to the electromagnets. It is to
be noted that an origin 0 in the position displacement of the object is
determined to be located at an equilibrium point of the object 10 as shown
in FIG. 1.
The function and operation of the magnetic bearing of this invention will
be explained below.
First, an explanation will be given of the optimal control theory in
general. A force acting on the object 10 is comprised of a gravitational
force and electromagnetic force. Now suppose only the electromagnetic
force with the gravitational force disregarded here. Then, the equation of
motion of the object 10 is given below:
##EQU1##
where m: the mass of the object 10 to be levitated; and
M(x): the function of the displacement x including the coil impedance of
the electromagnet.
Differentiating the velocity signal dx/dt of the velocity sensor 26 yields
an acceleration d.sup.2 x/dt.sup.2. The current levels on the respective
electromagnets 12 and 14 can be normally measured. M(x) is found through
the use of Equation (1). Since the function of M(x) is known beforehand,
it is possible to evaluate the displacement x. In this way, the
displacement x is evaluated from the current level I and velocity dx/dt.
In order to determine the switching timing of the changeover switch 20
through the use of the displacement x and velocity signal dx/dt, use is
made of the optimal control theory. According to the optimal control
theory, the switch changeover discriminant for minimum time control of
movement of the object in a field of motion of an equation:
##EQU2##
is given by:
F(x,dx/dt)=mdx/dt.vertline.dx/dt.vertline.+fx=0 (3)
Thus, the optimal control is realized if switching is effected to f at F(x,
dx/dt).gtoreq.0 or -f at F(x, dx/dt)<0. Here, f represents a constant.
In this way it is possible to realize the optimal control in general.
However, if the above-mentioned theory applies, the right-handed side of
Equation (1) is not a constant, but a function for the current level and
position, presenting a problem. For convenience, let us now consider a
bearing system satisfying, in place of Equation (1), the following
equation:
##EQU3##
This equation is obtained in the cases where permanent magnets are used
also as electromagnets and where the attractive force of the electromagnet
is linearized for the current level I. The essence of this invention is
not affected even if Equation (4) is used in place of Equation (1). A
subsequent explanation will be made based on this fact. The current level
I flowing upon the application of a voltage e across the electromagnets 12
and 14 is given by:
##EQU4##
where R: the resistance of the coil,
L=L(x): the inductance of the coil.
Since, in general, the coil has an inductance, the current level reveals a
time lag. The following explanation is based on the assumption that there
is, as an ideal state, no time delay of the current. If Equation (4) is
integrated once with L=0 and I=.+-.I0 (constant), then an equation:
##EQU5##
is obtained as an optimal switching discriminant.
A control system of a smaller extent of delay and control system of a
greater extent of delay were simulated using Equation (6). FIGS. 3A and 3B
and FIGS. 4A and 4B show the loci of the simulations in the phase plane.
When the coil has a small time constant (L/R) and then a time delay of the
current is small, the changeover switch 20 is switched to the S1 side at
an initial level point A to permit the energization of the electromagnet
12 as shown in FIG. 3A. At the level point C, the changeover switch 20 is
switched to the S2 side to permit the energization of the electromagnet
14. Thereafter, the displacement x of the object immediately reaches a
target position 0 and thus an optimal control is realized with minimum
time. This will be also well understood from the current level variation
of the electromagnets 12 and 14 as shown in FIG. 3B. After the
displacement x of the object has arrived at the 0 point, this state (the 0
state) is maintained by the frequent switching operations. However, this
system is not suitable for a smaller electric power, because the maximum
current normally flows through either of the electromagnets 12 ad 14. In
FIG. 3A, it is to be noted that the broken line P represents a switching
curve and that the time t1 corresponds to the point C.
FIGS. 4A and 4B show the operation of the object 10 and the time variation
of a current through the coil of the electromagnet (12, 14) when the time
constant (T=L/R) of the coil of the electromagnets is greater (L>>R>0) and
thus the time delay of the current is greater. In this case, use is made
of the same switching curve P as shown in FIG. 3. This system is subject
to an adverse influence resulting from the time delay of the current and
thus a final limit cycle is involved, causing a greater vibration and
positioning error. In other words, this system is not practical due to the
involving of the limit cycle.
According to this invention, these problems are solved through the positive
utilization of the phenomenon (FIGS. 3A and 3B and FIGS. 4A and 4B) which
results from the time delay of the current level. That is, the embodiment
of this invention adopts a switching line Q with a limited gradient with
respect to an origin 0 as shown in FIG. 5A. In FIG. 5A, the switching
curve P is shown for guidance only and can be derived from Equation (6).
This is of the same type as shown in FIGS. 3A and 4A. The switching curve
P has an infinite gradient at the origin 0 and is different from the
switching curve Q in this embodiment. Here, the switching curve Q becomes
a linear function for the displacement x and velocity signal dx/dt and
thus is represented as follows:
##EQU6##
Using the displacement signal x from the analog operating circuit 32 and
velocity signal dx/dt from the velocity sensor 26, the positive or
negative sign of H(x, dx/dt) is judged at the switching discriminator 34.
With H(x, dx/dt) at the positive sign the terminal S2 of the changeover
switch 20 is rendered ON and with H(x, dx/dt) at the negative sign the
terminal S1 is rendered ON. In consequence, as shown in FIG. 5A the
displacement of the object 10 converges from an initial level at a point A
to a target level at the point 0, leaving no limit cycle behind. FIG. 5B
shows a relation of the velocity signal dx/dt to the acceleration signal
d.sup.2 x/dt.sup.2. From this figure it is found that the object 10
converges toward an equilibrium point 0. FIG. 5C shows a relation of the
displacement x of the object to the time t. From FIG. 5C it is also found
that the displacement of the object 10 converges toward the equilibrium
point 0. FIG. 5D shows a variation of excitation currents iA, iB to the
time t. From a comparison between FIG. 5D and FIGS. 3B and 4B, it will be
appreciated that the system of this embodiment needs a very small level of
excitation current. Therefore, the embodiment of this invention, though
being somewhat lower in response characteristics, can achieve a stability
and a small power dissipation of practical importance. Furthermore, since
the switching discriminant H(x, dx/dt) is the linear function of the
displacement x and velocity signal dx/dt, it is possible to very readily
effect an analog composition. The value y in FIG. 5C is an estimated value
of the control system which is obtained by finding the position of the
object through the use of the equation of motion (1) under the assumption
that there is no disturbance. The actual displacement x of the object 10
is in agreement with the estimated value y, provided that there is no
disturbance.
An explanation has so far been made as to the embodiment of this invention
under the assumption that there is no disturbance. Now an explanation will
be given below with reference to FIGS. 6A through 6E where the system of
this invention is put under, for example, the gravitational field.
In this case, the object 10 is automatically moved to a position B where,
as shown in FIG. 6A, the electromagnetic force of the electromagnets 12
and 14 and gravity are in static equilibrium, with the result that the
object 10 is maintained at the position B. FIG. 6B shows a relation of the
velocity dx/dt to the acceleration d.sup.2 x/dt.sup.2. As shown in FIG.
6C, the estimated value y of the control system varies with the time t and
the object 10 reaches an equilibrium point as shown in FIG. 6D. The
excitation current of the coils varies as shown in FIG. 6E. It is to be
noted that the ON time of the electromagnet 12 is somewhat longer than
that of the electromagnet 14.
As evident from FIG. 6E, the dissipation current makes no appreciable
difference even in the presence of the disturbance as compared with that
in the absence of it. This is a marked advantage which can be obtained
from the system of this invention.
According to this invention, the object 10 can be magnetically levitated by
controlling the changeover switch 20 using a switching discrimination
signal generated in the operating circuit 24. In this case, the currents
through the electromagnets 12 and 14 are reduced to a very small level due
to the impedance of the coils, thus involving a very small dissipation
power. Therefore, the non-linear control type magnetic bearing of low cost
can be implemented without the need of providing any expensive position
displacement sensor and power amplifier.
A non-linear control type magnetic bearing according to a second embodiment
of this invention will be explained below by referring to FIGS. 7 and 8.
In this embodiment, a judging circuit 40 is connected between a terminal S0
of a changeover switch 20 and the end of a power supply 16 and adapted to
judge whether or not the current from the power supply 16 continues for
over a predetermined period of time at maximum. The judgment data is
supplied to a switching discriminator 34 in an operating circuit 24. The
object 10 can surely be levitated from an at-rest position under the
action of the judging circuit 40 and operating circuit 24. The other
arrangement of this embodiment is the same as that of the first embodiment
of this invention. Therefore, further explanation is omitted.
The operation of the second embodiment will be explained below. Now suppose
that, at rest, the object 10 is located, for example, on the side of the
electromagnet 14 and the changeover switch 20 is placed on the side of the
S2 terminal. Then, since the output of the velocity sensor 26 is 0, no
immediate switching occurs at the changeover switch 20. During this time
period, a maximum current continuously flows through the electromagnet 14
and the judging circuit 40 judges this state. The judgment data is
supplied to the switching discriminator 34, causing the changeover switch
20 to be forcedly switched to the S1 terminal of the switch 20.
Thereafter, the object 10 is automatically moved to a position 0 in the
same way as explained in connection with FIGS. 6A through 6E, and starts
to be levitated.
This invention is not restricted to the above-mentioned embodiments. For
example, a power relay may be used as the changeover switch in place of
the transistor. The judging circuit 40 is not required to be connected
between the power supply 16 and the changeover switch 20, and may be
placed anywhere in the associated circuit so long as it can judge that
maximum current flows through either of the electromagnets 12, 14 for over
a predetermined period of time. Various changes or modifications can be
made within the spirit and scope of this invention.
* * * * *
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