Method and switching arrangement for regulation of a two-dimensional vector of a segment by means of a value-discrete setting element with a limited switching frequency

A method and a switching arrangement are provided for regulating a two-dimensional vector of a controlled system with a discrete value final controlling element having a limited switching frequency. The discrete value final controlling element also has limited number of actual-value vectors. A switching arrangement, such as a microprocessor, receives actual values for the two-dimensional vector as well as a reference vector and selects three actual-value vectors that are adjacent to the reference vector. The switching arrangement generates a reference trajectory which represents a sequence of vector settings for the discrete value final controlling element. A switch from one actual-value vector to another takes place when the distance between the two-dimensional vector and a next corner point of the reference trajectory is at a minimum. The resulting mean value of the generated two-dimensional vector is equal to the reference vector.

BACKGROUND OF THE INVENTION 
The present invention relates to a method and apparatus for regulating a 
two-dimensional vector of a controlled system with a discrete-value final 
controlling element having a limited switching frequency. 
Discrete-value final controlling elements only have a limited number of 
different states which can be set at their outputs. If an intermediate 
value is to be generated, an approximation is achieved by switching back 
and forth between adjacent states. The approximate value to be generated 
is a value averaged over time. The low-pass effect which is generally 
present in the regulation segment in technical systems smoothes the 
alternating variable produced. 
If a two-dimensional vector of a controlled system is to be set, it is 
insufficient to consider each dimension separately, since they are in part 
dependent upon one another. Therefore, a change in the state of the final 
controlling element causes a change at more than one output. 
Known control methods generally proceed from a one-dimensional approach. 
However, such an approach fails to take into consideration the manner in 
which the different outputs of the final controlling element depend upon 
each other as well as the internal states of the final controlling 
element. 
Direct current regulation for rotary current drives using pulse inverters 
is described in "Regelungstechnische Praxis," Volume 24, 1983, Issue 11, 
pages 472 to 477. A very simple solution in terms of equipment is 
represented by two-point regulation. Such a regulation consists of three 
two-point regulators subjected to hysteresis. These regulators always 
cause a status change if the difference between the reference value and 
the actual value exceeds a certain limit. The system can therefore follow 
a changing reference value with maximum speed. This regulator type is used 
in intermediate circuit voltage inverters for regulating phase currents. 
Since the resulting phase currents are dependent upon one another, the 
switching states are not always optimal. 
If the reference value periodically passes through a certain function, then 
suitable switching time points (with reference to the period) can be 
calculated in advance. The dependence of several phases on one another can 
be possibly taken into consideration. The disadvantage of this method is 
that a separate pulse pattern has to be calculated for each operating 
point (period duration and level). Each shift of the operating point 
requires a transition between pulse patterns. This transition can take a 
long time, in some cases, since a change in pulse pattern is only 
permitted at certain times within a period. At the same time, errors occur 
during the dynamic process, since the pulse pattern is not operated at its 
optimum point. 
A one-dimensional process (two-point regulator with hysteresis) can be 
expanded to a multi-dimensional case. For this process, a hysteresis 
region around a reference value is predetermined instead of the hysteresis 
width. As soon as the actual value seeks to leave the region, a switch to 
another state takes place, which brings the actual value back to the 
inside part of this hysteresis region. With this method, the actual value 
is always kept within the vicinity of the reference value. Even if the 
reference value changes, quick following is guaranteed. The switching 
frequency can be influenced by the size of the hysteresis region, but 
cannot be precisely predetermined. 
Such a multi-dimensional method is described as predictive current 
regulation in "Messen-Steuern-Regeln," No. 13, June 1989, pages 20 to 23, 
and "EPE," 1987, pages 647 to 652, entitled "New Predictive Control 
Strategy for PWM-Inverters." A further example of predictive current 
regulation with optimization in reach time for a pulse inverter is 
described in "IPEC," Tokyo 1983, pages 1665 to 1675. A multi-dimensional 
method for a GTO-I inverter is described in the lecture "Four Quadrant 
AC-Motor Drive with A GTO Current Source Inverter With Low Harmonics And 
On Line Optimized Pulse Pattern" by O. Hintze and D. Schroder, printed in 
the "IPEC" Conference Minutes, Tokyo 1990, pages 405 to 412. 
With this method, the reference value can be reached by approximation, but 
it is not set as a mean value over time. It is possible that the 
trajectory of the actual value stays in a part of the hysteresis region 
for an extended period of time. This results in a mean value which is not 
equal to the reference value in every direction. This error has to be 
compensated with another regulator. This effect is particularly disruptive 
with large hysteresis regions and, therefore, small switching frequencies. 
With a multi-dimensional method, a relatively low switching frequency and 
good control dynamics are achieved. The disadvantages of the method with 
precalculated pulse patterns are avoided in this method. However, even in 
this method the switching frequency cannot be predetermined precisely 
because it depends on the operating point at each point in time and varies 
greatly. Another disadvantage of this method is that the voltage is 
regulated, for example, but not the integral above the voltage, i.e. the 
current. For example, it is possible that the mean value of the voltage 
only becomes zero in one direction, and thus the current in the orthogonal 
direction to the voltage deviates significantly from the reference value 
for an extended period of time. Simply by activating the inverter, an 
error in the current is caused, which has to be balanced out again with a 
regulator which acts on the hysteresis region. 
There is a need for a method and a switching arrangement for regulating a 
two-dimensional vector of a controlled system with a discrete-value final 
controlling element having a limited switching frequency that avoids the 
aforementioned disadvantages. 
SUMMARY OF THE INVENTION 
This and other needs are satisfied by the method and apparatus of the 
present invention. The method of the present invention comprises the 
regulation of a two-dimensional vector of a controlled system with a 
discrete-value final controlling element having a limited switching 
frequency. The reference value of this two-dimensional vector is 
approximated as an average value over time by selecting three manipulated 
variable actual values adjacent to a manipulated variable reference value. 
The three selected manipulated variable actual values are alternated in a 
predetermined sequence and at predetermined interval times during a 
switching period in such a way that the mean value of the actual vector, 
while passing over a reference trajectory formed by the manipulated 
variable actual values, is equal to its reference value. A switch from one 
manipulated variable actual value to the next manipulated variable actual 
value takes place precisely when the distance between the actual vector 
and a next corner point of the reference trajectory is at a minimum. 
In this manner, the optimum local curve for the actual vector is 
determined. The switching time points are determined in such a way that 
the actual vector of the controlled system exit runs into this local 
curve. This is achieved because the switch takes place precisely at that 
time. In other words, the next vector of the selected manipulated variable 
vector triplet is selected if the distance between the vector of the 
controlled system exit and the next corner point of the local curve 
(trajectory) reaches a minimum. 
With this method, a minimum deviation from the reference value is obtained 
with a given switching frequency, where the time mean coincides with the 
reference value, and the shortest possible control time is achieved. 
In a current impressing inverter having thyristors that can be shut off 
(e.g., a GTO-I inverter), where capacitors are arranged between the 
inverter and the machine, the capacitor voltage and the machine current 
are taken into consideration in the regulation. Since these variables are 
vectors, two-dimensional optimization takes place. In other words, not 
only the vector of the capacitor voltage is kept close to the reference 
vector, but also its mean value, and thus the machine current, is set 
correctly with only one variable (switching time point) being available. 
A second condition in this method is that the machine current should 
deviate as little as possible from the reference current. This is achieved 
when the voltage vector runs along the calculated reference trajectory and 
the integral over a voltage reference-actual difference over a pass 
becomes zero. 
The method of the present invention can also be used in the net-side 
current converter of the current impressing GTO-I inverter. Likewise, this 
method can be used for a voltage impressing inverter, also called a pulse 
inverter. 
The switching arrangement of the present invention includes a 
microprocessor which generates an interrupt signal, which is cyclically 
triggered in a fixed time grid. After the interrupt signal is generated, 
the actual values and the reference values are read into the 
microprocessor. The remaining time and the vector number of the related 
manipulated variable actual value are generated at the output of the 
microprocessor. Also, several switching signals are generated at the 
outputs of the microprocessor. 
As the switching arrangement, a microprocessor is provided, to which the 
reference values and actual values of the vectors are passed to the 
switching arrangement, which is coupled with the current converter valves 
of the discrete final controlling element on the output side, via a 
control set. The microprocessor processes the aforementioned method of the 
present invention, and generates a remaining time interval between 
scanning and the switching action to be performed, and the number of the 
next status of the final controlling element. While the control set 
carries out the status change, the microprocessor has already calculated 
the next calculation. 
Another feature of the method of the present invention is that the 
proportional interval times of the three selected manipulated variable 
actual values can be separately determined according to the following 
equations. 
##EQU1## 
A corner point of the generated reference trajectory is separately 
determined according to the following equations. 
##EQU2## 
The remaining time until a switching point is separately determined 
according to the following equations. 
##EQU3## 
If a negative value is determined for the remaining time, a switch to the 
next manipulated variable actual value immediately takes place. If a 
negative value is determined for a interval time, the related manipulated 
variable actual value is not taken into consideration for calculating the 
switching time.

DETAILED DESCRIPTION 
Referring to FIG. 1, an intermediate circuit current converter 2 is shown. 
A current impressed by approximation via a choke 4 is switched onto two of 
the three phases so that a closed circuit is always formed. The net-side 
inverter 6 and the machine-side inverter 8 are each structured with 
current converter valves that can be shut off, especially gate-turn-off 
thyristors (GTO thyristors). The machine-side inverter 8 can also be 
equipped with conventional thyristors. With these current converter 
valves, it is possible to interrupt the current directly. Due to the 
inductances of the machine 10, however, very high voltage peaks form, 
which can lead to destruction of the machine-side inverter 8. These 
voltage peaks can be significantly reduced by including capacitors 12 in 
the circuit. However, some other effects occur as a result: 
The inverter current i.sub.p (FIG. 2) no longer flows only into the machine 
10, but also into the capacitors 12, which means that the direct 
intervention in machine currents is lost. 
The block-shaped currents are smoothed by the capacitors 12, which causes 
the losses in the machine to become less and the torque waviness to be 
reduced. 
The capacitors 12 form an oscillating circuit with the leakage inductance 
.sigma.L.sub.S of the machine 10, which is only attenuated by the 
relatively small stator resistance R.sub.s. However, this oscillating 
circuit is excited by every switching action of the inverter 8. 
Reactive power is generated by the capacitors 12, which is consumed in the 
machine 10. At a certain frequency, the reactive power demand of the 
machine is completely covered by the capacitors 12 (self-excitation), 
which means that the system machine 10 is in idle and no longer takes any 
current i.sub.s from the inverter 8. 
Since the current i.sub.s is no longer impressed in the machine 10, the 
stator differential equation has to be considered for the regulation of 
the machine 10. 
Further differential equations result from the capacitors 12, i.e. from the 
capacitor voltages u.sub.s (stator voltages), which must be included in 
the calculation, and the order of the differential equation system is 
increased by a factor of 2. 
If the stator currents i.sub.s and stator voltages u.sub.s are included in 
the regulation and if the control of the drive is sufficiently quick, the 
oscillation circuit should not present any problem. In order to have 
sufficient intervention on the controlled system 10, 12, the only 
important consideration is that the switching frequency f.sub.s of the 
inverter 8 is greater than the inherent frequency of the oscillating 
circuit. The scanning frequency must also be sufficiently large. 
Referring to FIG. 2, the single-phase equivalent schematic of the drive of 
FIG. 1 with complex variables is shown. The net-side inverter 6 having the 
intermediate circuit is reduced to a simple current source 14, and the 
machine-side inverter 8 is symbolically replaced by a switch 16. The 
current source 14 and switch 16 generate the complex vector i.sub.p which 
lies in the spacial plane of the scalar variable of the intermediate 
circuit current i.sub.zw. Since the intermediate circuit current i.sub.zw 
is a constant variable, the inverter 8 must be controlled so hat there is 
always a closed and unambiguous current path through the inverter 8 and 
the machine 10. Consequently, one switch at the top of the bridge of the 
machine-side current converter 8 always has to be closed, and one switch 
has to be closed at the bottom. If more than two switches are closed, no 
definable state can be achieved. Therefore, there are nine possible switch 
settings. Of these, six of the switch settings are regular vectors, and 
three are zero vectors (i.e. bridge short circuits). 
Referring to FIG. 3, a first current path through the inverter 8 and the 
machine 10 for a switching which is shown as vector 1. If the current has 
flowed along this path for a sufficiently long period of time, the 
transient process has ended and the intermediate circuit current i.sub.zw 
only flows through the machine 10 and not through the capacitors 12. 
Because of the spacial arrangement of the machine windings, an unambiguous 
direction in vector space can be assigned to the machine current i.sub.s. 
Now if, starting from this state, a switch to the center inverter valve is 
made in the bottom half of the bridge, the intermediate circuit current 
i.sub.zw flows past the machine 10 (i.e., bridge short circuit) and is 
designated as vector O.sup.I. The machine current i.sub.s is maintained at 
first and finds a closed path via the capacitors 12 (see FIG. 4). 
Referring to FIG. 5, the directions of the current vectors for the 
different switching combinations and the current paths of the three 
possible bridge short circuits are shown. The current-carrying branches of 
the bridge of the inverter 8 are illustrated with a heavy line in each 
case. The magnitude of the vectors is determined by the magnitude of the 
intermediate circuit current i.sub.zw, while the direction of the vectors 
is determined by the activated current converter valves of the inverter 8. 
As described above, the method of the present invention for regulating a 
two-dimensional vector of a controlled system by a value-discrete final 
controlling element can also be used for a voltage impressing inverter, 
also called a pulse inverter. 
As with the net-side inverter, the pulse inverter can have a diode bridge 
or a self-guided inverter. The machine-side inverter is a self-guided 
inverter with free-run diodes. As with the intermediate circuit, an 
electrolyte capacitor is provided. The related single-phase equivalent 
schematic would be similar to the equivalent schematic of FIG. 2. Instead 
of the current source 14, a voltage source having a subsequent series 
circuit, comprising a net resistor and a net inductance, would be 
provided. The switch 16 in the equivalent schematic of the pulse inverter 
is only a switch for opening and closing the circuit and not for switching 
over. In addition, a second switch would also be present, which would be 
arranged in front of the series circuit comprising a leakage inductance 
and a stator resistance. 
Referring to FIG. 6, the drive of FIG. 1 having a regulation and control 
device is shown as a simplified block diagram. As with the current 
converter valves of the inverters 6 and 8, semiconductor valves which can 
be turned off are provided, particularly thyristors which can be shut off 
at high output, such as gate-turn-off thyristors (GTO thyristors). If GTO 
thyristors are used as the current converter valves, this intermediate 
circuit current converter 2 is also known as a GTO-I inverter. The 
load-side current converter 8 (value-discrete setting element), also 
called an inverter, is provided having capacitors 12 on the output side 
and supplies power to the machine 10, which can be provided with an R.P.M. 
transmitter 18. 
The regulation and control device works with a current model 20 and the 
R.P.M. transmitter 18. A transmitter-free regulation having a voltage 
model can also be provided, but regulated operation at smaller revolutions 
per minute (less than 5 Hz) is no longer possible. Regulation of the 
net-side current converter 6 is of subordinate importance in the method 
according to the invention. The regulation of the net-side converter 6 is 
shown in FIG. 6 as a rectifier regulation circuit 22. In addition, the 
regulation and control device comprises a flow regulator 24, an R.P.M. 
regulator 26, a valve generator 28, a current regulator 30 and a switching 
arrangement 32 for implementing the method according to the invention. 
The current model 20 generates an R.P.M. actual value .omega. and a flux 
actual value .psi. from the status actual values u and i of the machine 
10. The current model 20 receives a determined rotor position .epsilon., 
and generates the status vectors u.sub.s and i.sub.s, the angle .rho. of 
the flux .psi. and its angular velocity .rho.. To determine the 
derivations of the stator current vector u.sub.s, the inverter current 
i.sub.p of the load-side current converter B is also needed and is formed 
from the switching states S.sub..nu. and the intermediate circuit current 
i.sub.zw. The flux regulator 24 generates a field-oriented current 
component reference value i.sub.sd * of current i.sub.zw from a comparison 
of the flux reference value .psi.* and the flux actual value .psi.. The 
R.P.M. regulator 26 generates a field-oriented current component reference 
value i.sub.sq * from a comparison of an R.P.M. reference value .omega.* 
and the rpm actual value .omega.. The value generator 28 determines the 
amount of the intermediate circuit current reference value i.sub.zw *, 
which is converted into control signals S.sub..mu. for the net-side 
current converter 6, from the two field-oriented current component 
reference values i.sub.sd * and i.sub.sq * and the angular velocity .rho. 
of the flux .psi. multiplied by a Factor C. The field oriented current 
component reference values i.sub.sd * and i.sub.sq * are also supplied to 
the current regulator 30. The outputs of the current regulator 30 are the 
reference values u.sub.s * and i.sub.s * of the vectors u.sub.s, i.sub.s 
of the segment 10, 12. The switching arrangement 32 for implementing the 
method of the present invention can be a microprocessor and generates the 
control and switching signals S.sub..nu. (switching states) for the 
machine-side current converter 8 from the reference values u.sub.s *, 
i.sub.s * and the actual values .rho., .rho., i.sub. s, u.sub.s and 
i.sub.p generated by the current model 20. For a better understanding, the 
switching arrangement 32 is shown in greater detail in FIG. 7, as a 
hardware-type block schematic. 
The method for using a U inverter (i.e., pulse inverter) is discussed below 
based on the block schematic of the switching arrangement 32 of FIG. 7 and 
the single-phase equivalent schematic of the intermediate circuit current 
inverter 2 with subsequent capacitors 12 and the machine 10 of FIG. 2. 
For two-dimensional optimization (voltage with current as a secondary 
condition) there is only one variable (switching time point) available. In 
principle, six current vectors i.sub.p1 to i.sub.p6 are available, as well 
as three zero vectors i.sub.p0 I to i.sub.p0 III. These current vectors 
i.sub.p I to i.sub.p0 III are shown in more detail in FIG. 9, in a 
rectangular stator-fixed coordinate system. Referring to FIG. 5, the 
matching possible switch settings of the machine-side final controlling 
element 8 are shown. 
Assuming that the stator current indicator i.sub.s, as shown in FIG. 10, 
lies in a sector which is defined by the current indicators i.sub.p6, 
i.sub.p1 and i.sub.p0, each of these three current indicators i.sub.p6, 
i.sub.p1 and i.sub.p0 causes a movement (time change) u.sub.s6, u.sub.s1, 
and u.sub.s0 of the voltage indicator u.sub.s. This time change of the 
voltage indicator u.sub.s is proportional to the deviation i.sub.p6 
-i.sub.s, i.sub.p1 -i.sub.s, and i.sub.p0 -i.sub.s of the current 
indicator i.sub.s. The possible trajectories TR.sub.6, TR.sub.1, and 
TR.sub.0 are shown as dashed lines in FIG. 11, each started from the tip 
of the voltage indicator u.sub.s, by the selection of the three adjacent 
current indicators i.sub.p6, i.sub.p1 and i.sub.p0. Since each of the 
three vectors i.sub.p6, i.sub.p1 and i.sub.p0 causes a movement of the 
voltage vector u.sub.s, the voltage vector u.sub.s (actual value) will 
equal, in the long run, the reference value u.sub.s *. Also, it is not 
necessary for the voltage vector u.sub.s to equal the reference value 
u.sub.s *, even at specific points in time. Rather, a reduction in 
harmonic oscillations is more important. To minimize the harmonic 
oscillations, the voltage indicator u.sub.s is kept as close as possible 
to a reference value u.sub.s *, but need not be exactly reached. 
The reference trajectory TR* of the voltage indicator u.sub.s with the 
smallest deviation at a given switching frequency f.sub.s of the final 
controlling element 8 is a triangle composed of the trajectories TR.sub.n, 
TR.sub.n+1 and TR.sub.n+2 for three possible current vectors i.sub.pn, 
i.sub.pn+1 and i.sub.pn+2 The vector triplet i.sub.pn, i.sub.pn+1 and 
i.sub.pn+2 are adjacent indicators of an inverter current reference 
indicator i.sub.p *. This reference trajectory TR* of the voltage vector 
u.sub.s is shown in FIG. 12, in a rectangular stator-oriented coordinate 
system. The sequence of the current vectors and therefore the direction of 
passage should always be maintained and not changed after every period 
(which is usually done for known vector modulation with a voltage 
impressing pulse inverter). At no point in time will u.sub.s =u.sub.s *. 
For this triangle, only three commutations are necessary for current 
impressing inverters, while four are needed for a voltage impressing 
inverter. As an ancillary condition, it is necessary that the current 
i.sub.s deviate as little as possible from the reference current i.sub.s 
*. In a progression of the voltage us pursuant to FIG. 12, a closed curve 
forms as the trajectory of the current i.sub.s, represented in FIG. 13, if 
the integral over the voltage reference-actual difference over a pass 
becomes zero. 
The following assumptions are made for the calculation of a switching time 
point: 
There are three possible different current vectors i.sub.pn, i.sub.pn+1 and 
i.sub.pn+2 available at every moment, 
i.sub.s is constant during a switching cycle, 
i.sub.s *, u.sub.s * are constant values, 
.omega. is equal to zero. 
If the last two assumptions are not true, the method of the present 
invention will still yield good results, since the switching time points 
are calculated continuously and switching actions that have yet to be 
carried out can be corrected. The method of the present invention rapidly 
adjusts to changed input variables. 
In light of the description above, the requirements for effective operation 
are as follows: 
a) The trajectory of u.sub.s (triangle shape of FIG. 12) is closed upon 
itself, 
b) the size of the triangle is limited and can be predetermined, 
c) the trajectory i.sub.s (oval shape of FIG. 13) is also closed upon 
itself. 
If a voltage impressing inverter is used instead of the GTO-I inverter 2, 
the requirements for effective operation are as follows: 
aa) The trajectory of i.sub.s (triangle shape of FIG. 12) is closed upon 
itself, 
bb) the size of the triangle is limited and can be predetermined, 
cc) the trajectory u.sub.s (oval shape of FIG. 13) is also closed upon 
itself. 
Assuming that T.sub.n, T.sub.n+1 and T.sub.n+2 are the interval times of 
the individual vectors i.sub.pn, i.sub.pn+1 and i.sub.pn+2, or u.sub.pn, 
u.sub.pn+1 and u.sub.pn+2, respectively, and u.sub.sn, u.sub.sn+1 and 
u.sub.sn+2, or i.sub.sn, i.sub.sn+1 and i.sub.sn+2 are the gradients of 
the stator voltage u.sub.s or the stator current i.sub.s, respectively, 
for the different vectors i.sub.pu and u.sub.p, respectively, then the 
following equations apply for the aforementioned requirements: 
##EQU4## 
Assuming that i.sub.s is a constant value or that u.sub.s is a constant 
value in the interval T.sub.ges (switching period), a third order equation 
system is obtained from equations (1) and (2), or (1.1) and (2.2), 
respectively: 
##EQU5## 
The shape and the size of the triangle are determined by this equation 
system, but the location is not yet determined. If a corner point of the 
triangle is designated as u.sub.so or i.sub.so (see FIG. 12) then the 
following equations apply for the voltage vector u.sub.s or the current 
vector i.sub.s, respectively, during a switching period T.sub.ges : 
##EQU6## 
If equation (5) or (5.5), is inserted into equation (3) or (3.3), 
respectively, the following equation results from integration, and is 
resolved for the corner point u.sub.so or i.sub.so, respectively, which is 
being sought. 
##EQU7## 
In these equations it was assumed that the reference value u.sub.s * of the 
voltage indicator or the reference value i.sub.s * of the current 
indicator, respectively, not change at the pulse inverter during the 
switching period T.sub.ges. In that case, however, and it the derivation 
u.sub.s * is known, u.sub.s has to be replaced with (u.sub.s -u.sub.s *). 
In general, the voltage regulator u.sub.s or the current indicator i.sub.s 
does not lie on the reference trajectory TR* calculated in this way, but 
mostly in close proximity to it. With a suitable method, the voltage 
vector u.sub.s or the current indicator i.sub.s is made to run into this 
optimum cycle (reference trajectory TR*). 
This is achieved by selecting the switching time points in such a way so 
that the distance between the voltage vector u.sub.s or the current vector 
i.sub.s and the nearest corner point u.sub.so or i.sub.so to the switching 
time point is at a minimum. The following equations apply: 
##EQU8## 
To solve equation (7) or (7.7), the time progression of the stator voltage 
u.sub.s or the stator current i.sub.s is approximated for the 
instantaneous time point T.sub.0 by a Taylor series, and the higher order 
terms are ignored. The leakage of the voltage is known from the capacitor 
equation 
EQU u.sub.s =1/C(i.sub.p -i.sub.s). 
While the derivation of the current is known for the U inverter from the 
stator equation 
##EQU9## 
With the orthogonal components of the stator voltage u.sub.s1 and u.sub.s2 
or the stator current i.sub.s1 and i.sub.s2, the computation in Equation 
(7) or (7.7), respectively, can now be resolved. The minimum then lies at 
the point at which the derivation becomes zero. Finally, an equation of 
the first order is obtained, with the solution: 
##EQU10## 
and the time difference remains until the next switching point. In FIG. 
14, the voltage indicator u.sub.s is shown to run into the reference 
trajectory TR*. 
If .omega.=0, the reference trajectory TR* constantly deforms, and the 
trajectory segments are no longer straight lines. Since the trajectory is 
constantly being recalculated and only the corner point of the triangle at 
u.sub.s0 is important, these deformations do not result in any problems. 
Slight deviations occur only if the switching frequency f.sub.s becomes 
small in relation to .omega. R.P.M. 
Deviations in the voltage vector u.sub.s from the straight line segments 
TR.sub.n, TR.sub.n+1 and TR.sub.n+2 also occur due to the movement of the 
current vector i.sub.s in the GTO-I inverter due to the switching actions 
(harmonic oscillations). This effect can be circumvented by using the 
reference values (corresponding to the time mean of the current) for the 
calculation of the derivations u.sub.sn, u.sub.sn+1 and u.sub.sn+2, rather 
than the actual current values. 
In this manner, not only is the switching time point calculation 
independent of harmonic oscillations, but also, behavior similar to 
PT.sub.1 is obtained for the current, without an external regulator being 
present. 
Referring to FIG. 15, a situation where the actual current vector i.sub.s 
deviates significantly from the reference value i.sub.s * is shown. By 
using the reference vector, a reference trajectory TR* is calculated for 
the voltage indicator u.sub.s, through which the voltage indicator u.sub.s 
cannot pass (represented in FIG. 16). On the basis of the switching 
condition, the center point of the true trajectory TR according to FIG. 16 
lies offset from the reference value u.sub.s * in such a way that a 
voltage time area .DELTA.u.sub.II is formed, which drives the current 
i.sub.s in the direction of the reference current i.sub.s *. 
To estimate the behavior over time, the following equation is obtained from 
geometrical considerations and with the designations from FIGS. 15 and 16: 
##EQU11## 
From the stator differential equation, the following equation is obtained 
by approximation: 
EQU .DELTA.u.sub.ll =.sigma.L.sub.3 .DELTA.i.sub.3 (10) 
and thus inserting 
EQU .DELTA.i.sub.3 =-k.DELTA.i.sub.3 (11) 
into 
##EQU12## 
The equation (11) describes a differential equation of the first order with 
a negative, purely real pole (PT.sub.1 behavior). 
A suitable regulator for the current is therefore only necessary in order 
to eliminate stationary interference (e.g. in case of incorrectly set 
parameters or to improve the dynamics). 
Essentially, the following three equations must be solved: 
Equation (4) or (4.4), in order to determine the proportional times of the 
three vectors, 
Equation (6) or (6.6), for the corner point of the reference trajectory 
TR*, 
Equation (8) or (8.8), which reproduces the remaining time t.sub.s until 
the switching point. 
All three equations (4), (6) and (8) or (4.4), (6.6) and (8.8), 
respectively, can be solved purely by mathematics. However, as an 
ancillary condition, it is required that all time intervals are positive. 
EQU (t.sub.s -t.sub.0), T.sub.n, T.sub.n+1, T.sub.n+2 &gt;0 
If the equation (8) or (8.8) yields a negative solution, the optimum 
switching time point has already occurred. This can occur, for example, 
because of reference value changes. Immediate switching is still the best 
solution. 
Equation (6) or (6.6) always yields a useful solution if the time intervals 
T.sub.n, T.sub.n+1, T.sub.n+2 are positive. It is not always certain that 
these time intervals are all positive. If, for example, the intermediate 
circuit current i.sub.zw is too small, the stator current vector i.sub.s 
no longer lies within the triangle formed by the possible vectors 
i.sub.p1, i.sub.p0 and i.sub.p6 (FIG. 17). The stator voltage u.sub.s can 
therefore no longer be influenced in every direction, (see FIG. 18). In 
the indicator diagram according to FIG. 18, an increase in the voltage 
u.sub.s in the direction of the positive a-axis is no longer possible. A 
closed triangle as the local curve of the stator voltage u.sub.s of FIG. 
12 can only be achieved with a negative time (in this case, the time for 
the vector 0). This is not a logical result, and the equations have to be 
modified. 
The vector, which was evaluated with a negative time (in this case, the 
vector 0), is left out and no longer used for the calculation of the 
switching times. An influence in the direction perpendicular to the 
connection line of the remaining vectors i.sub.p1 and i.sub.p6 can no 
longer be achieved. It is satisfactory to correctly set only the 
proportion of the reference current tangential to the connection line. All 
variables for calculating the switching time points are therefore 
projected onto this connection line, and only one dimension remains. 
Assuming x.sup.t is the projection of the variable x onto the connection 
line of the vectors i.sub.pn and i.sub.pn+1, then the following equation 
applies according to FIG. 19: 
##EQU13## 
With this, the equation for the proportional time according to Equation (4) 
becomes a system of the second order: 
##EQU14## 
The next corner point of the reference trajectory, which is not 
one-dimensional, is obtained from: 
##EQU15## 
The remaining time until the next switch can therefore be calculated 
immediately: 
##EQU16## 
With this, it is possible to continuously generate correct switching times, 
but with the limitation that no influence exists any longer on the 
component perpendicular to the connection line of the possible current 
vectors i.sub.pn and i.sub.pn+1. 
In general, three vectors are used for calculating the switching time. It 
is necessary to determine which of the seven possibilities (FIG. 9) are 
the correct ones. 
In order to generate as few harmonic oscillations as possible at a given 
switching frequency f.sub.s, the vectors which produce the least possible 
deviation from the reference value should always be selected, in other 
words, the vectors which lie the closest to the reference inverter current 
i.sub.p * (FIG. 9). The following equation applies: 
EQU i.sub.p *=i.sub.s *+j.rho.C u.sub.s *. 
However, it should also be ensured that no unnecessary switching actions 
occur during the transition between two vector triplets. Only a transition 
between adjacent sectors is logical, and only if the vector i.sub.p, which 
connects adjacent sectors and is contained in both vector triplets, is 
switched on at that particular time. The zero vector cannot be viewed as a 
common vector contained in both vector triplets, since there are three 
different switching variants for it according to FIG. 5, and only one of 
them is logical in each sector, and adjacent sectors do not possess a 
common zero vector. It is shown in FIG. 9 that for a predetermined vector 
i.sub.p *, only the three adjacent vectors i.sub.p1, i.sub.p6 and i.sub.p0 
I are logical, since they have the smallest distance from the reference 
vector i.sub.p *. 
The sequence or the passage direction through the selected vector triplet 
remains as another degree of freedom in the determination of the switching 
time points. 
From the three different directions of the derivation of the stator voltage 
u.sub.sn, u.sub.sn+1 and u.sub.sn+2, a triangle is constructed as the 
local curve of the stator voltage u.sub.s. However, with the same mean 
value and the same content of harmonic oscillations, there are two 
orientations for the triangle, depending on the sequence of passage 
through the vector triplet. FIGS. 20 and 21 show two different 
possibilities. 
As mentioned above, it does not matter at all what sequence is selected, 
but it should be kept the same. As soon as a new vector triplet is 
selected, however, or if the reference values change, it may be better to 
change the sequence. A decision then has to be made as to what sequence 
fits the given initial values better. 
When passing through the triangle in the mathematically positive direction, 
represented in FIG. 20, the reference value u.sub.s * always lies to the 
left of the trajectory TR*, while it lies to the right otherwise, as shown 
in FIG. 21. At a given voltage u.sub.s it can therefore be immediately 
determined which sequence is better. The following equations apply: 
##EQU17## 
with the operator defined as: 
EQU x y=x.sub.1 .multidot.y.sub.2 -x.sub.2 .multidot.y.sub.1 (18) 
It has been assumed until now that all of the switching times can be 
implemented from the calculation. In fact, this cannot always be achieved. 
Difficulties occur at three points: 
When the reference value changes. 
Due to the minimum switch-on and switch-off times of the semiconductor 
elements. 
Due to the time-discrete method of operation of the processor. 
If the reference value u.sub.s * of the voltage changes suddenly, the 
reference vector u.sub.s * may no longer lie within the calculated 
reference trajectory TR.sub.alt * of the voltage u.sub.s. Also, the actual 
vector u.sub.s may move away from the reference value u.sub.s *, as seen 
in FIG. 22. In this case, a negative time difference until the next 
switching point is calculated. The best alternative, in this case, is to 
carry out the switching action immediately. This is represented by a 
trajectory TR.sub.Sch. This variation results automatically, since 
equation (8) yields a negative time, which can only be approximated by 
switching immediately. 
In order not to destroy the current converter valves of the setting element 
8 which can be turned off, the minimum switch-on and switch-off times must 
be observed. Otherwise the components are unevenly stressed by current 
constrictions and can become too hot. The minimum switch-on time is in the 
range between 20 and 200 .mu.sec, depending on the model and circuitry. As 
a minimum switch-off time, the double value can be assumed. Since the 
inverter currents i.sub.p are not measured, but are taken into 
consideration for regulation, the minimum times cannot be monitored only 
by the hardware, but rather, this limitation must also be reproduced in 
the processor, and the changed switching times have to be output to the 
control set and included in the further calculations. 
It was assumed until now that the calculation of the switching time points 
is carried out at infinite speed and that the switching times are always 
available. Due to the discrete method of operation of the processor and 
the finite scanning time, only one switching time point is calculated in 
each scanning interval. If the difference between two switching time 
points is less than the scanning time, at least two switching time points 
have to be calculated in a scanning interval t.sub.n+1 -t.sub.n (FIG. 23), 
even if these are not always implemented in the next scanning interval. 
Therefore, a two-stage calculation with the following sequence is 
necessary: 
Calculation of the first switching time point as described. 
Consideration of the minimum switching times. 
Estimation of the capacitor voltage at this time point, utilizing the 
previous results. 
Calculation of the reference value of the capacitor voltage at the next 
switching time point as in the first calculation. 
Calculation of the time difference until the switching point. 
Consideration of the minimum switching times. 
As long as the scanning frequency is at least twice as high as the 
switching frequency f.sub.s, good results are obtained with this two-stage 
advance calculation. At smaller scanning frequencies, however, additional 
calculation steps have to be added. With this extrapolation into the 
future, the uncertainty with regard to external influences increases. 
As mentioned above, the method of the present invention for regulating a 
two-dimensional vector u.sub.s, a controlled system 10, 12 by means of a 
value-discrete final controlling element 8 with a limited switching 
frequency f.sub.s is represented in greater detail in FIG. 7, as a 
hardware-type block schematic. The single-phase equivalent schematic of 
FIG. 2 has been used as the basis for the circuit of FIG. 7. The switching 
arrangement 32 calculates a corresponding sector, in which the reference 
value of the inverter current i.sub.p * is found, from the input variables 
u.sub.s *, i.sub.s * and i.sub.p with a scalar operator 34, defined in 
Equation (18), a comparator 36 and a sector counter 38. With the other 
input variables i.sub.s, i.sub.p, u.sub.s, u.sub.s *, .rho., and C of this 
switching arrangement 32, in combination with comparators 40 and 42, an 
adder 44, a scalar operator 46 and a comparator 48, the sequence of a 
vector triplet i.sub.p1, i.sub.p2 and i.sub.p3 selected by block 50 is 
determined. With the vector triplet i.sub.p1, i.sub.p2 and i.sub.p3, and 
the input variables u.sub.s *, i.sub.s * and .rho., three voltage 
gradients u.sub.s1, u.sub.s2 and u.sub.s3 are calculated, illustrated by 
block 52. With these voltage gradients u.sub.s1, u.sub.s2 and u.sub.s3 and 
equation (4), illustrated by a block 54, the interval times of the 
individual current indicators i.sub.p1, i.sub.p2 and i.sub.p3 of the 
selected vector triplet are obtained. With these interval times T.sub.1, 
T.sub.2 and T.sub.3 as determined, and the equation (6), embodied by the 
block 56, the location of the reference trajectory TR* is determined (i.e. 
a corner point u.sub.so or .DELTA.u.sub.so is determined). The switching 
time point must be selected in such a way that the distance between the 
voltage vector u.sub.s and the nearest corner point u.sub.so to the 
switching point becomes minimal so that a voltage vector u.sub.s runs into 
the reference trajectory TR*. With equation (8), embodied by the block 58 
and the corner point u.sub.so and a voltage gradient u.sub.s, the 
remaining time t.sub.s until the next switching point is obtained. This 
time t.sub.s is passed on to a control set 62, as well as a sector number 
by a gate 60, which includes the minimum switching time of a current 
converter valve, which can be turned off, of the final controlling element 
8. A hardware embodiment of the control set 62 is shown in greater detail 
in FIG. 8. At the outputs of the control set 62, the switching states 
S.sub..nu. for the setting element 8 are present. 
The control set according to FIG. 8 comprises a shift register 64 at its 
input, where a sector number and a time t.sub.s are continuously read in, 
and a transfer register 66 at its output. The sector numbers are passed to 
the transfer register 66 directly, where the time t.sub.s is passed to a 
comparator 68, the other output of which is connected with a counter 70. 
The counter 70 is set for a scanning time t.sub.n. If this counter status 
is reached, an interrupt signal S.sub.INT is generated. If the remaining 
time t.sub.s is greater tan the set scanning time t.sub.n, this time 
t.sub.s is not passed on to the transfer register 66. 
The method of the present invention and the switching arrangement 32 for 
implementing the method have the following advantages: 
The capacitor voltage u.sub.s is kept as close as possible to the reference 
value u.sub.s *. 
No resonances are excited, and existing oscillations are quickly regulated. 
The necessary machine current i.sub.s is already set correctly by the 
modulation. 
The harmonic oscillations are minimized. 
The adjustment time for the voltage u.sub.s and current i.sub.s is 
extremely short. 
The regulation strongly prevents parameter changes and marginal conditions 
of the inverter.