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Claims  |
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What is claimed is:
1. An apparatus for a discrete-time adaptive on-off switching control of a
continuous-time process by means of a binary switching actuator producing
two levels of actuating switching signals to on-off control said process,
said apparatus using for a determination of an on-off actuating signal a
prediction of a process output sequence over several future sampling
intervals as a response to a possible process input sequence that is
applied to a discrete-time linear process model, and estimating and
updating in every sampling interval parameters of a process model by means
of a parameter estimation means in order to adapt the parameters to the
process to be controlled, even when the process behavior changes, said
apparatus having a device to input and change a setpoint as well as a
measuring device for a periodic measurement of the process output,
characterized in that it comprises:
two alternatively working first and second control means, said first
control means being active in a stationary phase of the process and said
second control means being active in a start-up phase of the process or
after setpoint changes, each of said control means operating differently
to produce control signals for said actuator, and
switching means which activates one or the other of said control means in
dependence of the result of a measurement of the process output and a
current setpoint, said actuator being controlled by the control means
which is activated by the switching means.
2. An apparatus as claimed in claim 1, wherein said switching means
activates the second control means if the process has to be started or has
to follow a setpoint change and activates the first control means if the
process is run in the stationary phase.
3. An apparatus as claimed in claim 1, wherein said two control means are
provided with informtion about the process by means of a common parameter
estimation means.
4. An apparatus as claimed in claim 1, wherein said first control means
predicts all possible 2.sup.R future process output sequences, wherein the
prediction of the process output sequence on the basis of a certain
process input sequence is made by reusing the prediction of a portion of
the process output sequence made on the basis of at least another process
input sequence.
5. An apparatus as claimed in claim 1, wherein said first control means
decouples a prediction time from a sampling time by dividing each
discrete-time prediction step into a prefixed number of sampling
intervals.
6. An apparatus as claimed in claim 1, wherein a determination of an
evaluation parameter for each predicted process output and a resulting
selection of the actuating signal for a next sampling interval is carried
out together with the prediction.
7. An apparatus as claimed in claim 1, wherein the second control means
predicts an extremal point of a process input sequence within which only
one switching of an actuating signal occurs, the switching occurring after
a first sampling interval of said process input sequence within which only
one switching of an actuating signal occurs.
8. An apparatus as claimed in claim 7, wherein the actuating signal for a
next sampling interval is taken from the said process input sequence
within which only one switching of an actuating signal occurs, if a system
deviation in the extremal point of the future process output sequence is
of such a nature that no overshooting of the process output occurs.
9. An apparatus as claimed in claim 7, wherein, when process output
overshooting occurs, the actuating signal is taken for the next sampling
interval that results from a switching of the actuating signal to its
counteracting level that is for the first sampling interval of the said
process input sequence within which only one switching of an actuating
signal occurs.
10. An apparatus as claimed in claim 7, wherein a prediction time is
determined in such a manner that a time available within one sampling
interval is used entirely for determining the actuating signal to be
applied in the next sampling interval.
11. An apparatus as claimed in claim 1, further comprising a limit
supervisory control means for turning off the active control means to
switch the actuating signal of the process off or on if the process output
exceeds the preselected upper or lower limit of the process output,
respectively.
12. An apparatus as claimed in claim 1, further comprising manually
operable limit supervisory control means for controlling the process by
means of the actuator in such a way that the parameter estimation means
yields a process model which can be used for later adaptive control.
13. An arrangement as claimed in claim 1, wherein the actuator is used
synchronously with the measuring device.
14. An apparatus used in the transient phase of a process to be controlled
comprising:
means for generating a process input sequence within which only one
switching of an actuating signal occurs, the switching occurring after a
first sampling interval of the process input sequence,
means for predicting a process output sequence over a number of future
sampling intervals as a response to the process input sequence generated
that is applied to a discrete-time linear process model, and further
predicting the extremal point of the process output sequence, and
means for selecting an actuating signal for the next sampling interval from
the process input sequence generated, if a system deviation in the
extremal point of the future process output sequence is of such a nature
that no overshooting of the process output occurs, and for selecting,
where overshooting of a process occurs, an actuating signal for the next
sampling interval that results from a switching of such actuating signal
to its counteracting level that is for the first sampling interval of the
process input sequence generated.
15. A method of operating an apparatus for a discrete-time adaptive on-off
switching control of a continuous-time process by means of a binary
switching actuator producing two levels of actuating switching signals to
on-off control said process, said apparatus using for a determination of
an on-off actuating signal a prediction of a process output sequence over
several future sampling intervals as a response to a possible process
input sequence that is applied to a discrete-time linear process model,
and estimating and updating in every sampling interval parameters of a
process model by means of a parameter estimation means in order to adapt
the parameters to the process to be controlled, even when the process
behavior changes, said apparatus having a device to input and change a
setpoint as well as a measuring device for a periodic measurement of the
process output, said method comprising:
predicting in a stationary phase all possible 2.sup.R future process output
sequences and predicting in a transient phase an extremal point of one
future process output sequence which is caused by a process input sequence
within which only one switching of an actuating signal occurs, the
switching occurring after a first sampling interval of said process input
sequence, within which only one switching of an actuating signal occurs,
selecting one or the other prediction in dependence of the result of the
measurement of the process output and a current input setpoint, and
actuating the actuator in a next sampling interval based on a switching
level produced from said one or the other prediction. |
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Claims |
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Description |
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BACKGROUND OF THE INVENTION
The present invention relates to an arrangement for the predictive
discrete-time adaptive on-off control of continuous-time processes with
binary switching actuators.
Controllers with binary switching actuators excel in their reliability and
robustness. Parameter adjustment of conventional switching on-off
controllers is based on the results of empirical investigations on
standardized model processes specified by simple parameters. Due to the
non-linearity of the controller an analytical determination of its
parameters for an optimal control in the sense of an optimization
criterion can be done only with very high effort.
Difficulties in finding suitable controller parameters occur especially in
cases where the process to be controlled is not described precisely enough
by the parameters of the respective standardized model process or in cases
where its dynamics are not sufficiently known or time-variant.
Within the last decades adaptive controllers have been developed which
are--contrary to controllers with fixed parameters--able to adapt to the
momentary operating conditions of the process to be controlled, thus
increasing the quality of control of processes that are insufficiently
known or time-variant. By means of known parameter estimation methods a
process model is determined and used for finding out and establishing a
way of control which is optimal in the sense of a quality criterion.
The adaptive design methods known until recently are based on the
assumption that the controller is able to generate any actuating signal
level within the actuating range. Therefore they cannot be applied
directly for the design of an adjustable control arrangement in on-off
controllers, which allow only two possible switching levels. Concepts of
such discrete-time adaptive control arrangements as improved so that they
can have two switching levels as process input, have been known and
realized for several years. For the determination of the on-off actuating
signal, here, a prediction of process output sequences over several future
sampling intervals as reaction to possible process input sequences is used
to estimate the parameters by means of a parameter estimation method and
update them in every sampling interval in order to adapt the process model
to the process to be controlled, even when the process behaviour changes
(Breddermann, R.: Realization and Application of a Selftuning On-Off
Controller. Proceedings of the International Symposium on Adaptive
Systems, Bochum, FRG, 1980, and Hoffmann, U.; Breddermann, R.: Entwicklung
und Erprobung eines Konzepts zur adaptiven Zweipunktregelung, in:
Regelungstechnik 29. Jahrgang, 1981, no. 6, pp. 212-213).
Said publications describe a prior art of the invention which up to now has
been an imperfect realization of a concept for adaptive switching on-off
control which is still to be improved. The realization of the prior art
requires a high technical effort. The complex control arrangement has to
be operated by highly qualified staff. The control performance documented
in the publication mentioned first has a problem of excessive overshooting
of the process output in the start-up phase of the process or after
setpoint changes which the process is to follow. This problem is normally
undesired and even intolerable, in many applications.
SUMMARY OF THE INVENTION
In a discrete-time operating control arrangement for binary switching
actuators, the object of the invention is to enable even unskilled
personnel to operate it and to avoid overshooting to a great extent in the
start-up phase of the process or after setpoint changes without making the
additional technical effort which has been required.
In accordance with the present invention this objective is achieved by a
combination of an improved parameter estimation means and two
alternatively working first and second control means activated by a
switching means in dependence of the setpoint and the process output,
wherein the process output is measured periodically by a measuring device
and the setpoint can be given by means of a device for input and change of
data. Synchronously to the measurement of the process output the actuating
signal determined by the active control means is given out via an actuator
with two switching levels, e.g. a relay for the switching of electrical
heaters in thermal processes.
The switching between the said control means is advantageously used to
activate the first control means which is especially suited to control
disturbances or to follow changes of the process dynamics in the
stationary phase of the process or the second control means which is
especially suited to approach the desired setpoint without overshooting
and simultaneously estimate the process dynamics in the new operating
point at the start of the process or after a setpoint change,
respectively.
In one way of carrying out the invention the control means for the
transient phase, i.e. for the control at the start of the process and
after setpoint changes, is activated when the arrangement is turned on to
start the process or if a new setpoint is set, and the first control means
for the stationary phase is activated when in the transient phase the
measured process output reaches a prefixed distance from the setpoint for
the first time.
This is advantageous in order to immediately follow the new setpoint and in
that a fast and suitable reaction to disturbances is possible,
respectively, when there is a transition of the process from the transient
to the stationary phase near the setpoint.
In one way of carrying out the invention the parameter estimation means,
which works according to the known Least-Squares method with
U-D-factorization, yields the current values of the estimated process
parameters and the process output values measured currently or at previous
instants and the process input values determined currently or at previous
instants, which are necessary for the prediction of process output
sequences, and sends them as process model to the active control means.
The application of the above-mentioned estimation method is useful, as,
contrary to the methods applied up to now, it can easily realize fast
working numerical stable parameter estimation means. The same process
model can be used for the prediction in the same way by both control
means, so that a reduction in technical effort is possible.
In one way of carrying out the invention the prediction of the 2.sup.r
possible process output sequences over r prediction steps within the first
control means for the stationary phase can be performed in the following
way. The 2.sup.r process output sequences are successively predicted, as
responces of the process model to the 2.sup.r process input sequences
being different from each other. During these successive predictions, each
of the process input sequences for the current predictions of the process
output sequence is such that it has as many as possible switching levels
in common with ones of the previously used process input sequence within
the nearer future prediction steps and only the switching levels of each
of the process input sequences within the farther future prediction steps
are changed. And the corresponding values within the process output
sequence are predicted only with respect to the switching levels thus
changed. This is advantageous in so far as the information gained about
possible future process output sequences can be reused during the
successive prediction. Thus the technical effort and the necessary time
for processing the prediction within the first control means can be
reduced.
In one way of carrying out the invention each prediction step of the
prediction within the first control means for the stationary phase can be
divided into several sampling intervals, with the switching levels in the
sampling intervals of each prediction step remaining equal. This is
advantageous as with constant prediction time the sampling time and thus
the quantization of actuating power can be reduced and thereby the
necessary processing time only grows linearly and not any longer
exponentially with the ratio, prediction time vs. sampling time.
The determination of the evaluation parameter (cost-function) of each
predicted process output sequence is made in the known way directly with
the prediction of the corresponding process output sequences. In a further
carrying out of the invention the actuating signal to be given out in the
next sampling interval is selected in process input selection means by
comparison of the evaluation parameter of the just predicted process
output sequence with that of the process output sequence predicted before.
Thus a searching procedure for the minimal value from the 2.sup.r
evaluation parameters at the end of each sampling step can be avoided.
At the start of the process and after setpoint changes the number of
process input sequences that have to be considered for a prediction of
possible future process output sequences is smaller than 2.sup.r. It is
the aim of the control action in the transient phases of the process to
bring the process output near to the setpoint as fast as possible in order
to approach the setpoint with the process output with the least
overshooting by switching the actuating signal to its counteracting level
early enough. In a further way of carrying out the invention one can
advantageously consider that during the transient phases of the process
only the prediction of one single process output sequence is necessary.
In a further way of carrying out the invention the second control means
which is active at the start of the process or after setpoint changes is
designed in such a way that the extremal point of the future process
output sequence is predicted on the basis of one particular process input
sequence, which provides only such one single switching of the actuating
signal, that is made after the first sampling interval within that process
input sequence.
In a further way of carrying out the invention the second control means for
the transient phase contains a process input selection means which selects
the actuating signal for the first sampling interval, within the above
particular process input sequence, if the extremal point of the predicted
process output sequence lies below the new setpoint value after positive
setpoint changes and above the new setpoint value after negative setpoint
changes, so that no overshooting of the process output occurs.
In a further way of carrying out the invention process input selection
means applied for the transient phase of the process is designed in such a
way that in all other cases, i.e. where an overshooting is predicted when
maintaining the last actuating signal, the actuating signal to be given
out in the next sampling interval is favourably selected as the one which
results from switching such an actuating signal to its counteracting level
that is for the first sampling interval of the above particular process
input sequence used for the prediction.
In accordance with the invention, the second control means active at the
start of the process or after a setpoint change uses a number of
predictions of possible future process output sequence that is smaller
than that of the ones used by the first control means active for the
stationary phase. The thus saved processing time is advantageously used in
such a way that predicting means within the second control means is
enabled to make a prediction of the mentioned single process output
sequence which reaches further into the future.
That way the early recognition of the time for switching the actuating
signal to the counteracting level is ensured and a possibly too late
switching due to a too small number r of prediction steps can be avoided.
If the parameters of the process model used for the prediction are
incorrectly estimated an output of false actuating signal that leads to
intolerable operating conditions can occur more often with an arrangement
of that kind than with conventional control devices. In a way of carrying
out the invention therefore a superior limit supervisory control means is
applied in such a way that if the process output exceeds its preset upper
or lower limit, respectively, the just active first or the second control
means is turned off and the actuating signal of the process is switched
Off or On, respectively.
In a further way of carrying out the invention limit supervisory control
means, which can be activated manually, is applied for the control of the
process. According to the control by limit supervisory control means, by
means of the actuator the process is excited in such a way that the
process output periodically moves within a range of preset upper and lower
limits. This oscillation which in general is comparatively stronger than
the limit cycle in the stationary phase can advantageously be used to
determine a process model which matches as well as possible with the real
process. After the determination of the process model, when the first or
second control means is made active, predicting means within the first or
second control means can thus rely on a useful process model from the very
beginning of the control phase.
In a way of carrying out the invention the actuator for the output of the
actuating signal is driven at the same time with the same tact rate as the
measuring device for the acquisition of the process output.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram showing the structure of the concept of the
discrete-time adaptive on-off control.
FIG. 2 shows a process input, a process output, a deadtime and a prediction
time, wherein (a) represents the process inputs and the process outputs
required to describe a process model, and (b) the process input sequences
and the process output sequences in predictions of process outputs,
respectively.
FIG. 3 shows the relationship among a prediction time, a number of a
prediction steps and sampling intervals divided within one prediction
step.
FIG. 4 shows a prediction of the process output over three prediction
steps, where (a) depicts a tree structure with possible input sequences
and (b), resulting process output sequences, respectively.
FIG. 5 shows a tree structure denoting all 2.sup.r process input and output
sequences (r=3) and corresponding costfunctions.
FIG. 6 shows how the process output approaches a new setpoint in the
transient phase when the process input is once switched, (a) being in the
continuous-time case, while (b) in the discrete-time case.
FIGS. 7 and 8 show the predicted process output sequences and their
evaluation.
FIG. 9 is an NS chart showing the operation flow of the discrete-time
adaptive on-off switching controller.
FIG. 10 is a block diagram showing the structure of discrete-time adaptive
on-off switching controller.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
The structure of the concept
The discrete-time adaptive on-off switching control consists of the two
components, i.e. parameter estimation and predictive switching on-off
control. The predictive switching on-off control can be divided into the
prediction of process output and the determination of the optimal on-off
actuating signal.
FIG. 1 shows the structure of this concept (especially that of control in
stationary phase). The process 1 is, for example, a furnace provided with
an electric heater. In this case, heating electric current being fed to
the heater, process input is heating electric current. The heating
electric current fed to the heater is on-off controlled by means of an
actuator 2, for example, a relay. The temperature of the furnace is
measured at predetermined sampling intervals T by a sampling device 3.
Thus, in this case the process output is temperature.
For the parameter estimation, the process output Y(k) measured at
equidistant time instants and the process input U(k) actually applied to
the process 1 are used. As shown by Eqn. (1), from these values Y(k) and
U(k) the d.c.--values Y.sub.o and U.sub.o, respectively, are subtracted,
y(k)=Y(k)-Y.sub.o
u(k)=U(k)-U.sub.o (1)
wherein k is a parameter for discretely representing time, and time is
represented by k.multidot.T (k=0, 1, 2 . . . ) using sampling time
intervals T.
U.sub.o and Y.sub.o give the reference values of process input and output
in the operating point considered. They can be determined in the
standstill phase of the process as U.sub.o =0 and Y.sub.o =Y(0), for
instance.
These process input and output are used in an estimation algorithm (block
11 for parameter estimation) to determine a discrete-time process model
12. This process model 12 is represented by Eqn. (2).
##EQU1##
wherein G means transfer function of the process and z.sup.-d, deadtime
element, A(z.sup.-1) and B(z.sup.-1) being each given by the following
Eqns. with the sign " " denoting estimated value,
A(z.sup.-1)=1+a.sub.1 .multidot.z.sup.-1 +. . . +a.sub.n .multidot.z.sup.-n
B(z.sup.-1)=b.sub.1 .multidot.z.sup.-1-d +. . . +b.sub.n
.multidot.z.sup.-n-d (3)
wherein a.sub.1, . . . , a.sub.n and b.sub.1, . . . , b.sub.n are
parameters to be estimated. The model order n as well as the number of
deadtime steps d have to be chosen suitably depending on the process to be
controlled.
The process model at time k.multidot.T is completely described by its
parameters a.sub.i and b.sub.i ; i=1, . . . , n; and the process input ui
(i=(k-n-d+1), . . . , (k-d)) and output y.sub.i (i=(k-n+1), . . . , k)).
Though the process input u.sub.i is a value estimated and actually given
to the process, it is delayed by deadtime steps d by means of a deadtime
element 17. The process output y.sub.i is a value taken out by the
sampling device 3. In FIG. 2(a) the process input u.sub.i and the process
output y.sub.i to describe the model 12 at time k.multidot.T are indicated
on time axis.
These parameters a.sub.i and b.sub.i as well as the process input u.sub.i
and the process output y.sub.i can be written in vector form as follows.
The sign " " means vector. parameter vector
##EQU2##
and signal vector
x(k+1)=(-y(k) . . . -y(k-n+1).vertline.u(k-d) . . . u(k-d-n+1)).sup.T (5)
The recursive estimation of the model parameters a.sub.i and b.sub.i and
the updating of the signal vector are given in the section "parameter
estimation".
The thus gained process model 12 is used in the predictive on-off switching
control to determine the on-off process input to be applied to the process
1 at the next sampling interval from the view-point of a chosen
costfunction 15 (in the case of multi-step optimization, namely stationary
phase). As will be describe later, based on this actual process model 12
future process output sequence Y.sub.i are predicted (block 14). These
output sequences Y.sub.i are response of the process model 12 to the
future process input sequences U.sub.i generated in the block 13. Besides,
these process output sequences Y.sub.i are sequences which would be
possible within a prediction time T.sub.p ahead of the deadtime
d.multidot.T. Afterwards, the predicted process output sequences Y.sub.i
are evaluated by a costfunction 15. The process input sequence is said to
be optimal when the costfunction comes to minimum value owing to the
corresponding process output. The first element of the process input
sequence thus determined to be optimal (block 16) is used to switch on or
off the actuator 2 of the process 1 at the next sampling interval.
As shown in FIG. 3, the prediction over the prediction time T.sub.p is made
by dividing this T.sub.p into a certain suitable number of r prediction
steps, each prediction step consisting of q sampling intervals, within
which the process input u is assumed to have the same constant value. Thus
the prediction time T.sub.p is given by the following Eqn:
T.sub.p =r.multidot.q.multidot.T (6)
Thus there exist 2.sup.r process input sequences within the prediction time
T.sub.p. With respect to this prediction time T.sub.p, one optimal process
input sequence is determined.
The possibility of dividing each prediction step into q sampling intervals
serves to decouple the choice of the sampling interval T from the selected
prediction time T.sub.p and the number of prediction steps r. More
specifically, the sampling interval T can be optionally fixed irrespective
of the prediction time T.sub.p or the number of prediction steps r.
The multi-step-optimization over r>1 prediction steps is a more suitable
optimization method than the one-step-optimization (r=1), as is described
in the section "predictive on-off switching control".
The answer to the question as to which of the 2.sup.r process input
sequences are considered for prediction and evaluation, depends on the
control task. There are following two cases of control task:
(1) control of the process in the stationary (steady state) phase
(2) control of the process in the transient phase.
In the first case the "stationary phase control" is made, where all 2.sup.r
process input sequences are evaluated. In the second case, e.g. after
setpoint changes or in the start-up or shut down phase of the process, the
control is made via the "transient phase control". Only one process input
sequence is used for prediction here. In either case, according to the
process output sequence predicted it is decided whether the actual value
of the process input to be applied to the process 1 is kept constant at
the next sampling instant or has to be switched to its counteracting
level.
The decision as to which of the two switching controls is to be applied at
the moment depends on the set-point- and the process output sequence. The
change-over from the stationary phase control to the transient phase
control is made, e.g., after a setpoint change has occurred. The
change-over from the transient phase control to the stationary phase
control happens, e.g., when the absolute value of the deviation
(difference between setpoint w and the measured process output y)
.vertline.y.sub.d .vertline.=.vertline.y-w.vertline. is smaller than 0.5%
Y.sub.h (Y.sub.h is the possible full control range) for the first time
after the setpoint has changed. Thus the stationary phase control is not
used until the transient phase is settled to a full extent.
Parameter estimation
The Recursive Least Squares estimation (RLS-estimation) is a suitable
parameter estimation method within the adaptive on-off switching
controller. This method is applicable to any processes, and further
according to this method computer load can be reduced. The aim of the
parameter estimation is to determine the parameters a.sub.i and b.sub.i of
the process model (see Eqn. (2) and (3)) at any sampling instant
k.multidot.T from the acquired values y(k) and u(k). This aim is realized
by minimizing the so-called equation error (Eqn. (7)) of the loss function
(Eqn. (8)).
##EQU3##
The recursive estimation of the parameter vector .theta. is performed by
adding a correction term, the product of the equation error e(k) and a
correction vector g(k) (Eqn. (10)), to the latest actual parameter vector
.theta.(k-1). In other words, the recursive estimation equation is given
as
.theta.(k)=.theta.(k-1)+g(k).multidot.e(k). (9)
The correction vector g(k) (Eqn. (10)) includes the scalar (Eqn. (11)) and
the normalized covariance matrix of the parameter error (Eqn. (12)).
##EQU4##
The adaption factor .rho. in Eqns. (11) and (12) means the weight of data.
Owing to this .rho. a higher evaluation is given to the present data than
to the past data. The choice of .rho.<1 causes a greater change of
parameters, which results in giving a greater margin for parameter changes
and allowing an easier tracking of time-variant processes.
The above-mentioned method for determining model parameters is well known
in control engineering. A more general description of this estimation
method can be found among other in: Astrom/Eykhoff: System
Identification--A Survey. Automatica, Vol. 7, pp. 123-162, Pergamon Press,
1971 and V. Strejc: Least Squares Parameter Estimation. Automatica, Vol.
16, pp. 535-550, Pergamon Press, 1980.
The possibility to estimate the process parameters with a sufficient
exactness depends, among other things, on the numerical data processing on
a digital computer. The word length L (in Bit) of the internal
arithmetical data representation has an influence on the parameter
accuracy. Especially when using micro computers with L=32 Bit word length
for the representation of sign, mantissa and exponent rounding errors can
occur that lead to numerical instabilities of the recursive estimation.
Possibilities to avoid these problems are given by the U-D-Factorization.
This method was proposed by Bierman: Measurement Updating using the
U-D-Factorization. Automatica, Vol. 12, pp. 375-382, Pergamon Press, 1976.
This method is based on the calculation of the covariance matrix as matrix
product
P(k)=U(k).multidot.D(k).multidot.U(k).sup.T. (13)
U(k) is an upper triangular matrix, while D(k) is a diagonal matrix and can
be stored in vector form. This modification of the above-mentioned Least
Squares parameter estimation method is favourably used with the
discrete-time adaptive on-off switching controller in order to ensure
proper estimates when using a micro computer.
The predictive on-off switching control
(1) The stationary phase control
In the stationary phase control all 2.sup.r possible process input
sequences are evaluated over the given r prediction steps. The evaluation
of all possible process input sequences ensures that an optimal and not a
suboptimal switching behavior is determined for the next r prediction
steps.
The prediction and its evaluation in order to determine the optimal
switching behaviour are described below. For an easier understanding and
without loss of generality one prediction step is chosen as one sampling
interval, i.e. q=1. The process input can assume only two actuating levels
u.sub.max and u.sub.min so that all 2.sup.r process input sequences
resulting from block 13 (FIG. 1) are known beforehand. 2.sup.r process
input sequences over the future prediction steps are given by the
following equation.
U.sub.i (k+1)=(u(k+1) . . . u(j) . . . u(k+r)).sup.T ;
1.ltoreq.i.ltoreq.2.sup.r (14)
with
u(j).epsilon.{u.sub.max, u.sub.min }
The two process inputs u.sub.max and u.sub.min correspond to 1 (H level)
and 0 (L level), respectively when represented in terms of the switching
levels of the actuator 2. More specifically, when the actuator 2 is on,
the process input u.sub.max is given to the process 1 and when it is off,
u.sub.min is applied thereto. For a better understanding, all the process
input sequences are represented in terms of the switching levels of the
actuator 2 as follows:
##EQU5##
The process input sequences with r=3 are shown by means of a tree structure
at (a) in FIG. 4.
The future process output sequences Y.sub.i predicted (FIG. 1, block 14) as
response of the above-mentioned process model 12 to those process input
sequences U.sub.i are given by the following equation
##EQU6##
wherein the sign " " means a predicted value.
FIG. 4(b) indicates the predicted process output sequences Y.sub.i in the
case of r=3. Because of the deadtime element, the process output sequences
are delayed by (d+1) steps.
As seen from FIG. 4(a), the process input sequences U.sub.1 . . . , U.sub.4
each have a common value 1 (u.sub.max) at (k+1) and different values at
(k+2) and (k+3). As to the process input sequences U.sub.1 and U.sub.2, it
will be noticed that each of these sequences has a common value 1 at (k+1)
and (k+2) and different values merely at (k+3). Generally speaking, there
exist 2.sup.p of the 2.sup.r process input sequences which differ from
each other only within the last p prediction steps. All the process output
sequences are predicted by making use of such fact. Thus the information
about the future process output gained within the first (r-p) prediction
steps can be used for further predictions of (2.sup.p -1) process output
sequences once it has been calculated. It is sufficient to predict the
(2.sup.r+1 -2) possible values of the process output at equidistant time
instants in order to determine all 2.sup.r process output sequences within
the prediction time. With r=3, (2.sup.r+1 -2)=14. In FIG. 4(b) the number
of black dots is 14. Originally, (2.sup.r .times.r) process outputs, for
example in the case of r=3, 24 process outputs have to be predicted, but
according to this way of avoiding the duplication of the calculation for
prediction, far less predictions are sufficient.
The prediction of the process output is performed by calculating with use
of the estimated values, as shown by the following equations,
y(k+1)=x.sup.T (k+1).multidot..theta.(k) (17)
y(k+1+j)=x.sup.T (k+1+j).multidot..theta.(k) (18)
with
1.ltoreq.j.ltoreq.d+r.
The parameter vector .theta.(k) in Eqns. (17) and (18) is given by Eqn. (4)
and the signal vector x.sup.T (k+1) in Eqn. (17), by Eqn. (5).
Consequently, the process output y(k+1) of Eqn. (17) is predicted from the
process model at time k.multidot.T.
In Eqn. (18), the signal vector x.sup.T (k+2) has to be gained so as to
predict the process output y(k+2) in the case of j=1. The signal vector
x.sup.T (k+2) is acquired by subsubstituting k with (k+1) in Eqn. (5).
This substitution is equivalent to newly introducing y(k+1) and u(k-d+1)
as the values in the first and the (n+1)th rows respectively, shifting the
values in other rows to the following rows sucessively and further
removing the values in the nth and th 2nth rows, in the signal vector
x.sup.T (k+1) of Eqn. (5). In Eqn. (5) with the substitution of
k.fwdarw.k+1, y(k+1) is the predicted value derived from Eqn. 17. The
other values y(k), . . . , y(k-n+2) and u(k-d+1), . . . ,u(k-d-n+2) are
known ones.
Similarly, the signal vector x.sup.T (k+1+j)(j>1) is succesively derived
from the signal vector x.sup.T (k+j), by updating the first and the
(n+1)th elements with y(k+j) and u(K-d+j), respectively. With
(k-d+j)<(k+1), u(k-d+j) are known values and with (k-d+j).gtoreq.(k+1),
possible values are adopted as u(k-d+j).
Thus derived y(k+1), . . . ,y(k+d+1) of the predicted process outputs are
predicted values based on the already determined values. This is the
prediction over process deadtime (FIG. 10 block 54). See FIG. 2(b) as
well.
Within the following r prediction steps all process output sequences
Y.sub.i which are caused by the possible process input sequences U.sub.i
are derived by calculating Eqn. (18). This is the prediction over
prediction time (FIG. 10 block 34). See FIG. 2(b) as well.
The division of each prediction step into sampling intervals of constant
actuating level means the number of recursive solutions of Eqn. (18) which
is q-times larger than the above-mentioned case with q=1 appears.
Accordingly the vector Y.sub.i becomes q-times longer.
With the calculation of the predicted process output y goes the evaluation
by means of the costfunction. Although the predictive on-off switching
control is separated into prediction and determination of the optimal
on-off actuating level, it is sensible to combine prediction with
costfunction evaluation for enhancing the computational efficiency.
For the evaluation the predicted process output sequences Y.sub.i are
compared with the setpoint. In the stationary phase the setpoint is
assumed to be constant, so that future setpoint values are given by
w(k)=w(k+1)= . . . =w(k+d+r+1). (19)
The necessary setpoints are incorporated in the setpoint vector for the
comparison with the process output sequence vector Y.sub.i. The setpoint
vector is represented by
W(k+1)=(w(k+d+2) . . . w(k+d+r+1)).sup.T. (20)
The multi-step-costfunction thus reads:
J(k+1)=J(Y.sub.i (k+1 | | |
