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Adaptive and Predictive Strategies for Extruder Temperature Control
APPLICATIONS OF ADAPTIVE AND SELF-TUNING CONTROL
Copyright © IFAC Identification and System Parameter Estimation. Budapest. Hungary 1991
ADAPTIVE AND PREDICTIVE STRATEGIES FOR
EXTRUDER TEMPERATURE CONTROL R. Haber*, H. P. Jorgl**, T. Vecsei** and E. Haider·Schmid*** *Cologne Polytechnic (Fachhochschule Koln). Faculty oj Plant and Process Engineering, D-5000 Koln 21. Betzdorjer Str 2, Germany **lnstilUtejor Machine and Process Automation, Technical University ojVienna. A-I040 Vienna. Gusshausstrasse 27129. Austria ***KEBA Automatisierungselektronik. Gewerbehoj Urjahr. A-4041 Linz, Austria
Abstract. In the first part of the paper two self-tuning PID control algorithms are compared. Both algorithms are based on the estimated time delay and equivalent time constant of the process. While the first strategy uses on-line least squares parameter estimation, the second one computes the process parameters from the limit cycles induced by an on-off controller. The second part of the paper presents a predictive control which is especially suitable for the start-up phase. An extension of the algorithm assures that a prescibed temperature difference of the neighbouring zones is not exceeded during the start-up. Keywords. Adaptive control; predictive control; temperature control; extruder.
I NTRODUCTI ON
then they are much cheaper. Although the principles of adaptive control are well known the detai Is are usually kept confidential. Therefore, the objective of the authors was to develop and to examine some self-tuning control algorithms for temperature control in extruders.
The control task for the temperature zones of an extruder can be separated into two phases: the start-up and the stationary operation. A good start-up control requires, that o
there is no overshot in the temperature,
o
the final temperature is achieved fast,
o
the difference between the temperatures of the neighbouring zones is kept below a prescribed value.
In the literature three extruders are reported:
Once the temperature has achieved its steady-state value, it has to be kept constant. i.e.disturbances have to be rejected.
o
on-off control with feedback,
o
PID control and
o
predictive control.
algorithms
for
Hoffmann, MUller and SchUrmann (1983) report about the adaptive on-off controller with feedback. The parameters of the process are est imated by the recursive least squares method and the parameters of the PID-equi valent controller are set by rule of thumb . Gawthrop, Nomikos and Smith (1988) compare three adaptive algorithms. The so-called continuous-time self-tuning PID control is based on an on-line least squares estimation of the PID-paremeters. The so called Closed Loop Cycling Method relies on an on-off control. The parameters are calculated from the period of one oscillation in the operating point without any parameter estimation. The regulator parameters are designed according to the Ziegler-Nichols rules . An entirely other concept is realized by Hoffmann (1987), who used a predictive on-off regulator and calculated the cost function of the control for several steps ahead.
A self-tuning control algorithm can be of great help to the operator while putting a new extruder into ope rat ion, because each machine has its own set of parameters, and the time spent tuning the parameters manually can be saved. Furthermore, the self-tuning can be applied if the performance of the extrusion is not satisfactory as a consequence of e.g. changing of the material processed. Proposed is not a permanent adaptive control but a self-tuning algorithm which can be initialized by the operator. lhe extruders zone there is neighbouring practice, the
basic
are heated in several zones . In each a temperature sensor. Al though the zones influence each other in zones are controlled autonomously.
Several manufacturers sell PID-regulators with self-tuning possiblities . Most of them are based on the identification of the measured step response of the process . (A usual way is the graphoanalytical determination of the dead time and the time constant and the computation of the optimal regulator parameters based on rules of thumb.) The extruder producers implement the control algorithm in their control boards because
The paper presents the experiences of the authors with three different self-tuning controllers:
481
o
PID-controller estimation.
with
recursive
parameter
o
PlO-controller with process parameter calculat ion from the 1 imi t cycles caused by an on-off controller with hysteresis,
impossible. The limitation algorithm worked as follows : If the control signal would go beyond its limits, then the control signal was set to the 1 imi t and the actual control error was reset to the value which would cause the control signal to reach the limit.
requirements of the extruder control, i . e . o
it uses thumb in transient signal or
o
the limit cycle can be evaluated in any working point independent of the value of the maximal achievable temperature.
RECURSIVE LS ESTIMATION OF THE PROCESS PARAMETERS A start up of the extruder begins with a full power heat input This means that at the very beginning one does not need the parameters of either the controller or the process. On the other hand, there is large step in the heat input (100 :I.), and the step response of the temperature is well suited for the identification of the process .
The method is o
recursive least
squares
o
linear process mode I of order 3 or 4 and relative dealay time zero,
o
sampling time O. l-times the delay time of the VD-filter with forgetting factor of 0 . 99, the parameter estimation should be stopped before the temperature achieves its steady-state value (suggested is to stop at approximately 90 :I. of the steady state value).
the
evaluation of the
amplitude of the oscillations,
o
period of the oscillations and
o
relat ion of the periods of switching on to
Experiences have shown that the individual I imi t cycle experiments for the zones yield good results although the neighbouring zones influence each other .
process,
o
on
All these parameters are closely related to the delay time and the equivalent time constant of the process (Zeitz, 1986).
method,
o
based
switchi ng off .
Based on simulation and practical experiences Hoffmann (1987) recommends to perform the identification with the following parameters: o
the Chien-Hrones-Re swick rule of order to achieve a fast aperiodic for stepchanges of the reference a disturbance and
PREDICTIVE CONTROL IN STEADY-STATE With predictive control , the future temperature is computed on the prediction horizon for all possible future heating sequences. The predicted temperature sequence has to be compared wi th the reference signal and the heat ing sequence which causes the least deviation from the reference signal - usually in quadrat ic sense - has to · be chosen. Then, this heat input power has to be realized and the same computation has to be repeated in the next sampling interval .
The delay time and the time constant were computed from the simulated step response of the identified pulse-transfer function .
The long-range predictive control can be realized in an easy way because the number of the possible control sequences is restricted by the two choices (switching on or off) in each (control) sampling period. Consequently the number of the possible 1 sequences is 2P + where p is the lenght of the prediction horizon relative to the sampling time . Hoffmann (1987) recommended to choose p=6 and to take into account that computations can be further reduced if some auxiliary results are stored and are reused while computing the cost function of other sequences.
DETERMINATION OF THE PROCESS PARAMETERS FROM A LIMIT CYCLING EXPERIMENT CAUSED BY AN ON-OFF CONTROLLER Since least squares type parameter estimation algorithms need much computation efforts, attempts have been made in order to develop easier identification methods. Astrom and Hagglund (1984) design the PID-parameters according to the Ziegler-Nichols technique . The process is controlled first by an on-off controller and both the critical gain and the period of the oscillations are determined from the limit cycle .
An adaptive version of the above algorithm was used when the pulse-transfer function of the process was identified by the recursive least squares parameters estimation during the start-up period .
Gawthrop, Nomikos and Smith (1988) use a simi lar procedure which they call Closed Loop Cycling Method . The only difference to that of Astrom and Hagglund (1984) is that they do not need several oscillations but the process parameters needed are determined from one cycle .
PREDICTIVE START-UP CONTROL The aim of the start-up control is to achieve the reference value as fast as possible without any overshot . In the first sampling intervals the heat has to be switched on. Then it has to be checked when it has to be switched off the first time . Therefore only one heating sequence has to be checked :
Rosenberg (1966) designes an on-off controller with a first-order term feedback without identifying the process . Since this controller is equivalent to a continuous PD-controller, he uses the design rules of Ziegler-Nichols developed for first-order systems with delay time. The delay time and the time constant are determined from the oscillations caused by an on-off controller working with a reference signal being half of the maximal achievable temperature.
on, off, off, off, ... , off . As long as the predicted temperature is below the reference signal in the whole prediction horizon, the actual heating power is on, otherwise it shoul d be off.
The method developed by the authors is similar to the latter one but is better suitable for the
482
o
long-range predictive controller.
111
In addition, the predictive control algorithm was modified such that the temperature difference between the neighbouring zones does not exceed a prescribed value. The algorithms were simulated by means of a simulation program package, which modelled an industrial extruder. The simplest realizable algori thm, the PID-controller based on the 1 imi t cycling experiment, was realized on the real plant with succes.
1/2
MODELLING OF THE EXTRUDER
1/3 The extruder under investigation had four heating zones. The temperature of each zone was affected by the heat input to the zone and by the temperature difference between it and the neighbouring zones. In order to set up the model of the extruder the following symbols were introduced u (t);
i=l, ... 4
1
heat input to the i-th zone, (100 %
i=l, ... 4
G (s);
Fig.1: Extruder temperature zone model
full power, no heat input)
0 % Yl (t);
1/4
Al though the zones are coupled, controllers were used for the zones.
temperature of i-th zone,
i=l, ... 4 transfer function between the
11
input
heat
and
CONTINUOUS-TIME PID-CONTROLLER
the
As the objective was to extend a traditional cont inuous-t ime PID-control algorithm wi th a self-tuning possibility, the parameters of the continuous-time PID-algorithm were calculated . The recursive discrete-time algorithm was used in the digital simulations was:
temperature of the i-the zone, i=l, ... 4
transfer function between the temperature i-th
and
difference j-th
zone
of
the
and
the
autonomous
temperature of the i-th zone. The model of the extruder can be set up with the following equations:
where u(k) is the control signal and e(k) the control error . The static gain (K p )' derivative
y (s)
related to
(To) 1
and
integrating
(T
1
)
the parameters ql
time
constants
as follows
are
(t.T
=
sampling time): y (s)
G (S)U (S) + [Yl (s)-Y2(s) lG (S) + 2 22 12
2
+
y (s)
[Y3(s)-Y2(s)lG
32
(s)
G (S)U (S) + [Y2(s)-Y3(S) lG (s) + 3 33 23
3
+
Fig. 1
[Y4 (S)-Y3(s) lG
shows
the
43
q
(s)
model
The transfer functions were experiments:
of
the
from
two
1. separate heating of the zones,
In both cases the the heat input to the individual zones were changed stepwise
0 0 0
from from from from
p
0
Extruders have the specific feature that the the heating power can only be swithed on or off. Therefore the control signal of a PID-algorithm has to be transformed into a ratio, the value of which represents the portion of the sampling time when the full heating power is on.
2. simultaneous heating of the zones.
0
=KT/l1T.
The parameters of the controller were calculated according the rules of thumb by Chi en, Hrones und Reswick. The parameters were selected for a stepchange in the reference value aiming at a fast aperiodic transient behaviour. The process parameters used are the dead time and the equi valent time constant, which were updated in order to get a self-tuning control algorithm.
extruder
identified
o
0% to 40%, 40% to 30%, 30% to 40%, 40% to 0% .
A limitation of the control signal had to be realized, because without it, it happened that the control signal became negative after a stepwise decrease of the reference signal which is
483
The ppediction hopizon has to be chosen long enough in opdep to be able to decide about the futupe tempepatupe values with confidence. A long ppediction hopizon is easy pealiziable because only one sequence has to be computed . Hoffmann (1987) pecommended p=20, which ppoved to be satisfactopy to the authops as well .
ovepshot. In ppactice one usually has a-ppiopi knowledge about the ppocess at least at the second stapt-up on of the extpudep.
-.
-_ I• .
The ppedictive stapt-up can be executed adaptively because at the vepy beginning the heating powep is switched on and the pulse-tpansfep function of the ppocess can be estimated pecupsively by the least squapes method . Expepiences have shown that the ppocess papameteps ape coppectly estimated befope 10Y. of the pefepence value is peached.
WITH
..
,
-~
~: - - - - - - -u- -;:;
i --------------~ --------------,
.-
t
...
....
tine
[mini
...
Fig . 2 : Self-tuning PID-contpol
TEMEPERATURE
Fig. 3 shows a PID-contpol without tuning the paapmeteps . The tempepatures of all zones achieve theip pefepence values fast and aperiodically.
_.
..
,
1. Compute the heating poweps of the zones accopding to the ppedictive stapt-up stpategy .
2• .
2. Check wethep thepe is a lapgep tempepatupe diffepence between the ppedicted tempepatupes as allowed. If not, then go to Step 5.
,
..
r-----------~-------------~----------_,
1 loc I ,
---------------,------ - -.------, ----- - ------ --
.- .-
3. If the tempepatupe diffepence between to neighbouping zones is to lapge then change the pefepence signal of the wapmep zones to the lowest neighbouping tempepatupe plus the allowed diffepence .
l>. ....."." ., ."'""._. _t.·. __ .• u [ %]
u •
I t .~
....
.... ~ . . . . . . . . . . . . . . ...... t. . . . . .... _
2• •
...
,
ti me
PID-contpoll Fig.3 : papameters
4 . Compute the heating poweps of the zones accopding to the ppedictive steady-state stpategy using the modified pefepence value sequence of Step 3.
wi thout
tuning
[mi ni
of
the
Fig. 4 shows the self-tuning PID-contpol based on limit cycling induced by an on-off contpollep with hystepesis . The sampling time was 2 min . As can be seen, thepe is no ovepshot .
5. The computed actual heat ing powep has to be peal ized.
.....
Continue the computations in the next sampling intepval beginning with Step 1.
The above ppedictive stpategy can be pepfopmed adapti vely if the pulse-tpansfep function of the ppocess is pecupsively updated . ppactical expepiences have shown that a papametep estimation pepfopmed till 10Y. of the pefepence value. is peached alpeady yields good ppocess papameteps .
. --- -- -.----- ------- -- ---- -----,._--y [OC]
'oo .
",
~'
- - .. - .. - -
51.'
....
" ...1 ..
~
..
"
- .. - --.--- -- --"
'M .
....
SIMULATIONS
.-
The following simulations wepe pepfopmed on the model of the extpudep . The y values show the tempepatupes and the u values the heating poweps of the zones. The zones ape mapked as follows: 1:
f --.. ----.. . . . - , --------_. _. --:, -------------~
Fop techniological peasons thepemust not be to lapge a diffepence between the tempepatupes of the zones. This can happen duping the stapt-up, because the zones have diffepent time constants. In opdep to avoid an unwanted diffepence, cape has to be taken that the tempepatupe diffepence does not excess an a-ppiopi given value . The following algopithm can be applied:
6.
-. ':::i:::-:----::::: ~~~ --~~~~~~~~~~u~ , :~~ .... - .. , · ·• ·.. ·· ·. ·.1- . ,_'. ,. _ .. _..... _ .. ___. _ .~ . _ . _ . _ . _....... _. .. .• .• ..
2• .
The ppedictive stapt-up contpol has to be changed to the ppedicti ve steady-state contpol op to a PID-contpol at a given pepcentage (ca 90Y.) of the pefepence value. PREDICTIVE START-UP CONTROL DIFFERENCE LIMITATION
1 [OC]
Fig.4: Self tuning PID-contpol expepiment
, 2 : ....... , 3 : - - - - ; 4 : -.-.-.-
2" .
151 .
time
[mini
with limits c ycle
Fig. 5 ppesents the adaptive ppedictive contpo l of the extpudep. The sampling time was 0 .3 min and the numbep of ppedictions was 20 duping the stapt-up and 4 latep on. The stapt-up contpol was
Fig . 2 shows the self-tuning PID-contpol if no a-ppiopi knowledge about the ppocess papameteps was assumed . This is the peason fop the small
484
changed to steady-state control at 95% of the refence values of the ziones . The results are very good although no a-priori parameters known were used at the start of the identification. I..
CONCLUSIONS The following self-tuning control algorithms were developed respectively examined :
r-------~------~----------------~------,
oc
1,
/r'l~':7·::-1·:·:-~·:->,~>~ ~·~·. :~'--· --:~·~-i::~~:~:·:· :':
o
PlO-control with recursive parameter estimation,
y 2 Y3: • 1- ,. -----------------
o
PlO-control with limit cycle experiment with an on-off control with hysteresis,
o
predictive steady-state control,
o
predictive start-up control,
o
predictive start-up control with limitation of the temperature difference between the neighbouring zones .
/ ' .;;
~:/
I
- - -
- - -
~
-
- - - - -
- -
.'
"
:' , /
"J i. ____
I •.
I
J -
I
~
,
________ { _ __ • _ ___
~
_ • ___
: ,j "
,; ,
:;
.-
.:1
.-
....
..
1-'' -------:.-------......------~----- --+---------< 11.'
....
....
time
,
squares
The experiences of the authors are as follows:
[min)
o
There is practically no difference between the two adaptive PlO-controllers .
0
The self-tuning PlO-control based on the limit cycling needs much less operations than that of using a recursive least squares algori thm .
o
The predictive control works well both for the start-up phase and during the stationary phase.
o
The predictive control is better than the PlO control only during the start-up, if the temperature is near the reference signal then there is no remarkable difference between the PlO and predictive controls.
o
The computation requirements of the predictive control are much higher than those for the PlO-control .
o
Only the predictive control can take the limitations of the temperature differences between the neighbouring zones into account .
Fig . 5: Adaptive predictive control The adaptive predictive control with temperat~re difference limitation is shown in Fig . 6. l~.
least
r-------~~--------------------------------__,
' _________ !. ____ _ ___ l _ _ _ _ _ _ _ _ ... _ _ _ _ _ _ _ _
,
21" .
____
11'.
.-
~
,
,
_______ _ ! __ _ _ _ __ _
,
,
J _ _ __ __ _ _ -' ___ _ ____ _
,
....
.... ...
.•• ~------~------~------4_------~------~ ZI.'
11 . '
time
Fig . 6 : Adaptive predictive control temperature difference limitation.
[min]
with
It is up to the manufacturer which algorithm he wants to implement on his extruder.
No difference between the temperatures of the zones was allowed between 10% and 90% of the final value of the temperatures. (Under 10% the heating power was always swi tched on in order to get an excitation signal for the parameter estimation . Above 90% this limitation was lifted, because the reference val ues of the zones were different and otherwise all temperatures would have become equal in steady-state. )
Two versions are proposed : o
adaptive predictive start-up with temperature difference limitation and PlO control in staedy-state,
o
self-tuning PlO control cycling experiment.
based
on
Whi le the first version satisfies the stringent requirements, the second one is cheaper to implement.
APPLICATION The simplest realizable algorithm, the self-tuning PlO-controller was implemented on an 8-bit microprocessor board on an existing extruder with 13 heating zones (nozzle included).
1 imi t most much
REFERENCES Astrom, K. J., and T. Hagglung (1984) . "Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins", Automatica, 20, 5, 645-652 . Gawthrop, P. J . , P. E. Nomikos, and L. Smith (1988). "Adaptive Temperatur Control of Industrial Processes A Comparative Study. CONTROL'88 Conference, Osxford, UK, 42-48. Hoffmann, U. (1987). "An Self-tuning On-off Adaptive Controller for Processes with Swi tching Actuators" (in German). Automatisierungstechnik, 35 , 5, 184-191 .
Fig. 7 shows the results of the tuning experiments for the nozzle and the first two heating zones . The experiments for the different zones were performed one after another. As it turned out, one cycle was sufficient to tune the controller parameter satisfactorily. The respective cycle periods are indicated in the figure .
485
temperature
130°y---------r-------~~========~========~========c=====~==~======~========~========~======~~ 2 ! I T !
1------~-----~---- ~~ I . ~
- - - -I------f------•...:....+-~===_t.::J
~'"'4=_'==.=f:-:..:;:-:;::.-=.:--=-:=;~=-_=_=_=_=_.;::.L_=_=_=.:--=-=4=--=-:"::;-'=':-
--:r;==:T]
I
-----r -----
I I
,
j
-----~------ t ------r-----1-----II
300 ~------~--------~--------4_--------~------_+--------~--------~------_+--------4_------~
40
o
120
80
160
200
240
280
320
360
400 time
tempera ture
1300r---T---"""fj::;:::===::::!:::==~1
70° Ir-------.; 50° 30° L-------~__-------+--------~--------1_
360
400
440
480
time
520
Fig. 7: Results of the tuning exper1ments, l=nozzle temperature, 2=wall temperature of the first heat1ng zone, 3=wall temperature of the second heating zone. Hoffmann, U., U. MUller, and S. Schtirmann (1983). "A Microprocessor-based Self-tuning On-off Controller for Thermal Processes". PRP-5 Automation Symposium, Antwerpen, Selgien, 79-84. Rosenberg, W. (1966). "Switching Controllers" (in German). Automat1k, 11, 6, 222-226, 7, 254-257. Zeitz, K. H. (1986). "Control with On-off and Three-level Controllers" (in German). Oldenbourg, MUnchen, FRG.
486
Copyright © IFAC Identification and System Parameter Estimation. Budapest. Hungary 1991
ADAPTIVE AND PREDICTIVE STRATEGIES FOR
EXTRUDER TEMPERATURE CONTROL R. Haber*, H. P. Jorgl**, T. Vecsei** and E. Haider·Schmid*** *Cologne Polytechnic (Fachhochschule Koln). Faculty oj Plant and Process Engineering, D-5000 Koln 21. Betzdorjer Str 2, Germany **lnstilUtejor Machine and Process Automation, Technical University ojVienna. A-I040 Vienna. Gusshausstrasse 27129. Austria ***KEBA Automatisierungselektronik. Gewerbehoj Urjahr. A-4041 Linz, Austria
Abstract. In the first part of the paper two self-tuning PID control algorithms are compared. Both algorithms are based on the estimated time delay and equivalent time constant of the process. While the first strategy uses on-line least squares parameter estimation, the second one computes the process parameters from the limit cycles induced by an on-off controller. The second part of the paper presents a predictive control which is especially suitable for the start-up phase. An extension of the algorithm assures that a prescibed temperature difference of the neighbouring zones is not exceeded during the start-up. Keywords. Adaptive control; predictive control; temperature control; extruder.
I NTRODUCTI ON
then they are much cheaper. Although the principles of adaptive control are well known the detai Is are usually kept confidential. Therefore, the objective of the authors was to develop and to examine some self-tuning control algorithms for temperature control in extruders.
The control task for the temperature zones of an extruder can be separated into two phases: the start-up and the stationary operation. A good start-up control requires, that o
there is no overshot in the temperature,
o
the final temperature is achieved fast,
o
the difference between the temperatures of the neighbouring zones is kept below a prescribed value.
In the literature three extruders are reported:
Once the temperature has achieved its steady-state value, it has to be kept constant. i.e.disturbances have to be rejected.
o
on-off control with feedback,
o
PID control and
o
predictive control.
algorithms
for
Hoffmann, MUller and SchUrmann (1983) report about the adaptive on-off controller with feedback. The parameters of the process are est imated by the recursive least squares method and the parameters of the PID-equi valent controller are set by rule of thumb . Gawthrop, Nomikos and Smith (1988) compare three adaptive algorithms. The so-called continuous-time self-tuning PID control is based on an on-line least squares estimation of the PID-paremeters. The so called Closed Loop Cycling Method relies on an on-off control. The parameters are calculated from the period of one oscillation in the operating point without any parameter estimation. The regulator parameters are designed according to the Ziegler-Nichols rules . An entirely other concept is realized by Hoffmann (1987), who used a predictive on-off regulator and calculated the cost function of the control for several steps ahead.
A self-tuning control algorithm can be of great help to the operator while putting a new extruder into ope rat ion, because each machine has its own set of parameters, and the time spent tuning the parameters manually can be saved. Furthermore, the self-tuning can be applied if the performance of the extrusion is not satisfactory as a consequence of e.g. changing of the material processed. Proposed is not a permanent adaptive control but a self-tuning algorithm which can be initialized by the operator. lhe extruders zone there is neighbouring practice, the
basic
are heated in several zones . In each a temperature sensor. Al though the zones influence each other in zones are controlled autonomously.
Several manufacturers sell PID-regulators with self-tuning possiblities . Most of them are based on the identification of the measured step response of the process . (A usual way is the graphoanalytical determination of the dead time and the time constant and the computation of the optimal regulator parameters based on rules of thumb.) The extruder producers implement the control algorithm in their control boards because
The paper presents the experiences of the authors with three different self-tuning controllers:
481
o
PID-controller estimation.
with
recursive
parameter
o
PlO-controller with process parameter calculat ion from the 1 imi t cycles caused by an on-off controller with hysteresis,
impossible. The limitation algorithm worked as follows : If the control signal would go beyond its limits, then the control signal was set to the 1 imi t and the actual control error was reset to the value which would cause the control signal to reach the limit.
requirements of the extruder control, i . e . o
it uses thumb in transient signal or
o
the limit cycle can be evaluated in any working point independent of the value of the maximal achievable temperature.
RECURSIVE LS ESTIMATION OF THE PROCESS PARAMETERS A start up of the extruder begins with a full power heat input This means that at the very beginning one does not need the parameters of either the controller or the process. On the other hand, there is large step in the heat input (100 :I.), and the step response of the temperature is well suited for the identification of the process .
The method is o
recursive least
squares
o
linear process mode I of order 3 or 4 and relative dealay time zero,
o
sampling time O. l-times the delay time of the VD-filter with forgetting factor of 0 . 99, the parameter estimation should be stopped before the temperature achieves its steady-state value (suggested is to stop at approximately 90 :I. of the steady state value).
the
evaluation of the
amplitude of the oscillations,
o
period of the oscillations and
o
relat ion of the periods of switching on to
Experiences have shown that the individual I imi t cycle experiments for the zones yield good results although the neighbouring zones influence each other .
process,
o
on
All these parameters are closely related to the delay time and the equivalent time constant of the process (Zeitz, 1986).
method,
o
based
switchi ng off .
Based on simulation and practical experiences Hoffmann (1987) recommends to perform the identification with the following parameters: o
the Chien-Hrones-Re swick rule of order to achieve a fast aperiodic for stepchanges of the reference a disturbance and
PREDICTIVE CONTROL IN STEADY-STATE With predictive control , the future temperature is computed on the prediction horizon for all possible future heating sequences. The predicted temperature sequence has to be compared wi th the reference signal and the heat ing sequence which causes the least deviation from the reference signal - usually in quadrat ic sense - has to · be chosen. Then, this heat input power has to be realized and the same computation has to be repeated in the next sampling interval .
The delay time and the time constant were computed from the simulated step response of the identified pulse-transfer function .
The long-range predictive control can be realized in an easy way because the number of the possible control sequences is restricted by the two choices (switching on or off) in each (control) sampling period. Consequently the number of the possible 1 sequences is 2P + where p is the lenght of the prediction horizon relative to the sampling time . Hoffmann (1987) recommended to choose p=6 and to take into account that computations can be further reduced if some auxiliary results are stored and are reused while computing the cost function of other sequences.
DETERMINATION OF THE PROCESS PARAMETERS FROM A LIMIT CYCLING EXPERIMENT CAUSED BY AN ON-OFF CONTROLLER Since least squares type parameter estimation algorithms need much computation efforts, attempts have been made in order to develop easier identification methods. Astrom and Hagglund (1984) design the PID-parameters according to the Ziegler-Nichols technique . The process is controlled first by an on-off controller and both the critical gain and the period of the oscillations are determined from the limit cycle .
An adaptive version of the above algorithm was used when the pulse-transfer function of the process was identified by the recursive least squares parameters estimation during the start-up period .
Gawthrop, Nomikos and Smith (1988) use a simi lar procedure which they call Closed Loop Cycling Method . The only difference to that of Astrom and Hagglund (1984) is that they do not need several oscillations but the process parameters needed are determined from one cycle .
PREDICTIVE START-UP CONTROL The aim of the start-up control is to achieve the reference value as fast as possible without any overshot . In the first sampling intervals the heat has to be switched on. Then it has to be checked when it has to be switched off the first time . Therefore only one heating sequence has to be checked :
Rosenberg (1966) designes an on-off controller with a first-order term feedback without identifying the process . Since this controller is equivalent to a continuous PD-controller, he uses the design rules of Ziegler-Nichols developed for first-order systems with delay time. The delay time and the time constant are determined from the oscillations caused by an on-off controller working with a reference signal being half of the maximal achievable temperature.
on, off, off, off, ... , off . As long as the predicted temperature is below the reference signal in the whole prediction horizon, the actual heating power is on, otherwise it shoul d be off.
The method developed by the authors is similar to the latter one but is better suitable for the
482
o
long-range predictive controller.
111
In addition, the predictive control algorithm was modified such that the temperature difference between the neighbouring zones does not exceed a prescribed value. The algorithms were simulated by means of a simulation program package, which modelled an industrial extruder. The simplest realizable algori thm, the PID-controller based on the 1 imi t cycling experiment, was realized on the real plant with succes.
1/2
MODELLING OF THE EXTRUDER
1/3 The extruder under investigation had four heating zones. The temperature of each zone was affected by the heat input to the zone and by the temperature difference between it and the neighbouring zones. In order to set up the model of the extruder the following symbols were introduced u (t);
i=l, ... 4
1
heat input to the i-th zone, (100 %
i=l, ... 4
G (s);
Fig.1: Extruder temperature zone model
full power, no heat input)
0 % Yl (t);
1/4
Al though the zones are coupled, controllers were used for the zones.
temperature of i-th zone,
i=l, ... 4 transfer function between the
11
input
heat
and
CONTINUOUS-TIME PID-CONTROLLER
the
As the objective was to extend a traditional cont inuous-t ime PID-control algorithm wi th a self-tuning possibility, the parameters of the continuous-time PID-algorithm were calculated . The recursive discrete-time algorithm was used in the digital simulations was:
temperature of the i-the zone, i=l, ... 4
transfer function between the temperature i-th
and
difference j-th
zone
of
the
and
the
autonomous
temperature of the i-th zone. The model of the extruder can be set up with the following equations:
where u(k) is the control signal and e(k) the control error . The static gain (K p )' derivative
y (s)
related to
(To) 1
and
integrating
(T
1
)
the parameters ql
time
constants
as follows
are
(t.T
=
sampling time): y (s)
G (S)U (S) + [Yl (s)-Y2(s) lG (S) + 2 22 12
2
+
y (s)
[Y3(s)-Y2(s)lG
32
(s)
G (S)U (S) + [Y2(s)-Y3(S) lG (s) + 3 33 23
3
+
Fig. 1
[Y4 (S)-Y3(s) lG
shows
the
43
q
(s)
model
The transfer functions were experiments:
of
the
from
two
1. separate heating of the zones,
In both cases the the heat input to the individual zones were changed stepwise
0 0 0
from from from from
p
0
Extruders have the specific feature that the the heating power can only be swithed on or off. Therefore the control signal of a PID-algorithm has to be transformed into a ratio, the value of which represents the portion of the sampling time when the full heating power is on.
2. simultaneous heating of the zones.
0
=KT/l1T.
The parameters of the controller were calculated according the rules of thumb by Chi en, Hrones und Reswick. The parameters were selected for a stepchange in the reference value aiming at a fast aperiodic transient behaviour. The process parameters used are the dead time and the equi valent time constant, which were updated in order to get a self-tuning control algorithm.
extruder
identified
o
0% to 40%, 40% to 30%, 30% to 40%, 40% to 0% .
A limitation of the control signal had to be realized, because without it, it happened that the control signal became negative after a stepwise decrease of the reference signal which is
483
The ppediction hopizon has to be chosen long enough in opdep to be able to decide about the futupe tempepatupe values with confidence. A long ppediction hopizon is easy pealiziable because only one sequence has to be computed . Hoffmann (1987) pecommended p=20, which ppoved to be satisfactopy to the authops as well .
ovepshot. In ppactice one usually has a-ppiopi knowledge about the ppocess at least at the second stapt-up on of the extpudep.
-.
-_ I• .
The ppedictive stapt-up can be executed adaptively because at the vepy beginning the heating powep is switched on and the pulse-tpansfep function of the ppocess can be estimated pecupsively by the least squapes method . Expepiences have shown that the ppocess papameteps ape coppectly estimated befope 10Y. of the pefepence value is peached.
WITH
..
,
-~
~: - - - - - - -u- -;:;
i --------------~ --------------,
.-
t
...
....
tine
[mini
...
Fig . 2 : Self-tuning PID-contpol
TEMEPERATURE
Fig. 3 shows a PID-contpol without tuning the paapmeteps . The tempepatures of all zones achieve theip pefepence values fast and aperiodically.
_.
..
,
1. Compute the heating poweps of the zones accopding to the ppedictive stapt-up stpategy .
2• .
2. Check wethep thepe is a lapgep tempepatupe diffepence between the ppedicted tempepatupes as allowed. If not, then go to Step 5.
,
..
r-----------~-------------~----------_,
1 loc I ,
---------------,------ - -.------, ----- - ------ --
.- .-
3. If the tempepatupe diffepence between to neighbouping zones is to lapge then change the pefepence signal of the wapmep zones to the lowest neighbouping tempepatupe plus the allowed diffepence .
l>. ....."." ., ."'""._. _t.·. __ .• u [ %]
u •
I t .~
....
.... ~ . . . . . . . . . . . . . . ...... t. . . . . .... _
2• •
...
,
ti me
PID-contpoll Fig.3 : papameters
4 . Compute the heating poweps of the zones accopding to the ppedictive steady-state stpategy using the modified pefepence value sequence of Step 3.
wi thout
tuning
[mi ni
of
the
Fig. 4 shows the self-tuning PID-contpol based on limit cycling induced by an on-off contpollep with hystepesis . The sampling time was 2 min . As can be seen, thepe is no ovepshot .
5. The computed actual heat ing powep has to be peal ized.
.....
Continue the computations in the next sampling intepval beginning with Step 1.
The above ppedictive stpategy can be pepfopmed adapti vely if the pulse-tpansfep function of the ppocess is pecupsively updated . ppactical expepiences have shown that a papametep estimation pepfopmed till 10Y. of the pefepence value. is peached alpeady yields good ppocess papameteps .
. --- -- -.----- ------- -- ---- -----,._--y [OC]
'oo .
",
~'
- - .. - .. - -
51.'
....
" ...1 ..
~
..
"
- .. - --.--- -- --"
'M .
....
SIMULATIONS
.-
The following simulations wepe pepfopmed on the model of the extpudep . The y values show the tempepatupes and the u values the heating poweps of the zones. The zones ape mapked as follows: 1:
f --.. ----.. . . . - , --------_. _. --:, -------------~
Fop techniological peasons thepemust not be to lapge a diffepence between the tempepatupes of the zones. This can happen duping the stapt-up, because the zones have diffepent time constants. In opdep to avoid an unwanted diffepence, cape has to be taken that the tempepatupe diffepence does not excess an a-ppiopi given value . The following algopithm can be applied:
6.
-. ':::i:::-:----::::: ~~~ --~~~~~~~~~~u~ , :~~ .... - .. , · ·• ·.. ·· ·. ·.1- . ,_'. ,. _ .. _..... _ .. ___. _ .~ . _ . _ . _ . _....... _. .. .• .• ..
2• .
The ppedictive stapt-up contpol has to be changed to the ppedicti ve steady-state contpol op to a PID-contpol at a given pepcentage (ca 90Y.) of the pefepence value. PREDICTIVE START-UP CONTROL DIFFERENCE LIMITATION
1 [OC]
Fig.4: Self tuning PID-contpol expepiment
, 2 : ....... , 3 : - - - - ; 4 : -.-.-.-
2" .
151 .
time
[mini
with limits c ycle
Fig. 5 ppesents the adaptive ppedictive contpo l of the extpudep. The sampling time was 0 .3 min and the numbep of ppedictions was 20 duping the stapt-up and 4 latep on. The stapt-up contpol was
Fig . 2 shows the self-tuning PID-contpol if no a-ppiopi knowledge about the ppocess papameteps was assumed . This is the peason fop the small
484
changed to steady-state control at 95% of the refence values of the ziones . The results are very good although no a-priori parameters known were used at the start of the identification. I..
CONCLUSIONS The following self-tuning control algorithms were developed respectively examined :
r-------~------~----------------~------,
oc
1,
/r'l~':7·::-1·:·:-~·:->,~>~ ~·~·. :~'--· --:~·~-i::~~:~:·:· :':
o
PlO-control with recursive parameter estimation,
y 2 Y3: • 1- ,. -----------------
o
PlO-control with limit cycle experiment with an on-off control with hysteresis,
o
predictive steady-state control,
o
predictive start-up control,
o
predictive start-up control with limitation of the temperature difference between the neighbouring zones .
/ ' .;;
~:/
I
- - -
- - -
~
-
- - - - -
- -
.'
"
:' , /
"J i. ____
I •.
I
J -
I
~
,
________ { _ __ • _ ___
~
_ • ___
: ,j "
,; ,
:;
.-
.:1
.-
....
..
1-'' -------:.-------......------~----- --+---------< 11.'
....
....
time
,
squares
The experiences of the authors are as follows:
[min)
o
There is practically no difference between the two adaptive PlO-controllers .
0
The self-tuning PlO-control based on the limit cycling needs much less operations than that of using a recursive least squares algori thm .
o
The predictive control works well both for the start-up phase and during the stationary phase.
o
The predictive control is better than the PlO control only during the start-up, if the temperature is near the reference signal then there is no remarkable difference between the PlO and predictive controls.
o
The computation requirements of the predictive control are much higher than those for the PlO-control .
o
Only the predictive control can take the limitations of the temperature differences between the neighbouring zones into account .
Fig . 5: Adaptive predictive control The adaptive predictive control with temperat~re difference limitation is shown in Fig . 6. l~.
least
r-------~~--------------------------------__,
' _________ !. ____ _ ___ l _ _ _ _ _ _ _ _ ... _ _ _ _ _ _ _ _
,
21" .
____
11'.
.-
~
,
,
_______ _ ! __ _ _ _ __ _
,
,
J _ _ __ __ _ _ -' ___ _ ____ _
,
....
.... ...
.•• ~------~------~------4_------~------~ ZI.'
11 . '
time
Fig . 6 : Adaptive predictive control temperature difference limitation.
[min]
with
It is up to the manufacturer which algorithm he wants to implement on his extruder.
No difference between the temperatures of the zones was allowed between 10% and 90% of the final value of the temperatures. (Under 10% the heating power was always swi tched on in order to get an excitation signal for the parameter estimation . Above 90% this limitation was lifted, because the reference val ues of the zones were different and otherwise all temperatures would have become equal in steady-state. )
Two versions are proposed : o
adaptive predictive start-up with temperature difference limitation and PlO control in staedy-state,
o
self-tuning PlO control cycling experiment.
based
on
Whi le the first version satisfies the stringent requirements, the second one is cheaper to implement.
APPLICATION The simplest realizable algorithm, the self-tuning PlO-controller was implemented on an 8-bit microprocessor board on an existing extruder with 13 heating zones (nozzle included).
1 imi t most much
REFERENCES Astrom, K. J., and T. Hagglung (1984) . "Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins", Automatica, 20, 5, 645-652 . Gawthrop, P. J . , P. E. Nomikos, and L. Smith (1988). "Adaptive Temperatur Control of Industrial Processes A Comparative Study. CONTROL'88 Conference, Osxford, UK, 42-48. Hoffmann, U. (1987). "An Self-tuning On-off Adaptive Controller for Processes with Swi tching Actuators" (in German). Automatisierungstechnik, 35 , 5, 184-191 .
Fig. 7 shows the results of the tuning experiments for the nozzle and the first two heating zones . The experiments for the different zones were performed one after another. As it turned out, one cycle was sufficient to tune the controller parameter satisfactorily. The respective cycle periods are indicated in the figure .
485
temperature
130°y---------r-------~~========~========~========c=====~==~======~========~========~======~~ 2 ! I T !
1------~-----~---- ~~ I . ~
- - - -I------f------•...:....+-~===_t.::J
~'"'4=_'==.=f:-:..:;:-:;::.-=.:--=-:=;~=-_=_=_=_=_.;::.L_=_=_=.:--=-=4=--=-:"::;-'=':-
--:r;==:T]
I
-----r -----
I I
,
j
-----~------ t ------r-----1-----II
300 ~------~--------~--------4_--------~------_+--------~--------~------_+--------4_------~
40
o
120
80
160
200
240
280
320
360
400 time
tempera ture
1300r---T---"""fj::;:::===::::!:::==~1
70° Ir-------.; 50° 30° L-------~__-------+--------~--------1_
360
400
440
480
time
520
Fig. 7: Results of the tuning exper1ments, l=nozzle temperature, 2=wall temperature of the first heat1ng zone, 3=wall temperature of the second heating zone. Hoffmann, U., U. MUller, and S. Schtirmann (1983). "A Microprocessor-based Self-tuning On-off Controller for Thermal Processes". PRP-5 Automation Symposium, Antwerpen, Selgien, 79-84. Rosenberg, W. (1966). "Switching Controllers" (in German). Automat1k, 11, 6, 222-226, 7, 254-257. Zeitz, K. H. (1986). "Control with On-off and Three-level Controllers" (in German). Oldenbourg, MUnchen, FRG.
486