Note: Descriptions are shown in the official language in which they were submitted.
,r ~
Thc present invention relates to the tuning of a regulator of the
l''ll)-type for a process and more exactly defines a method and an
apl)arutus for Lringing - as a step in the mcthod of tunintJ the re-
~Julcltor - the process into a controlled self oscillation f`or deter-
mil-~intJ quantities whicl1 are essential for the tuning oF tht-' regula-
tnr. 'Ihe invention includes all variations and combinations (P,
Pl, PD, PID etc) Or the control functions of a PID-regulator.
Ihe PID-regulator is very common for the control of industrial
processes and provides for proportional, integrating and derivative
control. A process of larger scope employs a large number of such
regulators. PID-regulators are manufactured in large series as stan-
dard products. It is more and more common that the regulators arebased on microcompoters, and then more complicated control func-
tions can be used.
Even if the regulator is based on a microcomputer the principal
structure of a conventional PID-regulator is maintained since
persons in the industry skilled in the art have a long and expe-
rienced knowledge about and a feeling for the tuning of such PID-
regulators.
X
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1~IC~ C1~ r~ J~ O~S~ C.~J. ~ tl~od of Zic(~ arl(l
Nichols, ror thl mclllu.ll tur~ing of a PlD-rcgulltor in dcpen(Jcrlce of
the l~arallle~ers of the process. In spite of this many reytllatorc~ in
in~lu;trial processes aIe badly tunr!d in practise. rhis is ctue to
5 on one hancl the Fact that the manual tuning ~Yhich comprises manually
c~anging tl1e regulator amplification is tedious, on the othcr hand
the f`act that the parameters/properties of the process Elre changed
in course o r time.
rhere is also equipment for automatic tuning of PID-regulators but
such equiplnent is expensive and not quite simple to use. Moreover,
tl~ere are adaptive regulators but such regulators are much more
complicated than a simple PID-regulator and have not yet been used
at a large scale.
Thus, there is a need for a simple method of automatic tuning of a
PlD-regulator which method results in a non-expensive regulator. The
mrthod should be that simple that it can be applied on PID-regula-
tors realized by means of a microcomputer only by making a simple
chclnge of, or a mir~or addition to the program of the regulator.
The present invention provides a simple method of
tuning a PID-regulator and as a step thereof to provide a method
and an apparatus for bringing the system including the PID-regulator
into controlled self oscillation. Wl1en the system oscillates,
quantities of the process which are essential for the tuning can be
measured.
According to the present invention, there is provided a method where
the process and the regulator in comm~n have a transfer function G(s)
in a feed-back system and the system is ~L~uyllL into controlled self
oscillation for measuring the amplitude and the frequency of said
oscillation whereupon the regulator is tuned in dependence oF the
values measured for the amplitude and the frequency of said oscilla-
t:ion. In accordance with the invention the signal fed to the
X
5~
rrgu~all)r is sul)jcctc(J to thc e~`rcct Or a circuit. I`lJncLion (NL )
l1.1V; ~19 O non-linear cbarE~cteristic and havir)q a describirl~ f`~ c-
tinn N(~), A relatinn G (i~ N(A) = -1 is valid for at .least ooe
value of the anyular fre(luency ;and the amplitu(le A of saicl signal.
1he method facilitates simple automation of the tuning o~ l'ID-
re~ulators, particularly regulators based on a microcomputcr.
Brief description of` tl1c drawings
~he invention is described in greater detail below and with
reference to the ~ m~nyiny drawings in which
Fig. 1 is a block diagram of one embodiment illustrating the control
mcmbers of a PID-regulator as separate units;
rig. 2 is a diagram in the complex plane and illustrates tlie trans-
fer function of a process as a Nyquist curve, and shows the nega-
tive inverse of the so called describing function of a non linear
circuit function having an ideal relay characteristicJ
Fig, 3 is a block diagram showing the invention realized by means of
a regulator based on a microcomputerJ
Fig, 4 is a diagram of the same kind as Fig, 2 but in addition to
the Nyquist curve of a transfer function also shows the describing
function of a circuit function having an ideal relay characteristic
and a hysteresis~
Fig, 5 is a diagram defining the phase margin of a transfer func-
tion of a process; and
Fig, 6 is a diagram showing the bias of a non-linear circuit func-
tion to a predetermined working point.
~i5~
1l-e followinrJ dcscriptior1 of the invention incl~des all variatiorls
and combil13tior)s oF tl-le control functions of a PII)-regulator. For
instance the derivative control function of a regulator can be
omitted and only the P- and I- control functions be used.
First a prior art system is described for facilitating understandirlg
of the inven~ion. In Pig. 1 a block diagram shows a prior art system
based UpOI1 analog technique and prcvided with arl apparatus of the
invention for bringing the system into self oscillation.
A process 1 illustrated by means of its transfer function ll(i) is
controlled by means of a PI~-regulator 2 in respect of a Frocess
variable. The actual value y of the variable is obtained on an out-
put from the process 1 and is fed back over a negative feed-back
loop 3 to a sunlming ~unction 4 and there is combirled with a refe-
rence value Yre f for generating an error signal e which issupplied to the regulator 2.
Generally the following relationship holds between the error signal
e and the control signal u of the regulator:
u - k(e ~ T J e (t) dt * TD dt
where k TI and TD arè constants.
The regulator 2 is shown to include separate control function units
P I and D for analog control but can as shown below also be built
up by means of a microcomputer. Moreover switches 5 are shown for
the connection/disconnection of the P- I- and D-control functions
as well as by pass. The switches 5 are individually controlled by
mPans of a suitable control unit 6.
The transfer function of the regulator 2 combined with the process 1
;s designated G(s).
~2~5~
ior tul1irl(J tbe regulator by means o~ t~c prior arl metllod oS
Ziegler and Nichols the system is brough- into controlled sc~f
oscillation in that, at the same time as the integrating and deri-
vative units (I and D) of the regulator are disconnected the anlpli-
fication of the proportional control function unit P is increasedup to self oscillation by manually moving an adjusting mcans 9p.
Naintaininc3 the system in this state, the amplitudc and frequcrlcy
of the sLlf oscillationare determined by measuring by means of a
measuring unit 10 the system output siqnal v. The quantity values
resulting from said measuring are used for calculating the parameters
1<, TI and TD which are adjusted by means of the adjusting means
9p, 9i and 9d of the control function units P-, l- and D,respective-
ly. The parameters of the PID-regulator 2 are calculated and fixed
according to given formulas in the table below:
5 Regulator Amplification lntegration Derivative
(k) Time (TI) Time (TD)
P 0~5 kc
PI 0-4 kc û~8 Tc
PID U~6 kc 0~5 Tc 0~12 Tc
where k is the critical amplification, i.e. the amplification
of the system in self oscillation, and Tc is the period of time of
the self oscillation. The critical amplification obtained from the
measured quantity values in a known manner.
The method of Ziegler and Nichols for the tuning of a PlD-regulator
is a thumb rule based upon parameters of the Nyquist cur~e in the
complex plane, when this curve passes through the point(-1;0).
According to the Nyquist theorem a process is stable if the Nyquist
curve does not encircle the point (-1;0). The diagram of ~ig. 2
illustrates a Nyquist curve G(iu~ ) for positive values of the
angular frequency ~ .
In order to secure that the self oscillation occurs irrespecitve of
5~
small non-linearities, as a dead ~one and/or hysteresis, of thc
system thc input signal Yref can be subjected to a small disLurbance.
So far the feed-back system and the tuning method as described are
previously known.
Instead of the above mentioned method for determining the amplitudc
and the frequency of the self oscillation, there is according to the
invention introduced in series to and before the process 1 a non-
linear circuit 7 which has a describing function N(A) defined below.
Thus,a non-linear circuit function NL is introduced into the signal
path of the regulator 2 for processing the error signal e before
this signal is supplied to the process 1. This is illustrated in
Fig. 1 by means of a switch 8 which connects the circuit 7.
Said non-linear circuit function NL has a relay characteristic
which means that the output from the circuit 7 has a first low value
when the input e of the circuit is below a predetermined value and
has a second high value when the input signal exceeds said pre-
determined value. Thus, the output signal oscillates between two
values, e.g. the amplitudes +d and -d. Such a circuit can be reali-
zed by means of a simple comparator having a large internal ampli-
fication.
Although an ideal relay characteristic, i.e. right angled transitions,is preferred and is easily realized in a PID-regulator based on a
microcomputer, the invention operates also for less well defined
relay characteristics having a slope and/or overshoots.
A non-linear circuit function can be represented by a describing
function N(A), which is defined as the transfer function of the
circuit function when the input signal is a sine signal A sin (~ t3,
where A is the amplitude, uJ the angular frequency and t the time.
For bringing the system of Fig. 1 with the non-linear circuit
`` ~%~5~
lunctinn NL introduccd thcrein into ;el~ o;ci~laLioll ihe fol~u~!irlg
e4uation s~ e valid for at lcast one V~ UC' of the parametcrs A
and ul :
G (i ~ N(A) = -1
5 or
G (i ~
In the diagram of Fig. 2 the two functions G (i ~J ) and - ~ are
drawn in the complex plane. The amplitude and frequency of the self
oscillation are obtained from the parameter values in the crossing
point p of the depicited curves. By determining the amplitude and
frequency of the self oscillation the value of the transfer function
G (i ~ ) of the control system (including the PlD-regulator) in the
actual crossing point p can be determined and this information can
then be used for tuning the regulator.
A non-linear curcuit function NL having an ideal relay characteristic
has a describing function N(A) = 1Td where A is the amplitude of
the circuit function input signal e and d is the amplitude of the
output signal. The negative in~erse ~ N(A) of the describing
function becomes, drawn in the complex plane a straight line which
coincides with the negative real axis -Re.
In a non-linear circuit having a relay characteristic the Ziegler
and Nichols method is will suited for tuning a PID-regulator.
~Ihen the non-linear circuit 7 with the relay characteristic is con-
nected and the PID-regulator is entirely disconnected, i.e. by
passed, the system is brought into self oscillation. Possibly the
proportional unit P of the regulator can be connected for limiting
the amplitude of the oscillation. The amplitude A of the self
oscillation, being a measure of the crossing point p of the trans-
fer function G (i ~J ) with the negative real axis -Re, is
, ~
5~
dctermincd by mcasuring thc signal y aftcr the proccs-; b~ mcans o~`
the measuring unit 10. With a knowledge of this point, i.e. the
amplitude A, and the relay characteristics (the value d) of the~
non-linear circuit, the critical àmplification kc nf the system
can be calculated in accordance with the equation kc = ~rA
~loreover, the pcriod time Tc of the self oscillation i5 determined
t)y measurcment.
According to the formulas of Ziegler and Nichols the amplification,
integration time and derivation time are thereafter calculated,
and then the regulator is tuned in dependence of said calculated
parameters.
In this connection it should be mentioned that not only the P-unit
can be connected in the course of the oscillation and measuring.
hlso the I- and D-units can be connected individually or in combi-
nation - also with the P-unit. This is particular so if another
point on the Nyquist curve than the crossing point with the negative
real axis is to be identified. Reference is made to "ZiPgler Nichols
Auto-Tuners" by Karl Oohan Astrom, Department of Automatic, Lund
Institute of Technology, May 1982.
The above method can be performed manually or automatically in
dependence of how the regulator 2 and the non-linear circuit func-
tion NL is implemented.
The invention obviates the problem caused by small non-linearities
in the system which may obstruct self oscillation, since the intro-
duced non-linear circuit function NL largely eclipse any small non-
linearity.
The PID-regulators of today are usually built on the basis of a
microcomputer and Fig. 3 in a block diagram shows the system of
fig. 1 implemented with a regulator comprising a microcomputer.
On its input the microcomputer has an A/D-converter 11 and on its
~/
s~
cJ
ou~pu~ a D/A-converter 12. I~lorcover, ~l~erc is a micropr~cessor 1~,
a programable read only memory 14 (PROI~l)sorving as a program storage
14 and a random access melnory 15 (RA~) for buffering data. The
buffcr memory 15 has input and output registers as well as a clock
for generating output signals as pulses to the D/A-converter 12.
The units 13-15 of thc microcomputer are combined to cooperate in a
l;nnwn manner. The control functions for P-, I- and D-rcgulation are
stored in the program memory 14 together with any other soft ware
required by the microcomputer for its operation.
1û The analogously operating control function units shown in Fig. 1
as circuits can be illustrated by means of the circuit functions
k e for the proportional unit P, k/TI~J edt for the integrating
unit I and k ~TD ddt for the derivative unit D. In the embodiment
according to Fig. 3 these circuit functions are stored in the
program storage 14 asalgorithms for acting upon the regulator
input signal or error signal e or more specifically measured values
thereof in order to generate at the output of the regulator a control
signal u which is supplied to the process. Like the embodiment of
Fig. 1 the reference value Yref and the process actual value or
measured variable is y.
This known PID-regulator is tuned by means of not shown adjusting
means in that only the proportional control is involved, whereupon
the amplification is manually increased until self oscillation is
obtained. The amplification and the period of oscillation of the
self oscillation are measured and used for the calculation and
adjustment of the regulator parameters according to the formulas
of Ziegler and Nichols.
In order to bring the system into self oscillation for the purpose
of determining the amplitude and frequency of the self oscillation
there is, in accordance with the invention, introduced a circuit
function NL having a non-linear cl-aracteristic for processing the
regulator signal. This circuit function NL is implemented in the
5~
l(i
miCI`OCOlllpUtCr as a further a~(lnritllln and al;o coml)lics ~/ith thc
previously mentioned requirement f`or self oscillation. ~ilUs, for i~s
describing function N(A) it holds that G (i ~J ) N(A) = -1, where
G(s) does not include NL wl-ich is therefore shown within bracl<ets
in Fig. 3.
l~hen the PID-regulator is to be tuned, the sys~em for determining
the measured quantities of amplitude and frequency of the self
oscillation - is brought into self oscillation in that the non-
linear circuit function NL is introduced into the signal path of
the regulator signal,i.e. the error signal e, or more exactly
measured values of the input signal e to the regulator said values
being established by means of the microcomputer. Thus, the input
signal e to the regulator is processed by means of the non-linear
circuit function NL. The amplitude and the frequency of the self
oscillation are then determined in a suitable manner by measuring
nn the output signal y.
The measuring of the amplitude and frequency of said oscillation
is n~ part of the invention but any suitable method of measurement
can be used. For measurinq the amplitude three methods are mentioned:
1) The amplitude of consecutive nscillations is measured and the
amplitude value is accepted when the next amplitude value
differs less than a predetermined amount, e.g. 3 O of the
amplitude;
2) The method of recursive least squares identification is used;
3) Kalman filter is used.
The frequency can also be determined in several ways, three being
mentioned here:
1) The simplest procedure is to measure the time between consecu-
tive zero crossings of the oscillation;
~ ?) The method of recursive least squares can be used;
53~
1 1
3) A so call~ expanded ~alm!n filtcr call ~o uscd, ~Ihicl~ facilitaLcs
determination of both amplitude and freclucrlcy from the same
filter.
The block diagram of Fig. 3 illustrates the operation of the in-
vcntion. In practisc however, the error signal e is generated in~thcrcgulator itsclf` and so the fed back signcll -y can bc supplied to the
microprocessor 13 over a further A/D-converter. However, generally
a multiplexer is used on the regulator input before the A/D-con-
verter 11. These latter embodiments also facilitate measurements on
1û the output signal y for determining the amplitude and frequency
of the self oscillation.
By taking advantage of a non-linear circuit function NL, having a
relay characteristic, one application for tuning a PID-regulator
has been described. According to another application a PID-regulator
can be tuned to give a process system a desired phase margin. In
Fig. 5 the phase margin ~ m of a transfer function G~s) is shown.
This application is particularly appropriate if the non-linear
circuit function has a relay characteristic, preferably an ideal
characteristic with hysteresis. A circuit function having an ideal
relay characteristic and hysteresis processes an input signal in
such a way that the input signal when it decreases below a first
value -H results in a low output signal -d and when it increases
beyond a second value H, larger than said first value, results in
a high output signal +d. The output signal always is a square wave
signal. The value H is a measure on the hysteresis. It is realized
that the amplitude A of the input signal must exceed the hysteresis
H for correct operation.
The describing function N (A) of a circuit function having an ideal
relay characteristic and hysteresis is:
N (A) = ~rd . e i 0; 0 = arcsin H; A ~ H
~/hcre A like before is the amplitude of the input signal of the
non-lirlear circuit, d is tl~ a~)litudc o~ the outpuL siyna~ from
thc non-linear circuit, H is a measure on the hystcrcsis and 0
is a measure on the time delay bctween the input and the output.
The negative inverse of the describing function can he shown to
be:
A2 _ 112 ~ H
N'(A) 4d 4d
Since the imaginary member is independent of the amplitude A the
curvc of - ~7~ in the complex plane becomes a straight line
parallell to the negative real axis; cfr. Fig. 4.
ln the feed back system of Figs. 1 and 3 self oscillation will
occur if the curves of G(i ~)) and -1~N'(A) crosses as shown in
Fig. 4. Since the amplitude and frequency of the self oscillation
are obtained from the parameters of the curves at the crossing
point p, the transfer function C(i ~) can be determined at the
frequency of the self oscillation.
Thus, when a circuit function having a relay characteristic and
hysteresis is introduced into the signal path of the PID-regulator
self oscillation is caused to occur. By measuring the amplitude
and frequency of the self oscillation a desired phase margin of
the control system in question can be set. Reference is made to
"A PID Tuner based on Phase Margin Specification" by Tore Hagg-
lund, Department of Automatic Control, Lund Institute of Techno-
logy, Sept 1981.
Two embodiments which entails the introduction of a circuit
function of ideal relay characteristic have been disclosed for
the determination of parameters and the subsequent tuning o-F
a PID-regulator. The method of the invention is simple and can
be incorporated as a few program steps in a microcomputer. The
3û method can also be performe'd manually or entirely automatic. The
method entails interference into the normal control of a process
and therefore is performed intermittently. A program clork can
~2~5~
iniLia~e tuning of the l'l~-rcgulator at prcdclcrmincd in~eIvaIs
such as once every twenty-rour hours or once a wee~.
According to a requirement mentioned abovc for the describing
function of the non-linear circuit function NL the input signal of
thc describing function should be a sine signal. ûn the other hand
the output siynal of` said describing function is a square wave
signal. However, in most cases the transfer function of a process
is a low pass filter which results in that the process output signal
y which is fed back to the input of the regulator i8 filtered and
1û essentially only includes the fundamental frequency, i.e. harmonics
are filtered out.
Experiments have shown that processes having a relatively simple
or "good" transfer function which normally are controlled by means
of a conventional PID-regulator very well comply with the above
concept. Since the purpose of the invention is tn provide a simple
tuning method for use in simple PID-regulators the approximation
made is of a small significance.
In reallity the describing function of the non-linear circuit func-
tion holds also for input signals which differ considerably from the
2û sine shape. However, the input signal must be fairly symmetric. In
order to secure symmetry the non-linear circuit function is biased
to a suitable working point as shown in Fig. 6. A desired output
signal Yd corresponds to an input signal udes. The input signal
ude~ can be determined as that input signal for which the output
signal from the non-linear circuit function with an ideal relay
characteristic is symmetric. In its turn this can be determined by
measuring the positive and negative time periods T and T of the
output square wave signal resulting from the non-linear circuit
function NL. By means of successive measurements with different
input signals udes can be bstimated by interpolation. It is appre-
ciated that the parameters of the non-linear circuit function can
be chosen in different ways. It can be desirable to fix certain
5~L
1~
~ar~mc~crs while oti)cr paranl~ters are free ~o be cl~osen.
Illc invention is not limited to tlle rmbodimcnts describcd but can
bc modified within the scope of tlle pertaining claims.
