CA1201511A - Method and an apparatus in tuning a pid-regulator - Google Patents

Abstract

Abstract In tuning a regulator of the PID-type of a process in a feed back system where the process and the regulator has a transfer function G(s) in common, a method is provided for bringing the system into self oscillation for measuring the amplitude and fre-quency of the oscillation and tuning the regulator in dependence of the measurements obtained. A circuit function (NL) having non-linear characteristic and a describing function N(A) is intro-duced into the system in series to the process for acting on the regulator signal (e). Self oscillation is obtained if G(i .omega. )? N(A) = -1 for at least one value of the angular fre-quency .omega. and the amplitude A of the regulator signal (e). An apparatus for performing the method is disclosed.

Description

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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.
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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

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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.

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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).

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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 , ~

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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 ~/
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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;

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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

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~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.

Claims (10)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. In tuning a regulator of the PID-type of a pro-cess in a feed-back system where the process and the regula-tor have a transfer function G(s) in common, a method of bringing the system into self oscillation and whereupon the amplitude and the frequency of said oscillation and where-upon the amplitude and the frequency of said oscillation are determined and the regulator is tuned in dependence of the values determined for the amplitude and the frequency of the oscillation in which for achieving the self oscilla-tion a signal (e) fed to the regulator is subjected to the effect of a circuit function (NL) having a non-linear characteristic and having a describing function of n(A) such that G(i.omega.) N(A)=-l for at least one value of the angular frequency .omega. and the amplitude A of said signal and that the amplitude and the frequency of said oscillation are determined when the signal is subjected to the circuit function, whereupon the circuit function is removed.

2. A method as claimed in claim 1, in which said signal (e) fed to the regulator is subjected to the effect of a circuit function (NL) having a relay characteristic.

3. A method as claimed in claim 1, in which said signal (e) fed to the regulator is subjected to the effect of a circuit function (NL) having a relay characteristic and hysteresis.

4. In tuning a regulator of the PID-type of a pro-cess in a feed-back system where the process and the regu-lator have a transfer function G(s) in common, a method of bringing the system into self oscillation and whereupon the amplitude and the frequency of said oscillation are deter-mined and the regulator is tuned in dependence of the values determined for the amplitude and the frequency of the oscillation, in which for achieving the self oscillation a circuit function (NL) having a non-linear characteristic is introduced in series to the process, and said circuit function (NL) has a describing function n(A) such that G(i.omega.) N(A)= -1 for at least one value of the angular frequency .omega. and the amplitude of an input signal and that the amplitude and the frequency of said oscillation are determined when the signal is subjected to the circuit function, whereupon the circuit function is removed.

5. Means for bringing, in tuning a regulatory of the PID-type of a process in a feed-back system with a transfer function G (s) in common for the regulator and the process, the system into self oscillation for the purpose of measuring the amplitude and the frequency of said oscillation, including a circuit function (NL) with a non-linear characteristic and introduceable in series to the process, the circuit function (NL) having a describing function N(A) such that G(i.omega.) N(A= -1 for at least one value of the angular frequency .omega. and the amplitude A of an input signal.

6. Means as claimed in claim 5, in which the cir-cuit function (NL) has a relay characteristic.

7. Means as claimed in claim 5, in which the cir-cuit function (NL) has a relay characteristic and hysteresis.

8. Means as claimed in claim 5, 6 or 7, in which the circuit function (NL) is realized by an electrical cir-cuit and that a switch is provided for connecting the electrical circuit to the regulator.

9. Means as claimed in claim 5, 6 or 7, wherein the regulator comprises a microcomputer in which the con-trol functions of the regulator are realized by means of algorithms, in which the circuit function (NL) is realized by an algorithm in the microcomputer.

10. Means as claimed in claim 5, 6 or 7, in which the circuit function (NL) is biased to a predeter-mined working point of the process.