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A New Optimization Method of PID Control Parameters for Automatic Tuning by Process Computer
A NEW OPTIMIZATION METHOD OF PID CONTROL PARAMETERS FOR AUTOMATIC TUNING BY PROCESS COMPUTER T. Ohta*, H. Tanaka*, K. Tanaka*, N. Sannomiya** and Y. Nishikawa** *Fuji Facom Corporation, Kawasakl~ Japan **Department of Electrical Engineering, Kyoto University, Kyoto, Japan Abstract. The method for the optimization of PID control parameters is proposed to aid the tuning operation o~ fie~d engineera. The control system is restricted to the single-input single-output process, but may have time delay and non-self-regulation property. For the parameter estimation of the process, the characteristic areas are defined. In order to optimize control parameters, a new performance index is defined in the form of the weighted integral of squared error. The closed loop response obtained by the optimal settings has the desired damping factor and the stability margin. The present algorithm is implemented on a DDC system. The field tests carried out by using this system assure us that the method is applicable for the real processes. Keywords. PID control ; parameter optimization, weighted ISE ; parameter estimation ; ultimate sensitivity ; stability ; direct digital control.
1.
Fig. 1. Both self-regulation process and nonself-regulation process expressed by the following transfer functions are treated.
INTRODUCTION
DDC systems have been widely prevailed in Japan and the same PID control actions as analog controllers are adopted in them . In this paper, we develop the method for automatic tuning of FID control parameters and implement the procedure on a small process computer in order to aid the tuning operation due to the field engineers (Nishikawa 1978). We call this method of automatic tuning "auto-tuning" for short.
[self-regulation process] -Ls Ke Gp(s) = -n~-'---'---11 j = 1 (l+T j s)
(2.1)
[non-self-regulation process] Gp (s)
When we want to set the control parameters of the real process, the margin of the stability of the closed loop response must be considered. The margin guarantees us the stability in the case of the process parameter change and gives compensation for the error of the process parameter identification. The main theoretical part of this method is to search the optimal PID parameters with apprcpriate margin . For this purpose, we introduce, as a new perform2nce index, a weighted intergral of the squared error of the process output.
=
-Ls e - - - n -- - - Ts . 11 =l (l+TjS)
(2.2)
j
The following two forms of PID control equation are considered; namely, parallel form and series form. [Parallel form] Gc(s)
Kc(l+
G~(s)
1
1 Ti s
+ Td s )
[Series form] 1 Gc(s) = Kc(l+ - ) Td s Ti s G~ (s) = l+Tds
The weighting coefficient is given in terms of the period of the ultimate oscillation and the margin of the stability. 2. DESCRIPTION OF THE CONTROL SYSTEM
(2.3) (2.4) (2.5) (2.6)
The series form is utilized in the actual control systems and also adopted for our DDC system. However, the parallel form is used in this paper for the convenience of the theoretical investigation, because the action of PID has no interconnection.
In this paper, we consider the control system with single-input single-output, as shown in 133
T. Ohta et al.
134
PlO - C o ntro l ler
Pr oce;s
~ o lse
after the test pulse is cut. According to this discrimination, the process parameters are estimated, as follows.
.
:- ---- ------ i I
R I Reference
I.J
Gp ( S )
HI
[Self-regulation process]
'--P-,oc,,--'.
I
Y
O', (S)
L ______ ___ __ j
Fig. 1
Block diagram of PID-control system
3.
CONVENTIONAL TUNING METHOD
Typical tuning methods proposed previously are the ultimate sensitivity method, the method by the use of transient response and cut and try method. They have their own features but the application of the former two methods is difficult.
The process parameters of a self-regulation process are the gain K, the sum of the time constants TT and two characteristic areas 0,0'. They are calculated from the step response. The gain K is obtained by K=B/A (4.1) TT is given by (Nishikawa 1978) n ST (4.2) TT j~l Tj + L K'A the characteristic areas 0 , 0 ' are obtained by the area under the curve of the step response, shown in Fig. 2. These areas will be specified in section 5.1.
Ultimate sensitivity method is very simple and well known. However, we can rarely obtain the values of the ultimate sensitivity Ku and the period of the ultimate oscillation Pu of a real process, because the ultimate oscillation is not permitted for fear of danger.(Ziegler 194~ There are many kinds of methods which give the optimal settings from the parameters of the transfer function of the transient response. The method by Chien (1952) is well known among them. To apply these methods, there is some difficulty in finding the parameters of tne response. Therefore, these are rarely used. Cut and try method is easily understood. However, the strategy of the change of the settings is left to the field engineers. They must decide new settings carefully watching that the process never runs into the dangerous region. The aim of the development of the auto-tuning is to improve these unsatisfactory situation by calculating the optimal settings. We expect that this function makes the time for the tuning sufficiently short and also that it gives good closed loop responses. 4. 4.1
THE FUNCTION OF AUTO-TUNING
area surrounded by 0, TT'~
0'
area surrounded by 0, Ts, ~
-------------::,;-;0..0---
B
ST
o
step response
Ts
Fig. 2
TT
Characteristic areas of selfregulation process
[Non-self-regulation process] The process parameters T, TT and 0 of a nonself-regulation process are calculated directly from the pulse response (Fig. 3). T is calculated by (4.3) T = A·Tc/B in this case, the value B is used, because it is obtained from the pulse response. TT is given from n
TT
T' + L J
. 1:
J= 1
=
ST·T A'T
Tc
c
2
(4.4)
and 0 is defined as the area under the curve of step response, shown in Fig. 3.
Data sampling
The form of the test pulse must be carefully decided in order to obtain the suitable response for the estimation of the process parameters. Therefore, the form of the pulse is constructed in the computer, so that the sampled data are not exceeding out of the limit which the field engineer gives. Consequently, the amplitude A and the width Tc are automatically decided.
~ ~
a
The sampled data are stored in the memory of the computer. To economize the memory, the sampling interval is selected so that the whole sequence of the sampled data is stored in the prepared region of the computer memory. 4.2
o
pulse response
I I I
step response
Estimation of the process parameters
The response with or without self-regulation is distinguished on the basis of the offset
Fig. 3
Characteristic area of nonself-regulation process
A New Optimization Method 4.3
Tuning calculation
In the total procedure of the auto-tuning, the calculation of the optimal settings which will be described in chapter 6 is too complex to implement on a small scale computer. Therefore, the part of this calculation is simplified as follows. At first many kinds of transfer functions and combinations of time constants are picked up for getting various values of characteristic areas. In the next, the optimal settings are obtained to all the transfer functions by the aid of a large scale off-line computer. And finally, the relations between the characteristic areas and the optimal setting are approximated by polynomials. Because they include all the combinations of PI control or PID control, self-regulation process or nonself-regulation process, and process noise or reference change, we have 8(~2x2x2) polynomials. For the calculation of the polynomials, only a small program is needed. It takes very short time even by a ~-computer to get the settings from the estimated values of the process parameters. 5.
135
To get the more precise estimation, the partial characteristic area 0' is used in the calculation of the settings. The value 0' is small. 0' of the 1st order process is = 0.107 (5.8) for K = 0.5 (5.9)
oi
Although many other characteristic areas can be obtained, we should not use too many parameters. If we picked up many parameters to get the precise estimation and the differences between each others were not extinguishable, we would get rather incorrect estimation contradictory to the expectation. Since there exist many kinds of disturbances in the real process and they cause the error of the estimation, the small values or the samll differences of the values are not reliable. Considering these conditions, we select only two characteristic areas. 5.2
Characteristic area of non-self-regulation process
The characteristic area (0 ) is defined to the transfer function (2.2) with T=l just same as that of self-regulation process. The values of become as follows. For Gp(s) = l/ {s(l+Ts)} (5.10) 0 1 = 0.132 (5.11) For Gp(s) = e- Ls/{s(l+Ts) } (5.12) l 0L T '" T2 (0 . 5_e- )/(T+L)2 (5.13) As 0 1 ~hows, this value is so small that we do not pick up other characteristic areas. As a result, we can approximate the relations to the optimal settings by only using the low order polynomials with one argument in the case of non-self-regulation process.
°
THE CHARACTERISTIC AREAS OF PROCESS RES PONCES
The parameter estimation of the auto-tuning is based on the calculation of the areas, because the value of the integral (or sum) is stable to the noisy responses compared with the derivative or the value at a specified point.
°
5.1
Characteristic areas of self-regulation process
To the step response x(t) of the transfer function (2.1) with K=l, the area S( T) is defined as S(T) =
f
T
(5.1)
x(t) dt
o
by this S( T), = S(TT)
°
° and
0 ' are defined (5.2)
(5.3) 0 ' = S(TS) = S(KTT) where K is a constant less than 1. This takes the maximal value independently on the time constant
°
-1
(5.4)
01 = e = 0.368 for the first order process Gp(s) = l/(l+Ts)
-1
Calculation of ISE by the use of Hurwitz determinant
The error of the controlled variable is 6x(t) = r(t) - x(T) (6.1) We want to minimize the integral of the squared error (called ISE) :
t
J = {6x ( t) } 2 d t If the L~place transform of 6x is L {6x(t) } = Q(s)/M(s) n n-l M(s) ~ aos + als + .. . .. +
an_Is + an Q(s)· Q(-s) = boS
( 6 . 2)
(6.3)
(6 . 4)
2n-2
+ bls
2n-4
+ .....
2
(5.6) (5.7)
/( T+L)
As the above values show, a lower order process has the larger value of and infinite large gain (Kc) can be set if is equal to 01. Therefore, is thought a parameter of the process controlled .
°
°
6.1
METHOD FOR OPTIMIZING PID PARAMETERS
(5.5)
and for the delay process -Ls Gp(s) = e /(l+Ts) 0 L,T = Te
6.
°
+b _ s + b _ (6.5) n 2 n l then J is given by the ratio of two determi(6.6) where Hn is Hurwitz determinant of M(s) and Bn is obtained from Hn by changing the first row to (b o ' bl' "" b n _ 2 , b n _ l ).
T. Ohta et al .
136
Since Q(s) and M(s) contain PID parameters, we minimize the value J by adjusting these parameters. The minimum point is searched by the conjugate gradient method (Zangwill 1967). 6.2
Optimal results by this approximation are compared with those by the simple approximation by Ls = l/(l+L/m's)m e(6.8) The result shows that the second order approximation by Pad~ has enough accuracy but that the approximation by (6.8) converges very slowly. Optimal settings by ISE method
The control parameters which give the minimum value of ISE are used as the optimum settings and we call the optimization method based on the strategy of ISE minimum as ISE method. For the self-regulation process with the transfer function (6.9) the optimum settings of parallel PID control are given as (6.10) Kc = 19/8 + 8l/(16~) + 9~/16 Kc/Ti = 3( ~ +9)( ~+3)2/(16 ~2)
(6.11)
KcTd = (~2+8~+23)/4
(6.12)
Optimum PID Settings by ISE Method and USM Kc Ti 12.80 0.146 4.27 0.630 4.56 0.761 (P .... Parallel PID)
P-PNM P-RCM USM
Approximation of time delay
In order to apply the above procedure, we express time de~ay by the second order approximation of Pade -Ls l2-6Ls+L 2 s 2 e = l2+6Ls+L 2 s2 (6.7)
6.3
TABLE 1
Td 0.)33
0.774 0.190
to the step change of process noise (PNM). Table 1 shows the optimal settings to the reference step change (ReM) and those by the ultimate sensitivity method (USM) for ~=0.5. Figure 4 shows the closed loop responses for three kinds of PID settings. At a glance of the responses to process noise, the optimal settings by PNM seem to give a good result. But, watching it carefully, we find that the small oscillation remains. The effect of the slow damping of the oscillation by PNM is c learly shown in the response to the reference change. It means that the settings by PNM give undesirable oscillation. PNM has so high feedback gain that the margin of the stability is very little and the sensitivity of the process parameters is very high . Therefore , these settings are not applicable to the control of real processes. 6.4
Weighted ISE
To overcome the shortcomings of the settings by the ISE method, we intend to make the efficient use of the common sense of the field engineers, that is, the standard of 25% damping. This means that the latter amplitude of the successive peaks in the oscillation should be less than 25% of the former amplitude. We adopt a new weighted ISE of the form oo
J( 6) =
Jo {6x(t) .e 6t } 2
dt
(6.13)
where 6 is a positive number. It makes the value of ISE divergent to the 6x(t) with a smaller damping coeff icient than R. SO, we can get the optimal settings which satisfy the standard of 25 % damping, if we can predict the period of oscillation of the c losed loop response. Hereafter, we'll approve the next proposition. Proposition The period Pc of the oscillation of the closed loop response whose PID cont r o l parameters are set by a tuning strategy is proportional to the period Pu of the ultimate oscillation of the loop.
I'_ I'!'- "i
l' - iolC ."1
:
~
ff \
:
\
."
/
il
it !I iI
!
I o.ooJf.---_
Fig. 4
_ __ _ __ _ __ _ _ _ _
~;;;_,;;_;;
Results by the settings of Tab le 1
Acco rdin g to this proposition, we can set 6 by B = a/Pu (6.14) because we suppose Pc = Y'P u (6.15) where a and y are constants. In fact, the va lues of y of (6.15) by ISE listed in Table 2 are very closed to the others by the same tuning strategy. Therefore, the proposition i s not wrong. Weighted ISE is easily calcu-
A New Optimization Method TABLE 2
Y of eq. (6.15) for different )J
)J=0.5 1.20 1.41 1.40
P-PNM P-RCM USM
)J=1.0 1.19 1. 37 1.35
)J=2.0 1. 20 1. 35 1. 31
lated by the use of the relation
L{~x(t)eBt} = Q(s-B)/M(s-B) (6.16) from M(s) and Q(s).We get the coefficients of M(s-B) and Q(s-B) and construct the Hurwitz determinants by them. To use this weighted ISE method, it is not necessary to worry about giving the weighting coefficient. This is the exceeding merit. Now, we can get the optimal settings with the desired damping ratio of a closed loop response by only giving one value of a . As we expect, the larger value of a gives the larger damping factor and the more stable closded loop response. To the different values of a, the optimal settings of PI or PID parameters of the process expressed by the transfer function (6.9) with )J=0.5 are listed in Table 3, 4. The corresponding closed loop responses are shown in Fig. 5. The values of a are selected from 0 to 1.0 for PI control and from 0 to 2.0 for PID, because the shorter periods of the oscillation can be expected for PID than PI.
TABLE 3
P-PNM P-RCM
TABLE 4 P-PNM P-RCM
137
For the same values of a , the similar responses are obtained also in the case of nonself-regulation processes. Above all, it is recognized that the optimal settings by the larger value of a have the smaller sensitivity of the error of the parameter estimation of a process. That is, the closed loop responses are very little different when the optimal parameters by the larger value of a are set. Thus, we can avoid the unstability caused by the error of the parameter estimation.
7.
RESULTS OF THE FIELD TESTS
The automatic tuning algorithm is based on the parameter estimation by the characteristic areas and the approximate relation between the characteristic areas and the optimum settings. Besides these errors of the estimation and approximation, the other kinds of errors exist in the data sampling and processing. In order to investigate the va lidity and the effectiveness of the present algorithm, some field tests are executed in the power plant of a chemical work ; KAPROLACTAM Work of UBE Industries Co . , Ltd. Among them, two pulse responses are shown.
r- -- - - -
Optimal parameters of PI-control Kc Ti Kc Ti
a=O.O 3.20 2.67 2.07 2.92
a=0.25 2.62 2.23 1. 76 2.36
a =0.5 2.12 1.92 1.49 1. 99
a =1.0 1. 35 1.51 1.03 1.52
---e
TAR _
TEMPERATURE
Fig. 6
Schematic diagram of the control loop of a heat exchanger
Optimal parameters of PID-control Kc Ti Td Kc Ti
Td
a=O.O 8.00 0.267 1.00 2.87 0.804 1. 36
a =0.5 7.63 0.403 0.781 3. 27 1.21 0 . 937
a =1.0 6.73 0.576 0.640 3 . 28 1. 52 0.715
a =2.0 4.21 1.01 0.468 2.59 1. 73 0.484
auto-tuni;1 g
t
steam valve opening
tar temperature t ime DDC by the s ettin gs of auto-tu n i ng
steam valve opening reference
----------------Fig. 7
-
Fig. 5
P-PNM optimal PI- contro l
tar temperature
Resul ts of the field tes ts of a temperature control
T. Ohta e t a l .
138
8.
steam drain
drain tank
Fig. 8
Schematic diagram of the control loop of a tank
,., .OIIJ'EH II()CJ'1'I..fif ~T[IR '"''
,,...,,,
,\ - -----'.
CONCLUSION
A simple method for auto-tuning of PID controller is proposed. This algorithm is eas! 1y implemented on a process computer. This algorithm is applicable to the self-regulation process, to the non-self-regulation process and also to the process with time delay. In the use of this method, one does not need some apriori information about the characteristics of the process response. This is a big merit. This method has been implemented on the DDC system (FUJI MICREX). In order to apply the method, it is sufficient to install one card of auto-tuning in the standard system. As the field tests show, the settings of PID parameters can be used for actual control loops and give the stable closed loop responses.
2S.0·~
auto-tuning
Fig. 9
Results of the field tests of a level control
If a large scale off-line computer can be used, the approximation of the relation between the characteristic areas and the optimal settings is not necessary, and a complex transfer function with many constants can be dealt directly. This method is expected to aid the control engineers who want to get the process parameters from the sampled data and to give the optimal settings of the PID control parameters. Acknowledgements -- He would like to express our gratitude to Mr. M. Horiki, Deputy Manager, and Mr . M. Yamanaka of Engineering Department of UBE CAPROLACTAM Hork of UBE Industries Co., Ltd, for providing the opportunities to execute the field tests of this tuning method. He thank Dr. H. Itakur~ Mr. M. Okudaira, Mr. S. Hayashibe and Mr. Y. Imaizumi, who participated in the development of this method. REFERENCES
A self-regulation process (heat exchanger) is represented schematically in Fig. 6. Tar is used for fuel after it is warmed by steam. Controller (TIC) adjusts the tar temperature at a fixed set point. The auto-tuning was executed in this control loop. The computer output to the position of the valve and the temperature response is recorded (Fig. 7). As this figure shows, the disturbance to the process is small. From this small response, auto-tuning calculated the settings (K c =l.32, Ti=74 sec. Td=l8 sec.) which gave the stable closed loop response as shown in Fig. 7.
Chien, K.L., J.A. Hrones and J.B. Reswick (1952) On the Automatic Control of Generalized Passive Systems. Trans. ~ l!i... 175-185. Nishikawa, Y., N. Sannomiya, H. Itakura, T. Ohta, Y. Imaizumi, H. Tanaka and K. Tanaka (1978) Auto-tuning of PID Control Parameters in FUJl MlCREX. FUJI Electrical Journal 2]l, 203-208. (in Japanese) Nishikawa, Y., N. Sannomiya, T. Ohta and M. Okudaira (1978) Determination of the optimal PID parameters under consideration of the ultimate oscillation charaAnother example of non-self-regulation process cteristics of processes.Preprints of (drain tank level) is represented in Fig. 8. 7th SICE (Society of Instrument and The level is controlled by the controller (LIC). Control Engineers) Symp. on control The results of auto-tuning and controlled theory. 265-270. (in Japanese) behavior are shown in Fig. 9. Zangwi1l, H.I. (1967) Minimizing a function without calculating derivatives. CompuAs these results show, we can get the suffiter Journal ~ 293-296. ciently good settings without knowing process Ziegler, J.G., N.B. Nichols and N.Y. Rochescharacteristics. At least, the field engineer ter (1942) Optimum Settings for Autocan start tuning from this recommendation. matic Controllers. Trans. ASME. ~, 759-768 .
1.
Fig. 1. Both self-regulation process and nonself-regulation process expressed by the following transfer functions are treated.
INTRODUCTION
DDC systems have been widely prevailed in Japan and the same PID control actions as analog controllers are adopted in them . In this paper, we develop the method for automatic tuning of FID control parameters and implement the procedure on a small process computer in order to aid the tuning operation due to the field engineers (Nishikawa 1978). We call this method of automatic tuning "auto-tuning" for short.
[self-regulation process] -Ls Ke Gp(s) = -n~-'---'---11 j = 1 (l+T j s)
(2.1)
[non-self-regulation process] Gp (s)
When we want to set the control parameters of the real process, the margin of the stability of the closed loop response must be considered. The margin guarantees us the stability in the case of the process parameter change and gives compensation for the error of the process parameter identification. The main theoretical part of this method is to search the optimal PID parameters with apprcpriate margin . For this purpose, we introduce, as a new perform2nce index, a weighted intergral of the squared error of the process output.
=
-Ls e - - - n -- - - Ts . 11 =l (l+TjS)
(2.2)
j
The following two forms of PID control equation are considered; namely, parallel form and series form. [Parallel form] Gc(s)
Kc(l+
G~(s)
1
1 Ti s
+ Td s )
[Series form] 1 Gc(s) = Kc(l+ - ) Td s Ti s G~ (s) = l+Tds
The weighting coefficient is given in terms of the period of the ultimate oscillation and the margin of the stability. 2. DESCRIPTION OF THE CONTROL SYSTEM
(2.3) (2.4) (2.5) (2.6)
The series form is utilized in the actual control systems and also adopted for our DDC system. However, the parallel form is used in this paper for the convenience of the theoretical investigation, because the action of PID has no interconnection.
In this paper, we consider the control system with single-input single-output, as shown in 133
T. Ohta et al.
134
PlO - C o ntro l ler
Pr oce;s
~ o lse
after the test pulse is cut. According to this discrimination, the process parameters are estimated, as follows.
.
:- ---- ------ i I
R I Reference
I.J
Gp ( S )
HI
[Self-regulation process]
'--P-,oc,,--'.
I
Y
O', (S)
L ______ ___ __ j
Fig. 1
Block diagram of PID-control system
3.
CONVENTIONAL TUNING METHOD
Typical tuning methods proposed previously are the ultimate sensitivity method, the method by the use of transient response and cut and try method. They have their own features but the application of the former two methods is difficult.
The process parameters of a self-regulation process are the gain K, the sum of the time constants TT and two characteristic areas 0,0'. They are calculated from the step response. The gain K is obtained by K=B/A (4.1) TT is given by (Nishikawa 1978) n ST (4.2) TT j~l Tj + L K'A the characteristic areas 0 , 0 ' are obtained by the area under the curve of the step response, shown in Fig. 2. These areas will be specified in section 5.1.
Ultimate sensitivity method is very simple and well known. However, we can rarely obtain the values of the ultimate sensitivity Ku and the period of the ultimate oscillation Pu of a real process, because the ultimate oscillation is not permitted for fear of danger.(Ziegler 194~ There are many kinds of methods which give the optimal settings from the parameters of the transfer function of the transient response. The method by Chien (1952) is well known among them. To apply these methods, there is some difficulty in finding the parameters of tne response. Therefore, these are rarely used. Cut and try method is easily understood. However, the strategy of the change of the settings is left to the field engineers. They must decide new settings carefully watching that the process never runs into the dangerous region. The aim of the development of the auto-tuning is to improve these unsatisfactory situation by calculating the optimal settings. We expect that this function makes the time for the tuning sufficiently short and also that it gives good closed loop responses. 4. 4.1
THE FUNCTION OF AUTO-TUNING
area surrounded by 0, TT'~
0'
area surrounded by 0, Ts, ~
-------------::,;-;0..0---
B
ST
o
step response
Ts
Fig. 2
TT
Characteristic areas of selfregulation process
[Non-self-regulation process] The process parameters T, TT and 0 of a nonself-regulation process are calculated directly from the pulse response (Fig. 3). T is calculated by (4.3) T = A·Tc/B in this case, the value B is used, because it is obtained from the pulse response. TT is given from n
TT
T' + L J
. 1:
J= 1
=
ST·T A'T
Tc
c
2
(4.4)
and 0 is defined as the area under the curve of step response, shown in Fig. 3.
Data sampling
The form of the test pulse must be carefully decided in order to obtain the suitable response for the estimation of the process parameters. Therefore, the form of the pulse is constructed in the computer, so that the sampled data are not exceeding out of the limit which the field engineer gives. Consequently, the amplitude A and the width Tc are automatically decided.
~ ~
a
The sampled data are stored in the memory of the computer. To economize the memory, the sampling interval is selected so that the whole sequence of the sampled data is stored in the prepared region of the computer memory. 4.2
o
pulse response
I I I
step response
Estimation of the process parameters
The response with or without self-regulation is distinguished on the basis of the offset
Fig. 3
Characteristic area of nonself-regulation process
A New Optimization Method 4.3
Tuning calculation
In the total procedure of the auto-tuning, the calculation of the optimal settings which will be described in chapter 6 is too complex to implement on a small scale computer. Therefore, the part of this calculation is simplified as follows. At first many kinds of transfer functions and combinations of time constants are picked up for getting various values of characteristic areas. In the next, the optimal settings are obtained to all the transfer functions by the aid of a large scale off-line computer. And finally, the relations between the characteristic areas and the optimal setting are approximated by polynomials. Because they include all the combinations of PI control or PID control, self-regulation process or nonself-regulation process, and process noise or reference change, we have 8(~2x2x2) polynomials. For the calculation of the polynomials, only a small program is needed. It takes very short time even by a ~-computer to get the settings from the estimated values of the process parameters. 5.
135
To get the more precise estimation, the partial characteristic area 0' is used in the calculation of the settings. The value 0' is small. 0' of the 1st order process is = 0.107 (5.8) for K = 0.5 (5.9)
oi
Although many other characteristic areas can be obtained, we should not use too many parameters. If we picked up many parameters to get the precise estimation and the differences between each others were not extinguishable, we would get rather incorrect estimation contradictory to the expectation. Since there exist many kinds of disturbances in the real process and they cause the error of the estimation, the small values or the samll differences of the values are not reliable. Considering these conditions, we select only two characteristic areas. 5.2
Characteristic area of non-self-regulation process
The characteristic area (0 ) is defined to the transfer function (2.2) with T=l just same as that of self-regulation process. The values of become as follows. For Gp(s) = l/ {s(l+Ts)} (5.10) 0 1 = 0.132 (5.11) For Gp(s) = e- Ls/{s(l+Ts) } (5.12) l 0L T '" T2 (0 . 5_e- )/(T+L)2 (5.13) As 0 1 ~hows, this value is so small that we do not pick up other characteristic areas. As a result, we can approximate the relations to the optimal settings by only using the low order polynomials with one argument in the case of non-self-regulation process.
°
THE CHARACTERISTIC AREAS OF PROCESS RES PONCES
The parameter estimation of the auto-tuning is based on the calculation of the areas, because the value of the integral (or sum) is stable to the noisy responses compared with the derivative or the value at a specified point.
°
5.1
Characteristic areas of self-regulation process
To the step response x(t) of the transfer function (2.1) with K=l, the area S( T) is defined as S(T) =
f
T
(5.1)
x(t) dt
o
by this S( T), = S(TT)
°
° and
0 ' are defined (5.2)
(5.3) 0 ' = S(TS) = S(KTT) where K is a constant less than 1. This takes the maximal value independently on the time constant
°
-1
(5.4)
01 = e = 0.368 for the first order process Gp(s) = l/(l+Ts)
-1
Calculation of ISE by the use of Hurwitz determinant
The error of the controlled variable is 6x(t) = r(t) - x(T) (6.1) We want to minimize the integral of the squared error (called ISE) :
t
J = {6x ( t) } 2 d t If the L~place transform of 6x is L {6x(t) } = Q(s)/M(s) n n-l M(s) ~ aos + als + .. . .. +
an_Is + an Q(s)· Q(-s) = boS
( 6 . 2)
(6.3)
(6 . 4)
2n-2
+ bls
2n-4
+ .....
2
(5.6) (5.7)
/( T+L)
As the above values show, a lower order process has the larger value of and infinite large gain (Kc) can be set if is equal to 01. Therefore, is thought a parameter of the process controlled .
°
°
6.1
METHOD FOR OPTIMIZING PID PARAMETERS
(5.5)
and for the delay process -Ls Gp(s) = e /(l+Ts) 0 L,T = Te
6.
°
+b _ s + b _ (6.5) n 2 n l then J is given by the ratio of two determi(6.6) where Hn is Hurwitz determinant of M(s) and Bn is obtained from Hn by changing the first row to (b o ' bl' "" b n _ 2 , b n _ l ).
T. Ohta et al .
136
Since Q(s) and M(s) contain PID parameters, we minimize the value J by adjusting these parameters. The minimum point is searched by the conjugate gradient method (Zangwill 1967). 6.2
Optimal results by this approximation are compared with those by the simple approximation by Ls = l/(l+L/m's)m e(6.8) The result shows that the second order approximation by Pad~ has enough accuracy but that the approximation by (6.8) converges very slowly. Optimal settings by ISE method
The control parameters which give the minimum value of ISE are used as the optimum settings and we call the optimization method based on the strategy of ISE minimum as ISE method. For the self-regulation process with the transfer function (6.9) the optimum settings of parallel PID control are given as (6.10) Kc = 19/8 + 8l/(16~) + 9~/16 Kc/Ti = 3( ~ +9)( ~+3)2/(16 ~2)
(6.11)
KcTd = (~2+8~+23)/4
(6.12)
Optimum PID Settings by ISE Method and USM Kc Ti 12.80 0.146 4.27 0.630 4.56 0.761 (P .... Parallel PID)
P-PNM P-RCM USM
Approximation of time delay
In order to apply the above procedure, we express time de~ay by the second order approximation of Pade -Ls l2-6Ls+L 2 s 2 e = l2+6Ls+L 2 s2 (6.7)
6.3
TABLE 1
Td 0.)33
0.774 0.190
to the step change of process noise (PNM). Table 1 shows the optimal settings to the reference step change (ReM) and those by the ultimate sensitivity method (USM) for ~=0.5. Figure 4 shows the closed loop responses for three kinds of PID settings. At a glance of the responses to process noise, the optimal settings by PNM seem to give a good result. But, watching it carefully, we find that the small oscillation remains. The effect of the slow damping of the oscillation by PNM is c learly shown in the response to the reference change. It means that the settings by PNM give undesirable oscillation. PNM has so high feedback gain that the margin of the stability is very little and the sensitivity of the process parameters is very high . Therefore , these settings are not applicable to the control of real processes. 6.4
Weighted ISE
To overcome the shortcomings of the settings by the ISE method, we intend to make the efficient use of the common sense of the field engineers, that is, the standard of 25% damping. This means that the latter amplitude of the successive peaks in the oscillation should be less than 25% of the former amplitude. We adopt a new weighted ISE of the form oo
J( 6) =
Jo {6x(t) .e 6t } 2
dt
(6.13)
where 6 is a positive number. It makes the value of ISE divergent to the 6x(t) with a smaller damping coeff icient than R. SO, we can get the optimal settings which satisfy the standard of 25 % damping, if we can predict the period of oscillation of the c losed loop response. Hereafter, we'll approve the next proposition. Proposition The period Pc of the oscillation of the closed loop response whose PID cont r o l parameters are set by a tuning strategy is proportional to the period Pu of the ultimate oscillation of the loop.
I'_ I'!'- "i
l' - iolC ."1
:
~
ff \
:
\
."
/
il
it !I iI
!
I o.ooJf.---_
Fig. 4
_ __ _ __ _ __ _ _ _ _
~;;;_,;;_;;
Results by the settings of Tab le 1
Acco rdin g to this proposition, we can set 6 by B = a/Pu (6.14) because we suppose Pc = Y'P u (6.15) where a and y are constants. In fact, the va lues of y of (6.15) by ISE listed in Table 2 are very closed to the others by the same tuning strategy. Therefore, the proposition i s not wrong. Weighted ISE is easily calcu-
A New Optimization Method TABLE 2
Y of eq. (6.15) for different )J
)J=0.5 1.20 1.41 1.40
P-PNM P-RCM USM
)J=1.0 1.19 1. 37 1.35
)J=2.0 1. 20 1. 35 1. 31
lated by the use of the relation
L{~x(t)eBt} = Q(s-B)/M(s-B) (6.16) from M(s) and Q(s).We get the coefficients of M(s-B) and Q(s-B) and construct the Hurwitz determinants by them. To use this weighted ISE method, it is not necessary to worry about giving the weighting coefficient. This is the exceeding merit. Now, we can get the optimal settings with the desired damping ratio of a closed loop response by only giving one value of a . As we expect, the larger value of a gives the larger damping factor and the more stable closded loop response. To the different values of a, the optimal settings of PI or PID parameters of the process expressed by the transfer function (6.9) with )J=0.5 are listed in Table 3, 4. The corresponding closed loop responses are shown in Fig. 5. The values of a are selected from 0 to 1.0 for PI control and from 0 to 2.0 for PID, because the shorter periods of the oscillation can be expected for PID than PI.
TABLE 3
P-PNM P-RCM
TABLE 4 P-PNM P-RCM
137
For the same values of a , the similar responses are obtained also in the case of nonself-regulation processes. Above all, it is recognized that the optimal settings by the larger value of a have the smaller sensitivity of the error of the parameter estimation of a process. That is, the closed loop responses are very little different when the optimal parameters by the larger value of a are set. Thus, we can avoid the unstability caused by the error of the parameter estimation.
7.
RESULTS OF THE FIELD TESTS
The automatic tuning algorithm is based on the parameter estimation by the characteristic areas and the approximate relation between the characteristic areas and the optimum settings. Besides these errors of the estimation and approximation, the other kinds of errors exist in the data sampling and processing. In order to investigate the va lidity and the effectiveness of the present algorithm, some field tests are executed in the power plant of a chemical work ; KAPROLACTAM Work of UBE Industries Co . , Ltd. Among them, two pulse responses are shown.
r- -- - - -
Optimal parameters of PI-control Kc Ti Kc Ti
a=O.O 3.20 2.67 2.07 2.92
a=0.25 2.62 2.23 1. 76 2.36
a =0.5 2.12 1.92 1.49 1. 99
a =1.0 1. 35 1.51 1.03 1.52
---e
TAR _
TEMPERATURE
Fig. 6
Schematic diagram of the control loop of a heat exchanger
Optimal parameters of PID-control Kc Ti Td Kc Ti
Td
a=O.O 8.00 0.267 1.00 2.87 0.804 1. 36
a =0.5 7.63 0.403 0.781 3. 27 1.21 0 . 937
a =1.0 6.73 0.576 0.640 3 . 28 1. 52 0.715
a =2.0 4.21 1.01 0.468 2.59 1. 73 0.484
auto-tuni;1 g
t
steam valve opening
tar temperature t ime DDC by the s ettin gs of auto-tu n i ng
steam valve opening reference
----------------Fig. 7
-
Fig. 5
P-PNM optimal PI- contro l
tar temperature
Resul ts of the field tes ts of a temperature control
T. Ohta e t a l .
138
8.
steam drain
drain tank
Fig. 8
Schematic diagram of the control loop of a tank
,., .OIIJ'EH II()CJ'1'I..fif ~T[IR '"''
,,...,,,
,\ - -----'.
CONCLUSION
A simple method for auto-tuning of PID controller is proposed. This algorithm is eas! 1y implemented on a process computer. This algorithm is applicable to the self-regulation process, to the non-self-regulation process and also to the process with time delay. In the use of this method, one does not need some apriori information about the characteristics of the process response. This is a big merit. This method has been implemented on the DDC system (FUJI MICREX). In order to apply the method, it is sufficient to install one card of auto-tuning in the standard system. As the field tests show, the settings of PID parameters can be used for actual control loops and give the stable closed loop responses.
2S.0·~
auto-tuning
Fig. 9
Results of the field tests of a level control
If a large scale off-line computer can be used, the approximation of the relation between the characteristic areas and the optimal settings is not necessary, and a complex transfer function with many constants can be dealt directly. This method is expected to aid the control engineers who want to get the process parameters from the sampled data and to give the optimal settings of the PID control parameters. Acknowledgements -- He would like to express our gratitude to Mr. M. Horiki, Deputy Manager, and Mr . M. Yamanaka of Engineering Department of UBE CAPROLACTAM Hork of UBE Industries Co., Ltd, for providing the opportunities to execute the field tests of this tuning method. He thank Dr. H. Itakur~ Mr. M. Okudaira, Mr. S. Hayashibe and Mr. Y. Imaizumi, who participated in the development of this method. REFERENCES
A self-regulation process (heat exchanger) is represented schematically in Fig. 6. Tar is used for fuel after it is warmed by steam. Controller (TIC) adjusts the tar temperature at a fixed set point. The auto-tuning was executed in this control loop. The computer output to the position of the valve and the temperature response is recorded (Fig. 7). As this figure shows, the disturbance to the process is small. From this small response, auto-tuning calculated the settings (K c =l.32, Ti=74 sec. Td=l8 sec.) which gave the stable closed loop response as shown in Fig. 7.
Chien, K.L., J.A. Hrones and J.B. Reswick (1952) On the Automatic Control of Generalized Passive Systems. Trans. ~ l!i... 175-185. Nishikawa, Y., N. Sannomiya, H. Itakura, T. Ohta, Y. Imaizumi, H. Tanaka and K. Tanaka (1978) Auto-tuning of PID Control Parameters in FUJl MlCREX. FUJI Electrical Journal 2]l, 203-208. (in Japanese) Nishikawa, Y., N. Sannomiya, T. Ohta and M. Okudaira (1978) Determination of the optimal PID parameters under consideration of the ultimate oscillation charaAnother example of non-self-regulation process cteristics of processes.Preprints of (drain tank level) is represented in Fig. 8. 7th SICE (Society of Instrument and The level is controlled by the controller (LIC). Control Engineers) Symp. on control The results of auto-tuning and controlled theory. 265-270. (in Japanese) behavior are shown in Fig. 9. Zangwi1l, H.I. (1967) Minimizing a function without calculating derivatives. CompuAs these results show, we can get the suffiter Journal ~ 293-296. ciently good settings without knowing process Ziegler, J.G., N.B. Nichols and N.Y. Rochescharacteristics. At least, the field engineer ter (1942) Optimum Settings for Autocan start tuning from this recommendation. matic Controllers. Trans. ASME. ~, 759-768 .