- Email: [email protected]
New Methods for Process Identification and Design of Feedback Controller
-
0960-3085/97/$10.00+0.00
lChemE
0 Institution of Chemical Engineers
NEW METHODS FOR PROCESS IDENTIFICATION AND DESIGN OF FEEDBACK CONTROLLER A. JAHANMIRI and H. R. FALLAHI Department of Chemical Engineering, Shiraz University, Shiraz, Iran
N
ew methods for identification of a process and the design of a feedback controller are proposed. In the process identification, the predictive model consists of a second-order transfer function plus time delay, and the related parameters are evaluated from open-loop response. The feedback controller considered includes a PID controller and a standard lead-lag. The controller parameters are easily evaluated from the parameters of the process model. The method of tuning is illustrated by applying it to a number of systems, and the results are compared to those obtained using the previous methods. ’
Keywords: identijication; IMC; tuning
INTRODUCTION Feedback control with a PID controller is widely used in industry because of its simplicity, robustness and successful practical applications. Two methods-Ziegler and Nichols’ and Cohen and Coon2 were used until 1953. After that new techniques were introduced for setting the parameters of the PID controller, such as those of Yuwana and Seborg3. This method is based on a closed-loop test; a second order, time delay transfer function is assumed as thetransfer function of the closed loop and the parameters are estimated by a small change in the controller set point. Then, by assuming a first order, time delay transfer function for the process, the parameters are evaluated by back calculation. Lee-Chen4 modified the above method and Astrom and Hagglund’ used the limit cycles of relay feedback to find the Nyquist points of the process and obtained the PID controller setting by applying the modified’method or a dominant pole design method. Lee et aL6 modified the method of Yuwana and Seborg3; they considered a second order plus dead time transfer function and the use of a closed-loop test. In this paper, new methods for the process identification and tuning of a PID controller with a lead-lag part are introduced. In the design of the controller, the concept of Internal Model Control (IMC) is observed for improved performance of the new controller. IDENTIFICATION An open loop test is used for identifying the process model GJs), and for a step change in the set point or to manipulate a variable of magnitude A, the open loop response is obtained. For the process model a second order transfer function with dead time is considered as:
overdamped state to a critically damped or underdamped form. General methods must be proposed for each state to identify the process and these methods are considered in the following section. It must be noted that all methods of process identification usedin this paper, are basedon theopen loop test with a unit step change asan input variable.
Overdamped Response The identification of this form of response is difficult using the open loop test, as there are no simple, accurate means available to calculate the parameters of the model. For this reason, a set of time dependent parameters has been selected in this case. The selected time dependent parameters are optimized from 2% to 95% of those required; the related response is an appropriate percentage of the final steady state value of the response. Theoptimumvalues arrived at are the times required to reach 90%, 70% and5% of the steady state value. For greater simplicity, the solution of the optimization and set of non-linear equations are proposed by some fitted equations. If it is assumed that tu, t,and tf are the required times to reach 90%,70% and 5% of the steady state value, respectively, the parameters of the model can be estimated by equations (2) to (5), as follows: B K=A
-U
0.48446561 - 0.753234997 1 - 2.09464647 13.9352
Considering the ‘l’value and/or the open loop response of the process, the state of the process may vary from an 519
~
o.4771
520
FALLAHI
and tn - e
JAHANMIRI
(5)
-0.424301 + 4.62533c - 2.65412e-c Considering equations (2) to ( 9 , a main parameter tf can be seen. This parameter can vary from therequired time for 2% to 5% of steady state response value, and the optimum valuemustbe selected from the minimizationofIAE between the model and the process response. The conventional value of tf is therequired time forthe response of the process to reach 5% of its steady state value. The parameters K and 8 are obtained directly from equations (2) and (3). With the auxiliary parameter v, the valueofcanbe estimated using equation (4b), and finally, the value of r is obtained directly by means of equation (5). 7 =
Underdamped Response In this case, becauseof the oscillatory natureof the response, a set of relations can be found whichconsist of the damping factor with the second order transfer function model. In fact, with useful parameters such as overshoot or rise time, the timeconstantanddamping factor of the proposed model can be found, withthe estimated time ofthe first inflection point as well as the dead time. In this section, whole equations formodelling are exact and wholeauxiliary and model parameters have been obtained from the exact solution of the original set of linear equations. Assuming that 2, tR, and tI are the overshoot, rise time and first inflection time of response, respectively, then the
‘c’,
TUNING OF PID CONTROLLER The Internal Model Control (IMC) structure was introduced as an alternative to the classic feedback structure7. Its main advantage is that closed loop stability is assured simply by choosinga stable IMC controller. Also, closed loop performance characteristics (such as setting time) are related directly to the controller parameters, which makes on-line tuning of the IMC controller very convenient. The manipulation necessary to transform the block diagram of the IMC controller into the one classic feedback control system leave the manipulation signals and response unaffected. If this criterion is applied, the IMC controller can be changed to a classic feedback controller with the same performance. The following discusses these concepts in more detail. For several decades only the so-called linear quadratic (H2-) optimal control was available. The controller (G,) is determined such that the integral square error is minimized for a particular input (v): m
0
Also, the measure of minimization is the 2-Norm of the error (Ilel12) and the ‘input set’ is just one specific input ‘v’. Using Parseval’S theorem, the optimization problem can be stated in the frequency domain as: 1 minIlell; = min2a
1
le(iw)12dw
-cm
For single-input single-output (SISO) systems the IMC controller ‘Q’ is generally chosen, this is H2 optimal for a particular input ‘v’ where ‘v’ could be the disturbance or the setpoint. Finally, ‘Q’ could be calculated as: min Ilel12 = min 11(1 - G,,,Q)vl12 (15) Q
Q
In the case whereinput ‘v’ is stable and in a non-minimum phase, the IMC controller is obtained by multiplying ‘Q’ by the filter ‘f’ as: where: If the IMC structure and the classic feedback loop are combined, where G,,, and Qfare the both part of the control system, into ‘Gc’, then a classic feedback control system is achieved as:
Finally, by use of a first order Pade approximation, controller transfer function is obtained as:
the
where: where: r=J2/2*$2=aJ2/4 Usingequation (7), the parametercanbe calculated easily. Thenthe parametersand $2 are calculated, and finally, the parameters 7 and 8 can be directly calculated from equations (8) and (9).
Trans IChemE, Vol75, Part A, July 1997
d
hod
NEWMETHODSFORPROCESS
IDENTIFICATION AND DESIGN OF FEEDBACKCONTROLLER
521
Table I . Fitted models and IAE for different methods. Model parameters Process
I
I1
I11
Ys3-zN' Chen4-ZN' LCE~ Present
0.255 0.343 0.754 0.261
Y S-ZN Chen-ZN LCE FJ
0.244 0.275 0.759
Y S-ZN Chen-ZN LCE Present
0.213 0.250 0.265
bl
a2
a1
r
b2
0.255 0.342 0.172
0.173
0.972 3.72
0.208
16
0.244 0.276 0.1 1.97
1.25 0.109
0.213 0.253 0.027
0.209 1.738
e
IAE
4.69 4.50 4.19
1.600 0.477 0.137 0.125
3.18 2.69
2.170 1.230 0.258 0.189
4.20 3.38
5.890 4.520 0.334 0.000
7
1.722
0.908
0.1 10 0.400
1.oo
3.000
Table 1 shows that theIAEfor proposed method are much lower than those of previous r n ~ d e l s ' , ~ As , ~ .expected, ~. the advantageof this work is obtained for the underdamped process, which is difficult to approximateby afirst order plus dead-time model. The tuning results are shown in Tables 2a, 2b. Also, for a step change in set-value, the response of each method is plotted in Figures 1, 2, and 3.
For the process identification described in this paper, a 'good' estimation for X may defined as:
x = 0.258 + 0. lcT
(21)
Equation (18) shows that the introduced controller is a PID controller coupled by a standard lead-lag. With alternate results in equations (18) to (21), the controller parameters are easily evaluated. In the above, it has beenshownthat IMC provided a convenient design of the classical feedback controller. However, there are significant advantages to not only designing the controller via IMC but also implementing in the IMC structure as a model 'G,' and a controller 'Q'.The effect of a parallel path with the model in IMC structure is to subtract the effect of the manipulated variables from the process output. If it is assumed for the moment that the model is a perfect representation of the process, then the feedback signal is equal to the influence of the disturbance and is not affected by the action of the manipulated variable. Thus, the system is effectively open loop and the usual stability problems associated with feedback have disappeared.
RESULTS A comparison hasbeenmade between the proposed method (FJ) and the previous methods, such as Yuwana and Seborg3, Chen4 and Lee et aL6 To show the reliability of this method, three different processes which have been used by the previous investigators are used.
I ) Overdamped process with dead time: e-3s
Gp(4 =
(S
+ 1I2(2s+ 1)
II) Overdamped with high-order process:
111) Underdamped process with dead time: e-' = 9s2 2.4s l
+
+
1.4
Table 2a. Tuning results for three closed-loop tests. Process
KC
71
70
1.0
YS3-ZN' Chen4-ZN' LCE~
I
11 1.47
I11 1.51
7.04
YS3-ZN' Chen4-ZN' L C E ~3.43 YS-ZN' Chen4-ZN' 2.74 LCE~
l .21 1.04 1.62 0.812 1.62 l .68 1.41
6.62 4.14
-
1.76 6.47 4.33
1.30
5.13 4.36
1.28 1.09
5.38
l .65 1.34
1.47
Q
)
0
-
2 0.8 a
4
-
0.6 :
0.4
-
0.2
-
l .77
Table 2b. Tuning results for proposed method. Process
I
rI 111
KC
71
TD
a
P
0.6948 1.2042 1.7518
3.3467 3.1556 2.4000
0.8863 0.9569 3.7500
1.8599 0.9851 0.5000
0.4237 0.2445 0.1350
Trans IChemE, Vol75, Part A, July 1997
0.0 0 ~
l
l
l
10l
l
i
20 l
l
l
l
30 '
'
'
'
'
~
Time Figure 1. Closed-loop responses ofPID control system for process I. this work; - - - - Lee et aL6; - - - Yuwana and Seborg3 and Ziegler and Nichols' ; - - - Chen4 and Ziegler and Nichols'.
522
FALLAHI 1.2
and
JAHANMIRI 1
I
1.0 -
0.8 -
80.6
I
VI-
2 ; 0.4
-
1. As shown in Figures 1, 2, and 3, the response time of the proposedmethod is very small, especially for the underdamped processes. 2. As shown in Figure 3, the proposed method gives a very small response time for substantially the second order processes with or without dead time. For the cases where there is no dead time, the response will act in the same way as the response of a first order system with a time constant of one-tenth of the actual process (i.e., the response is changed to first order such that Y ( t )= l - e(-10t/r7) 1. 3. The lead-lag, coupled with a PID controller, always creates the phaselead for response,because,basedon equation (20), alp = (1 @/X) > 1. Therefore, the phase lead of the numerator on the lead-lag part is greater than the phase lag in the denominator of the lead-lag part. This means that this method compensates, to some extent, the dead-time of the process.
+
0.2 -I
4
5
0.0 0
10
15
1
20
Time Figure 2. Closed-loop responses of PID control system for process 11. this work; - - - - Lee et aL6; - - - Yuwana and Seborg3 and Ziegler and Nichols'; - - - Chen4 and Ziegler and Nichols'.
For process identification, the integral of absolute error (IAE) is used as a comparison between the open-loop step response of the model and the process. The integral time in IAE calculation is set to 30, 2 1, and 40 for process I, 11, and I11 respectively. The final results of the models are shown in Table 1. Note that the optimum values of Kc for closed-loop tests of other methods are 1S , 2.0, and 2.0 for process I, 11, 111, respectively.
CONCLUSIONS This paper presents a general algorithm for the process identification and design of a feedbackcontroller. From the results whichwereshown in Figures 1, 2, and 3, it is concluded that the proposed methods are more accurate than previous ones. There are afew points whichmustbe emphasized.
1.0 -
0.8 -
%!0.6 CA
I
2 : 0.4
-
o * 2 ~ ~ , l l l , , , l l , , l l l , ,, , l, l, i , , l l ~ , l l l i , ~ I
Greek Letters a constant a (defined in equation 20) P constant a (defined equation in 20) CL constant a (defined in equations 11 and 12a) constants (defined in equations 10, 11, 12a, 12b) 7 time constant model for rl, 70 integralandderivativetime of PIDcontroller damping coefficient for second-order model 5'
REFERENCES 1. Ziegler, J. G. and Nichols, N.B., 1942, Optimum setting for automatic controllers, Trans ASME, 6 4 : 759. 2. Cohen, G. H.andCoon, G . A., 1953,Theoreticalinvestigations of retarded control, Trans ASME, 75: 827. 3. Yuwana, M. andSeborg,D.E.,1982, A newmethodforon-line controller tuning, AZChE J, 28: 434. 4. Chen, C. L., 1989, A simplemethodforon-lineidentificationand controller timing, AZChE J , 35: 2073. 5. Astrom, K. J. andHaggland, T., 1988, Automatic tuning of PZD controllers (Instrument Society of America). 6. Lee, J., Cho, W. and Edgar, T. F., 1990, An improved technique for PID controller tuning from closed loop test, AZChE J, 36 (1 2): 189 1. 7. Morari, M. and Garcia, C. E., 1982, Internal model control 1, Znd Eng Chem Proc Des Dev,21: 308. 8. Chen, C. L., 1978, Analysis and Synthesis of Linear Control Systems (Pond Wood Press, New York). 9. Edgar, T. F.,Heeb, R. andHougen, J. O., 1981,Computeraided process control system design using interactive graphics, Comput & Chem Eng, 5 : 225. 10. Lee, J., 1989, On-line PID controller tuning from a single closed-loop test, AZChE J, 35: 329.
ADDRESS
0.0
0
NOMENCLATURE magnitude of setpoint change steady state value of the response error f IMC filter IAE Integral absolute of error K , Kc model,PIDcontrollergain 9,t,,, ts required times for 5%, 90% and 70% of steady state value tR timerise tl inflection time V input typical z overshoot A B e
5
10
15
20
Time Figure 3. Closed-loop responses of PID control system for process 111.thiswork; - - - - Lee et - - - YuwanaandSeborg3andZieglerand Nichols' ; - - - Chen4 and Ziegler and Nichols'.
Correspondence concerning this paper should be addressed to Dr A. Jahanmiri, Departmentof Chemical Engineering, Shiraz University, Shiraz, Iran. Fax: 98-71-52725. The manuscript was received 25 January 1996 and accepted for publication after revision 2 August 1996.
Trans IChemE, Vol75, Part A, July 1997
0960-3085/97/$10.00+0.00
lChemE
0 Institution of Chemical Engineers
NEW METHODS FOR PROCESS IDENTIFICATION AND DESIGN OF FEEDBACK CONTROLLER A. JAHANMIRI and H. R. FALLAHI Department of Chemical Engineering, Shiraz University, Shiraz, Iran
N
ew methods for identification of a process and the design of a feedback controller are proposed. In the process identification, the predictive model consists of a second-order transfer function plus time delay, and the related parameters are evaluated from open-loop response. The feedback controller considered includes a PID controller and a standard lead-lag. The controller parameters are easily evaluated from the parameters of the process model. The method of tuning is illustrated by applying it to a number of systems, and the results are compared to those obtained using the previous methods. ’
Keywords: identijication; IMC; tuning
INTRODUCTION Feedback control with a PID controller is widely used in industry because of its simplicity, robustness and successful practical applications. Two methods-Ziegler and Nichols’ and Cohen and Coon2 were used until 1953. After that new techniques were introduced for setting the parameters of the PID controller, such as those of Yuwana and Seborg3. This method is based on a closed-loop test; a second order, time delay transfer function is assumed as thetransfer function of the closed loop and the parameters are estimated by a small change in the controller set point. Then, by assuming a first order, time delay transfer function for the process, the parameters are evaluated by back calculation. Lee-Chen4 modified the above method and Astrom and Hagglund’ used the limit cycles of relay feedback to find the Nyquist points of the process and obtained the PID controller setting by applying the modified’method or a dominant pole design method. Lee et aL6 modified the method of Yuwana and Seborg3; they considered a second order plus dead time transfer function and the use of a closed-loop test. In this paper, new methods for the process identification and tuning of a PID controller with a lead-lag part are introduced. In the design of the controller, the concept of Internal Model Control (IMC) is observed for improved performance of the new controller. IDENTIFICATION An open loop test is used for identifying the process model GJs), and for a step change in the set point or to manipulate a variable of magnitude A, the open loop response is obtained. For the process model a second order transfer function with dead time is considered as:
overdamped state to a critically damped or underdamped form. General methods must be proposed for each state to identify the process and these methods are considered in the following section. It must be noted that all methods of process identification usedin this paper, are basedon theopen loop test with a unit step change asan input variable.
Overdamped Response The identification of this form of response is difficult using the open loop test, as there are no simple, accurate means available to calculate the parameters of the model. For this reason, a set of time dependent parameters has been selected in this case. The selected time dependent parameters are optimized from 2% to 95% of those required; the related response is an appropriate percentage of the final steady state value of the response. Theoptimumvalues arrived at are the times required to reach 90%, 70% and5% of the steady state value. For greater simplicity, the solution of the optimization and set of non-linear equations are proposed by some fitted equations. If it is assumed that tu, t,and tf are the required times to reach 90%,70% and 5% of the steady state value, respectively, the parameters of the model can be estimated by equations (2) to (5), as follows: B K=A
-U
0.48446561 - 0.753234997 1 - 2.09464647 13.9352
Considering the ‘l’value and/or the open loop response of the process, the state of the process may vary from an 519
~
o.4771
520
FALLAHI
and tn - e
JAHANMIRI
(5)
-0.424301 + 4.62533c - 2.65412e-c Considering equations (2) to ( 9 , a main parameter tf can be seen. This parameter can vary from therequired time for 2% to 5% of steady state response value, and the optimum valuemustbe selected from the minimizationofIAE between the model and the process response. The conventional value of tf is therequired time forthe response of the process to reach 5% of its steady state value. The parameters K and 8 are obtained directly from equations (2) and (3). With the auxiliary parameter v, the valueofcanbe estimated using equation (4b), and finally, the value of r is obtained directly by means of equation (5). 7 =
Underdamped Response In this case, becauseof the oscillatory natureof the response, a set of relations can be found whichconsist of the damping factor with the second order transfer function model. In fact, with useful parameters such as overshoot or rise time, the timeconstantanddamping factor of the proposed model can be found, withthe estimated time ofthe first inflection point as well as the dead time. In this section, whole equations formodelling are exact and wholeauxiliary and model parameters have been obtained from the exact solution of the original set of linear equations. Assuming that 2, tR, and tI are the overshoot, rise time and first inflection time of response, respectively, then the
‘c’,
TUNING OF PID CONTROLLER The Internal Model Control (IMC) structure was introduced as an alternative to the classic feedback structure7. Its main advantage is that closed loop stability is assured simply by choosinga stable IMC controller. Also, closed loop performance characteristics (such as setting time) are related directly to the controller parameters, which makes on-line tuning of the IMC controller very convenient. The manipulation necessary to transform the block diagram of the IMC controller into the one classic feedback control system leave the manipulation signals and response unaffected. If this criterion is applied, the IMC controller can be changed to a classic feedback controller with the same performance. The following discusses these concepts in more detail. For several decades only the so-called linear quadratic (H2-) optimal control was available. The controller (G,) is determined such that the integral square error is minimized for a particular input (v): m
0
Also, the measure of minimization is the 2-Norm of the error (Ilel12) and the ‘input set’ is just one specific input ‘v’. Using Parseval’S theorem, the optimization problem can be stated in the frequency domain as: 1 minIlell; = min2a
1
le(iw)12dw
-cm
For single-input single-output (SISO) systems the IMC controller ‘Q’ is generally chosen, this is H2 optimal for a particular input ‘v’ where ‘v’ could be the disturbance or the setpoint. Finally, ‘Q’ could be calculated as: min Ilel12 = min 11(1 - G,,,Q)vl12 (15) Q
Q
In the case whereinput ‘v’ is stable and in a non-minimum phase, the IMC controller is obtained by multiplying ‘Q’ by the filter ‘f’ as: where: If the IMC structure and the classic feedback loop are combined, where G,,, and Qfare the both part of the control system, into ‘Gc’, then a classic feedback control system is achieved as:
Finally, by use of a first order Pade approximation, controller transfer function is obtained as:
the
where: where: r=J2/2*$2=aJ2/4 Usingequation (7), the parametercanbe calculated easily. Thenthe parametersand $2 are calculated, and finally, the parameters 7 and 8 can be directly calculated from equations (8) and (9).
Trans IChemE, Vol75, Part A, July 1997
d
hod
NEWMETHODSFORPROCESS
IDENTIFICATION AND DESIGN OF FEEDBACKCONTROLLER
521
Table I . Fitted models and IAE for different methods. Model parameters Process
I
I1
I11
Ys3-zN' Chen4-ZN' LCE~ Present
0.255 0.343 0.754 0.261
Y S-ZN Chen-ZN LCE FJ
0.244 0.275 0.759
Y S-ZN Chen-ZN LCE Present
0.213 0.250 0.265
bl
a2
a1
r
b2
0.255 0.342 0.172
0.173
0.972 3.72
0.208
16
0.244 0.276 0.1 1.97
1.25 0.109
0.213 0.253 0.027
0.209 1.738
e
IAE
4.69 4.50 4.19
1.600 0.477 0.137 0.125
3.18 2.69
2.170 1.230 0.258 0.189
4.20 3.38
5.890 4.520 0.334 0.000
7
1.722
0.908
0.1 10 0.400
1.oo
3.000
Table 1 shows that theIAEfor proposed method are much lower than those of previous r n ~ d e l s ' , ~ As , ~ .expected, ~. the advantageof this work is obtained for the underdamped process, which is difficult to approximateby afirst order plus dead-time model. The tuning results are shown in Tables 2a, 2b. Also, for a step change in set-value, the response of each method is plotted in Figures 1, 2, and 3.
For the process identification described in this paper, a 'good' estimation for X may defined as:
x = 0.258 + 0. lcT
(21)
Equation (18) shows that the introduced controller is a PID controller coupled by a standard lead-lag. With alternate results in equations (18) to (21), the controller parameters are easily evaluated. In the above, it has beenshownthat IMC provided a convenient design of the classical feedback controller. However, there are significant advantages to not only designing the controller via IMC but also implementing in the IMC structure as a model 'G,' and a controller 'Q'.The effect of a parallel path with the model in IMC structure is to subtract the effect of the manipulated variables from the process output. If it is assumed for the moment that the model is a perfect representation of the process, then the feedback signal is equal to the influence of the disturbance and is not affected by the action of the manipulated variable. Thus, the system is effectively open loop and the usual stability problems associated with feedback have disappeared.
RESULTS A comparison hasbeenmade between the proposed method (FJ) and the previous methods, such as Yuwana and Seborg3, Chen4 and Lee et aL6 To show the reliability of this method, three different processes which have been used by the previous investigators are used.
I ) Overdamped process with dead time: e-3s
Gp(4 =
(S
+ 1I2(2s+ 1)
II) Overdamped with high-order process:
111) Underdamped process with dead time: e-' = 9s2 2.4s l
+
+
1.4
Table 2a. Tuning results for three closed-loop tests. Process
KC
71
70
1.0
YS3-ZN' Chen4-ZN' LCE~
I
11 1.47
I11 1.51
7.04
YS3-ZN' Chen4-ZN' L C E ~3.43 YS-ZN' Chen4-ZN' 2.74 LCE~
l .21 1.04 1.62 0.812 1.62 l .68 1.41
6.62 4.14
-
1.76 6.47 4.33
1.30
5.13 4.36
1.28 1.09
5.38
l .65 1.34
1.47
Q
)
0
-
2 0.8 a
4
-
0.6 :
0.4
-
0.2
-
l .77
Table 2b. Tuning results for proposed method. Process
I
rI 111
KC
71
TD
a
P
0.6948 1.2042 1.7518
3.3467 3.1556 2.4000
0.8863 0.9569 3.7500
1.8599 0.9851 0.5000
0.4237 0.2445 0.1350
Trans IChemE, Vol75, Part A, July 1997
0.0 0 ~
l
l
l
10l
l
i
20 l
l
l
l
30 '
'
'
'
'
~
Time Figure 1. Closed-loop responses ofPID control system for process I. this work; - - - - Lee et aL6; - - - Yuwana and Seborg3 and Ziegler and Nichols' ; - - - Chen4 and Ziegler and Nichols'.
522
FALLAHI 1.2
and
JAHANMIRI 1
I
1.0 -
0.8 -
80.6
I
VI-
2 ; 0.4
-
1. As shown in Figures 1, 2, and 3, the response time of the proposedmethod is very small, especially for the underdamped processes. 2. As shown in Figure 3, the proposed method gives a very small response time for substantially the second order processes with or without dead time. For the cases where there is no dead time, the response will act in the same way as the response of a first order system with a time constant of one-tenth of the actual process (i.e., the response is changed to first order such that Y ( t )= l - e(-10t/r7) 1. 3. The lead-lag, coupled with a PID controller, always creates the phaselead for response,because,basedon equation (20), alp = (1 @/X) > 1. Therefore, the phase lead of the numerator on the lead-lag part is greater than the phase lag in the denominator of the lead-lag part. This means that this method compensates, to some extent, the dead-time of the process.
+
0.2 -I
4
5
0.0 0
10
15
1
20
Time Figure 2. Closed-loop responses of PID control system for process 11. this work; - - - - Lee et aL6; - - - Yuwana and Seborg3 and Ziegler and Nichols'; - - - Chen4 and Ziegler and Nichols'.
For process identification, the integral of absolute error (IAE) is used as a comparison between the open-loop step response of the model and the process. The integral time in IAE calculation is set to 30, 2 1, and 40 for process I, 11, and I11 respectively. The final results of the models are shown in Table 1. Note that the optimum values of Kc for closed-loop tests of other methods are 1S , 2.0, and 2.0 for process I, 11, 111, respectively.
CONCLUSIONS This paper presents a general algorithm for the process identification and design of a feedbackcontroller. From the results whichwereshown in Figures 1, 2, and 3, it is concluded that the proposed methods are more accurate than previous ones. There are afew points whichmustbe emphasized.
1.0 -
0.8 -
%!0.6 CA
I
2 : 0.4
-
o * 2 ~ ~ , l l l , , , l l , , l l l , ,, , l, l, i , , l l ~ , l l l i , ~ I
Greek Letters a constant a (defined in equation 20) P constant a (defined equation in 20) CL constant a (defined in equations 11 and 12a) constants (defined in equations 10, 11, 12a, 12b) 7 time constant model for rl, 70 integralandderivativetime of PIDcontroller damping coefficient for second-order model 5'
REFERENCES 1. Ziegler, J. G. and Nichols, N.B., 1942, Optimum setting for automatic controllers, Trans ASME, 6 4 : 759. 2. Cohen, G. H.andCoon, G . A., 1953,Theoreticalinvestigations of retarded control, Trans ASME, 75: 827. 3. Yuwana, M. andSeborg,D.E.,1982, A newmethodforon-line controller tuning, AZChE J, 28: 434. 4. Chen, C. L., 1989, A simplemethodforon-lineidentificationand controller timing, AZChE J , 35: 2073. 5. Astrom, K. J. andHaggland, T., 1988, Automatic tuning of PZD controllers (Instrument Society of America). 6. Lee, J., Cho, W. and Edgar, T. F., 1990, An improved technique for PID controller tuning from closed loop test, AZChE J, 36 (1 2): 189 1. 7. Morari, M. and Garcia, C. E., 1982, Internal model control 1, Znd Eng Chem Proc Des Dev,21: 308. 8. Chen, C. L., 1978, Analysis and Synthesis of Linear Control Systems (Pond Wood Press, New York). 9. Edgar, T. F.,Heeb, R. andHougen, J. O., 1981,Computeraided process control system design using interactive graphics, Comput & Chem Eng, 5 : 225. 10. Lee, J., 1989, On-line PID controller tuning from a single closed-loop test, AZChE J, 35: 329.
ADDRESS
0.0
0
NOMENCLATURE magnitude of setpoint change steady state value of the response error f IMC filter IAE Integral absolute of error K , Kc model,PIDcontrollergain 9,t,,, ts required times for 5%, 90% and 70% of steady state value tR timerise tl inflection time V input typical z overshoot A B e
5
10
15
20
Time Figure 3. Closed-loop responses of PID control system for process 111.thiswork; - - - - Lee et - - - YuwanaandSeborg3andZieglerand Nichols' ; - - - Chen4 and Ziegler and Nichols'.
Correspondence concerning this paper should be addressed to Dr A. Jahanmiri, Departmentof Chemical Engineering, Shiraz University, Shiraz, Iran. Fax: 98-71-52725. The manuscript was received 25 January 1996 and accepted for publication after revision 2 August 1996.
Trans IChemE, Vol75, Part A, July 1997