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Ind. Eng. Chem. Res. 1998, 37, 482-488
Process Identification Using Pulse Response and Proportional-Integral Derivative Controller Tuning with Combined Guidelines Tae W. Ham and Young H. Kim* Department of Chemical Engineering, Dong-A University, 840 Hadan-dong, Saha-gu, Pusan 604-714, Korea
A procedure for the identification of a chemical process as a second-order plus dead time model using square pulse response is proposed, and the model is utilized to obtain the tuning parameters of a PID controller for the process. The tuning technique employs combined open-loop and closedloop tuning guidelines, of which a new guideline for the maximum closed-loop modulus is suggested. The proposed identification method requires short identification time and results in no significant disturbance in the process from the test. As a closed-loop tuning guideline, a maximum modulus is used, while gain and phase margins are utilized as open-loop guidelines. For the improved implementation of the maximum closed-loop modulus, the relationship between the maximum closed-loop modulus and the damping coefficient of the process is analyzed to find a linear relationship between them and to apply in the tuning. In four example processes, the performance of the proposed technique is compared with that of two other methods employing simple tuning guidelines, and it is found that the proposed method gives a better performance. 1. Introduction Though advanced control is extensively used in chemical processes, the conventional proportionalintegral derivative (PID) control has not been replaced by the modern control techniques. It is mostly implemented as basic control of the process owing to its robustness and simplicity. As the PID controller is popular in chemical processes, the tuning of the controller has been widely studied to improve the control performance. The controller tuning techniques are classified into two groups, open-loop and closed-loop techniques. The process reaction curve method (Cohen and Coon, 1953) is one of open-loop techniques. It is a simple and fast procedure, and a first order plus dead time model (FOPDT) is obtained to be employed in tuning. However, it leaves an off-set from the initial steady state during the test. On the other hand, closed-loop techniques, such as the Ziegler-Nichols tuning (Ziegler and Nichols, 1942) and the relay feedback method (A° stro¨m and Ha¨gglund, 1984), guarantee stability without the off-set problem. However, they take long test time and give only gain margin information unless the test input is manipulated for other than a phase angle of -180° (Kim, 1995). While gain and phase margins are based on an openloop characteristic of the control system, the maximum closed-loop modulus is the closed-loop system guideline. Though the closed-loop modulus is preferred for the controller tuning (Luyben, 1990), the open-loop guideline is often employed in many tuning procedures, e.g., the Ziegler-Nichols tuning and the relay feedback method, since its implementation is easier than the closed-loop guideline and no process model is necessary. In the meantime, for the improved performance, a combination of the open-loop and closed-loop guidelines * To whom all correspondence should be addressed. Tel.: 8251-200-7717. Fax: 82-51-200-7728. E-mail: yhkim@ seunghak.donga.ac.kr.
has been adopted in the studies of Harris and Mellichamp (1985) and Lee et al. (1990). In order to utilize the process model in the controller tuning, the identification of an unknown process has been investigated by many studies. Yuwana and Seborg (1982) employed proportional control with step change in set point and the closed-loop response was used to identify the process model of an FOPDT model. Then, the ultimate values are computed from the model to lead to the parameters of the PID controller with the Ziegler-Nichols tuning guideline (Ziegler and Nichols, 1942). Lee (1989) improved the identification method using the model reduction technique (Ouyang et al., 1987), and Chen (1989) utilized the same model to develop a procedure to compute the ultimate values directly from the identified model. Though most of the chemical processes are represented with an FOPDT model, an underdamped response is obtained in some cases, such as the cascade control system, and the FOPDT model does not handle the response. A second order plus dead time (SOPDT) model was employed by Lee et al. (1990) to result in the improved identification. Also, formulas to calculate the PID control parameters were developed using combined tuning guidelines in the study. Other step response identification techniques (Huang and Chou, 1994; Krishnaswamy and Rangaiah, 1996) were also utilized for the identification of the SOPDT model. However, these methods take long test time, which makes them more susceptible to disturbance during the test period and leads to identification error. In this study, a process identification procedure utilizing square pulse response is introduced. Though it is an open-loop procedure, no output off-set is generated from the test and the initial steady state is recovered. Moreover, test time is so short that no difficulty is encountered in field application. From the response, an SOPDT model is made to be applied in controller tuning, and a combined tuning guideline of gain margin, phase margin, and maximum closed-loop
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Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 483
modulus is employed. For the improved implementation of the closed-loop modulus in the tuning, the relationship between the modulus and the damping coefficient of the process model is analyzed and the adjusted value of the modulus according to the damping coefficient instead of its suggested constant value is applied. Finally, the set-point tracking and regulatory performance of the proposed tuning procedure are compared with those of the previous procedures in four different example processes. 2. Process Identification In the identification of chemical processes frequency response techniques, such as the Ziegler-Nichols tuning and the relay feedback method, are preferred to the step test because input of the process is controlled as a closed-loop system. However, the procedures take more time to identify the process and give single information at the phase angle of -180° which can only be used to compute gain margin. When a square pulse is introduced in an open-loop system, a simple pulse response is obtained in a relatively short period of time. Unlike the step test, this pulse test returns the output to the initial steady state unless a too large pulse is applied or the system has a very small damping coefficient. Most of chemical processes have damping coefficients close to 1. If the response is fitted to a second order plus dead time (SOPDT) model, eq 1, with the input of a square pulse, eq 2, a time domain solution of the response is obtained as eq 3 using partial fractions expansion (Stephanopoulos, 1984) and inverse Laplace transformation
G(s) )
Kpe-τds
u(s) ) y(t) ) K
(1)
τ2s2 + 2ζτs + 1 A(1 - e-Bs) s
[
(2)
]
e-a(t-τd) - e-a(t-τB) e-b(t-τd) - e-b(t-τB) + a(a - b) b(b - a)
(3)
where
a)
ζ 1 2 + xζ - 1 τ τ
b)
ζ 1 2 - xζ - 1 τ τ
K ) AKp/τ2 τB ) τd + B Also, A and B are pulse height and width as shown in Figure 1a. The height and width of the injected pulse depend upon the process gain and time constant, but provided that an identifiable output is obtained, small values should be implemented not to disturb the process. A square pulse of a height of 1 and a width of 1 min is applied to all four example processes of this study. Then, differentiation of eq 3 leads to
Figure 1. Illustrations of square pulse input and process response: (a) input; (b) output.
dy ) e-at[eaτB - eaτd] - e-bt[ebτB - ebτd] dt
(4)
The information of two points, p1and p2, from the response curve in Figure 1b and eqs 3 and 4 can lead to a system of three algebraic equations, of which the solution gives the values of the parameters in the process model (eq 1). Note that the derivative at the peak point is zero. Therefore, two algebraic equations are found from eq 3 and one from eq 4. However, there are four parameters to find, and three equations are used in the identification. Namely, 1 degree of freedom exists, and therefore an optimization procedure has to be utilized. Moreover, since not all the unknown processes fit the standard model, the parameters can only be obtained through optimization. The minimizing objective function of the identification is the sum of absolute values of differences between calculated values of eqs 3 and 4 and the values of p1 and p2. Different weightings can be introduced among the absolute values, but the same weight is applied in this study. Also, no constraint is implemented in the optimization. The minimization is conducted by the Nelder-Mead simplex search method (1964) through the function “fmins” in MATLAB. The procedure requires initial values of the model parameters, but any reasonable values are good for the initial guess. The point p2 is arbitrarily chosen, while p1 is the peak point. Though Figure 1b demonstrates the response of an overdamped system, the points p1 and p2 are selected in the same manner for both overdamped and underdamped systems. In this study, the point p2 is obtained at t ) 10 min for all systems. Since any change in all four parameters of the SOPDT model has influence on the output of the model, the parameters can be uniquely estimated (Himmelblau, 1978). However, only three data, two from p1 and one from p2, are used to find four parameters, so that there is 1 degree of freedom of the identification. Using one more point information eliminates the degree of freedom, which leads to complexity in the optimization. The model structure is fixed and process gain only changes the height of the peak without affecting the shape of the peak; therefore, three equations are enough to find four model parameters from the minimization of the sum of model fitting errors. Often prior knowledge of the process gain and delay time is available and helps the identification. The validity of the identified process model is examined by comparing the response curve of the process output with the computed output from the identified model. This identification procedure needs only one response peak. Therefore, the required identification time is
484 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998
much less than that of the frequency response techniques and some step response procedures (Yuwana and Seborg, 1982; Lee et al., 1990) as well. Also, output returns to its initial value and no off-set results from the test unlike the step test. The unknown and unmeasured disturbances affect the estimation of true process model parameters. However, the ultimate aim of the estimation is to find the proper tuning parameters of the control system, and the tuning guideline contains some safety margin for uncertainty. Unless the identification includes too much uncertainty or the performance index of the tuning is too large, the effect of the unknown and unmeasured disturbance has no significant impact on the controller tuning. In addition, noise in the measured output is easily observed and eliminated using various digital filters. 3. Tuning In the design of PID controllers, open-loop characteristics, gain and phase margins, are simply computed and the control parameters, proportional gain, reset time, and rate time, are easily obtained using a simple formulation, e.g., the Ziegler-Nichols settings (Ziegler and Nichols, 1942). However, the closed-loop characteristic, maximum modulus, of the control system gives better information on the control performance, though its computation is difficult. For the improvement of the control performance, multiple tuning guidelines are implemented in Harris and Mellichamp (1985). Since all the guidelines cannot be satisfied at the same time, a performance index is proposed by combining the guidelines and minimized to find the control parameters giving the best performance. A similar performance index is introduced in this study. Namely, three tuning guidelines, two from the open-loop characteristic and one from the closedloop characteristic, are utilized in combination as
IP ) |AR1 - 1| + |AR2 - 0.5| + |Mp - Mr|
The minimization of the performance index (eq 5) makes the control system satisfy more closely the three tuning guidelines, phase margin, gain margin, and maximum closed-loop modulus. Since all the three guidelines can be satisfied at the same time in most control systems, the best tuning is found by minimizing the index. As the reference value of the maximum closed-loop modulus, the value of 1.26 corresponding to 2 of the log modulus is recommended in several literatures (Edgar et al., 1981; Harris and Mellichamp, 1985; Luyben, 1990). For improved performance, however, Lee et al. (1990) used different values of the reference amplitude for systems having different damping coefficients. Through simulation in the study, they found that a low value (near 1.02) gives a better performance for overdamped processes while a high value (near 1.25) is better for underdamped processes. This means that the maximum modulus of the closed-loop system is related to the damping coefficient of the process. In this study, an adjusted reference value from the following equation is utilized in the tuning of PID controllers
Mr ) 1.433 - 0.433ζ
where the coefficients are from the suggestions of Lee et al. (1990) (Mp ) 1.0 for ζ ) 1.0 and Mp ) 1.26 for ζ ) 0.4) and there is linear relationship between the damping coefficient and the reference amplitude. The linear relationship is found from the following analysis. The relationship between the damping coefficient and the maximum closed-loop modulus is not simply and explicitly obtained. Instead, it is investigated for a simple system and the result is exercised in this study. Lee et al. (1990) indicates that the best performance is obtained when the maximum modulus has a phase angle of -135° in the closed-loop frequency response. At the frequency, the open-loop frequency characteristic is represented as
Gc(jωp) Gp(jωp) ) Moe-j(3/4)π
(5)
where AR1 is the amplitude ratio, modulus, of the openloop frequency response at a phase angle of -135° and AR2 is the amplitude ratio at a phase angle of -180°. Mp is the maximum modulus of the closed-loop frequency response, and its reference value is Mr. The modulus of the closed-loop transfer function of a control system varies with frequency, and the maximum closedloop modulus is the largest value of the modulus having a specific frequency. The maximum closed-loop modulus is the most useful frequency-domain tuning guideline. In the system having an unusual frequency response curve, the controller tuning with the gain and phase margins can give a poor control performance (Luyben, 1990). Therefore, the tuning guideline should include the maximum closed-loop modulus. The Nichols chart is a graphical tool to find the value, but a numerical search is conducted in this study. Since a phase margin of 45° and a gain margin of 2 are suggested for the chemical process control (Luyben, 1990), the first and second terms of eq 5 are arranged for the margins. By using an amplitude ratio in all three guidelines instead of degrees in the phase margin, no adjustment of weights among them is necessary, unlike other combined tuning guidelines (Harris and Mellichamp, 1985).
(6)
(7)
From the relationship between open-loop and closedloop moduli and open-loop phase angle,
Mo )
1 x2
x
1 1 Mp2 2 1 1Mp2
(8)
Meanwhile, when a PID control is applied to a SOPDT model as a control system, the open-loop modulus and phase angle are found as
x(
τDω -
KcKp Mo )
)
1 τIω
2
+1
x(1 - τ2ω2)2 + (2ζτω)2
(9)
and
(
φo ) tan-1 -
)
(
)
2ζτω 1 - τdω + tan-1 τDω 2 2 τ 1-τ ω Iω (10)
The phase angle in eq 10 found from eq 7 shows the value of ω, and eqs 8 and 9 give the relationship between
Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 485
the maximum closed-loop modulus and damping coefficient. Since the explicit formulation is not available, numerical numbers are obtained for the case of Kp ) 1, τ ) 1, and τd ) 1 having a PID controller of Kc ) 3, τI ) 4, and τD ) 1. The result is illustrated in Figure 2 showing a nearly linear relationship between 0.4 and 1 of the damping coefficient which most of the chemical processes have. Meanwhile, the minimization of eq 5 is carried out by the Nelder-Mead simplex search method (1964) through the function “fmins” in MATLAB as done in the parameter estimation of the process model. The initial values are more sensitive to finding the minimum than the search of model parameters, and several different sets of the initial values have to be tried to see if they give a minimum. 4. Examples Four example systems are used for the performance test of the proposed tuning method of this study. Also, the performances are compared with the results of Yuwana and Seborg (1982) and Lee et al. (1990). The simulation of the control system is carried out with the MATLAB.
Process I Gp(s) )
e-3s (s + 1)2(2s + 1)
Process II Gp(s) )
1 (s + 1)5
Process III Gp(s) )
e-s 9s2 + 2.4s + 1
Process IV Gp(s) )
e-s (2s + 2s + 1)(s + 1) 2
The first three processes are adopted from Yuwana and Seborg (1982), and the last process is newly introduced to see the effect of the damping coefficient on controller tuning. While the first two are overdamped systems, the remaining two are underdamped systems. Especially, the last system has an intermediate damping coefficient between the first (and second) and the third processes. 5. Results and Discussion A square pulse having a height of 1 and a width of 1 min is introduced, and the response is analyzed to find the parameters of an SOPDT model (eq 1). Table 1 shows model parameters of example processes. Note that process III is a real SOPDT system and the estimated parameters are very close to the actual values. In order to see the sensitivity of selection of p2 in Figure 1, the model is identified for the values of p2 at t ) 6-20 min in example process III having a standard SOPDT model, and the ranges of identified model
Figure 2. Relationship between the damping coefficient and maximum closed-loop modulus in the PID control system. Table 1. Estimated Model Parameters process I II III IV
Kp
τ
ζ
τd
0.992 0.996 1.000 1.054
1.863 1.721 3.000 1.966
1.000 1.000 0.400 0.620
3.398 1.477 1.004 1.323
parameters are Kp ) 0.998-1.003, τ ) 2.995-3.001 min, ζ ) 0.400-0.403, and τd ) 1.004-1.009 min. Therefore, no significant difference is observed with different locations of p2. Though the dead-time identification in this study does not produce any problem for the example processes of various dead times, some step response techniques of model identification show difficulty for large dead-time processes (Jutan, 1989). Provided that an output signal is available, a significant amount of dead time does not affect the identification of the process model. A procedure similar to this study is applied to a large deadtime process (Luyben, 1986), and no difficulty is observed. The proposed identification technique is also applied to an inverse response process (Chen, 1991), (-s + 1)/ (2s2 + 3s + 1), and the identified parameters are Kp ) 1.146, τ ) 1.464 min, ζ ) 1.037, and τd ) 0.515 min. Applying the first-order Pade´ approximation to the identified model leads to close formulation with the original model, and the validity of the identified model is easily yielded. Using the parameters of the identified model, the optimization of the performance index (eq 5) is conducted to find the control parameters which are given in Table 2. The parameters of Yuwana and Seborg (1982) and Lee et al. (1990) are also listed in the table. From the control parameters, the gain and phase margins and the maximum closed-loop modulus are computed and included in the table. Conventional tuning procedures, such as the ZieglerNichols tuning and Cohen-Coon tuning, give a rate time one-quarter or less of the reset time. However, many closed-loop procedures (Edgar et al., 1981; Jutan and Rodriguez, 1984; Lee et al., 1990; Peebles et al., 1994) including this study yield more than one-quarter of the reset time for the rate time. For the comparison of control performances, the responses of step set-point change and regulatory performances are illustrated in Figures 3-6 for processes I-IV, respectively. In order to examine regulatory performance, an off-set value of 1 is added to the
486 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 Table 2. Tuning Results and Integral of Absolute Errors IAE process I II III IV
YS LCE HK YS LCE HK YS LCE HK YS LCE HK
Kc
τI
τD
GM
PM
Mp
set point
regulation
1.21 0.812 0.931 1.62 1.41 1.41 1.47 1.77 1.95 1.26 0.960 1.15
7.04 4.33 4.34 5.13 3.43 3.50 6.62 4.14 3.33 4.77 2.69 2.01
1.76 1.30 1.54 1.28 1.47 1.50 1.65 2.74 3.02 1.19 1.15 1.69
1.39 2.24 1.95 2.47 2.82 2.81 4.52 2.72 2.30 1.93 2.57 1.99
72.2 60.1 58.3 64.5 63.1 64.7 44.5 47.8 45.1 67.8 62.7 54.0
2.68 1.00 1.15 1.17 1.01 1.01 1.34 1.23 1.33 1.50 1.00 1.31
7.77 5.90 5.76 3.38 3.01 3.00 4.57 2.83 2.57 3.90 3.21 3.16
5.67 5.38 4.85 3.15 2.60 2.64 4.31 2.43 2.12 3.76 2.93 2.61
Figure 3. Comparison of control performances in process I: (top) set-point tracking; (bottom) regulating performance.
Figure 5. Comparison of control performances in process III: (top) set-point tracking; (bottom) regulating performance.
Figure 4. Comparison of control performances in process II: (top) set-point tracking; (bottom) regulating performance.
Figure 6. Comparison of control performances in process IV: (top) set-point tracking; (bottom) regulating performance.
input of the process as a disturbance and output deviation is investigated. The output of process I is shown in Figure 3. The solid line indicates the outcome of this study, the dashdotted line is of Yuwana and Seborg (1982), and the dashed line is of Lee et al. (1990). In the top figure, set-point tracking is demonstrated. This study and the tuning of Lee et al. show similar performances, while the tuning of Yuwana and Seborg leads to a highly oscillatory response. The outcome of this study gives a
faster response than that of Lee et al. The regulatory performances are illustrated in the bottom figure which indicates a performance comparison similar to the setpoint tracking. In process II as shown in Figure 4, comparable responses of all three tunings are found in both set-point tracking and regulation, though the highest deviation is observed in the result of Yuwana and Seborg. Notice that this process has no process delay. As the highest deviation is attained from the tuning of this study in the top of Figure 5 of process
Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 487
III, the fastest response and settlement are found. A similar outcome is yielded for process IV in Figure 6. In regulatory performances of the two processes, the tuning of this study apparently leads to the smallest deviation among three tunings. Since the tuning of process IV is not investigated in other studies, it is conducted by the instructions given in the studies and the results are included in Table 2. This process has the intermediate damping coefficient between processes I (and II) and III. Therefore, in Lee et al.’s tuning, Mp ) 1.02 is used because it gives higher ωp. Though fast response produces high deviation, the fast tracking to a new set point is favorable unless the deviation is too large. In other words, the tracking has to be compromised for proper deviation, leading to the best performance. One way to measure the performance of different tunings is to check the sum of deviations. Numerical comparison of the performance is carried out by computing the integral of absolute errors (IAE) between the set point and output. The computed IAE’s for set-point tracking and regulatory performance are listed in Table 2. The outcome of this study has the smallest errors except the regulatory performance in process II. Not only the numerical IAE comparison but also the consistency of overshoot in servo performances of four example processes indicates the improvement of the proposed tuning procedure. Since this study is based on a continuous system, the identification and tuning procedures may not be applied to the digital PID control system widely used in the chemical, petrochemical, and petroleum industries. However, the sampling time of the digital system in field application, often one-quarter of a second, is much shorter than the process time constant so that the procedures can be implemented to the digital systems. During the identification of a process, an unstable response may arise due to the open-loop test procedure when the size of the input pulse is excessive; therefore, the size should be small provided that the response is measurable and the identification is possible. The stability of the closed-loop system of tuning outcome is examined using the Routh stability criterion or the Nyquist stability criterion (Seborg et al., 1989). In order to ensure the stability of the closed-loop system, the maximum value of the performance index of the tuning can be suggested and applied to the tuning optimization. But, no clear value is found in the example processes of this study. 6. Conclusion A process identification method using a square pulse response is proposed, and the identified model is employed to find the parameters of the PID controller by applying combined open-loop and closed-loop tuning guidelines. Also, a new tuning guideline for the maximum closed-loop modulus is suggested. The proposed identification method takes a short test time and leaves no disturbance to process. Moreover, since the process model is available, any controller tuning guideline can be applied. The analysis between the maximum closed-loop modulus and the damping coefficient of the process for a typical system results in the linear relationship which is implemented in the combined guideline including gain and phase margins.
The performance of the proposed tuning procedure is examined in four different processes, and the numerical comparison with other tuning techniques indicates that the tuning of this study gives the best performance. Acknowledgment Financial support from the Dong-A University and the Korea Science and Engineering Foundation through the Automation Research Center at POSTECH is gratefully acknowledged. Nomenclature A ) amplitude of pulse input a ) parameter in eq 3 B ) width of pulse input, min b ) parameter in eq 3 Gc ) transfer function of the controller Gp ) transfer function of the process j ) fundamental imaginary number K ) parameter in eq 3 Kc ) proportional gain Kp ) process gain Mo ) open-loop modulus Mp ) maximum closed-loop modulus Mr ) reference value of Mp s ) Laplace parameter t ) time, min u ) input y ) process output Greek Letters ω ) angular velocity, rad/min ωp ) angular velocity at maximum modulus, rad/min φo ) open-loop phase angle, rad τ ) time constant, min τB ) parameter in eq 3, min τd ) delay time, min ζ ) damping coefficient Abbreviations AR ) amplitude ratio FOPDT ) first order plus dead time GM ) gain margin IP ) index of performance PID ) proportional-integral derivative PM ) phase margin, deg SOPDT ) second order plus dead time
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Ouyang, M.; Liaw, C. M.; Pan, C. T. Model Reduction by Power Decomposition and Frequency Response Matching. IEEE Trans. Autom. Control 1987, AC-32, 59. Peebles, S. M.; Hunter, S. R.; Corripio, A. B. Implementation of a Dynamically-Compensated PID Control Algorithm. Comput. Chem. Eng. 1994, 18, 995. Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley & Sons: New York, 1989; pp 260, 365. Stephanopoulos, G. Chemical Process Control: An Introduction to Theory and Practice; McGraw-Hill: New York, 1984; p 145. Yuwana, M.; Seborg, D. E. A New Method for On-Line Controller Tuning. AIChE J. 1982, 28, 434. Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 64, 759.
Received for review May 21, 1997 Revised manuscript received November 12, 1997 Accepted November 13, 1997 IE970361W
