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Ind. Eng. Chem. Res. 2001, 40, 2654-2659
Tuning Smith Predictors Using Simple Formulas Derived from Optimal Responses Ibrahim Kaya† Engineering Faculty, Electrical and Electronics Department, Inonu University, 44069 Malatya, Turkey
Good control of processes with long dead time is often achieved using a Smith predictor configuration. However, not much work has been carried out on obtaining simple tuning rules for a Smith predictor scheme. This paper develops optimal analytical tuning formulas for proportional-integral-derivative (PID) controllers in a Smith predictor configuration assuming perfect matching. Exact limit cycle analysis has been used to estimate the unknown parameters of a first-order plus dead time (FOPDT) or second-order plus dead time (SOPDT) plant transfer function. Simple analytical tuning rules based on these FOPDT and SOPDT are then derived which can be used to tune a PID controller in a Smith predictor scheme. Some examples are given to show the value of the approach presented. 1. Introduction Plants with long time delays can sometimes not be controlled effectively using a proportional-integralderivative (PID) controller in the conventional singlefeedback loop structure. The main reason for this is that the additional phase lag contributed by the time delay tends to destabilize the closed-loop system. The stability problem can be solved by decreasing the controller gain. However, in this case the response obtained is very sluggish. The Smith predictor, shown in Figure 1, is well-known as an effective dead-time compensator for a stable process with long time delays.1 The closed-loop transfer function for the system of Figure 1 is given by
Gc(s) G(s) e-Ls C(s) T(s) ) ) R(s) 1 + Gc(s) [Gm(s) + Ge(s)]
(1)
where Ge(s) ) G(s) e-Ls - Gm(s) e-Lms. Gm(s) e-Lms, G(s) e-Ls, and Gc(s) are respectively the plant’s dynamic model and the transfer functions of the plant and the controller, which is usually a PI or a PID controller. The stability of the Smith predictor is affected by the accuracy with which the model represents the plant. Based on the assumption that the model used matches perfectly, the plant dynamics, Ge(s) ) 0, and the closedloop transfer function of eq 1 reduces to
T0(s) )
Gc(s) Gm(s) e-Lms 1 + Gc(s) Gm(s)
(2)
According to eq 2, the parameters of the primary controller, Gc(s), which is typically taken as PI or PID, may be determined using a model of the delay free part of the plant. Many possible approaches for determining or tuning the parameters of an appropriate controller, Gc(s), have been given in the literature, and recent contributions include refs 2-5. The method proposed by Kaya and Atherton2 replaces the conventional controller, Gc(s), by a PI-PD or PI-P structure where the PD or P part is †
Fax: +90 422 3401046. E-mail:
[email protected].
Figure 1. Smith predictor control scheme.
implemented in an inner feedback loop. The choice of selection of the controller structure in the inner loop, that is, to choose a PD or only a P, depends on the model order. The tuning parameters of the PI-PD or PI-P controllers have been obtained using standard forms,6 and it was shown that the PI-PD or PI-P structure can give superior performance. The difficulty with the design is to involve a tradeoff between selected values of Kp and Ti, the gain and integral time constant of the PI controller in the forward path, respectively. Ha¨gglund’s method is based on a first-order plus dead time (FOPDT) model which is obtained from a step-response test. The step-response test is an open-loop test; therefore, if there is any external disturbance during the identification procedure, it may lead to large errors. Many studies have been devoted to the development of tuning rules based on optimization. The main disadvantage of using optimization as a design criterion is that the transfer function of the plant must be known. To eliminate this disadvantage, a possible approach is to use the relay autotuning method to estimate an FOPDT model7,8 and then optimize the controller parameters based on this assumed model transfer function.9,10 In section 3, this approach is carried out for the delay free part of the FOPDT plant transfer function which can be used to find tuning parameters of a PI or PID controller in a Smith predictor configuration if perfect matching is assumed. In refs 9 and 10, a FOPDT model was also used to approximate higher order plant transfer functions. A better approximation may be achieved with a second-order plus dead time (SOPDT) plant transfer function, which is considered in this paper, and it is shown how simple tuning formulas can be obtained. If the plant dynamics changes, the performance of the closed loop may deteriorate. To maintain
10.1021/ie000194r CCC: $20.00 © 2001 American Chemical Society Published on Web 05/19/2001
Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2655
Figure 2. SISO relay feedback system.
an improved closed-loop performance for mismatching cases, a filter with a steady-state gain of 1 in the outer loop of the Smith predictor can be used.11 2. Parameter Estimation To obtain analytical expressions for use in finding the controller tuning parameters, a model of the plant transfer function must be available. The relay feedback method (see Figure 2) has gained much attention in recent years because Astro¨m and Ha¨gglund successfully applied the technique to automatic tuning of PID controllers.12,13 It is used here to estimate the unknown parameters, using exact limit cycle analysis, of a stable FOPDT or SOPDT plant transfer function. There are several methods14-18 in the literature to estimate parameters of FOPDT or SOPDT plant transfer functions using the relay feedback method. However, most of them use the approximate describing function method. Because only two measurements of the limit cycle amplitude and frequency are made for use with the describing function analysis and there are three unknowns for a FOPDT and four for a SOPDT plant transfer function, two relay tests must be performed. Kaya and Atherton7,8 suggested parameter estimation under asymmetric limit cycle oscillations so that all of the unknown parameters of a FOPDT or SOPDT plant transfer function can be obtained in one relay test. The detailed analysis of this estimation method7,8 is not given here, but interested readers may refer to the cited publications. 3. Simple Tuning Formulas Using Optimization Optimization has always been a powerful design method to determine controller parameters. The integral of squared error, ISE, criterion is one of the most wellknown criteria, but it generally results in a significant overshoot and a relatively long settling time. The timeweighted versions of the ISE criterion give relatively smaller overshoots and comparable settling times. In the next subsections, simple tuning formulas are obtained using the ISTE and IST2E criteria for a PI or PID controller in a Smith predictor configuration based on either FOPDT or SOPDT models. 3.1. Tuning for a PI Controller. First, a FOPDT model, Gm(s) e-Lms ) Ke-Lms/(Ts + 1), and a PI controller with the ideal transfer function
Gc(s) ) Kp(1 + 1/Tis)
(3)
are considered. Assuming a perfect matching between the plant and model dynamics, the error for the Smith predictor structure from Figure 1 is
E(s) )
R(s) 1 + Gc(s) Gm(s)
(4)
Repeated optimizations were carried out on this error for R(s), a step input, using eq 3 for Gc(s) and K/(Ts + 1)
Figure 3. PI controller parameters when a FOPDT model is used over the range κ ) 2.50-5.00. The continuous curve shows the fitting produced by the formula, and asterisks show T/Ti as a function of KKp. Table 1. PI Tuning Formulas Based on a FOPDT Model κ ) KKp range 1.50-2.50 a b
2.60-5.00
5.10-15.00
ISTE
IST2E
ISTE
IST2E
ISTE
IST2E
1.9443 -0.4722
1.3037 -0.2426
1.5556 -0.2173
1.1209 -0.0666
1.2149 -0.0679
1.0196 -0.0075
for Gm(s), for different values of normalized gain, κ ) KKp. Figure 3 shows the relationship between the normalized gain κ and T/Ti for the ISTE and IST2E criteria for a set-point change over the range of 2.505.00 for KKp. The formula
1 a ) (KKp)b Ti T
(5)
was obtained to fit the graphical results, using a leastsquares fitting technique. In the graph, the continuous curve shows the fitting produced by the formula and the asterisks show T/Ti as a function of KKp. To obtain a good fit to the data for κ between 1.5 and 15, the formula was used over three ranges and the coefficients are listed in Table 1 for the ISTE and IST2E criteria. Once the controller gain is specified, which is chosen so that the normalized gain κ falls in one of the ranges given in Table 1, the controller integral time constant Ti can be calculated from eq 5. When a SOPDT model, Gm(s) e-Lms ) Ke-Lms/[(Tis + 1)(T2s + 1)], is used, calculations can be carried out in a way similar to that used for the FOPDT model. Using Kp(1 + 1/Tis + Tds) for Gc(s) and K/[(T1s + 1)(T2s + 1)] for Gm(s), the error given by eq 4 was minimized for the ISTE and IST2E criteria, and some of the results for T1/Ti versus κ ) KKp are given in Figure 4. The equation
()
T2 a 1 ) (KKp)b T i T1 T1
c
(6)
was used to fit the graphical data using a least-squares curve-fitting method. The coefficients in the equation are listed in Table 2 for various ranges of normalized gain, κ. A close investigation of the table reveals that the value of c is always around -0.50 for both criteria; thus,
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Figure 4. PI controller parameters when a SOPDT model is used. The continuous curve shows the fitting produced by the formula, and asterisks show T/Ti as a function of KKp.
eq 6 can be rearranged as
()
T1 1 a ) (KKp)b Ti T1 T2
1/2
(7)
The required a and b values are given in Table 2. 3.2. Tuning for a PID Controller. When the controller in a Smith predictor scheme is chosen as a PID controller, then the process must be modeled by a SOPDT transfer function. The reason for this is that when the delay free part of the FOPDT, K/(Ts + 1), and the ideal PID form are used in optimization to minimize the error given by eq 4, the optimization procedure may not converge to a solution. Some of the results for T1/Ti and L/T1 for set-point change, again using the ISTE and IST2E criteria, are given in Figure 5. Curve fitting in a least-squares sense gives
() ()
a1 T2 1 ) (KKp)b1 Ti T 1 T1 Td ) T1a2(KKp)b2
Table 2. PI Tuning Formulas Based on a SOPDT Model
c1
T2 T1
() ()
κ ) KKp range
(8) 1.50-2.50 c2
(9)
The a, b, and c coefficients are given in Table 3 for different ranges of κ ) KKp. Again the value of c was found to be around -0.50 for T1/Ti and 0.50 for L/T1 for both criteria and ranges. Therefore, eqs 8 and 9 can be rearranged as
a1 T1 1 ) (KKp)b1 Ti T1 T2
Figure 5. PID controller parameters. The continuous curve shows the fitting produced by the formula, and asterisks show T/Ti as a function of KKp.
a b c
ISTE
IST2E
0.6873 -0.4022 -0.5000
0.6341 -0.2112 -0.4999
2.60-5.00
5.10-15.00
ISTE
IST2E
ISTE
IST2E
0.6345 -0.3051 -0.5008
0.6315 -0.1965 -0.5000
0.6647 -0.3295 -0.4998
0.6638 -0.2274 -0.4998
Table 3. PID Tuning Formulas Based on a SOPDT Model κ ) KKp range 1.50-2.50 ISTE
IST2E
2.60-5.00 ISTE
IST2E
5.10-15.00 ISTE
IST2E
(11)
a1 0.6781 0.6338 0.5916 0.5554 0.5169 0.4950 b1 -0.2709 -0.2410 -0.1144 -0.0888 -0.0323 -0.0184 c1 -0.5000 -0.5000 -0.5000 -0.5002 -0.5001 -0.5001 a2 0.1058 0.1057 0.2446 0.2453 0.4049 0.3978 b2 1.3371 1.3041 0.3877 0.3637 0.0767 0.0640 0.5003 0.4998 0.5002 0.4999 0.5000 0.4999 c2
where the a1, a2, b1, and b2 values are again given by Table 3. It should be pointed out that the fits used to find the a, b, and c values have some degree of bias (see Figures 3-5). However, in all cases those bias values do not exceed 1% or 2%. It should also be pointed out that eqs 5-11 are obtained using either a FOPDT or SOPDT transfer
function model. However, they can be used for a wide range of process transfer functions because most plant transfer functions used in the industry can be modeled by either a FOPDT or SOPDT model. The Smith predictor structure performs well for a process with a large time delay. However, it has a drawback of requiring an accurate matching between the plant and model dynamics; otherwise, the closedloop performance of the system, usually, deteriorates,
Td ) T1a2(KKp)b2
1/2
T2 T1
(10)
1/2
Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2657
Figure 6. ISTE criterion value for different percentage changes in the plant time delay.
Figure 7. Step responses for example 1 for a -30% change in the time delay with and without a filter.
especially for time delay mismatches. Hence, to guarantee a better performance in the case of a mismatch between the plant and model dynamics, a filter with low-pass characteristic can be used.11 This has been illustrated later by an example. 4. Illustrative Examples In this section several examples are given to show the use of the method. The first example considers a FOPDT transfer function to illustrate the performance of the system with and without the filter. In the second example, a third-order system is considered to show that second-order modeling gives a better approximation for processes with higher order transfer functions. In the last example, a process with a second-order transfer function is given to compare the performance of the proposed design method with some existing methods. Example 1. A first-order transfer function
G(s) )
1 e-10s 5s + 1
is considered. The model transfer function is obtained accurately using the relay feedback method. When the controller gain Kp is constrained to 1.5, then the integral time constant is calculated as Ti ) 3.1143 using Table 1 and eq 5. With these controller parameters, the performance of the system is investigated for different percentage changes in the plant time delay. Figure 6 shows the value of the ISTE criterion for different percentage changes in the plant time delay when no filter and a filter with two different values are used in the feedback loop. When a filter is used in the feedback loop, the minimum value of the integral decreases, illustrating a better performance, The larger the percentage change, the larger the effect of the filter. Figures 7 and 8 illustrate the step responses for +30% and -30% change in the time delay with and without a filter. It is seen that a better closed-loop performance can be achieved with the filter in the feedback. Example 2. A third-order transfer function of
G(s) )
2 e-5s (4s + 1)(3s + 1)(2s + 1)
is given. Using the identification method given by Kaya and Atherton,7,8 the FOPDT model parameters were obtained as K ) 2.001, T ) 7.325, and L ) 8.328 and
Figure 8. Step responses for example 1 for a +30% change in the time delay with and without a filter.
the SOPDT model parameters were K ) 2.001, T1 ) 4.068, T2 ) 4.068, and L ) 6.164. The control design methods given in section 3 were implemented to compare the performance in each case. The controller gain was limited to 2.0 in each of the three cases. Then using the equations given in section 3.1, the integral time constants for a PI controller with FOPDT and SOPDT models were calculated as Ti ) 6.369 and 9.792, respectively. The parameters of a PID controller were obtained as Ti ) 8.062 and Td ) 1.707, using the equations given in section 3.2. The responses of the closed-loop system to a step input for the three controllers in a Smith predictor structure are given in Figure 9. The best result, as expected, was achieved with a PID controller in the Smith predictor structure. A PI controller in the Smith predictor configuration, when the FOPDT model is used, gives the worst response. This makes sense because a Smith predictor is sensitive to modeling errors and a FOPDT transfer function will not give the best model for the higher order plant. A PI controller in a Smith predictor structure with the SOPDT model results in a fast response, but the settling time is slightly longer when compared to the response with a PID controller in the Smith predictor structure. Figure 10 shows control signals for the example. It is seen that a PID controller with a SOPDT model in the Smith predictor has also the smallest initial control signal.
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Figure 9. Step responses for example 2.
Figure 11. Step responses for example 3.
Figure 10. Control signals for example 2.
Figure 12. Control signals for example 3.
Example 3. A process with a SOPDT transfer function given by
have been given. The method gives fast closed-loop responses with a small overshoot and short settling time for perfect matching, but in the case of a mismatch between the plant and model dynamics, the response of the system may deteriorate. This is particularly true when the time delay to time constant ratio is large and a mismatch occurs in the time delay. To achieve a better closed-loop response and stability, a low-pass filter with a steady-state gain of 1 can be used in the outer feedback loop. If the closed-loop performance is not satisfactory even when a filter is used in the feedback loop, the relay feedback test can be reperformed to find new model parameters. New controller parameters can then be recalculated using the new model parameters to maintain the closed-loop performance.
G)
e-5s (10s + 1)(5s + 1)
is considered. The model of the plant is obtained accurately from an asymmetric limit cycle data using the relay feedback method. Constraining the controller gain Kp to 2.5 results in the remaining tuning parameters of Ti ) 13.273 and Td ) 2.467, using the ISTE criterion and eqs 10 and 11. Response of the Smith predictor with these controller parameters to a unit step and a disturbance of d ) -0.1 at t ) 80 s is shown in Figure 11. Similar results for the design methods of Palmor,4 Hang,3 and Ha¨gglund5 are also given in this figure for comparison. The design methods proposed by Ha¨gglund and Hang give slow closed-loop responses. Palmor’s method gives good responses but results in a slightly longer settling time than the proposed method. The control signals for the design methods are shown in Figure 12, which shows that Palmor’s method gives a larger initial control signal. 5. Conclusions In this paper simple tuning rules for a PID controller in a Smith predictor structure using the ISTE and IST2E criteria based on FOPDT or SOPDT plant models
Literature Cited (1) Smith, O. J. A Controller to Overcome Dead Time. ISA J. 1959, 6, 28-33. (2) Kaya, I.; Atherton D. P. A new PI-PD Smith Predictor for Processes with Long Dead Time. 14th IFAC World Congr. 1999, 282-288. (3) Hang, C. C.; Wang, Q. G.; Cao, L. S. Self-tuning Smith Predictors for Processes with Long Dead Time. Int. J. Adapt. Signal Process. 1995, 9, 255-270. (4) Palmor, Z. J.; Blau, M. An Auto-tuner for Smith Dead Time Compensator. Int. J. Control 1994, 60, 117-135. (5) Ha¨gglund, T. A Predictive PI Controller for Processes with Long Dead-Time Delay. IEEE Control Syst. Mag. 1992, 12, 5760.
Ind. Eng. Chem. Res., Vol. 40, No. 12, 2001 2659 (6) Graham, D.; Lathrop, R. C. The Synthesis of Optimum Response: Criteria and Standard Forms. II. Trans. AIEE 1953, 273-288. (7) Kaya, I.; Atherton, D. P. An Improved Parameter Estimation Method using Limit Cycle Data. Int. Conf. Control ’98 1998, 1, 682-687. (8) Kaya, I.; Atherton, D. P. Parameter Estimation from Relay Autotuning with Asymmetric Limit Cycle Data. J. Process Control 2001, in press. (9) Smith, A. C.; Corripio, B. A. Principles and Practice of Automatic Process Control; John Wiley & Sons: New York, 1985. (10) Zhuang, M.; Atherton, D. P. Automatic Tuning of Optimum PID Controllers. IEE Proc. D 1993, 140, 216-224. (11) Normey-Rico, J. E.; Bordons, C.; Camacho, E. F. Improving the Robustness of Dead-Time Compensating PI Controllers. Control Eng. Pract. 1997, 5, 801-810. (12) Astro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of Simple Regulators. 9th IFAC World Congr. 1984, 1867-1872. (13) Astro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of Simple Regulators with Specification on Phase and Amplitude Margins.
Automatica 1984, 20, 645-651. (14) Luyben, W. L. Derivation of Transfer Functions for Highly Nonlinear Distillation Columns. Ind. Eng. Chem. Res. 1987, 26, 2490-2495. (15) Li, W.; Eskinat, E.; Luyben, W. L. An Improved Autotune Identification Method. Ind. Eng. Chem. Res. 1991, 30, 1530-1541. (16) Shen, S. H.; Wu, J. S.; Yu, C. C. Use of Biased-Relay Feedback for Autotune Identification. AIChE J. 1996, 42, 11741180. (17) Wang, Q. G.; Hang, C. C.; Zou, B. Low Order Modelling from Relay Feedback Systems. Ind. Eng. Chem. Res. 1997, 36, 375-381. (18) Kaya, I. Relay Feedback Identification and Model Based Controller Design. D. Phil. Thesis, University of Sussex, Brighton, Sussex, U.K., 1999.
Received for review February 7, 2000 Revised manuscript received October 2, 2000 Accepted February 27, 2001 IE000194R
