Ind. Eng. Chem. Res. 2000, 39, 973-976
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Tuning Proportional-Integral Controllers for Processes with Both Inverse Response and Deadtime William L. Luyben† Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
Many tuning methods have been proposed over the last half-century. Most apply to deadtime/ lag processes, but several have studied either integrating processes or inverse response processes. Very few have explored processes in which both deadtime and inverse responses occur. This type of response is observed in adiabatic tubular reactors when reactor outlet temperature Tout is the controlled variable and reactor inlet temperature Tin is the manipulated variable. Typically the control structure uses a cascade arrangement in which the secondary loop uses furnace heat input or heat-exchanger bypassing to control Tin. This paper addresses the problem of controlling processes that exhibit both inverse response and deadtime. The Ziegler-Nichols tuning method recommended in the literature is shown to give poor performance at both small and large values of deadtime and positive-zero time constant. A new tuning method is proposed in which the PI tuning constants are presented as functions of the positive zero τz and the deadtime D. Results are given in the form of easy-to-use tuning charts. 1. Introduction Many chemical processes can be modeled for the purpose of feedback controller tuning by a transfer function containing only a gain, a deadtime, and a firstorder lag.
Chien and Fruehauf4 suggested IMC tuning. Tyreus and Luyben5 (TL) developed rules that can be used for both the integrating and lag processes.
KTL ) Ku/3.2
(5)
τTL ) 2.2Pu
(6)
-Ds
GM(s) )
Kpe τos + 1
(1)
This type of model is able to adequately represent the dynamics of many processes over the frequency range of interest for feedback controller design, i.e., near the ultimate frequency where the total open-loop Nyquist plot approaches the (-1, 0) point. There have been many papers over the last 5 decades that present tuning rules for these types of processes. One of the most frequently referenced methods is that of Ziegler-Nichols (ZN).
KZN ) Ku/2.2
(2)
τZN ) Pu/1.2
(3)
However, ZN tuning is known to be too underdamped for small deadtimes and too sluggish for large deadtimes (Hang et al.1 and Ho et al.2). Probably the most effective rules for the deadtime/lag process are those presented by Marlin.3 Tuning rules have also been developed for integrating processes with deadtime.
GM(s) ) Kpe-Ds/s
(4)
† E-mail:
[email protected]. Telephone: 610-758-4256. Fax: 610-758-5297.
They are more conservative than those of ZN and therefore give a better performance with small values of deadtime. However, the TL settings result in a very sluggish performance when deadtimes are large. Controller tuning for processes with inverse response has been discussed by several workers. The literature up to 1975 was reviewed by Waller and Nygardas,6 who concluded that the complex pole-zero compensation control structures “offered no significant advantage” over standard ZN proportional-integral-derivative (PID) settings. It should be noted, however, that they did not cover a wide range of parameter values. Tyreus and Luyben7 studied a reactor/preheater process that had deadtime, inverse response, and open-loop instability. They demonstrated the unexpected beneficial effect of using integral action to stabilize the system. Ogunnaike and Ray8 reviewed several tuning methods for inverse-response processes. There appears to be very little consideration of processes with both deadtime and inverse response. This paper explores this problem. It was motivated by some recent studies of adiabatic tubular reactors that exhibited this type of behavior. We wish to control reactor outlet temperature Tout by changing reactor inlet temperature Tin. The control structure uses a cascade arrangement in which the secondary loop uses furnace heat input or heat-exchanger bypassing to control Tin. The primary controller holding Tout changes the setpoint of the Tin secondary controller.
10.1021/ie9906114 CCC: $19.00 © 2000 American Chemical Society Published on Web 02/25/2000
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Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 Table 1. Ultimate Gain and Period deadtime D 0
0.2
0.4
0.8
1.6 Figure 1. Open-loop step response of process with positive zero.
2. Process Studied The transfer function for a process with inverse response has a positive zero at s ) +1/τz.
GM(s) )
Y(s) U(s)
-Ds
)
Kp(-τzs + 1)e
(7)
(τo1s + 1)(τo2s + 1)
The rest of the transfer function can consist of several dynamic elements, depending on the complexity of the response. In this paper we study the inverse-response process given in the equation above in which there are two lags and deadtime, in addition to the positive zero. The open-loop unit step response of this process is shown in Figure 1. The parameter values used in this figure are Kp ) τo1 ) τo2 ) 1. The larger the magnitude of τz, the larger the initial drop in the output Y. For example, when τz ) 1.6, the response drops to a negative value that is almost 40% of the final positive steadystate value, and it takes almost two time constants (τo1) to come back to the initial steady-state value of zero. Thus, large values of τz give open-loop responses that look somewhat like large deadtimes. Because the ZN settings are quite sluggish for large deadtime processes, we would expect that ZN tuning would give sluggish responses when τz values are large. We demonstrate that is indeed true in the next section. Therefore, there is a need for an improved tuning procedure. The process given in eq 7 has a phase angle that drops below -180° even when D ) 0 because the positive zero drops the phase angle an additional 90°. A normal negative zero (a “lead”) would increase the phase angle, not decrease it. Therefore, this process has an ultimate gain and ultimate frequency, despite the fact that the order of the denominator is only 2. When there is no deadtime (D ) 0), the ultimate gain and ultimate frequency can be calculated analytically. Using a proportional controller (GC(s) ) Kc) gives the closed-loop characteristic equation
1 + GC(s)GM(s) ) 1 +
KcKp(-τzs + 1)e-Ds (τo1s + 1)(τo2s + 1)
) 0 (8)
(τo1τo2)s2 + (τo1 + τo2 - KcKpτz)s + 1 + KcKp ) 0
(9)
τz
Pu
Ku
0.2 0.4 0.8 1.6 0.2 0.4 0.8 1.6 0.2 0.4 0.8 1.6 0.2 0.4 0.8 1.6 0.2 0.4 0.8 1.6
1.89 2.56 3.36 4.19 2.81 3.37 4.15 4.99 3.53 4.06 4.77 5.73 4.77 5.23 6.00 6.97 6.81 7.21 8.00 9.08
10.0 5.00 2.50 1.25 5.49 3.58 2.10 1.15 3.92 2.89 1.88 1.09 2.65 2.20 1.61 1.03 1.82 1.66 1.37 0.991
Substituting s ) iω into eq 9 and separating into real and imaginary parts give
[1 + KcKp - τo1τo2ω2] + i[ω(τo1 + τo2 - KcKpτz)] ) 0 + i(0) (10) At the ultimate frequency (ω ) ωu), the controller gain is the ultimate gain (Kc ) Ku). Equation 10 gives two equations, which can be solved for the two unknowns (ωu and Ku).
KuKp ) (τo1 + τo2)/τz ωu )
x
1 + (τo1 + τo2)/τz τo1τo2
(11) (12)
When the deadtime is not zero, the ultimate gain and frequency can be found numerically using an iterative procedure. The phase angle is
arg G ) -ωD - arctan(ωτz) - arctan(ωτo1) arctan(ωτo2) (13) At the ultimate frequency, argG ) -π, giving eq 14, which can be solved for ωu given the values of D, τo1, and τo2.
-π ) -ωuD - arctan(ωuτz) - arctan(ωuτo1) arctan(ωuτo2) (14) The ultimate period is Pu ) 2π/ωu. Then the ultimate gain can be calculated.
1 + (ωuτo1)2x1 + (ωuτo2)2 x ) Ku ) |G(iω )| Kpx1 + (ωuτz)2 1
(15)
u
Table 1 gives results of these calculations for a range of deadtimes and positive zeros. The values of the other parameters used in the table are Kp ) τo1 ) τo2 ) 1. As expected, ultimate gains decrease and ultimate periods increase as τz increases and as D increases.
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Figure 3. Closed-loop log modulus with ZN and proposed tuning.
In the left part, the deadtime and positive-zero time constant are both small. The ZN tuning gives a large peak of about +7 dB, which indicates a small damping coefficient. In the right part, the deadtime and positivezero time constant are both large. The ZN tuning gives small values of Lc at low frequencies, indicating a sluggish return to setpoints. It is clear that an improved tuning procedure is needed. 4. Proposed Tuning Method
Figure 2. Closed-loop response with ZN and proposed tuning: (A) D ) 0.2; (B) D ) 1.6.
3. Simulation Results with ZN Tuning The open-loop process given in eq 7 is simulated with a PI controller using the ZN tuning rules.
GC(s) ) Kc
τIs + 1 τIs
(16)
Waller and Nygardas6 recommended using ZN tuning with PID controllers. In this paper we use PI controllers because in many real industrial applications the presence of noise prevents the use of derivative action. Using PI controllers also gives a more conservative design, which is more robust. Figure 2A gives results when D ) 0.2 for a range of τz values. The responses are for unit step disturbances in setpoint. The left part shows that the ZN settings give quite oscillatory responses for small values of τz but give sluggish responses for large values of τz. When the τz values are about the same as the lag time constants (τo ) 1), the ZN settings work well. This is the range explored by Waller and Nygardas.6 Figure 2B gives results when D ) 1.6. Now the ZN response is sluggish for all values of τz. Figure 3 gives a frequency-domain representation of these results. The servo closed-loop log modulus Lc is given for two sets of conditions.
Lc ) 20 log
| | Y(s)
set Y(s)
) 20 log
|
GC(s)GM(s)
1 + GC(s)GM(s)
|
(17)
The sluggish responses for large values of D and/or τz suggest that reset time constants need to be smaller than those given by the ZN method to increase the rate at which the process is driven to the setpoint. Also for small values of τz the reset values should be larger than the ZN values to reduce the oscillatory response. A simple empirical equation was developed that varied the reset time constant as a function of both deadtime and the value of the positive zero. Figure 4A gives a graph of the relationships. The ordinate is the Pu/τI ratio. The smaller this ratio, the larger the reset time and the more conservative the tuning. Remember that ZN tuning sets this ratio at 1.2 (see eq 3). The TL tuning method sets this ratio at 1/2.2 ) 0.45, which gives larger reset values. For small values of either D or τz, the Pu/τI ratio is reduced. When either D or τz becomes larger, a larger Pu/τI ratio is used. In equation form, the relationship is
Pu/τI ) [0.5 + 1.56τz] + [3.44 - 1.56τz]D
(18)
The first term on the right-hand side of the equation is the intercept on the y axis when D ) 0. It increases as τz increases, giving smaller reset values. The second term gives the slopes of the lines times D. The dependence on deadtime varies with τz, with the slope becoming bigger as τz increases. Once the reset time was established, the controller gain was varied to find the value that gave a peak in the closed-loop log modulus curve of +2 dB. The results of these calculations are given in Figure 4B. The ordinate is the Ku/Kc ratio. The smaller this ratio, the larger the controller gain relative to the ultimate gain. Remember that ZN tuning sets this ratio at 2.2 (see eq 2). The TL tuning method sets this ratio at 3.2, which gives smaller gain values. These results show that the larger the value of τz, the smaller the ratio (the larger the gain). There is a
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Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000
) τo1 ) τo2 ) 1, so the values of the parameters can be considered to be approximately dimensionless with respect to τo and Kp. So, all the reported values of Kc are really KpKc, and all the reported values of τI, Pu, and time are really τI/τo, Pu/τo, and t/τo. 5. Conclusion A new controller tuning procedure is proposed for processes whose dynamics include both inverse response and deadtime. The method has been tested for a wide range of parameter values. It is simple to use and gives a performance that is much better than the ZN tuning method, which has been recommended in the literature for inverse-response processes. Nomenclature D ) deadtime GM ) process transfer function GC ) feedback controller transfer function Kc ) controller gain Kp ) process gain KTL ) Tyreus-Luyben gain KZN ) Ziegler-Nichols gain Lc ) closed-loop log modulus Pu ) ultimate period t ) time s ) Laplace transform variable U ) manipulated variable Y ) process output Yset ) setpoint τI ) reset or integral time constant τo ) process open-loop time constant τTL ) Tyreus-Luyben reset time constant τz ) positive-zero time constant τZN ) Ziegler-Nichols reset time constant ωu ) ultimate frequency (rad/time) Figure 4. Proposed tuning: (A) reset ratio; (B) gain ratio.
nonmonotonic dependence on deadtime. As D increases from zero, the Ku/Kc ratio initially decreases. It reaches a minimum at about D ) 0.2-0.4 and increases thereafter. These results could be put into equation form by fitting the curves with quadratic equations, but we have not attempted this job. Returning to Figure 2, we can see a direct comparison between the proposed tuning method and the ZN results. The sluggish responses for large D’s and τz’s are eliminated. The underdamped responses for small D’s and τz’s are also eliminated. A word of caution is appropriate at this point. This tuning method was developed for a range of deadtimes and positive-zero time constants from 0 to 1.6. From our studies to date, we think this is the range of practical interest. We have tested the method on deadtime values up to 3.2 and found that it gives good performance up to this point. However, the user is cautioned not to extrapolate this correlation beyond this point because it may fail at some point. A final note is also in order about the generic applicability of the results. We have used values to Kp
Literature Cited (1) Hang, C. C.; Astrom, K. J.; Ho, W. K. Refinements of the Ziegler-Nichols tuning formula; IEE Proceedings D; IEE: Herts, U.K., 1991; Vol. 138, No. 2, pp 111-118. (2) Ho, W. K.; Hang, C. C.; Zhou, J. Performance and gain and phase margins of well-known PI tuning formulas. IEEE Trans. Control Syst. Technol. 1995, 3, 245-248. (3) Marlin, T. E. Process Control; McGraw-Hill: New York, 1995. (4) Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve performance. Chem. Eng. Prog. 1990, Oct, 33-41. (5) Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/deadtime processes. Ind. Eng. Chem. Res. 1992, 31, 2625-2628. (6) Waller, K. V. T.; Nygardas, C. G. On inverse response in process control. Ind. Eng. Chem. Fundam. 1975, 14 (3), 221-223. (7) Tyreus, B. D.; Luyben, W. L. Unusual dynamics of a reactor/ preheater process with inverse response, deadtime and openloop instability. J. Proc. Control 1992, 3 (4), 241-251. (8) Ogunnaike, B. A.; Ray, W. H. Process Dynamics, Modeling and Control; Oxford University Press: Oxford, U.K., 1994; p 608.
Received for review August 10, 1999 Revised manuscript received December 6, 1999 Accepted January 11, 2000 IE9906114
