3480
Ind. Eng. Chem. Res. 1996, 35, 3480-3483
Tuning Proportional-Integral-Derivative Controllers for Integrator/Deadtime Processes William L. Luyben† Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
Chien and Fruehauf proposed the use of a simple integrator/deadtime transfer function to model many chemical processes, particularly those with large time constants. Tyreus and Luyben presented tuning rules that give the optimal reset time and controller gain for proportionalintegral (PI) control of this type of process. This paper extends the previous work with PI control to proportional-integral-derivative (PID) controllers. Tighter control is possible with PID control, provided signals are not noisy. Frequency domain methods are used to show that the derivative tuning constant should be set equal to the reciprocal of the ultimate frequency. The controller gain is then set equal to 0.46 times the ultimate gain. This process has unusual dynamic behavior when PID control is used, which makes controller tuning nontrivial. The system exhibits conditional stability: at low controller gains the loop is unstable, and at high controller gains the loop is again unstable. Contrary to conventional tuning, a decrease in gain results in an unexpected decrease in closed-loop damping coefficient over a certain range of controller gains. Introduction Chien and Fruehauf (1990) suggested that 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 pure integrator.
GM(s) ) (Kpe-Ds)/s 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. This simple model is very convenient for process identification because it contains only two parameters: Kp and D. These can be calculated directly from the results of a relayfeedback test, which give the ultimate gain Ku and the ultimate frequency ωu.
D ) π/(2ωu) Kp ) ωu/Ku Friman and Waller (1994) demonstrated that this type of model can also be used in multivariable systems. The tuning rules given by Tyreus and Luyben (1992) set the integral and gain tuning constants at the following:
Kc ) Ku/3.2 τI ) 2.2Pu where Pu ) 2π/ωu. No tuning rules for proportionalintegral-derivative (PID) controllers were given in their paper. The current paper extends this work to PID controllers, which are capable of giving smaller closedloop time constants for the same closed-loop damping coefficient. Of course, the use of derivative action is limited to processes in which the signals of the controlled variables are not noisy. The use of PID control †
E-mail:
[email protected]. Telephone: 610-758-4256. FAX: 610-758-5297.
S0888-5885(96)00069-3 CCC: $12.00
is not recommended for most loops because tuning is more difficult, changes in manipulated variables are larger, and the loops are less robust, i.e., more sensitive to changes in process parameters. Therefore, derivative action should only be used when high performance is truly required and large changes in manipulated variables can be tolerated. One typical example is temperature control in an open-loop unstable reactor by manipulating coolant flow rate. Process Studied The numerical case considered is the one used by Tyreus and Luyben in which Kp ) 0.0506% per min and D ) 6 min. The ultimate gain of this process is 5.17, and the ultimate frequency is 0.2618 rad/min. The recommended PI controller settings are Kc ) 1.6 and τI ) 52.5 min. The procedure we follow is to set the reset time equal to that recommended by Tyreus and Luyben and then find the values of derivative time τD and controller gain Kc that give a +2 dB peak in the closed-loop log modulus curve. Frequency domain methods are used because this permits the deadtime in the system to be handled rigorously. Figure 1 gives Nyquist plots of the total open-loop transfer functions (GCGM) for three controllers:
1. Proportional-only (P-only) with Kc set equal to half the ultimate gain (Kc ) 2.6) 2. Proportional-integral (PI) with the settings given by Tyreus and Luyben
3. PID with τI ) 52.5, τD ) 3.82 (the reciprocal of the ultimate frequency), and Kc ) 2.39. These settings are those recommended later in the paper These three curves have their closest approach to the (-1, 0) point at different frequencies. Figure 2 shows © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3481
Figure 1. Nyquist plots for P, PI, and PID control of integrator/ deadtime process.
Figure 2. Closed-loop log modulus plots for P, PI, and PID control.
this more clearly. The closed-loop log modulus LC is plotted versus frequency.
LC ) 20 log10
|
|
GC GM 1 + GCGM
The improvement in performance (wider bandwidth giving smaller closed-loop time constant) in going to PID control is clearly illustrated. These curves also demonstrate the deterioration when a PI controller is used. Note the interesting resonant peaks at higher frequencies, which are particularly apparent in the PID curve. This is due to the multiple approaches to the (-1, 0) point caused by the spiraling of the deadtime element around the origin. Notice also that the low-frequency asymptotes of all these curves are 0 dB, even for P-only control. This is due to the integrator in the process driving the steady-state error to zero even with P-only control. The three controllers have the following transfer functions:
P: GC(s) ) Kc τIs + 1 PI: GC(s) ) Kc τIs
[
PID: GC(s) ) Kc
][
]
τIs + 1 τDs + 1 τIs 0.1τDs + 1
Results Figure 3 gives Nyquist plots for the process with PID controllers for different values of controller gain. The values of reset and derivative tuning parameters are fixed at 52.5 and 3.82 min, respectively. Notice that the contours are close to the (-1, 0) point at low frequencies for low values of gain. They are again close to the (-1, 0) point for high values of gain, but now the closest approach occurs at high frequency. These results indicate that the effect of controller gain on closed-loop damping (as reflected in the closeness of the Nyquist plots to the (-1, 0) point) is different than in conventional processes. We normally expect the closedloop damping coefficient to decrease as controller gain is increased. In the frequency domain, this translates
Figure 3. Nyquist plots showing the effect of controller gain with PID control.
into the peak in the closed-loop log modulus increasing as the controller gain is increased. These results show that this process with PID control has some unusual dynamics. The system exhibits conditional stability. Therefore controller tuning is not straightforward. If the loop is too oscillatory, we might try to decrease the controller gain, but this could make the loop even more oscillatory. Figure 4 shows this effect clearly. The functional relationship between controller gain and maximum closed-loop log modulus LCmax shows a minimum. The curve goes to infinity as gain goes to 0 and again goes to infinity as gain approaches the ultimate gain. Note that there are two values of controller gain that give the same maximum closed-loop log modulus. The lowgain point gives a peak in the closed-loop log modulus curve at low frequencies, and the high-gain point gives a peak at high frequencies. Naturally we would choose the high-gain point to achieve the smaller closedloop time constant (the reciprocal of the resonant frequency ωr). Results for four values of derivative time τD are shown in Figure 4. The table below gives the controller gains and resonant frequencies that give +2 dB LCmax for each value of derivative time.
3482 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996
Figure 4. Effect of controller gain on maximum closed-loop log modulus.
Figure 5. Effect of derivative time on closed-loop log modulus.
These results show that the optimal value of τD is the τD
Kc
ωr
0.5/ωu 1/ωu 2/ωu 3/ωu
2.66 2.39 1.44 0.63
0.183 0.36 0.40 0.018
reciprocal of the ultimate frequency. For larger values, the range of gains that give LCmax values equal to or less than +2 dB is very narrow. This implies that controller tuning would be very difficult and loop robustness would be poor. For smaller values of τD, the resonant frequency ωr decreases, indicating larger closed-loop time constants. Figure 5 shows the LC plots for the four cases with the controller gains given above. As τD is increased, the resonant frequency moves to the right initially, but eventually a low-frequency peak limits the controller gain. Figure 6 compares the time-domain dynamic responses of the process for the three controllers: P, PI, and PID. The disturbance is a unit step change entering through a first-order lag with a 10 min time constant. The improved performance of the PID and P controllers over the PI controller is clear. The larger changes in the manipulated variable are also demonstrated.
Figure 6. Load response for P, PI, and PID control.
Figure 7. Closed-loop log modulus plots for Ziegler-Nichols settings and proposed settings.
A numerical case has been used to explore the process, to demonstrate effects of parameters, and to draw conclusions. However, these results apply to any integrator/deadtime process because all tuning constants are expressed in terms of ultimate gains and ultimate frequencies. This implicitly accounts for different deadtimes and different process gains. The controller gain found in the numerical example was 2.39. We can express this in terms of the ultimate gain to arrive at a general tuning equation.
Kc ) 0.46Ku It is interesting to compare the tuning rules presented in this paper with the tradition Ziegler-Nichols settings. The following table lists the two methods.
Kc τI τD
Ziegler-Nichols
proposed
Ku/1.7 Pu/2 Pu/8
Ku/2.2 2.2Pu Pu/6.3
Figure 7 compares the closed-loop log modulus plots using the two tuning methods. The Ziegler-Nichols settings give much more underdamped behavior (high peak in the closed-loop log modulus curve) for a slightly
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3483
larger closed-loop time constant (smaller resonant or breakpoint frequency).
to achieve the tightest possible control under a fixed set of process parameters.
Conclusion
Nomenclature
The unusual dynamics of the integrator/deadtime process with PID control have been demonstrated. The recommended PID controller tuning procedure is as follows:
Determine the ultimate gain and ultimate frequency, typically using the relay-feedback method
D ) deadtime (min) GM ) process transfer function GC ) feedback controller transfer function Kc ) controller gain Kp ) process gain m ) manipulated variable t ) time (min) s ) Laplace transform variable y ) process output Greek Letters
Set the integral time constant equal to 2.2 times the ultimate period. Set the derivative time constant equal to the reciprocal of the ultimate frequency. Set the controller gain equal to 0.46 times the ultimate gain. It is important to remember that the controller tuning recommended uses a fairly conservative criterion of +2 dB maximum closed-loop log modulus, which corresponds to a closed-loop time constant of about 0.4. This provides a controller that is quite robust (insensitive to changes in process parameters). Improvements in performance (smaller closed-loop time constants) can be obtained, but at the expense of reduced robustness. We have found that most chemical engineering processes perform well using the +2 dB criterion. It is more important to prevent a loop from going unstable than
τD ) derivative time (min) τI ) integral time (min) ω ) frequency (rad/min) ωr ) resonant frequency (rad/min)
Literature Cited Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve performance. Chem. Eng. Prog. 1990, October, 33-41. Friman, M.; Walker, K. V. Autotuning of Multiloop Control Systems. Ind. Eng. Chem. Res. 1994, 33, 1708-1717. Tyreus, B. D.; Luyben, W. L. Tuning PI Controllers for Integrator/ Deadtime Processes. Ind. Eng. Chem. Res. 1992, 31, 26252628.
Received for review February 5, 1996 Revised manuscript received June 18, 1996 Accepted June 19, 1996X IE9600699
X Abstract published in Advance ACS Abstracts, September 1, 1996.
