Ind. Eng. Chem. Res. 1997, 36, 5339-5347
5339
Design of Low-Frequency Compensators for Improvement of Plantwide Regulatory Performance Paul W. Belanger and William L. Luyben* Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Iacocca Hall, 111 Research Drive, Bethlehem, Pennsylvania 18015
One of the major weaknesses of classical feedback controllers is their inability to track ramplike disturbances without offset. This hinders their performance in plantwide environments where load-transfer functions with large load time constants transform step load disturbances into ramp-like disturbances. This paper presents a solution to this problem. Two types of lowfrequency compensators are introduced: double integral compensators and lag compensators. Tuning rules are developed for these compensators for the case where the base controller is of the PI variety. The effectiveness of low-frequency compensation in improving the performance of PI controllers is demonstrated first through the use of a simple transfer function model and then through a rigorous simulation involving a more realistic plantwide system. The idea of low-frequency compensation has been used in other engineering disciplines for many years for tracking ramping setpoint signals (i.e., in the fields of mechanical and electrical engineering, “type 2” servomechanisms are often used when ramps in setpoint signals are anticipated (Chestnut and Mayer, 1951; Nixon, 1954; Franklin et al., 1994), but it has not been used in the process control field for rejecting the effects of frequently occurring ramp-like disturbances. Introduction One of the weaknesses of feedback controllers is that they cannot provide strong control action in the neighborhood of the crossover frequency (the frequency at which oscillations can exist in the absence of external signals). This weakness is necessary in order to ensure stability. It is an inherent limitation of feedback control that cannot be changed, regardless of the type of feedback controller used. No feedback controller (classical, model predictive, fuzzy logic, etc.) can reject completely the effects of disturbance components in the neighborhood of its crossover frequency. Most controllers are tuned as tightly as possible without violating certain robustness criteria (i.e., peak log modulus, gain margin, phase margin, etc.). Increasing the control effort (transfer function magnitude) in the neighborhood of the crossover frequency eventually leads to closedloop instability. There is another weakness, however, that affects the performance of certain classical feedback controllers (i.e., P, PI, and PID controllers) at lower frequencies. In a plantwide environment control systems are often subjected to ramp-like disturbances. This occurs whenever there is a large separation in process time constants (of an order of magnitude or higher). Whenever a “slow” system (a system having at least one large time constant) and a “fast” system (a system having only small time constants) interact, the output of the slow system will be a ramp-like disturbance to the fast system. The output of the slow system will almost never be a pure ramp; however, with respect to the time scale of the fast system, it can be approximated closely by a series of ramps. Large separations in time constants are typical in a plantwide environment. Large process holdups and the presence of material and thermal recycle streams introduce very large time constants. In addition, many inventory control loops are loosely tuned in order to smooth flow rate disturbances. This also * To whom correspondence should be addressed. Phone: (610)758-4256. Fax: (610)758-5297. E-mail:
[email protected]. S0888-5885(97)00287-X CCC: $14.00
introduces large time constants to the process. Quality controllers are often tuned as tightly as possible to prevent the production of offspec product. The closedloop time constants of these loops are small. The standard set of classical controllers mentioned above cannot ordinarily handle ramp-like disturbances well. It is for this reason that the use of standard classical controllers in a plantwide environment will usually result in sluggish performance. To illustrate the points made above, consider a simple transfer function model. Let the transfer function from the manipulated variable to the controlled variable be given by
GM(s) )
1 (s + 1)3
Assume that a PI controller tuned with the TL settings (Tyreus and Luyben, 1992) is used to control the process. The transfer function describing the controller is given by
7.99s + 1 B(s) ) 2.48 7.99s Assume that the time constant of the load-transfer function is so large that it can be approximated by an integrator over a large range of frequencies. Thus:
GL(s) ≈ 1/s The time response of this system when forced by a unit step in load is shown as the solid curve in Figure 1. Note that, although a PI controller is used, the steady-state offset is not 0. This is because the loadtransfer function transforms the unit step in load into a ramp. The PI controller cannot reject the effects of the ramp disturbance, as will be explained below. The dashed curve in Figure 1 illustrates the response of the system to the same disturbance when the gain of the controller is doubled. As expected, the initial response of the system is much more oscillatory (doubling the © 1997 American Chemical Society
5340 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
Figure 1. Time response of the simple example process when forced by a unit step change in load.
The only way to drive this offset to zero would be to alter the structure of the controller such that the product GMB contains a double integrator. There are other tricks, as will be shown later in this paper, that can also be employed to reduce the magnitude of this offset substantially. Even if the load-transfer function is not an integrator, in most plantwide systems the load-transfer function time constant will be so large that it resembles an integrator over a very wide range of frequencies (the breakpoint frequency of the lag will be very low). Thus, while a closed-loop system controlled with a PI controller will exhibit no offset at infinite time when forced with a step in load, the closed-loop performance of the system will still be exceptionally poor (there will still be a high peak in the regulator log modulus curve that will span a wide range of frequencies). Fortunately, the weakness of feedback controllers at low frequencies is a problem that can be easily corrected. In this paper the use of low-frequency compensators is introduced as a tool for improving performance in a plantwide environment. Tuning rules are developed to aid in the design of these compensators. It is shown that the use of compensators is especially effective at rejecting the effects of disturbances in a plantwide environment. Of course, similar or superior benefits may be attained through the application of feedforward compensation; however, since feedforward control requires available load disturbance measurements and an accurate process model, it will not always be a practical solution. Design of a Double-Integral Compensator
Figure 2. Closed-loop regulator log modulus plot of the simple example process.
The first type of compensator considered is the double integrator. The addition of this type of compensator in parallel with an ordinary PI controller forces the error in tracking a ramp disturbance to eventually go to zero. The resulting controller (the PII controller) has the form
[
CO(t) ) Kc (t) + gain reduces the stability of the system); however, the steady-state offset is reduced. Thus, there is a dynamic tradeoff here between the ability of the PI controller to reject the effects of the ramp and the closed-loop stability of the system. The closed-loop regulator log modulus plot of this process is given in Figure 2. Note the sharp peak at high frequency. The magnitude of this peak cannot be increased to improve performance without compromising the robustness of the controller. As mentioned, this is a limitation of all feedback controllers and it cannot be changed. The second limitation noted above can be observed by looking at the low-frequency portion of the log modulus plot. Note that the load disturbance actually excites the controlled variable more at low frequencies than at the resonant frequency. This poor performance at low frequencies is due to the structure of the controller. The controller is not capable of following the ramp disturbance that it sees without some offset (as can be seen by applying the final value theorem of Laplace transforms for a unit step in load):
[
]
GL 1 ) 3.22 Error(tf∞) ) lim s sf0 1 + GMB s
(1)
Equation 1 shows that this system will exhibit offset at infinite time when forced with a unit step in load.
∫
]
∫∫
1 1 (t) dt + ( (t) dt) dt τI1 τI2
(2)
which can be expressed in the Laplace domain as
[
]
τI1τI2s2 + τI2s + τI1 CO(s) ) B(s) ) Kc (s) τI1τI2s2
(3)
With three adjustable parameters, tuning could require a considerable amount of trial and error, so a tuning heuristic is needed. This section focuses on the development of a tuning heuristic that yields a closedloop system with a small time constant and a reasonable damping coefficient (approximately 0.4). The development is very similar to the development of the TL tuning rules for PI controllers (Tyreus and Luyben, 1992). Since most chemical processes have large time constants, it is reasonable to assume that the process model can be approximated with an integrator-deadtime model:
GM(s) ) KM
exp(-Ds) s
(4)
One of the major advantages of using this type of model is that it has only two adjustable parameters, and these can be directly related to the ultimate gain and ultimate period. Of course, the tuning heuristics developed will be applicable only to systems that are open-
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5341
loop stable and have at least one large time constant. For systems with small time constants the tuning rules will be too conservative. The application of the tuning heuristics to an open-loop unstable process could lead to a process that is too oscillatory or even unstable. One assumption that is made up front is that the second reset time constant is selected such that the transfer function given in eq 3 has repeated zeros. This is analogous to saying that we would like to have the double-integral action act over as large a frequency range as possible without affecting the stability of the closed-loop system (without contributing significantly at frequencies near the resonant frequency of the closedloop system). This is a reasonable assumption and greatly simplifies analysis. The transfer function (3) can only have repeated roots if the second reset time (τI2) is related to the first (τI1) by
τI2 ) 4τI12
(5)
Equation 3 simplifies to
B(s) ) Kc
(2τI1s + 1)2
(6)
4τI12s2
The combination of eqs 4 and 6 gives for the total open-loop transfer function
GMB(s) ) KMKc
(2τI1s + 1)2 exp(-Ds)
(7)
4τI12s3
Since the integral reset time dictates what the closedloop time constant will be, it is desirable to set the integral time constant to the smallest value possible. It is still desired, however, to maintain a desired degree of robustness (damping coefficient of 0.4). There will be a limit for the reset time below which a damping coefficient of 0.4 cannot be achieved for any choice of controller gain. The question at this point is, what is this limit? The answer to this question can be found by analyzing the phase angle of the open-loop transfer function. According to a Nichols chart, in order for the closed-loop system to have a peak servo log modulus of +2 dB (analogous to having a damping coefficient of 0.4), the open-loop phase angle must rise to at least -128° or -2.23 rad. The open-loop phase angle as a function of frequency is given by
3 arg(GMB) ) - π - ωD + 2 arctan(2τI1ω) 2
(8)
By differentiating eq 8 with respect to ω and setting the derivative equal to zero, we can solve for the frequency that maximizes the open-loop phase angle. Substituting this value of ω back into eq 8 results in an analytic expression for the peak open-loop phase angle:
(
)
D τI1 1 3 arg(GMB)peak ) - π 2 τI1 D 4
0.5
+
[(
2 arctan 2
)]
τI1 1 D 4
0.5
The minimum reset time that gives an open-loop phase angle peak of -2.23 rad can be found numerically:
τI1 ) 9.05D
(9)
Both reset times can be determined analytically as functions of the process deadtimes from eqs 5 and 9. What remains is to find an expression for the controller gain that gives the desired peak in the closed-loop servo log modulus curve. For a damping coefficient of 0.4, the peak should be close to +2 dB. Arriving at an analytical expression for Kc is difficult since the presence of the deadtime results in an infinite number of poles in the closed loop system. Through the analysis of several numerical cases with different process deadtimes, it was found that the correlation
Kc ) 0.516/KMD
(10)
results in peak log moduli of roughly +2.0 dB. Equations 5, 9, and 10 can be reexpressed in terms of the ultimate gain and ultimate period of the assumed integrator plus deadtime process transfer function:
Kc ) Ku/3.04
(11)
τI1 ) 2.26Pu
(12)
τI2 ) 20.5Pu2
(13)
With the aid of eqs 11-13, the tuning parameters of the PII controller can be found from the results of a single relay-feedback test. Some adjustment of the parameters may be required in order to achieve the desired performance for systems that behave significantly differently from an integrator plus deadtime model; however, these settings provide a good initial estimate. In the sections that follow, the performance and robustness of a PII controller with the given tuning rule are analyzed for a simple system and for a more realistic plantwide system. Design of a Lag Compensator The second type of compensator that is considered is a lag compensator placed in series with a PI controller. The goal here is to increase the amount of control action (i.e., higher gain) at lower frequencies without affecting the behavior of the controller in the neighborhood of the resonant frequency. The resulting lag-compensated PI controller, or LCPI controller, has the same transient response as a PI controller but has a much smaller error in tracking ramp disturbances. Before developing rules for the design of a lag compensator, it is useful to analyze its form in the frequency domain. The lag compensator is given in the Laplace domain by
BLC(s) )
τLCs + 1 RτLCs + 1
(14)
The bode plot of this transfer function for various values of R is given in Figure 3. At very low frequencies the compensator does not contribute to the magnitude or phase of the controller. At very high frequencies the compensator decreases the magnitude of the controller by a factor of R and does not contribute significantly to the phase of the controller. Thus, if the compensator is designed properly (the time constant of the compensator is made large enough), the gain of the controller can be increased by a factor of R without affecting the behavior of the controller in the neighborhood of the resonant frequency. The magnitude of the control action at low frequencies, however, will be increased by
5342 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
for the lag-compensated controller of the form
B(s) ) Kc
Figure 3. Bode plot of the lag compensator.
a factor of R. This decreases the error of the controller in tracking ramp disturbances. It is important to note that, over a certain range of frequencies, the compensator contributes a significant amount of phase lag. This phase lag has a destabilizing effect (decreasing the open-loop phase angle moves the Nichols plot to the left, increasing the height of the resonant peak). It is necessary to design the compensator such that it does not contribute a significant amount of phase lag at frequencies in the neighborhood of the resonant peak. The phase contribution of the lag compensator is given by
)(
τLC ) 11.42/ωresonant will result in a small (5° at most) decrease in the openloop phase angle at the resonant frequency. According to a Nichols chart, this will not significantly increase the height of the closed-loop resonant peak. The heuristic that the resonant frequency in a typical PI closed-loop system is approximately half of the ultimate frequency allows a simple expression for τLC to be obtained:
(15)
With the time constant of the compensator set, all that remains is the selection of a proper R. It was found that setting R ) 10 (equivalent to reducing the error in ramp tracking by a factor of 10) results in fairly good ramp disturbance tracking. Thus, the tuning parameters should be (assuming the PI controller was originally tuned with the TL heuristics)
In this section the performance and robustness of a PII controller and a LCPI controller tuned with the given tuning heuristics are analyzed for a simple analytic process. Consider a system where the process transfer function is
(16)
Kc ) RKu/3.22
(17)
τI ) 2.2Pu
(18)
1 (s + 1)3
(20)
and the load-transfer function is given by
GL(s) )
1 500s + 1
(21)
where the time scale is in minutes. This model is an example of the behavior of a plantwide system. The manipulated variable has an immediate effect on the controlled variable (this is a direct consequence of proper control system structure selection). The system is third order, giving the process a crossover frequency. Deadtime could also have provided a crossover frequency; however, it is of interest to investigate the effectiveness of PII and LCPI control on a system that is significantly different from the integrator plus deadtime model used to develop the tuning heuristics. The load-transfer function has a large time constant to reflect the fact that many loads enter the process at a location where the capacitance is high. Even if this were not the case, the presence of a material or thermal recycle can significantly increase the dominant load time constant. For this system the ultimate gain is 8.0 and the ultimate period is 3.63. The tuning parameters for the PII controller are given by
Kc ) 2.63 R ) 10
(19)
Application to a Simple Transfer Function Model
GM(s) )
For all reasonable values of R the lag compensator contributes less than 5° of phase lag when ωτLC > 11.42 (arctan(11.42) ) 85°). This is reflected in Figure 3. Thus, the assignment
)
τIs + 1 τLCs + 1 τIs RτLCs + 1
With the aid of eqs 15-19, a lag-compensated PI controller can be designed by knowing only the ultimate gain and ultimate frequency. The parameters may need to be adjusted somewhat for systems that deviate significantly from the integrator plus deadtime system used to determine the TL tuning parameters; however, for most chemical process applications it is expected that these rules will lead to satisfactory results. In the sections that follow the performance and robustness of a LCPI controller with the given tuning heuristics are analyzed for a simple transfer function model and for a more realistic plantwide system. Note that the lag compensator is the inverse of a conventional derivative unit (lead/lag). Derivative action is used to provide high-frequency compensation. The lag/lead compensator proposed in this paper provides low-frequency compensation.
arg(BLC(s)) ) arctan(ωτLC) - arctan(RωτLC)
τLC ) 3.64Pu
(
τI1 ) 8.20
τI2 ) 270.1
For a LCPI controller the tuning parameters are given by
Kc ) 24.8
τI ) 7.99
R ) 10
τLC ) 13.2
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5343
Figure 4. Closed-loop regulator log moduli plots: a comparison of PI, PII, and LCPI control.
Figure 5. Comparison of PI, PII, and LCPI responses to ramp disturbances with different slopes.
The tuning parameters of a PI controller tuned with the TL settings are
Kc ) 2.48
τI ) 7.99
Figure 4 gives the closed-loop regulator log moduli plots of the simple system when PI (tuned with the TL settings), PII (tuned with the given heuristics), and LCPI (tuned with the given heuristics) controllers are used. It is clear that the use of the proposed compensators does not affect the behavior of the system at high frequencies. The higher frequency resonant peaks are the same height and are located at the same frequency. The big difference in performance is seen at lower frequencies. While the lower frequency resonant peak is slightly higher when the double-integral mode is used, the frequency range over which the peak is significant is much smaller. The lower frequency peak for the LCPI controller is both smaller and less flat than that for the PI controller. Thus, it can be expected that the performance of both the PII controller and the LCPI controller will be significantly better for ramp disturbances (and will consequently perform better in processes with large load time constants) than that of the PI controller. Figure 5 illustrates the differences in response when various ramp disturbances are introduced directly at the
Figure 6. Response of PII control to a ramp disturbance when Ku and Pu are in error.
output of the system with the third-order transfer function controlled by PI, PII, and LCPI controllers. This approximates what would happen if step changes of various magnitudes were passed through a load-transfer function with a large time constant. As expected, PI control cannot reject the low-frequency effects of the ramp disturbance. The higher the slope of the ramp disturbance is, the worse the performance of the PI controller becomes. The PII controller, on the other hand, is able to eventually bring the controlled variable back to its setpoint. The peak error in the controlled variable increases as the slope of the ramp disturbance increases; however, it can be seen that in all cases the peak error for the PII controller is smaller than the peak error for the PI controller. The peak error of the LCPI controller is even smaller than that of the PII controller. The main disadvantage of the LCPI controller over the PII controller is that it is incapable of driving the error to zero. The final value of the tracking error is, however, 10 times smaller than that of the PI controller. This is because the R factor was set equal to 10. One important question arises with the use of the compensators: does the presence of the compensators cause robustness problems? One favorable characteristic of the compensated controllers is that they function well when applied to a system that is substantially different from the integrator plus deadtime system for which they were designed. The only point that the model predicts exactly is the crossover point. Aside from model mismatch there are other factors that could cause stability problems. For example, when the operating conditions change, the location of the crossover point will change as well. This leads to a situation where the tuning of the controller is inappropriate for the operating conditions. Consider the case where either the ultimate gain or the ultimate period is in error by 50%. This case is illustrated for the compensated controllers applied to the third-order system in Figures 6 (PII) and 7 (LCPI). For comparative purposes the responses of standard PI controllers tuned with the Tyreus-Luyben settings are given in Figure 8. As can be seen in Figures 6 and 7, both of the compensated controllers remain stable despite the large error in ultimate gain and ultimate period. When the ultimate gain of the system is higher than the ultimate gain used to tune the controllers, the control performance becomes sluggish and overdamped because the
5344 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
Figure 7. Response of LCPI control to a ramp disturbance when Ku and Pu are in error.
Figure 9. Comparison of PI, PII, and LCPI control for various load time constants (solid error curves, PI; dashed error curves, PII; dash-dotted error curves, LCPI).
Figure 8. Response of a PI controller with TL settings to a ramp disturbance when Ku and Pu are in error.
controllers are more conservative than they need to be under these conditions. When the ultimate gain is lower than the ultimate gain used to tune the controllers, the closed-loop systems become snappier and more oscillatory; nevertheless, they remain stable. Errors in the ultimate period produce similar responses. Underpredicting the ultimate period results in moving the system closer to instability, while overpredicting the ultimate period results in sluggish performance. A comparison of Figures 6 and 7 with Figure 8 shows that the robustness values (degree of oscillation when the parameters are in error) of the PII and LCPI controllers are about the same as that of a PI controller tuned with the TL settings. All three controllers are tuned to yield a closed-loop resonant peak of about +2 dB (damping coefficient of 0.4). It was found that the PI controller tuned with the ZN settings is unstable when the ultimate period is underpredicted by 50% and is close to being unstable when the ultimate gain is overpredicted by 50%. Thus, the PI controller tuned with the ZN settings is not nearly as robust as the other three controllers. The fact that both of the compensated controllers remain stable despite large errors in ultimate gain and period leads to the conclusion that the PII and LCPI controllers are robust, at least with the tuning heuristics given above.
The major advantage of compensated PI control over conventional PI control is that the compensated PI controller is able to reject the effects of ramp disturbances. This does not mean that the use of a compensator will lead to better control with respect to all types of load disturbances. It is important to study the weaknesses of compensated PI control in order to determine when it should be used and when it should not be used. Consider the effects of making a unit step change in load to a process described by eqs 20 and 21 with various load time constants. The responses of these systems under PI, PII, and LCPI control are illustrated in Figure 9. It can be seen that when τL ) 1, the initial responses of the systems under PI, PII, and LCPI control are nearly the same. The PI controller brings the system gradually back to steady state, while the PII and LCPI controllers bring the system back toward steady state and overshoot the setpoint slightly. It takes a significant amount of time (about 80 min) for the PII controller to bring the system back to steady state. This is because the double integrator gives the controller a considerable amount of momentum. Before the system under PII control can reach steady state, the integral of the error must be driven to zero. This means that the area between the positive portion of the error curve and zero must be equal to the area between the negative portion of the error curve and zero. In the case where τL ) 1, the tail portion of the curve is only slightly less than zero; however, the error remains below zero for a significant amount of time. The LCPI controller does not take as long as the PII controller to recover from the overshoot. This is because the LCPI controller does not require that the integral of the error be driven to
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5345
zero. This is a favorable characteristic of the lag compensator. The differences between the controllers are more significant for the case where τL ) 10. The PII controller still causes the output variable to overshoot its setpoint; however, the overshoot is larger in this case than when τL ) 1. The duration of the overshoot is again large enough to drive the integral of the error to zero. In this case the portion of the error curve below zero is not as flat as it was in the case for τL ) 1. The LCPI controller overshoots its setpoint as well; however, the overshoot is smaller than that for the case of the PII controller, and it takes less time for the LCPI controller to recover. When τL ) 100, the major advantages of compensated PI control become apparent. The peak error corresponding to PI control is significantly higher than that of the systems controlled via PII or LCPI. The peak error is the smallest for LCPI control. This illustrates one of the advantages of using compensators in an environment where the load time constants are large. The PII error curve still exhibits some overshoot; however, the magnitude of the peak overshoot in relation to the peak error is smaller than was the case where τL ) 10. It still takes some time for the PII controller to compensate for the overshoot, but in this case it takes less time for the PII controller to drive the error back to zero than it takes the PI controller. It is obvious in this case that the integral-squared error for the PII error curve is smaller than the integral-squared error for the PI error curve. The response of the system controlled with the LCPI controller can be seen to be superior to both PI and PII in this case. The LCPI controller has the smallest peak error, it drives the error to zero faster than the other two controllers, and there is almost no overshoot. For the case where τL ) 1000 the differences in the peak errors are even larger than before. The error curve for the PII controller still exhibits some overshoot; however, it is only slight. The LCPI controller does not overshoot its setpoint in this case. It can be seen that the PI controller has a significant amount of difficulty under these conditions in driving the error back to zero (it requires approximately 4000 min). During this time period the controlled variable is significantly far away from setpoint. In comparison, the PII and LCPI controllers are able to bring the controlled variable back to its setpoint fairly quickly. It is interesting to compare the shapes of the signals to the controllers to the load time constants. For the case where τL ) 1 the signal seen by the controller almost resembles a step with respect to the shapes of the other curves. It was noted that under these conditions the responses of the three controllers were similar except for the sustained overshoot caused by the PII controller and the slight overshoot caused by the LCPI controller. The signal to the control system for the case where τL ) 10 has a significant amount of curvature when compared to the signals corresponding to the larger time constants but has significantly less curvature than the signal corresponding to the small load time constant. It was noted that under these conditions both the PII and LCPI controllers produce a significant amount of overshoot. For the larger time constants (τL ) 100 and τL ) 1000) the signals to the controllers start to resemble ramp disturbances over short time spans. Under these conditions the PII and LCPI controllers were able to drive the error to zero much faster than
the PI controller. This is mainly due to the inadequacy of PI control with respect to ramp-like disturbances. As far as making quantitative comparisons of performance, the integral-squared errors for the systems using PI, PII, and LCPI control are nearly identical when the load time constant is 10 or less. Large differences in the ISE are seen when the load time constant is 100 or greater. For small load time constants the amount of undershoot resulting from the use of PII and LCPI control is undesirable. For this reason it is recommended that if the load time constant is less than a factor of 10 above the process time constant, the use of compensators should not be considered. If the load time constant is a factor of 100 or higher than the process time constant, then compensation should be used. Lag compensation was found to produce superior results over doubleintegral compensation; however, either compensator results in a dramatic improvement in control performance for large load time constants. With regards to making setpoint changes, the ability of the compensated controller to track very slow ramps in setpoint will be fairly good while the ability of the controller to track step changes in setpoint will be poor. This conclusion can be reached by considering the signals to the controller shown in Figure 9 to be setpoint disturbances rather than as the output of the load-transfer function. Thus, if compensated control is used and frequent setpoint changes are anticipated, then it may be necessary to turn off the compensator when setpoint changes are made. Improvement in Plantwide Regulatory Performance Since many plantwide processes have load-transfer functions with very large time constants, it can be expected that the use of compensated PI control will significantly improve performance with respect to load rejection. A good example of a plantwide process with large load time constants has been presented in Belanger (Belanger and Luyben, 1997). This process is illustrated in Figure 10. A brief description of the process is given here. For more details the reader is referred to the given citation. In this system a CSTR is used to convert reactant “A” into product “B” via a first-order irreversible reaction. The temperature of this reactor is such that the reaction rate constant is k ) 0.34 h-1. Fresh feed is introduced to the CSTR at a rate of Fo lb mol/h. The composition of the feed stream is ZI0, ZA0, and ZB0 where “I” is an inert component that has a volatility between that of A and B. The relative volatilities of the three components are RI ) 1.33, RA ) 2.0, and RB ) 1.0. The reactor effluent stream is fed to a distillation column. The overhead of the column (which contains most of the unreacted reactant) is recycled back to the CSTR. The bottoms product of the first column is fed to a second distillation column where inert is separated from product. The bottoms product is withdrawn at a rate of 239.5 lb mol/h and has a purity of 98.85 mol % B. The inert that is separated from the product is removed from the column as the overhead product. In this system the reactor effluent flow rate is manipulated to regulate reactor inventory. A loosely tuned proportional-only controller is used. The flow rates of the bottoms stream of each column are used to regulate base levels. The accumulator inventories are controlled by manipulating reflux flow rates. The recycle is manipulated to maintain a constant reflux
5346 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
Figure 10. Plantwide system under consideration.
ratio in the first column. Column pressures are maintained through the manipulation of cooling water flows. All loops governing inventories (i.e., liquid levels and column pressures) are assumed “perfect” with the exception of the reactor inventory control loop. The flow rate of steam to the first column reboiler is used to control the amount of A in the bottoms product of the first column (xAB1). The flow rate of steam to the second column reboiler is used to control the purity of the product stream. The purge flow rate is used to regulate the amount of I in the purge. All compositions are measured directly, and there is a 3 min deadtime associated with each composition measurement. Originally, in the cited paper, PI composition controllers were used. These controllers were tuned with the TL settings (Tyreus and Luyben, 1992). In this section the benefits of using low-frequency compensators on this system are demonstrated. Consider the case where all of the PI composition controllers are replaced with PII and LCPI controllers. A comparison of responses to a 50 lb mol/h step increase in fresh feed flow rate and to a 2 mol % step increase in
feed inert level is given in Figure 11. As can be seen, the improvement in load rejection performance is significant. A comparison between Figure 11 and Figure 9 illustrates the relative magnitudes of the load-transferfunction time constants. The time constant of the loadtransfer function relating composition disturbances to product composition is very large. The response of the product composition with respect to the step in inert level resembles the case where the load time constant is 1000 times that of the process time constant. The use of compensated control results in a large improvement with respect to this type of load disturbance. Similarly, the time constant relating the effects of inert changes to purge composition is also large. There is a strong similarity between the response of the purge composition to the step change in inert levels and the response of the third-order system with respect to a step change in load with a load time constant 100 times that of the process time constant. Compensated control is effective at rejecting the effects of the inert change on purge composition. Thus, it can be seen that the use of
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5347
shown that for systems where the load time constant is less than a factor of 10 larger than the process time constant the use of compensators is undesirable. This is because of the overshoots caused by the use of the compensators. It was mentioned that for setpoint changes the compensator should be turned off. The effectiveness of the PII and LCPI controllers in a plantwide environment was demonstrated. It was shown that compensated control resulted in better inert load rejection as well as better rejection of the effects of flow rate disturbances on product composition. The success of compensated control in the given plantwide system was largely due to the presence of large load time constants. In general, it is expected that the application of compensators will result in similar improvements in plantwide performance in other systems. Nomenclature Figure 11. Comparison of PI, PII, and LCPI control for various load time constants (solid error curves, PI; dashed error curves, PII; dash-dotted error curves, LCPI).
compensators leads to a significant improvement in the rejection of disturbances in inert loading. The trajectories of the product composition when a step in feed flow rate is made resemble a cross between the trajectories seen in the analytic case where the load time constant is 100 times the process time constant and where the load time constant is 1000 times the process time constant. Application of compensated control results in a significant improvement in the rejection of the effects of feed flow rate changes on product composition. The effects of flow rate changes on purge composition are not significantly improved. The time constants relating changes in feed flow rate to purge composition are smaller than the other time constants. The overshoot caused by the double-integral action is not large, however, and since the control of purge composition is not as critical as the control of product composition, it is worth the price for the improved control over the other effects. Since flow rate can be easily measured, there is a possibility that a feedforward controller could be designed to compensate for this effect. Conclusions Two types of compensated PI controllers have been studied in this paper. The tuning heuristics for the controllers have been presented. The superiority of the compensated controller over the traditional PI controller with respect to rejecting the effects of ramp-like disturbances has been demonstrated. This has been related to the load rejection performance of systems with large load time constants using each type of controller. It was shown that the use of compensated control results in significant improvements in load rejection performance over the use of PI control when the load time constant of a process is a factor of 100 or more larger than the process time constant. It was also
B ) transfer function of the controller CLLM ) closed-loop regulator log modulus CO ) controller output D ) deadtime GL ) load-transfer function GM ) manipulated variable transfer function Kc ) controller gain Ku ) ultimate gain Pu ) ultimate period s ) Laplace domain variable t ) time Greek Symbols R ) “gain” of the lag compensator ) error τI ) integral reset time (ordinary PI) τI1 ) integral reset time (PII) τI2 ) double integral reset time (PII) τLC ) time constant of the lag compensator ω ) frequency
Literature Cited Belanger, P. W.; Luyben, W. L. Ind. Eng. Chem. Res. 1997, in press. Chestnut, H.; Mayer, R. W. Servomechanisms and Regulating System Design; John Wiley & Sons: New York, 1951. Franklin, G. F.; Powell, J. D.; Emami-Naeini, A. Feedback Control of Dynamic Systems; Addison-Wesley Publishing Co.: New York, 1994. Nixon, F. E. Principles of Automatic Controls; Prentice-Hall: New York, 1954. Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/ dead time processes. Ind. Eng. Chem. Res. 1992, 31, 2625-2628.
Received for review April 11, 1997 Revised manuscript received July 30, 1997 Accepted July 31, 1997X IE970287H
X Abstract published in Advance ACS Abstracts, November 1, 1997.
