Ind. Eng. Chem. Res. 1999, 38, 4113-4119
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Partitioned Error Control Kenneth A. Debelak* and Mark L. Rutherford Department of Chemical Engineering, Box 1604 Station B, Vanderbilt University, Nashville, Tennessee 37235
We propose a new two-degrees-of-freedom control structure. Two-degrees-of-freedom controllers are usually implemented when there are stringent requirements on both setpoint tracking and disturbance rejection. Two-degrees-of-freedom controllers are not new; however, the structure of our controller is novel and has several desirable features. It is also a model-based two-degreesof-freedom controller, but is not quite the same as an internal model control controller. Under the perfect model assumption, the characteristic equations for the setpoint change and load disturbance are distinct and include the process model and the controller designed to handle each disturbance. This control structure allows for separate controllers to be designed to handle setpoint changes and load disturbances. Another desirable feature is that it allows for simple closed-loop determination of the process model. For processes where both setpoint changes and load disturbances occur, controller design is a compromise for a single feedback controller (one-degree-of-freedom controller). For example, if setpoint changes are implemented by ramp inputs and disturbances enter in the form of step changes, then the controller has to be chosen for either a good response to ramp changes in the setpoint or a good response to step changes in the disturbance, or else some compromise must be found. In some cases, an acceptable compromise may not exist. For these types of control systems, where both setpoint changes and load rejections are important, a twodegrees-of-freedom controller is required. Horowitz1 first proposed and studied the two-degrees-of-freedom controller. There are very many examples of two-degreesof-freedom controllers in the literature, too many to list. A few examples from chemical engineering include Morari and Zafirou2 who extended two-degrees-offreedom controllers into the internal model control (IMC) structure, and Lundstrom and Skogestad,3 Limbeer et al.,4 Van Digglen and Glover,5 and Skogestad et al.6 who have used two-degrees-of freedom controllers for distillation column control. A typical block diagram of a two-degrees-of-freedom controller is shown in Figure 1. The controller shown in Figure 1 has two components, a feedback controller, Gc, and prefilter, Gr. The feedback controller, Gc, handles the disturbance inputs, and Gr shapes the setpoint tracking response. Gc and Gr can be designed simultaneously. However, in practice, Gc is designed first to handle disturbances, and then Gr is designed to track the setpoint. A twodegrees-of-freedom control system in which the controllers can be independently designed would be desirable. The controllers should be capable of distinguishing between setpoint changes and load disturbances. We propose the following control structure (see Figure 2). This control structure includes a model of the process, G/p, and two controllers, Gc1 and Gc2, which can be independently tuned to handle both setpoint changes and load disturbances. We call this configuration partitioned error control (PEC). Partitioned error control belongs to the family of model-based controllers, which * To whom correspondence should be addressed. Phone: (615)322-2088. E-mail:
[email protected].
Figure 1. Two-degrees-of-freedom controller.
Figure 2. Partitioned error control structure.
includes direct synthesis control, internal model control, and generic model control. It is also a two-degrees-offreedom controller. The closed-loop transfer function for this system is given by
C)
[
]
(Gc1 - Gc2)Gp Gc2Gp + (R) + 1 + Gc2Gp (1 + G G )(1 + G G/ ) c2 p c1 p GL (L) (1) 1 + Gc2Gp
If the plant and the model match, i.e., Gp ) G/p, then the closed-loop transfer function reduces to
C)
Gc1Gp GL (R) + (L) 1 + Gc1Gp 1 + Gc2Gp
(2)
The two controllers can be tuned independently for setpoint changes and load disturbances. In addition, the characteristic equation for each response is only dependent on the controller designed for each response.
10.1021/ie990220p CCC: $18.00 © 1999 American Chemical Society Published on Web 09/09/1999
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Figure 3. Comparison of closed-loop setpoint response for the process given by eq 4 for PEC, a ZN-tuned controller, and a TLCtuned controller.
Figure 4. Comparison of closed-loop load response for the process given by eq 4 for PEC, a ZN-tuned controller, and a TLC-tuned controller.
Table 1. Controller Tunings
use Murrill’s settings for a load disturbance in a PI controller. A prefilter, Gr, would be designed to improve the setpoint tracking.
Murrill Kc/τ1 setpoint load 0.28/7.39
Ziegler-Nichols Kc/τ1
TLC Kc/τ1
0.43/9.98
0.3/26.4
0.42/6.48
If Gc1 ) Gc2, then the closed-loop transfer function reduces to
C)
Gc2Gp GL (R) + (L) 1 + Gc2Gp 1 + Gc2Gp
4e-3.5s 7s + 1
c
Solving for Gr,
(3)
The system behaves like an ordinary feedback system. Partitioned error control allows for different controller designs and tunings for the setpoint controller and the load controller and is applicable for general single-input/ single-output (SISO) systems. As an example, consider the SISO system where
Gp ) GL )
GcG/p C ) Gr R 1 + G G/
(4)
Table 1 gives the proportional-integral (PI) controller settings according to the tuning rules of Murrill,7 Lopez et al.,8 and Rovira et al.,9 which minimize the ITAE performance index for setpoint changes and load disturbances, along with the Zielger-Nichols (ZN) settings and the TLC (“tender loving care”) settings of Tyreus and Luyben.10 The closed-loop setpoint and load responses are given in Figures 3 and 4. PEC uses the different load and setpoint tunings suggested by Murrill and assumes Gp ) G/p. The ITAE values are calculated for unit step changes in the setpoint and load and are included in the figures. For the ZN and TLC responses, one set of controller tunings is used for setpoint changes and load disturbances. On the basis of integral of the time-weighted absolute error (ITAE) values, PEC performs better than the controllers tuned with the ZN and TLC settings. This is expected since the advantage of the PEC system is that the control structure allows for different controller settings and/or different controller structures. Now let us compare the PEC design with a conventional two-degrees-of-freedom controller. A conventional two-degrees-of-freedom controller design would
Gr )
(
1+
)() -1
GcG/p GcG/p
(5)
p
C R
(6)
Ref
where (C/R)Ref is a desired reference trajectory. It should be possible to design Gr for any desired reference trajectory. However, Gr could be unstable if G/p has right-half-plane zeros or unrealizable if G/p contains a time delay. The design of Gr as well as the parameters in Gc require that G/p ) Gp. A suitable choice of reference trajectory for the transfer function given in eq 4 would be
(RC)
) Ref
1 e-τrs λs + 1
(7)
where λ is chosen to obtain the desired speed of response and τr is chosen to allow Gr to be realizable. If Gc is a PI controller, and G/p is a first-order-plus-dead-time transfer function with a gain, Kp, a time constant, τ, and a time delay, τd, then Gr is given by
Gr )
τI s(τs + 1) KcKp (τIs + 1)(λs + 1)
e-(τd-τr)s +
e-τrs (λs + 1)
(8)
Let G/p to be equal to the transfer function given in eq 4, where Kp ) 4, τ ) 7, and τd ) 3.5, and Gc be a PI controller with Murrill’s settings. The prefilter, Gr, is given by
6.48 s(7s + 1) (0.42)(4) e-3.5s + Gr ) (6.48s + 1)(λs + 1) (λs + 1)
(9)
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Figure 5. Comparison of closed-loop setpoint response for the process given by eq 4 for PEC and two-degrees-of-freedom controller with a prefilter design.
Figure 7. Process response using setpoint controller design.
For setpoint changes a typical plant-inverse controller would be
Gc1(s) ) (ωc/s)G-1 p
(11)
Skogestad and Postlewaite’s setpoint controller design is based on the plant inverse as described by eq 11. They choose the crossover frequency ωc ) 10 rad/s. and approximate the term (0.05s + 1)2 by (0.1s + 1). Then, in the controller, they allow this term to be effective over 1 decade by using a lag-lead, (0.1s + 1)/(0.01s + 1), to give a realizable design. The controller is Figure 6. Process model from Skogestad and Postlewaite.
A digital implementation of this prefilter would be feasible; however, it is a nonstandard controller form. The values of Kc and τI in Gc for the different controller tunings, Murrill’s, Ziegler-Nichols, and TLC would result in the same setpoint response since the desired response is specified by eq 7. The prefilter, Gr, would be different for each case. An alternative approach is to design the prefilter as a lag-lead element:
Kr )
τleads + 1 τlags + 1
(10)
If we want a quick response τlead > τlag, or for a slower response τlead < τlag. Figure 5 compares the PEC design with the prefilter design given by eq 8 for values of λ ) 3.5 and 7 for a setpoint change. With λ ) 3.5, the ITAE for the prefilter design is sligthly better than that of the PEC design. Decreasing λ results in a more tightly tuned controller, which would be more susceptible to model error. Consider a second example, shown in Figure 6 from Skogestad and Postlewaite.11 Their control objectives are as follows: (1) Setpoint Tracking: The rise time should be less than 0.3 s and the overshoot should be less than 5%. (2) Disturbance Rejection: The output in response to a unit step disturbance should remain within the range [-1, 1] at all times, and it should return to 0 as quickly as possible (|y(t)| should at least be less than 0.1 after 3 s). (3) Input Constraints: u(t) should remain in the range (-1, 1) at all times to avoid input saturation.
Gc1 )
10 10s + 1 0.1s + 1 s 200 0.01s + 1
(12)
Using loop-shaping design methods, Skogestad and Postlewaite designed the following controller for disturbance rejection:
Gc2 ) 0.5
s + 2 0.05s + 1 s 0.005s + 1
(13)
Implementing either of the controller designs as a onedegree-of-freedom controller will not satisfy all of the control objectives. Skogestad and Postlewaite indicate that controller Gc2 satisfies all the requirements for disturbance rejection, but the overshoot is 24% for the setpoint tracking. When the inverse-based controller is applied to a disturbance input, the response is very sluggish and the output is at 0.75 at time ) 3 s, where it should be less than 0.1. The two cases are shown in Figures 7 and 8. However, if we implement these two controllers in the PEC control structure shown in Figure 2, the above criteria are met for both setpoint and load disturbances (Figure 9). Let us now compare the PEC controller with a prefilter design. Skogestad and Postelwaite designed a two-degrees-of freedom prefilter controller using eq 6. From a step-response test, they approximated (GcG/p)/(1 + GcG/p) by the sum of two transfer functions:
0.5 0.7s + 1 1.5 ) 0.1s + 1 0.5s + 1 (0.1s + 1)(.5s + 1)
(14)
They chose a reference trajectory, (C/R)Ref ) 1/(0.1s + 1), a first-order response with no overshoot. The result-
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Figure 10. Comparison of PEC controller with a two-degrees-offreedom prefilter for the process described in Figure 6.
Figure 8. Process response using load controller design.
an inverse response or a delay, the setpoint controller could be tuned more aggressively. In all model-based systems the question of process model mismatch arises. How does PEC handle plantmodel mismatch? Control of setpoint changes involves both the partitioning and feedback loops as indicated in eq 1. Qualitatively, under model mismatch, PEC will perform like a feedback controller that uses a weighted output of Gc1 and Gc2 to control the process. The stability of the PEC system is determined by the characteristic equation of eq 1. For a load disturbance, the characteristic equation involves only the load controller, Gc2, and the process transfer function, Gp. This is true for both the PEC and the conventional configuration. Equation 1 can be rewritten for a setpoint change as
C) Figure 9. PEC controller using both setpoint and load controllers.
ing prefilter for the two-degrees-of-freedom controller is
Gr )
0.5s + 1 0.7s + 1
(15)
which was slightly modified to prevent the control signal from exceeding 1 to
Gr )
0.5s + 1 (0.65s + 1)(0.03s + 1)
(16)
A comparison between the setpoint response between the prefilter design and the PEC design is shown in Figure 10. The ITAE for the PEC response is much less than that for the prefilter design. It reaches the setpoint quicker and is less oscillatory. The load response for the prefilter design and the PEC design would be the same since they both use the controller given in eq 13. PEC would be applicable to processes, which for example exhibit an inverse response or a response with a significant delay. The controller settings for these two responses significantly detune the controller in comparison to those for undelayed or noninverse responses. If the effect of the manipulated variable did not have
Gc1Gp(1 + Gc2G/p)
R
(1 + Gc2Gp)(1 + Gc1G/p)
(17)
The characteristic equation includes both controllers, the actual process model, and the estimate of the process model. One way to estimate the effect of process model mismatch is to plot the closed-loop frequency diagram for the conventional control structure and for the PEC structure for different errors in the estimates of the process gain, time constant, and time delay. As the process approaches its limit of stability, the maximum closed-loop log modulus will approach infinity. Essentially, as the maximum closed-loop log modulus approaches infinity, the process is approaching the point (-1, 0) of a Nyquist plot. After the limit of stability is reached, the point (-1, 0) is encircled and the maximum closed-loop modulus will decrease. This occurs because the maximum log modulus is only an indication of how close we are to the point (-1, 0). Figure 11 is a closedloop frequency plot for the conventional control structure in which errors in the process gain of 10%, 25%, and 40% were used to determine the PI controller settings for the process described by eq 4, using the setpoint settings suggested by Murrill from Table 1. The maximum closed-loop modulus increases as the error in the process gain increases, characteristic of an increasingly oscillatory response. The process does not reach its limit of stability until the error in the process gain reaches about 63%. Similar plots are shown in
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Figure 11. Peak of the closed-loop log modulus for conventional PI controllers using Murrill’s setpoint tunings with errors in the estimate of the gain for the process given in eq 4.
Figure 13. Peak of the closed-loop log modulus for conventional PI controllers using TLC tunings with errors in the estimate of the gain for the process given in eq 4.
Figure 12. Peak of the closed-loop log modulus for conventional PI controllers using ZN tunings with errors in the estimate of the gain for the process given in eq 4.
Figure 14. Peak of the closed-loop log modulus for PEC controllers using Murrill’s setpoint tunings with errors in the estimate of the gain for the process given in eq 4.
Figures 12 and 13 for conventional controllers using ZN and TLC settings. The controller tuned with the ZN settings is the most sensitive to errors in the process gain. The peak in the closed-loop modulus increases the fastest for a controller tuned with ZN settings. The TLCtuned controller is the least sensitive. Figure 14 is a closed-loop frequency plot for the PEC structure with the same errors in the estimates of the process gain. Murrill’s controller settings are used for the setpoint and load controllers. For an error in the estimate of the gain of 10%, the peak in the closed-loop log modulus, -0.0014, is less for the PEC system than the peak in the closed-loop log modulus, 0.4867, for the conventional controller. However, as the error increases, the maximum in the closed-loop log modulus increases at a faster rate for the PEC structure than for the conventional control structure. The PEC structure is more sensitive to larger errors in the estimates of the gain than is the conventional control structure. If one examines the characteristic equation, eq 17, one can see that there is a multiplicative effect, since both Gc1 and Gc2 appear in
the characteristic equation. Their values are determined by the estimate of the process, G/p. Similar plots could be made for errors in the estimates of the time constant and dead time. When compared with the ZN-tuned conventional controller, the peak in the closed-loop log modulus is lower for the PEC controller than for the conventional controller using ZN tunings at the same level of error in the estimate of the process gain. Figure 15 shows the closed-loop log modulus for the example of Skogestad and Postelwaite for errors in the process gain of 10%, 25%, and 40% using eq 13 for the controller and eq 16 for the prefilter. The peak in the closed-loop log modulus is minimal, even at an error of 40% in the process gain. Figure 16 shows the peak in the closedloop log modulus using the PEC structure, with the setpoint controller, eq 12, and the disturbance controller, eq 13 of Skogestad and Postlewaite, for errors in the process gain of 10%, 25%, and 40%. At a 10% error in the process gain, the difference between the two structures is minimal. However, as the error increases, the PEC structure is more sensitive to errors in the gain,
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Figure 15. Peak of the closed-loop log modulus for the twodegrees-of-freedom controller design of Skogestad and Postlewaite for the process given in Figure 6.
Figure 16. Peak of the closed-loop log modulus for PEC controllers using Skogestad and Postlewaite’s setpoint and disturbance controllers for the process given in Figure 6.
as indicated by the increasing size of the peak in the closed-loop log modulus curve. If you look at the outputs from the two controllers for a setpoint change, you will find that when G/p ) Gp, the output from controller Gc2 is zero. As the mismatch between G/p and Gp increases, the output from controller Gc2 increases. The control action becomes the sum of the outputs from controllers Gc1 and Gc2. The output of controller Gc2, m, can be given by the following transfer function:
Gc1Gc2(G/p - Gp) m ) R (1 + G G )(1 + G G/ ) c2
p
c1
(18)
p
If G/p ) Gp, then the output from controller Gc2 will be equal to zero, for any input R. The output from controller Gc2 can be monitored during setpoint changes. If Gc2’s output deviates from zero, it is an indication of plantmodel mismatch, i.e., G/p * Gp. The deviation of Gc2’s output from zero can be used as an indicator by an
Figure 17. Comparison of the setpoint response for the process described by eq 4 with no error in the estimate of the process gain and noise in the measurement signal for a control system using PEC and Murrill’s tunings with a control system using a prefilter and ZN tunings.
Figure 18. Comparison of the setpoint response for the process described by eq 4 with a 10% error in the estimate of the process gain and noise in the measurement signal for a control system using PEC and Murrill’s tunings with a control system using a prefilter and ZN tunings.
adaptive controller, e.g., Foxboro’s Exact and Control Soft’s Intune, of when to update the model. This feature of the PEC structure could be easily incorporated into an adaptive controller. Astrom and Wittenmark12 have established techniques for identifying models, which can be used by an adaptive controller. Adjustments of the model parameters in G/p can be made through a series of on-line setpoint tests until the output from controller Gc2 is zero. Noise in process control can arise from the measuring devices, the surroundings, or the process. How does PEC compare with a conventional controller in a noisy environment? To compare the effect of noise on a PEC and a conventional controller, band-limited white noise was included in the feedback signal to the controller. The primary difference between band-limited white noise and white noise is that the band-limited white noise block produces output at a specific sample rate, which is related to the correlation time of the noise.
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Similar results are also found when comparing the prefilter design with the PEC design for Skogestad and Postlewaite’s process model. Partitioned error control offers the opportunity to implement separate conventional-type controllers for setpoint changes and load disturbances with little additional computational effort. PEC’s advantage is that both controllers can be designed and tuned separately to optimize the process response. In addition, the PEC structure provides a means by which one can determine when process/model mismatch has occurred. The partitioned error control structure becomes more sensitive to process modeling errors than the conventional control structure as the magnitude of the errors increase. Literature Cited
Figure 19. Comparison of the setpoint response for the process described by eq 4 with a 40% error in the estimate of the process gain and noise in the measurement signal for a control system using PEC and Murrill’s tunings with a control system using a prefilter and ZN tunings.
Theoretically, continuous white noise has a correlation time of 0, a flat power spectral density (PSD), and a covariance of infinity. In practice, physical systems are never disturbed by white noise, although white noise is a useful theoretical approximation when the noise disturbance has a correlation time that is very small relative to the natural bandwidth of the system. Figures 17-19 compare setpoint changes for the process described by eq 4 for a control system using PEC and a PI controller using Murrill’s settings, with a control system using a prefilter and PI controller with ZN tunings for different levels of error in the estimate of the process gain. The noise power and sampling time for the band-limited white noise are set at 0.005 and 0.1 minutes, respectively. When Gp ) G/p, PEC performs better than a prefilter with ZN tunings, both with respect to ITAE and rise time (Figure 17). With a 10% error in the estimate of the process gain, PEC still performs better than a prefilter with ZN tunings (Figure 18). However, as the error in the estimate of the gain increases to 40%, the performance of the PEC deteriorates below that of the prefilter with ZN tunings (Figure 19). The controller performance with a prefilter and TLC tunings will be similar, since the setpoint response is determined from the prefilter design equation, eq 8.
(1) Horowitz, I. M. Synthesis of Feedback Systems; Academic Press: London, 1963. (2) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (3) Lundstrom, P.; Skogestad, S.; Doyle, J. C. Two-Degrees-ofFreedom Controller Design for an Ill-Conditioned Distillation Process Using µ-Synthesis. IEEE Trans. Control Sys. Technol. 1999, 7, 12. (4) Limbeer, D. J. N.; Kasenally, E. M.; Perkins, J. D. On the Design of Robust Two-Degrees-of-Freedom Controllers. Automatica 1993, 29, 157. (5) Van Digglen, F.; Glover, K. A. Hadamard Weighted Loop Shaping Design Procedure. Proc. IEEE Conf. Decision Control (Tucson, AZ) 1992, 2193. (6) Skogestad, S.; Morari, M.; Doyle, J. C. Robust Control of Ill-Conditioned Plants: High Purity Distillation. IEEE Trans. Automat. Control 1988, 1092. (7) Lopez, A. M.; Murrill, P. W.; Smith, C. L. Controller Tuning Relationships Based on Integral Performance Criteria. Instrum. Technol. 1967, 14, 57. (8) Murrill, P. M. Automatic Control of Processes; International Textbook: Scranton, PA, 1967. (9) Rovira, A. A.; Murrill, P. W.; Smith, C. J. Tuning Controllers for Set-point Changes. Instrum. Control Syst. 1969, 42, 67. (10) Tyreus, B. D.; Luyben, W.L. Tuning PI Controllers for Integrator/Dead Time Processes. Ind. Eng. Chem. Res. 1992, 32, 2625. (11) Skogestad, S.; Postlewaite, I. Multivariable Feedback Control, Analysis and Design; John Wiley and Sons Ltd.: New York, 1996. (12) Astrom, K. J.; Wittenmark, B. Computer Controlled Systems; Prentice Hall: Englewood Cliffs, NJ, 1984.
Received for review March 22, 1999 Revised manuscript received July 22, 1999 Accepted August 11, 1999 IE990220P
