Ind. Eng. Chem. Res. 2005, 44, 4617-4624
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PROCESS DESIGN AND CONTROL A New Ratio Control Architecture† Antonio Visioli‡ Dipartimento di Elettronica per l’Automazione, University of Brescia, Via Branze 38, 25123 Brescia, Italy
In this paper, a new ratio control architecture is proposed. It aims at providing high performance in both set-point following and load disturbance rejection specifications. A tuning procedure is given in order to set all the parameters in the control scheme so that the design effort required by the user is kept at a minimum level. Simulation and experimental results show the effectiveness of the methodology. 1. Introduction Proportional-Integral-Derivative (PID) controllers are the controllers most adopted in industry because they are able to provide satisfactory performances for a wide range of processes, despite their relative simplicity. Further, they can be employed effectively as the basis of more complex control schemes where couplings between simple control systems are exploited. A significant example in this context is ratio control, which consists of keeping a constant ratio between two process variables.1 There are a variety of industrial applications where this is of concern, such as chemical dosing, water treatment, chlorination, mixing vessels, and waste incinerators. For example, in combustion systems it is necessary to control accurately the air-to-fuel ratio in order to obtain high efficiency, and in blending processes a selected ratio of different flows has to be maintained to keep a constant product composition.2,3 It is well-known that in the past 60 years, a major effort has been made by researchers to develop useful techniques for the implementation of the basic PID algorithm (tuning and automatic tuning methods) and of additional functionalities such as antiwindup, gain scheduling, and adaptive control.4 However, nowadays the increase of computational capability available in Distributed Control Systems (DCS) as well as in singlestation industrial controllers motivates the development of new methodologies for the design and the implementation of more complex PID-based control systems. Obviously, in addition to the achievement of high performances, the ease of understanding and of use is a major requirement for any new technique to be suitable for application in industrial environments. For example, operators should not need to select values of parameters whose physical meaning is not clear, and in any case their work should be simplified as much as possible by the presence of automatic tuning techniques. In this context, to improve the classic architectures, a new ratio control structure (the so-called “Blend station”) has been proposed recently in ref 5 and this † This work was supported in part by MIUR scientific research funds. ‡ Tel.: +39-030-3715460. Fax: +39-030-380014. E-mail:
[email protected].
idea has been further developed in ref 6. However, in these works only the set-point following task is addressed and should load disturbances occur in the plant the devised methods are not appropriate. In this paper, a new ratio control architecture is proposed. In contrast to the approaches mentioned above, it aims at achieving good set-point following and load disturbances rejection performances at the same time. A tuning procedure for the overall control scheme is devised in order to avoid any additional design effort from the user. The paper is organized as follows. In section 2 a short introduction to ratio control is provided. In section 3 the new ratio control architecture is proposed. The tuning procedure is revealed in section 4. Simulation and experimental results are presented in sections 5 and 6, respectively, while conclusions are drawn in section 7. 2. Overview of Ratio Control The aim of a ratio control system is to keep the ratio between the values of two process variables y1 and y2 equal to a constant value a, to meet some higher level requirements, irrespective of possible set-point changes and load disturbances that might occur on the plant. For this purpose, the control scheme shown in Figure 1 (also termed series metered control) is generally implemented. Each variable is controlled by two separate controllers C1 and C2 (typically of PI type) and the output y1 of the first process is multiplied by a and adopted as the set-point signal of the closed-loop control system of the second process, i.e., it is r2(t) ) ay1(t).1 The main disadvantage of this scheme is related to its transient response to a change in the set-point r1 since the output y2 is necessarily delayed with respect to y1, due to the closed-loop dynamics of the second loop. In general, the second loop is chosen as the one with the fastest dynamics. However, to keep the ratio close to the desired value, it might be necessary to detune the first loop and therefore the performance obtained in the set-point following task and in the rejection of the load disturbance d1 decreases. A possible alternative scheme is the one shown in Figure 2 (termed parallel metered control). In this case, provided that the two closed-loop systems have the same
10.1021/ie048893h CCC: $30.25 © 2005 American Chemical Society Published on Web 05/18/2005
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Figure 1. The typical ratio control scheme (series metered control).
Figure 4. The new ratio control architecture.
the constant parameter γ in the expression (1) with a dynamic system F. The control scheme is shown in Figure 4. The transfer function F(s) is determined in such a way that the transfer function from r1 to y1 is the same as the transfer function from r1 to y2 scaled by a. When the control scheme shown in Figure 4 is analyzed, it can be deduced that Figure 2. An alternative ratio control scheme (parallel metered control).
y2 )
C2P2 F + C1P1 ar 1 + C2P2 1 + C1P1 1
(2)
C1P1 r 1 + C1P1 1
(3)
and
y1 ) Figure 3. The cross-limited control scheme.
dynamics, high performance can be achieved in the setpoint following task, but obviously, a disturbance acting on the first process can cause a large error in the ratio value. For this reason, this approach is not often adopted in practical cases. To overcome the drawback of the scheme of Figure 1, alternative approaches have been proposed recently. In particular, in ref 5 the use of the so-called Blend station has been devised; namely, the reference signal r2(t) is calculated as
r2(t) ) a(γr1(t) + (1 - γ)y1(t))
(1)
where γ is a constant parameter that provides a compromise between the two schemes mentioned above. This approach has been further extended in ref 6 where a time-varying parameter γ(t), determined by means of a PI control law based on the ratio error, is employed. However, while these techniques allow the increase of the set-point following performances, they are detrimental in the load disturbance rejection capabilities of the control scheme. Finally, it is worth remembering that in the particular case of combustion control, where a selected air-to-fuel ratio has to be maintained, it is often essential to prevent the occurrence of a fuel-rich environment since this might lead to a furnace explosion. In this context, the so-called cross-limited control (also known as leadlag control) shown in Figure 3 can be adopted.7 The two loops are interlocked by using low and high selectors that force the fuel to follow the air flow when the setpoint increases and that force the air to follow the fuel when the set-point decreases. The new architecture presented in this paper can be straightforwardly accommodated, according to this idea, to address this case. 3. New Ratio Control Architecture The ratio control architecture proposed in this paper extends the idea of the Blend station by substituting
After trivial calculations, it results that the two loops have identical dynamic response if
F(s) )
C1(s)P1(s)
(4)
C2(s)P2(s)
As is common practice in industrial environments, both the processes are assumed to have first-order plus dead time (FOPDT) dynamics; i.e., they are modeled according to the following transfer functions:
P1(s) )
K1 e-L1s T1s + 1
(5)
P2(s) )
K2 e-L2s T2s + 1
(6)
Further, again according to industrial practice, the two controllers are of PI type; i.e., we have
( (
) )
C1(s) ) Kp1 1 +
1 Ti1s
(7)
C2(s) ) Kp2 1 +
1 Ti2s
(8)
Taking into account the expression of F(s) (4) and those of the processes (5)-(6) and of the controllers (7)-(8), it turns out that, in order for the system F(s) to be causal, it must be L1 g L2. This relation gives a guideline on how to select the first loop; i.e., the first loop has to be selected as the one with the process with the larger dead time. For the purpose of tuning the overall control system (see section 4), it is worth determining the zeros, the poles, and the gain of the transfer function F(s). They are reported in Table 1.
Ind. Eng. Chem. Res., Vol. 44, No. 13, 2005 4619 Table 1. Zeros, Poles, and Gain of Transfer Function F(s) 1 1 ,Ti1 T2 1 1 ,Ti2 T1 K1Kp1Ti2 K2Kp2Ti1
zeros poles gain
4. Tuning It is assumed that an estimate of the FOPDT transfer functions of the two processes is available; for example, it can be obtained by the well-known area method.4 As already mentioned in section 3, the first loop has to be selected as the one with the larger dead time. Then because the new architecture guarantees that the desired ratio is obtained along the whole transient response when a set-point change is required (provided that the two processes have actually a FOPDT dynamics), the selection of the parameters of the two controllers has to be done according to the following intuitive guidelines: • The parameters of C2(s) have to be chosen in order to provide the best rejection of a load disturbance d2. • The parameters of C1(s) have to be chosen in order to provide the best rejection of a load disturbance d1. • The parameters of C1(s) and C2(s) have to be chosen in order to have a frequency response of F(s) as low as possible. In this way, the reference r2 of the second loop is determined mainly by output y1 of the first loop instead of the reference r1 of the first loop (note that the transfer function from y1 and r2 is a(1 - F(s))). Thus, high performance on the desired ratio is obtained when a load disturbance is acting on the first process. To achieve the mentioned goals, the following procedure can be adopted. First, the PI controller C1 is tuned according to the method proposed in ref 8, which is devoted to obtaining a desired specification on the load disturbance rejection task. In this context, it is convenient to choose the value of the time constant of the desired transfer function between the load disturbance d1 and the process output y1 equal to the value of the time constant T1 of the process. In this way, in addition to a good degree of robustness, a low value of the ratio Kp1/Ti1 is achieved,8 which is important in order to ensure a low-frequency response of F(s) (see Table 1). Note also that when the process time constant is chosen as the desired time constant of the load disturbance to process output transfer function, the method employed aims at canceling the process pole, i.e., Ti1 ) T1. Subsequently, the PI controller C2 is tuned by first imposing again a pole-zero cancellation in the second loop, i.e., by setting Ti2 ) T2. In other words, with the previous choices, we have simply
F(s) ) Ke-(L1-L2)s
(9)
where
K)
K1Kp1Ti2 K2Kp2Ti1
stable and that the largest value Ms of the sensitivity function is constrained. It should be noted that Ms can be considered as a useful tuning parameter because it allows one to handle the tradeoff between performance and robustness. In general, typical values of Ms are chosen in the range 1.2-2.0, to ensure a sufficient damping of the closed-loop system. However, since in this case the set-point response is not of concern, because it is equal to the one of the first loop (due to the ratio control architecture), it might be convenient to choose a higher value of Ms, e.g., Ms ) 2.5, to obtain a higher value of Kp2 and therefore a lower value of the gain of F(s). Actually, choosing a value of Ms ) 2.5 is still sensible, as a higher value is generally obtained by applying the well-known Ziegler-Nichols tuning formula.4 Remark 1. It is worth emphasizing that the proposed tuning procedure is not claimed to be optimal. However, it is sensible and provides satisfactory results (see sections 5 and 6), despite the simple transfer function (9) that is eventually obtained. Actually, to obtain a lower value of |F(jω)|, it might be reasonable to decrease the value of Kp1 and/or to increase the value of Kp2 (see Table 1). However, this would result in excessively sluggish responses of the first loop and in excessively oscillatory responses for the second loop, respectively. The same considerations apply with respect to the choice of the integral time constants Ti1 and Ti2. Remark 2. It has to be stressed that by exploiting expression (4), the proposed control architecture (i.e., the scheme shown in Figure 4) can be adopted with different tuning strategies for the two controllers C1 and C2, achieving in any case the desired ratio in the presence of a set-point change. Thus, the user might retain its know-how in tuning the two controllers, without impairing the effectiveness of the methodology. For example, detuning the controller of the first loop implies that when a load disturbance occurs on the first process, a slower rejection is obtained, but the desired ratio is kept better during the transient. In addition, if a process is lag-dominant, the use of a pole-zero cancellation in the feedback control design yields to a slow load disturbance response. In this case an alternative tuning rule can be conveniently employed, for example, the one proposed in ref 10. Further, an available more accurate model of the processes can be fully exploited by adopting expression (4). Remark 3. It is worth noting that with the proposed tuning strategy, in cases where the dead times of the processes are negligible, the devised method can be interpreted as a specific choice of the parameter γ in the Hagglund’s Blend station (see (1) and (9)) (the one that provides the desired ratio during the set-point response). Remark 4. Note also that, provided that the two PI controllers are designed in order to make the two closedloops systems asymptotically stable, the overall control system is asymptotically stable since it is the parallel and cascade composition of asymptotically stable systems.
(10)
Then, parameter Kp2 is selected by following basically the same idea described in ref 9. Thus, to have good load disturbance rejection performances, Kp2 is fixed, after solving an optimization problem, since the maximum value that guarantees that the closed-loop system is
5. Simulation Results Two illustrative examples are shown in order to demonstrate the effectiveness of the devised methodology. For the sake of clarity, in both examples the desired ratio a is set equal to unity. To evaluate the results from
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Figure 5. Process outputs with the new schemesexample 1.
Figure 6. Control signals with the new schemesexample 1.
a quantitative point of view, the following performance index will be calculated:
J)
∫0∞ |ay1(t) - y2(t)| dt
(11)
Obviously, a low value of J implies that a good performance is achieved. Note that the approaches proposed in refs 5 and 6 will not be considered as they are not devoted to load disturbances rejection task. 5.1. Example 1. Consider the following two FOPDT systems:
P1(s) )
1 e-3s 4s + 1 (12)
P2(s) )
1 e-s 6s + 1
Figure 7. Process outputs with the classic schemesexample 1.
When the proposed method and the proposed tuning procedure (with Ms ) 2.5) are applied, the following results: Kp1 ) 0.57, Ti1 ) 4, Kp2 ) 2.58, Ti2 ) 6. Consequently, we obtain
F(s) ) 0.33e-s Then, a unit step has been applied to the set-point signal at time t ) 0 s and then to the load disturbance signals d1 and d2 at time t ) 30 s and t ) 70 s, respectively. The two process outputs are shown in Figure 5, while the two manipulated variables are plotted in Figure 6. It appears that a perfect ratio control is achieved in the presence of a set-point change, as expected, but performances are very satisfactory as well, even in the presence of load disturbances. For the sake of comparison, results obtained by the classic ratio control scheme (see Figure 1) are reported in Figures 7 and 8 (note that the PI controllers have been tuned as for the new method). It appears that the worst performance obtained for the set-point change is not counterbalanced by a better performance in the presence of load disturbances. The value of the performance index J calculated over the whole experiment is 4.80 for the new method (actually, it is J ) 0 for just the set-point step response and J ) 2.46 for the load disturbance d1 transient response) and 8.55 for the classic one (with J ) 2.44 for just the set-point step
Figure 8. Control signals with the classic schemesexample 1.
response and J ) 3.77 for the load disturbance d1 response). Indeed, the new scheme provides in this case a better load disturbance rejection than the classic one. This is explained by the fact that a somewhat aggressive tuning has been selected for the second loop, and when a load disturbance occurs in the first loop, the new scheme provides for the second loop a lower reference
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Figure 9. Reference signal r2 with the new and the classic schemessexample 1.
signal r2 than the classic scheme (see Figure 9, where obviously r2 is equal to y1 for the classic scheme). This is also the reason for the lower control effort that is required by the new scheme. It should be stressed that it cannot be claimed that the proposed technique provides in general a better performance than the classic one when a load disturbance occurs in the first loop. With proper tuning (i.e., by detuning the second loop, at the expense of performance in the set-point following task and in the rejection of a load disturbance affecting process P2) the typical scheme could provide a smaller value of J. However, it can be noted that the tuning method presented in section 4 provides practically the best performance with the devised control scheme of Figure 4. Indeed, if the gain K is selected in order to minimize J when a step load disturbance occurs on the process P1, then K ) 0.30 and J ) 2.44, which is very close to the values obtained with formula (10). To demonstrate better the effectiveness of the proposed approach, a comparison has been made also by evaluating the performance obtained by applying, within the classic approach context, other tuning methods. In particular, the SIMC method proposed in ref 10, which is also based on pole-zero cancellation, has been applied to both loops, resulting in Kp1 ) 0.67, Ti1 ) 4, Kp2 ) 1.5, and Ti2 ) 6. The obtained process outputs are shown in Figure 10 (the plots of the control variables and of the reference signal r2(t) are not reported for the sake of brevity). The results are J ) 4.44 for the set-point step response, J ) 4.93 for the load disturbance d1 response, and J ) 4.00 for the load disturbance d2 response. Further, the Kappa-Tau tuning method,4 based on a dominant pole design, has also been considered. Specifically, the first loop has been tuned by applying the tuning rules for which the expected maximum sensitivity Ms is 1.4 (i.e., the robustness is privileged with respect to the aggressiveness), while, on the contrary, the second loop has been tuned by applying the tuning rules for which the expected maximum sensitivity Ms is 2.0 (i.e., the aggressiveness is privileged with respect to the robustness). The results are Kp1 ) 0.24, Ti1 ) 2.74, Kp2 ) 1.20, and Ti2 ) 4.12. The obtained process outputs are plotted in Figure 11. The results are J ) 3.78 for the set-point step response, J ) 6.31 for the load disturbance d1 response, and J ) 3.70 for the load disturbance d2 response. The effectiveness of the new approach, with respect to the others, appears.
Figure 10. Process outputs with the classic scheme and SIMC tuning rulessexample 1.
Figure 11. Process outputs with the classic scheme and KappaTau tuning rulessexample 1.
5.2. Example 2. As a second example, two systems that are not of first order are considered:
P1(s) )
1 (s + 1)4 (13)
1 P2(s) ) e-s 2 (s + 1) An estimate of a FOPDT transfer function has been obtained by means of the area method. The results are K1 ) 1, T1 ) 1.84, L1 ) 1.92, K2 ) 1, T2 ) 1.39, and L2 ) 1.57. It appears that these processes are quite difficult to control as they have a large normalized dead time (i.e., the ratio between the dead time and the time constant of the process). The tuning procedure described in section 4 (again with Ms ) 2.5) has been applied by considering the estimated process models, resulting in Kp1 ) 0.51, Ti1 ) 1.84, Kp2 ) 0.77, Ti2 ) 1.39, and
F(s) ) 0.48e-0.35s A unit step has been applied to the set-point signal at time t ) 0 s and then to the load disturbance signals d1
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Figure 12. Process outputs with the new schemesexample 2.
Figure 15. Control signals with the classic schemesexample 2.
Figure 13. Control signals with the new schemesexample 2.
Figure 16. Reference signal r2 with the new and the classic schemessexample 2.
Figure 14. Process outputs with the classic schemesexample 2.
and d2 at time t ) 30 s and t ) 70 s, respectively. The two process outputs are reported in Figure 12, while the two control variables are shown in Figure 13. It turns out that, despite the fact that the two processes are not FOPDT, and therefore a perfect ratio control cannot be achieved, performances are still very satisfactory. Again, a comparison with the classic scheme of Figure 1 has
Figure 17. Process outputs with the classic scheme and SIMC tuning rulessexample 2.
been performed. Related results are shown in Figures 14-16. The same considerations done for example 1 apply also in this case. In fact, with respect to the whole experiment, for the proposed technique it is J ) 5.77 (where the set-point step response contributes for 0.51 and the load disturbance d1 response for 2.33), while
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Figure 18. Process outputs with the classic scheme and KappaTau tuning rulessexample 2.
Figure 20. Process outputs with the new schemesexperimental results.
Figure 21. Reference signal r2 with the new schemesexperimental results.
6. Experimental Results Figure 19. The two tanks experimental setup.
for the classic one it is J ) 9.31 (where the set-point step response contributes for 2.86 and the load disturbance d1 response for 3.52). Note that the value of K that minimizes J in the presence of a step load disturbance d1 only is K ) 0.53, which yields to J ) 2.32. As for example 1, also in this case the SIMC tuning rules, as well as the Kappa-Tau ones, have been employed. When the SIMC tuning rules are applied, the results are Kp1 ) 0.50, Ti1 ) 1.84, Kp2 ) 0.45, and Ti2 ) 1.39. The corresponding process outputs (with the classic scheme) are shown in Figure 17. The resulting value of J for the whole experiment is J ) 11.84 (with J ) 4.09 for the set-point response and J ) 4.44 for the load disturbance d1 response). By applying the KappaTau tuning rules, we obtain Kp1 ) 0.19, Ti1 ) 1.29, Kp2 ) 0.39, and Ti2 ) 0.98 and the resulting process outputs are those shown in Figure 18. The results are J ) 2.72 for the set-point step response, J ) 4.50 for the load disturbance d1 response, and J ) 3.63 for the load disturbance d2 response. Hence, the effectiveness of the proposed tuning strategy appears again.
To prove the effectiveness of the devised technique in practical applications, a laboratory experimental setup (made by KentRidge Instruments) has been employed (see Figure 19). Specifically, the apparatus consists of two separated small Perspex tower-type tanks (whose area is 40 cm2) in which a level control is implemented by means of a PC-based controller. Each tank is filled with water by means of a pump whose speed is set by a dc voltage (the manipulated variable), in the range 0-5 V, through a PWM circuit, and is fitted with an outlet at the base in order for the water to return to a reservoir. The measure of each level is given by a capacitive-type probe that provides an output signal between 0 (empty tank) and 5 V (full tank). Since the apparent dead time of the system is very small with respect to its dominant time constant, to provide a significant result, a time delay of 10 and 5 s has been added via software at the input of the first and second process, respectively. Despite the model being nonlinear (as the flow rate out of the tank depends on the square root of the level), a FOPDT model has then been estimated for each process by applying the area method to the open-loop
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is biased due to the noise) is J ) 26.7 for the new method (note that it is J ) 7.4 for the set-point step response and J ) 9.5 for the load disturbance d1 response) and J ) 32.7 for the classic method (in this case it is J ) 13.2 for the set-point step response and J ) 9.9 for the load disturbance d1 response). Thus, experimental results confirm the effectiveness of the proposed methodology. 7. Conclusions
Figure 22. Process outputs with the classic schemesexperimental results.
In this paper a new ratio control architecture that improves the typical series metered control architecture has been proposed. Remarkably, in case a perfect model of the process is available, the process outputs ratio is kept at the desired value during the whole set-point transient response. High performance is in any case obtained in the presence of model uncertainties. Despite this, also the load disturbance rejection capabilities are very satisfactory. A tuning procedure has been devised so that no tuning effort from the user is needed. The methodology is easy to implement (note that no extra measurements are required with respect to the standard ratio controllers) and it is based on the use of classical PI controllers so that it can be easily understood by operators who retain their know-how. Thus, the overall methodology appears to be suitable for implementation in the industrial context. Literature Cited
Figure 23. Reference signal r2 with the classic schemes experimental results.
responses with a step from 2 to 2.5 V at each process input. The FOPDT models obtained are
P1(s) )
2.27 -11s 1.98 -6s e e P2(s) ) 25s + 1 25s + 1
According to the procedure described in section 4, the PI controller of the first process has been tuned with Kp1 ) 0.31 and Ti1 ) 25, while that of the second process with Kp2 ) 1.82 and Ti2 ) 25. Results are shown in Figures 20 and 21 for the new scheme and in Figures 22 and 23 for the classic one. The resulting value of the performance index (obviously the result in both cases
(1) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process dynamics and control; Wiley: New York, 2004. (2) Shinskey, F. G. Process Control Systems - Application, Design, and Tuning; McGraw-Hill: New York, 1996. (3) Shinskey, F. G. Feedback Controllers for the Process Industries; McGraw-Hill: New York, 1994. (4) A° stro¨m, K. J.; Ha¨gglund, T. PID controllers: theory, design and tuning; ISA Press: Research Triangle Park, NC,1995. (5) Ha¨gglund, T. The Blend station - a new ratio control structure. Control Eng. Pract. 2001, 9, 1215. (6) Visioli, A. Design and tuning of a ratio controller. Control Eng. Pract. 2005, 13, 485. (7) Gomes, V. G. Controlling fired heaters. Chem. Eng. 1985, 63. (8) Chen, D.; Seborg, D. E. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 2002, 41, 4807. (9) A° stro¨m, K. J.; Panagopoulos, H.; Ha¨gglund, T. Design of PI controllers based on non-convex optimization. Automatica 1998, 34, 585. (10) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291.
Received for review November 16, 2004 Revised manuscript received April 14, 2005 Accepted April 22, 2005 IE048893H
