An Efficient Structure for Parallel Cascade Control - Industrial

IMC-Based Control Strategy for Open-Loop Unstable Cascade Processes. Tao Liu, Weidong Zhang, and Danying Gu. Industrial & Engineering Chemistry ...
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Ind. Eng. Chem. Res. 1996, 35, 1845-1852

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PROCESS DESIGN AND CONTROL An Efficient Structure for Parallel Cascade Control Daniele Semino* and Alessandro Brambilla Department of Chemical Engineering, University of Pisa, Via Diotisalvi 2, 56126 Pisa, Italy

A control structure capable of avoiding interactions between the two loops of a parallel cascade system is developed. A conventional controller is used in the inner loop and an IMC controller is inserted in the outer loop in order to reject only the effects of the perturbations that are not taken care of by the inner controller. Various possible IMC structures are compared in order to show the advantages of the proposed structure. This control scheme outperforms conventional controllers and other control structures that are designed to avoid interactions between the two loops both in the nominal case and in the presence of uncertainties, provided that the IMC controller is carefully designed. The proposed structure is shown to be even more advantageous when the primary measurements are infrequently sampled. 1. Introduction It is quite common in the control of product quality in chemical processes to control strictly some process variable which is closely related to the property of interest and is readily available from on-line measurements and use scarce and delayed quality measurements for minor corrections of the control action. This is what is typically done in practice for the control of product composition in a distillation column. A tray temperature, whose measurement is available with no delay at a very high sampling rate, is chosen and controlled; at the same time, the composition measurements, available at a slow sampling rate after the analyzer dead time, are used to correct the temperature setpoints. Product samples are typically taken once every 20-40 min, and their measures are therefore both delayed and rare. This kind of control structure has been known for a long time as cascade control. It is common practice to represent such a structure as a series cascade scheme even if a number of authors, starting from Luyben (1973), have realized the parallel nature of most cascade systems. A parallel cascade system is one in which both the manipulated variable and the disturbance affect the primary and the secondary output through parallel actions, while in a series cascade both actions on the primary output take place through the secondary one. A parallel cascade example is the above-mentioned distillation column composition control. It is clear therefore that the series or parallel nature of a cascade is related to the characteristics of the process under analysis and not to the control system to be designed. Examples of series cascade and parallel cascade processes are described by Luyben (1973). Analysis and design of conventional controllers for parallel cascade control structures have been the subject of a work by Yu (1988). Shen and Yu (1990) applied these results to the selection of the secondary measurement when a number of different disturbances are present. Brambilla and Semino (1992) introduced a * Author to whom correspondence should be addressed. Telephone: +39 (50) 511238. Fax: +39 (50) 511266. E-mail: [email protected].

S0888-5885(95)00710-X CCC: $12.00

nonlinear filter between the two controllers in order to partially decouple the two loops and improve the performances of the control system. Brambilla et al. (1994) proposed some nondimensional parameters to address the choices which are required to design parallel cascade controllers for multicomponent distillation columns with dual control. McAvoy and Ye (1995) discussed a nonlinear inferential parallel cascade control structure which originates both from parallel cascade control and from inferential sensing. There are two issues, however, that are not addressed by the usual parallel cascade control structures. The first one concerns the interactions between the primary and the secondary loop. In particular, either the outer controller is detuned so that most of the work is done by the inner one and a slow return of the primary output to steady-state takes place or it interacts with the inner controller by imposing delayed corrections to errors that have already been partially eliminated by the inner loop so that oscillations in the output response take place. A partial solution to this problem is the use of a nonlinear filter between the two controllers (Brambilla and Semino, 1992); however, the performances of this control structure have been shown to deteriorate when the parallel cascade scheme differs relevantly from a series cascade one. The second issue concerns the slow sampling rate of the primary measurement. In the usual approach the two loops are considered as continuous: this is reasonable for the secondary measurement which is usually available at an arbitrary high sampling rate but is a strong approximation for the quality measurement, which is often available quite rarely. This makes the former issue even more relevant since inappropriate control actions of the primary controller are kept constant for the long sampling time of the primary output. In the present paper a parallel cascade control structure is proposed that decouples the actions of the two controllers. In particular the signal which is fed back to the primary controller accounts only for the effect of the disturbance on the primary output that has not already been taken care of by the secondary controller. This is accomplished by introducing an ad-hoc IMC © 1996 American Chemical Society

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as the ratio between the steady-state variation of the manipulated variable in the presence of the disturbance under perfect control of the secondary output and the above variation under perfect control of the primary output. Its definition and value are as follows:

γ)

Figure 1. Parallel cascade control structure.

Figure 2. Series cascade equivalent of the parallel cascade control structure.

structure in the outer loop. The parallel cascade scheme is compared initially to the series cascade one in order to show differences and similarities; this comparison is useful to understand how to extend control structures which are effective for the series cascade case to the parallel cascade one. Thereafter the proposed structure is compared with the available ones both in the case of continuous measurements and in the case of rarely sampled primary measurements. Moreover, it is taken into account how the proposed control structure addresses the choice of the secondary measurement in comparison with other developed criteria (Shen and Yu, 1990; Brambilla and Semino, 1992; Brambilla et al., 1994). 2. Parallel Cascade The parallel cascade control structure is shown in Figure 1. It is apparent that both the manipulated variable and the disturbance affect the primary and secondary outputs through parallel transfer functions. Block diagram transformation rules can be applied to turn the structure into one that resembles the usual series cascade (Figure 2). The schemes in Figure 1 and in Figure 2 are equivalent when G*p1 ) Gp1/Gp2 and G*d1 ) Gd1 - Gd2Gp1/Gp2. It is clear, however, that the physical relationships between input and output variables are partly lost in the transformed scheme where cause-effect relationships that do not have any physical sense (effects of d and u on y1 through y2) have been introduced. The importance of the structure in Figure 2 will come clear in the following paragraph; here it suffices to note that it is useful in making clear that when Gd1 ) Gd2Gp1/Gp2 the parallel scheme is perfectly equivalent to a series cascade; in this case after the disturbance has been rejected by the inner loop, y1 returns to its setpoint also when C1 is in open loop. Moreover, if the only concern is about the absence of offset on y1 when only the inner controller is active, the gain equivalence Kd1 ) Kd2Kp1/ Kp2 is all that is needed. The same result can be obtained through the “interaction measure” γ introduced by Yu (1988). γ is defined

[∂u/∂d]y2 [∂u/∂d]y1

)

Kd2Kp1 Kd1Kp2

(1)

The value of γ is appropriate to understand the role of the external controller since it gives information about the required variation in the secondary output setpoint for a unit step disturbance (∆r2 ) (1 - 1/γ)Kd2), when the primary output is back to its steady state. When γ > 1, less power is required to control y1 than to control y2 and a setpoint change of y2 in the direction of the effect of the disturbance on y2 is appropriate. Analogously, when 0 < γ < 1, a setpoint change of y2 in the opposite direction is required. When γ < 0, however, the variation in the manipulated variable required by the inner controller to counteract the effect of the disturbance on y2 is opposite to what is needed to control y1 and the effect of the parallel cascade on the control of the primary output is detrimental. Yu (1988) suggests also a criterion for the tuning of the controller parameters based on the value of γ. An internal controller only with proportional action is suggested if γ > 1 and stability of the secondary loop is not violated. IMC tuning methods for PI controllers (Rivera et al., 1986; Morari and Zafiriou, 1989; Brambilla et al., 1990) are suggested for all other values of γ and for the primary controller. However, some arbitrariness on the choice of the controller parameters remains, and some trial and error needs to be accomplished to find parameters that work well for the disturbances of interest. Moreover, performances are strongly dependent on the controller parameters, and interactions between the primary and secondary controller are usually present. The validity of these statements which are qualitatively reasonable will come clear from the case studies. The performance is made less dependent on the external controller parameters and some interaction is eliminated if a nonlinear filter is inserted between the two controllers (Brambilla and Semino, 1992). However, the performances of this control structure tend to deteriorate when γ differs considerably from 1. Moreover, it is not straightforward how to choose the value of the filter constant so that some arbitrariness remains. These are the reasons why a new control structure that avoids interactions and does not suffer from the same limitations has been developed. 3. Proposed Structure The basic idea to eliminate interactions between the two control loops and make tuning easy comes from the introduction of an IMC controller in the outer loop. This has been shown to be very effective for the series cascade (Scali and Brambilla, 1990); however, extension to the parallel cascade scheme is not straightforward. The easiest way to see how it can be accomplished comes from the equivalence between the structure in Figure 1 and the one in Figure 2. The system in Figure 2 can be seen as a series cascade control scheme with two different disturbances. Let us consider the series cascade scheme in the IMC structure represented in Figure 3 (the tilde is used to

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Figure 3. Series cascade in the IMC structure. Figure 6. Series cascade equivalent of the parallel cascade in the IMC structure.

Figure 4. Series cascade in the IMC structure modified in the inner loop.

Figure 7. Parallel cascade in the IMC structure. Figure 5. Alternative IMC structure for series cascade.

refer to the process models). The main advantage of such a control structure is that the design of the two controllers is completely decoupled. Being in the nominal case:

y1 ) P2P1q1q2r1 + (1 - P2q2)P1d1 + (1 - P1P2q1q2)d2 (2) one can design q2 for d1 and q ) q1q2 for d2 and r. The outer controller q1 ()q/q2) therefore has no effect on the performance in the case of a disturbance on the inner loop. In the common case of a fast internal loop, the inner IMC can be replaced by a conventional controller (as in Figure 4 where C2 is an approximation of q2/(1 - P ˜ 2q2)) with almost no changes in performances. Still no interactions are present but only the outer loop requires an advanced controller. Other IMC alternatives for the series cascade can be proposed like the one in Figure 5, which, however, loses the advantage of the decoupling, makes the structure more intricate (P ˜ 1* ) P ˜ 1P ˜ 2C2/(1 + P ˜ 2C2)), and has been shown to give worse performances (Scali and Brambilla, 1990). Given the equivalence between the structures in Figure 1 and in Figure 2, one can consider how to extend the structure in Figure 4 to the parallel cascade case. Keeping the representation in Figure 2, one can propose the structure in Figure 6. Here the controller C2 can be designed to reject the effect of d on y2 through Gd2, while q1 is used to reject the remaining effect of d on y1 through Gd1*. The actions of the two controllers in accomplishing these two tasks are decoupled given that the control structure is completely equivalent to the one in Figure 4 (and also in Figure 3). In the nominal case the primary output will be:

y1 ) Gp2G*p1q1q2r1 + (1 - Gp2q2)G* p1Gd2d + (1 - Gp2G*p1q1q2)G*d1d (3)

where q2 ) C2/(1 + Gp2C2). It is relevant to recall that all the criteria for the selection of the secondary measurement suggest choosing it so that the effect of Gd1* is kept small for the main disturbances of interest (Shen and Yu, 1990; Brambilla and Semino, 1992; Brambilla et al., 1994); this suggestion is confirmed in this study, given that when Gd1* is close to 0, the outer controller work is scarcely relevant. It is easy at this point to go back to the proper parallel cascade scheme and see how the structure in Figure 6 can be implemented. The resulting structure is shown in Figure 7. It is important to realize that the feedback signal in the outer loop is d(Gd1 - Gd2Gp1/Gp2) in the nominal case. This means that the action that is required to the primary controller q1 is used only to counteract the effect of the disturbance on y1 that is not already being rejected by C2. Moreover, it is peculiar of this control structure that the usual IMC path originates from the secondary output and not from the manipulated variable. The relationship in eq 3 becomes:

Gp1 y1 ) Gp1q1q2r1 + (1 - Gp2q2) G d+ Gp2 d2

(

(1 - Gp1q1q2) Gd1 -

)

Gd2Gp1 d (4) Gp2

where again q2 ) C2/(1 + Gp2C2). To make the analysis clear and complete, it is worth remembering what the response would have been with the simple IMC structure without cascade:

y1 ) Gp1qr1 + (1 - Gp1q)Gd1d

(5)

The advantage of the proposed control structure is in rejecting part of the effect of the disturbance ((Gd2Gp1/ Gp2)d) depending on the fast dynamics of the secondary output (Gp2) and only the remaining effect ((Gd1 Gd2Gp1/Gp2)d) with the slow dynamics of Gp1.

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Figure 8. Alternative IMC structure for parallel cascade (A).

There are two conditions that need to be verified in order for the structure in Figure 7 to be advantageous over an IMC structure without cascade: (1) Gp1/Gp2 is nonminimum phase: this condition is completely equivalent to the one for the series cascade (Morari and Zafiriou, 1989) and indicates that there are advantages in inverting Gp2 over inverting Gp1. (2) It is advantageous in terms of performance to reject (Gd1 - Gd2Gp1/Gp2)d over rejecting Gd1d: in the common case (see examples) in which the dynamics of Gd1 and Gd2Gp1/Gp2 are similar, this corresponds to 0 < γ < 2 (|Kd1*| < |Kd1|). Both conditions are intuitively obvious: cascade control is appropriate if the internal loop is able to make the dynamic response faster; moreover, if the system is a parallel cascade one, it is required that the rejection of the disturbance on the internal variable does not lead the primary output farther away from its steady-state value than the disturbance itself. The former condition is usually satisfied when cascade control is an issue (e.g., composition control in a distillation column cascaded to the control of a tray temperature). The latter depends both on the selected secondary measurement and on the disturbance to be rejected. A careful analysis of the values of γ for different disturbances and different secondary measurements is therefore appropriate. In any case, a choice of the secondary measurement that corresponds to values of γ close to one for the disturbances of interest (Shen and Yu, 1990; Brambilla and Semino, 1992; Brambilla et al., 1994) is the most advantageous. Once the two conditions are satisfied, the control structure in Figure 7 can be implemented by tuning C2 on Gp2 (IMC-like tuning techniques are suggested) and designing q1 for the disturbance Gd1*d and the process Gp1* (rigorously q ) q1q2 should be designed for Gp1 with disturbance Gd1*d, but for a fast internal loop, this is exactly the same); the filter time constant can be tuned to give robustness to the control structure. In this way the control scheme is advantageous given that it does not suffer from interactions, it can be easily designed with little arbitrariness, and it can be easily implemented with only one advanced controller. After developing and justifying the structure in Figure 7, it can be of interest to understand why more obvious extensions of the IMC structure to the parallel cascade scheme either do not work or are less advantageous. Let us consider the structure in Figure 8, which apparently is the more reasonable extension of the IMC scheme in this case. In the nominal case, the output y1 is related to r1 and d by the following equation:

y1 ) Gp1q1q2r1 + (1 - Gp1q1q2)Gd1d - Gd2Gp1q2d (6) with q2 ) C2/(1 + Gp2C2). It is clear that, if q ) q1q2

Figure 9. Alternative IMC structure for parallel cascade (B).

has been designed to follow without offset a setpoint variation, a steady-state offset comes out in the presence of the disturbance; this is due to the fact that the primary controller does not take into account that a partial rejection of the effect of the disturbance is accomplished by the secondary controller. A feasible control structure is the one in Figure 9 where G ˜p ) G ˜ p1C2/(1 + G ˜ p2C2). The nominal response is:

y1 ) Gp1q1q2r1 + (1 - Gp1q1q2)(Gd1 - Gd2Gp1q2)d (7) where q2 ) C2/(1 + Gp2C2). When the dynamics of the -1 secondary loop is very fast so that q2 ≈ Gp2 , this is completely equivalent to the result in eq 4. The simplicity of design and implementation has, however, been lost in the general case. Given that the structure in Figure 9 is the parallel cascade equivalent of the one in Figure 5, no advantages in performance are expected. Some final considerations may be needed about the details of the practical implementation of the control structure in Figure 7. The inner loop is a traditional conventional controller; the outer loop instead contains two nonconventional blocks: G ˜ p1/G ˜ p2, which transforms a secondary measurement into an equivalent primary signal, and q1, which computes the variation in the secondary output setpoint. Both these blocks are in most practical cases (see examples below) simple leadlag structures with at most an additional delay (in G ˜ p1/ G ˜ p2) or a filter (in q1). The implementation of the controller with any computer-based control system would therefore be possible without any difficulties. Even if probably the average practicing control engineer would not be able to accomplish the design of the whole control structure, once the design has been developed, he should not have problems in following its operation due to the simplicity of the structure and to the presence in the outer loop of a single tuning parameter (the filter time constant) with a clear effect both on performance and on robustness. 4. Case Studies Case Study No. 1. In the first example the system is characterized by the same dynamics in Gp1 and Gp2 and in Gd1 and Gd2 with the exception of a time delay (24 min) which corresponds to a delay in the measurement of the primary output. The transfer functions are as follows:

Gp1 )

1.24e-33s 30s + 1

(8)

3.1e-9s 30s + 1

(9)

Gp2 )

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Gd1 )

(1 + k)e-24s 15s + 1

(10)

2.5 15s + 1

(11)

Gd2 )

which correspond to (Figure 2):

Gp1* ) 0.4e-24s

(12)

ke-24s 15s + 1

(13)

Gd1* )

Variations in the parameter k allow one to change γ (γ ) 1/(1 + k)) in order to test different parallel cascade systems. Performances of four different control schemes, which are described in the following, are compared. In the conventional scheme (Figure 1) C2 has been tuned for Gp2 and C1 for Gp1* (control scheme no. 1). The parameters are chosen following Morari and Zafiriou (1989) and Brambilla et al. (1990):

kc2 )

τp2 + θp2/2 Kp2θp2(c + 1)

) 0.77 (c ) 0.6)

(14)

τI2 ) τp2 + θp2/2 ) 34.5 kc1 )

θp1*/2 Kp1*θp1*(c + 1)

(15)

) 0.78 (c ) 0.6)

τI1 ) θp1*/2 ) 12

(16) (17)

In the proposed scheme (Figure 7) C2 is the same as above, while q1 is designed for Gp1* and a step disturbance (control scheme no. 2) or for Gp1* and a step disturbance entering Gd1* (control scheme no. 3). In the former case:

q˜ 1 ) 2.5;

q1 )

2.5 ; λs + 1

λ)3

q1 )

2.5(12s + 1) ; λs + 1

Figure 11. Case study no. 1. Dynamic behavior of the four control structures for a step disturbance when k ) 0.3: (1) conventional scheme, (2) IMC scheme (step design), (3) IMC scheme (disturbance design), (4) nonlinear filter scheme.

(18)

and in the latter:

q˜ 1 ) 2.5(12s + 1);

Figure 10. Case study no. 1. Dynamic behavior of the four control structures for a step disturbance when k ) -0.1: (1) conventional scheme, (2) IMC scheme (step design), (3) IMC scheme (disturbance design), (4) nonlinear filter scheme.

λ)3 (19)

where the parameter of the filter has been chosen for robustness. The last control structure (control scheme no. 4) is the one with a nonlinear filter between the two conventional controllers (Brambilla and Semino, 1992). The conventional controllers are the same as above. Brambilla and Semino propose slight modifications of the outer controller when the value of γ is known; in this case, as γ varies according to the value of k, this modification is not taken into account. The value of the nonlinear filter parameter R has been chosen in order to avoid wide oscillations when both loops are closed (R ) 8). The different schemes have been tested for different values of the parameter k. Comparison of performances are reported here for k ) -0.1 (Figure 10) and k ) 0.3 (Figure 11). Performances of the partly advanced controller are better in all cases even if the improvement is more relevant in particular when k < 0 (γ > 1 ). To understand what happens, it is enough to examine the

first actions of the outer controller in the conventional scheme. When the outer controller gets the first information, it counteracts variations in the primary output that have already been partially or totally rejected by the secondary controller. In particular, when k ) -0.1, the first action of the primary controller that corresponds to the error in the primary output at time t ) 24 starts affecting the output itself at t ) 57, when the error has already changed sign. On the other side in the partly advanced structure, the information that gets to the primary controller is immediately related to the part of the effect of the disturbance that the outer controller has to reject. A partial decoupling can be accomplished also by making use of the nonlinear filter. However, if the parameter R is chosen in order to damp the oscillations relevantly, performances tend to deteriorate when γ differs significantly from 1. As Brambilla and Semino (1992) showed in their work, this structure is successful for series cascade schemes or for parallel cascades that do not differ relevantly from a series cascade. Moreover, if a large value of R is needed, the dynamics of setpoint tracking are very slow. If the two control schemes that use the proposed structure (nos. 2 and 3) are compared, it is clear that further improvements in performance can be obtained if q1 is designed based on Gd1*. However, for the dynamics in the present example, such improvements

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are not highly relevant. As far as the response to a step setpoint variation is concerned, the appropriate design is the one in control scheme no. 2 (see eq 3); anyway, even if control scheme no. 3 shows a larger overshoot, performance is very good for both schemes that use the proposed structure. Case Study No. 2. In the second example, a GPL splitter (Lorenzi et al., 1995) is analyzed to see how performances are affected by large time constants in the transfer functions. The top composition is controlled with cascade on the temperature of tray no. 8. A delay of 20 min due to the composition analyzer is present. Performances of different control structures are compared in the case of a disturbance on the feed flowrate. The transfer functions are as follows:

Gp1 )

-0.0067e-20s 105.8s + 1 -5.217 101.6s + 1

(21)

0.05843e-20s 115.5s + 1

(22)

44.15 109.5s + 1

(23)

Gp2 ) Gd1 )

(20)

Gd2 )

Figure 12. Case study no. 2. Dynamic behavior of the three control structures for a step disturbance in the nominal case: (1) conventional scheme, (2) IMC scheme (step design), (3) IMC scheme (disturbance design).

which correspond to (Figure 2):

0.0067 -20s e 5.217

(24)

-0.0018e-20s 110s + 1

(25)

Gp1* ≈ Gd1* ≈

In the conventional scheme (no. 1), the controllers are tuned following the same guidelines as above with the exception of the integral time of C2 which is overtuned to speed up the dynamics of the inner loop which has no delay but a very slow time constant. The parameters are as follows:

kc2 )

kc1 )

τp2 ) -19.5 (λ ) 1) Kp2λ

(26)

τI2 ) τp2/5 ) 20.3

(27)

θp1*/2 Kp1*θp1*(c + 1)

) 194.6 (c ) 1)

τI1 ) θp1*/2 ) 10

(28) (29)

In the proposed control structure, the same C2 is used and two designs of the IMC controller are considered. In the former (control scheme no. 2), the IMC controller is designed for Gp1* and a step disturbance as follows:

|

Gp2 Gp1

(1) q˜ 1 is designed for Gp1* and Gd1*:

q˜ 1 )

|

Gp2 (τ *(1 - e-θp1*/τd1*)s + 1) ) Gp1 m d1 5.217 105.8s + 1 (18.3s + 1) (31) 0.0067 101.6s + 1

(2) A type II filter is used given that the disturbance is similar to a ramp:

q1 ) q˜ 1f;

f)

2λs + 1 ; (λs + 1)2

λ )10

(32)

1 ; λs + 1 λ ) 1 (30)

The filter parameter has been tuned in order to achieve robustness for 10% errors on the process gains.

However, to properly design the IMC controller, it has to be taken into account that the process has a large time constant (and therefore a small θ/τ ratio) and is affected by a disturbance with a large time constant. Scali et al. (1992) showed that the correct design in this case is as follows (control scheme no. 3):

Performances of the three control structures have been compared in the nominal case (Figure 12) and in the case of a 10% error on Kp2 (Kp2 ) -5.74; Figure 13) and on Kp1 (Kp1 ) -0.00737; Figure 14). Both IMC structures perform better in all cases. However, control scheme no. 2 shows a slow return to the steady state,

q˜ 1 )

m

)

5.217 105.8s + 1 ; 0.0067 101.6s + 1

Figure 13. Case study no. 2. Dynamic behavior of the three control structures for a step disturbance in the uncertain case (Kp2 ) -5.74): (1) conventional scheme, (2) IMC scheme (step design), (3) IMC scheme (disturbance design).

f)

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Figure 14. Case study no. 2: Dynamic behavior of the three control structures for a step disturbance in the uncertain case (Kp1 ) -0.00737): (1) conventional scheme, (2) IMC scheme (step design), (3) IMC scheme (disturbance design).

Figure 15. Case study no. 3. Dynamic behavior of the two control structures for a step disturbance when k ) -0.1: (1) conventional scheme, (2) IMC scheme (step design).

and its performances are only marginally better than those of the conventional scheme particularly in the nominal case (when γ ≈ 1 and the conventional scheme suffers less from interactions). The properly designed IMC controller outperforms the other schemes in all cases; its performances are much better also in the uncertain case even if this control structure makes the largest use of the model. The proper design of the IMC controller (Scali et al., 1992) is therefore strongly suggested when both the process and the disturbance time constants are large. As far as robustness is concerned, it is relevant to point out which process knowledge is needed to build a correct controller in order to understand how sensitive the controller is to process nonlinearities and to other causes of process/model mismatch. The controller in control scheme no. 2 requires in practice only the ratio of the two process gains. It is known for distillation columns that the process nonlinearity may cause a relevant variation in the gains particularly when high purities are concerned. However, such variations take place for the gains of both the primary and the secondary processes in a similar way so that the effect on the gain ratio is marginal. The above analysis on what happens for errors on one gain at a time is therefore surely conservative. If a disturbance specific controller is built (control scheme no. 3), it is required in addition the knowledge of the disturbance time constant (for which variations due to nonlinearities are less strong)

Figure 16. Case study no. 3. Dynamic behavior of the two control structures for a step disturbance when k ) 0.3: (1) conventional scheme, (2) IMC scheme (step design).

and of the analyzer delay (which is usually known with small errors). The analyzer delay is also required in both cases in the block G ˜ p1/G ˜ p2. Finally, as usual in IMC control, it is one of the purposes of the controller filter to avoid inconveniences due to process and measurement noises. It is clear in this case as well that the correct design for a step setpoint change is the one in control scheme no. 2; being disturbance specific, control scheme no. 3 gives a quite large overshoot in the presence of a step setpoint variation. Case Study No. 3. The third case study is aimed at showing the effect of long sampling times on the measurement of the primary output. The same system as in case study no. 1 is analyzed. However, in place of a delay of 24 min on the measurement, it is assumed that a delay of only 6 min is present but that the measurement is available once every 24 min. The control action computed by the outer controller once every 24 min is therefore maintained at the calculated value for the entire following sampling time. As far as the tuning of the conventional controllers is concerned, the effect of the sampling time Ts is taken into account with a delay θs ) Ts/2 in the primary process. The only controller parameter that changes with respect to case study no. 1 (eqs 14-17) is τI1 ) (θp1* + θs)/2 ) 9. The outer controller is implemented in the conventional control scheme (no. 1) in the velocity form as follows:

uk+1 ) uk + KC(1 + Ts/τI)ek - KCek-1

(33)

In the proposed scheme (no. 2) the IMC controller is designed in the discrete domain based on Gp1* and a step disturbance (this design has been shown to give, for this process, performances comparable to those of the more involved design):

q˜ 1 ) 2.5;

q1 )

2.5(1 - a)z ; z-a

a ) 0.135

(34)

Performances of the two control structures are compared when k ) -0.1 (Figure 15) and when k ) 0.3 (Figure 16); for the sake of clarity, the continuous behavior of the primary controlled variable after the analyzer dead time is reported. The detrimental effect of keeping constant a wrong control action is clear. When the outer controller starts acting in the conventional scheme, it works to suppress an effect of the

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disturbance that has already been rejected by the internal controller; given that this control action is not changed for the long sampling time, it causes even wider oscillations than in case study no. 1 despite the relevantly smaller delay. The proposed structure accounts for the correct disturbance effect also in this case and performances are as good as those in case study no. 1. 5. Conclusions A new control structure has been proposed in order to avoid interactions between the two loops for a parallel cascade system. The control structure makes use of a conventional controller in the inner loop and of an IMC controller in the outer loop which is used to reject only those effects of the perturbations that have not been eliminated by the action of the inner controller. The motivations of the different elements that appear in the control structure are presented starting from the similarities between series and parallel cascade and analyzing various possibilities of advanced structures. Performances of the proposed structure are compared with those of conventional controllers and of other control schemes that are designed to avoid interactions between the loops. The proposed controller outperforms the other controllers both in the nominal case and in the presence of uncertainties. It is relevant to note that the design of the IMC controller must follow the guidelines of Scali et al. (1992) in the case when both the process and the disturbance are characterized by a very large time constant. In the frequent case of rarely sampled primary measurements, the proposed structure is even more advantageous in avoiding misleading control actions.

Brambilla, A.; Scali, C.; Chen, S. Robust Tuning of Conventional Controllers. Hydrocarbon Process. 1990, 11, 53. Brambilla, A.; Semino, D.; Scali, C. Design and Control Selection of Cascade Loops in Distillation. Paper presented at the IFAC Workshop on Integration of Process Design and Control, Baltimore, MD, June 27 and 28, 1994; p 171. Lorenzi, G.; Semino, D.; Scali, C.; Brambilla, A. Analysis and Design of Cascade Control Schemes for Distillation. Paper presented at the 4th IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes, Helsingor, Denmark, June 7-9, 1995; p 255. Luyben, W. Parallel Cascade Control. Ind. Eng. Chem. Fundam. 1973, 12, 463. Mc Avoy, T. J.; Ye, N. Nonlinear Inferential Parallel Cascade Control. Paper presented at the 4th IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes, Helsingor, Denmark, June 7-9, 1995; p 441. Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: Englewood Cliffs, NJ, 1989. Rivera, D. E.; Morari, M.; Skogestad, S. Internal Model Control 4: PID Controller Design. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 252. Scali, C.; Brambilla, A. Analysis of Cascade Control Schemes for Chemical Processes. Chim. Ind. (Milan) 1990, Q9. Scali, C.; Semino D.; Morari, M. Comparison of Internal Model Control and Linear Quadratic Optimal Control for SISO Systems. Ind. Eng. Chem. Res. 1992, 31, 1920. Shen, S. H.; Yu, C. C. Selection of Secondary Measurement for Parallel Cascade Control. AIChE J. 1990, 36, 1267. Yu, C. C. Design of Parallel Cascade Control for Disurbance Rejection. AIChE J. 1988, 34, 1833.

Received for review November 28, 1995 Revised manuscript received March 19, 1996 Accepted March 23, 1996X IE950710R

Literature Cited Brambilla, A.; Semino, D. Nonlinear Filter in Cascade Control Schemes. Ind. Eng. Chem. Res. 1992, 31, 2694.

X Abstract published in Advance ACS Abstracts, May 1, 1996.