Advanced Control Structures in Multivariable Cascade

Copyright ~, IFAC Dynamics and Control of Process Systems. Corfu, Greece, 1998

ADVANCED CONTROL STRUCTURES IN MULTIVARIABLE CASCADE D. Semino*, S. Porcari* , A.Brambilla*

*Dipariimento di Ingegneria Chimica, Univer,itci degli Studi di Pila - ITALY

Abstract. The issues related to the use of cascade structures in multivariable controllers are addressed in this paper. An advanced structure, which makes use of a control scheme belonging to the IMC family is proposed. This control structure, which extends to the multivariable case the one of Semino and Brambilla (1996), is applied to a distillation column example in order to show both its benefits and its simplicity in practical cases. Some issues related to the development of other controllers belonging to the class of predictive control are finally mentioned. Copyright @1998 IFA C Key Words. Cascade Control, Multivariable Systems, Model-based Control, Distillation Columns, Disturbance Rejection.

Before pointing out the purpose of this paper, it is worthwhile mentioning a few works that address some of these issues. Yu (1988) analysed the design of conventional controllers for parallel cascade control structures; thereafter 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 nonlinear filter between the two controllers in order to partially decouple the two loops and improve the performances of the control system. Mc Avoy and Ye (1995) discussed a nonlinear inferential parallel cascade control structure which originates both from parallel cascade control and from inferential sensing. Wo1ft' and Skogestad (1996) applied a temperature controller below the two point control scheme of a binary distillation column with conventional controllers in all the loops to show the advantages in terms of interactions and disturbance rejection. Recently, a cascade control structure has been proposed (Semino and Brambilla, 1996) which maintains the simplicity in the design while addressing directly issues (1-4). Such a structure, which will be briefiy reviewed in the next section, belongs to the wide class of Internal Model Control (IMC) structures and has been developed for single input-two outputs (SITO) systems in which there is one manipulated variable, one primary

1. INTRODUCTION Cascade structures are well-known for facilitating the rejection of disturbances which affect secondary measurements with a more favourable dynamics than those of the primary outputs. However, some basic issues that are primary to make this control structure effective and that are also very common in practice are often overlooked in most of the works that can be found in the literature. Such issues are the following:

1. the parallel nature of the cascade structure (mentioned for the first time by Luyben, 1913), i.e. both the manipulated variable and the disturbance affect the primary and the secondary output through parallel actions (see Figure 1); 2. the appropriate choice of the secondary measurements; 3. the dynamic interactions between the loops that usually take place due to the delayed corrections of the outer controller to errors that have already been partially eliminated by the inner loop; 4. a large difference in the sampling times at which the primary and the secondary outputs are available; 5. the interactions of the cascade structure with other loops of the overall plant control scheme.

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output and one secondary output. In plant-wide control, however, it is quite uncommon that such a SITO structure works independently of the other control loops in the system. In particular, there may be more primary outputs that are affected by the fast internal loop or more internal loops that return different sec.ondary variables to their set-points. In both cases ' mteractions among the different parts of the control structure may end up reducing the advantages of the introduction of the cascades. This behaviour has motivated the study of advanced control structures for multi variable cascade systems. In particular, the structure by Semino and Brambilla (1996) has been extended to multi variable systems including reptangular systems in which the number of primary and secondary outputs are different. This extension is the subject of section 2. Applications to distillation columns are presented in section 3; such case studies demonstrate also how the improvements in performance of the proposed structure with respect to simplified ones are related to the value of the parameters suggested by Lorenzi et al. (1995) for the choice of the secondary measurement. The issues related to the use of predictive controllers with multivariable cascade systems are described in section 4 and possible solutions are proposed.

d1

d2

"~~ Fig. 2. Series cascade in the !MC structure (SITO case)

where gd2(gpl/gp2)d represents the effect of the disturbance on the primary variable which is corrected with a very fast internal loop. This avoids the feedback of already rejected disturbances to the outer loop, which would end up causing interactions between the loops and oscillations. It is this concept which will facilitate the extension to multi variable cascade systems. It is relevant to notice that in the case when 9dl = 9d29pI!9p2, the parallel cascade reduces to a series cascade structure and only the internal controller works for the rejection of the disturbance. Let us consider the multivariable system in which there are n primary outputs Yl, m secondary outputs Y2, m input variables U2 used to dose the internal loops, n - m input variables Ul used to control the primary outputs together with the m set-points r2 of the secondary outputs. The disturbances acting on the system are contained in the r-dimensional vector d. The block diagram of the multivariable cascade system with all the involved matrices of transfer functions is depicted in Figure 3. d

2. MULTIVARIABLE CASCADE CONTROL SCHEME The control structure by Semino and Brambilla (1996) stems from the idea of using the secondary measurement as the input to the process model of an IMC structure (Figure 1). d

t-----EI--r-

Yl Fig. 3. Multivariable cascade with the outer loop open

Perfect rejection of the disturbance by the internal loop with the outer loop open implies U2 = -G221Gd2d from which Yl = (Gdl-G12G221Gd2)d follows. This is the disturbance that needs to be rejected by the outer loop through manipulation of Ul and r2. This can be accomplished by the structure in Figure 4 being:

Fig. 1. Proposed control structure in the SITO case

This configuration originates from the extension of the IMC structure for series cascade SITO systems (Figure 2) to parallel cascade SITO systems. The basic effect of the control structure in Figure 1 is the separation of the effect of the disturbance which is rejected by the inner loop from what is fed back to the outer loop. In the nominal case the signal which is fed back to the outer loop is:

d out = (g,ll - gd2 gpl )d gp2

(2)

(3)

(1)

314

G12 and G21 reduce to a column and a row vector respectively. The analysis of the minimum and nonminimum phase elements of G 22 is therefore straightforward as is the check of the possibility of cancellation with the nominimum phase elements in G 12 •

(4)

dout

= Yl - y~ - y~ =

= (G,1l -

(5)

G 12 G 221 GIl2)d

2.3. SITO C(Ucade SY6tem6

d __- - - . . . , t--,--r--

This last subsection is added to make the reader aware that the control structure by Semino and Brambilla (1996) can be seen as a subcase of the more general structure in Figure 4 when n = m = 1. In this case the number of inputs is clearly one while the number of outputs (primary+secondary) is n + m 2. Being n m (square system) no double path in parallel to the process is required.

Y2

=

=

Fig. 4. Proposed scheme for multi variable cascade

3. APPLICATIONS TO DISTILLATION COLUMN

A condition for this structure to be applicable is G 22l being stable and realizable or at least its unrealizable parts being cancealed by the nonminimum phase parts in G 12 • If this is not the case, the nonminimum phase elements in G 22 that do not appear in G12 need to be separated before inversion. The design of the IMC controller Q is to be done for the n x n process

The most typical problem for which cascade control structures are designed is the control of distillation columns. The dual control of a distillation column (control of compositions at both ends) with the addition of the faster control of a tray temperature is a common occurrence in distillation column control. In this section, the performances of differently designed control structures are compared for different distubances and different selections of the secondary measurement. The four control structures are the following:

(6) and the disturbance (Gill - G 12 G 221 GIl2)d. This control structure may appear complex in this general block diagram representation but turns out to be particularly simple and straightforward in practical cases as will be shown in the applications. In the last part of this section it is shown how the structure is simplified in specific cases.

1. a 2 x 2 diagonal controller without cascade 2. a 2 x 2 diagonal controller with a conventional cascade 3. an advanced SITO structure designed for one composition loop and a conventional controller for the other composition loop 4. the proposed multivariable control structure.

2.1. Square Ca6Cade SY6tem6

It is desired to understand how much the additional complexity in design and control that is involved going from control structure # 1 to control structure # 4 is payed back by an improvement in performance. The differences in performance are closely related to the choice of the secondary measurement and to how this is related to the interactions among the loops and to the capability of rejection of the disturbances on the compositions by means of the cascade. Such properties can be quantitatively expressed by means of the non-dimensional parameters introduced by Lorenzi et al. (1995):

By square cascade systems, it is (maybe improperly) referred to cascade systems in which the number of secondary outputs equals the number of primary outputs (n = m). If such is the case, the vector Ul is void (as the matrices G 11 , G 2 t), only a single path parallel to the process is present, the feedback signal ofthe outer loop is Yl -y~ and the controller is an approximate inverse of G 12 G 22l •

2.2. Single Ca6cade SY6tem6 By single cascade systems, it is referred to cascade systems in which there is a single secondary output around which the inner loop is closed (m = 1). If such is the case, the square matrix G 22 reduces to a single element so that its inverse can be factored out of all the expressions in which it appears.

1. 11

[(~)ml

-

(~lri] /(~)ml

k11ki2/1:12k01 which expresses the fraction of

the effect of the manipulated variable m2 on the output variable Yl, which is recovered

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by returning the internal variable y; at its steady-state value as the result of closing the cascade loop;

A delay of 6 minutes in the compositions is due to the analyser which supplies measurements with a sampling time of 24 minutes. As it is typical of many distillation columns the dynamics of the different elements are very similar to each other. This simplifies greatly the design of the advanced controllers since in all the ratios of first order systems the dynamic elements can be cancealed out and only a delay remains. In this way, the matrix in (6) has a very simple form:

2. 1'2 [(~)m; - (~),;.m.] /(~)m; kllkid/k1dkil which represents the fraction of the effect of the disturbance on the primary output which is recovered by dosing the internal loop; 3. 1'3 =

[(~)m; - (~)".m.] /(~)ml.m. =

k21k1d/kllk2d which expresses the fraction of

the effect of the disturbance on the output variable Y2 which is recovered as the result of perfect control on output variable Yl.

(12)

and the controller can be obtained by inverting its invertible part and multiplying by a diagonal first order filter (both elements are discrete with the appropriate sampling time). All the details ofthe design of the 4 controllers can be found elsewhere (Porcari, 1998). Figure 5 shows the performance of the 4 controllers. Control structure # 1 has a slow behaviour due to the lack of the fast temperature measurement; control structure # 2 responds faster but suffers from the interactions between the internal and the external loop and from the slow sampling time. Control structures # 3 and # 4 respond similarly as far as Y1 is concerned but # 4 is the only one which is able to return quicky also Y2 to the desired value.

Here the pedices 1 and 2 for y indicate the composition cascaded to the temperature and the one at the other end of the column respectively and i indicates the internal variable while the pedices 1 and 2 for m indicate the two manipulated variables. The ideal situation is the one where the three parameters have a value dose to one for all disturbances of interest (no interactions between the composition loops, perfect rejection by means of the cascade). This may not happen for two reasons: the secondary measurement has not been properly chosen; there is no secondary measurement which satisfies all the criteria for all possible disturbances so that a compromise solution must be accepted. These are the reasons why it is interesting to compare performances both in the (almost) ideal case and when at least one of the parameters has a value which is not dose to one.

3.2. 'Yi

:f:.

1

For the sake of simplicity, the analysis of what happens in the nonideal case when one of the 'Yi differs relevantly from 1 is carried out by analysing 3.1. 'Yi :::::: 1 the fictitious case in which only one gain at a time of the 9 transfer functions in equations (7-10) is The process transfer functions in the first case changed in order to change only one of the 'Yi pastudy are as follows (gains are nondimensional, rameters. This may correspond to the analysis of times are in minutes): a different disturbance or to a different choice of the secondary measurement. 1.24e- 15 , 3.1e- 9 • In order to change 1'2, variations in kid have been (7) 911 = 30. + 1 ; 9il = 30. + 1 considered. Figure 6 shows the behaviour of the 4 controllers when kid = 3.6 and 1'2 = 1.6. The same comments as for Figure 5 are still valid. 0.81e- 15 , -0.91e- 15 , However, the return of the temperature to its (8) 912 921 = 30. + 1 ' 27. + 1 steady-state value corresponds to a too strong correction in Yl which takes place for all cascade controllers (as indicated by the value of 1'2). When the composition measurements become available, (9) 9i2 = 922 control structure # 4 guarantees a quick recovery of both controlled variables to their desired value. 0.ge-15, 2.5e- 9 , 0.5e- 15 , In order to change 1'3, variations in k 2d have been 91d 15. + 1 ; 9id = 15. + 1; 92d = 15. + 1 . (10) considered. Figure 7 shows the behaviour of the 4 controllers when k 2d 0.37 and 1'3 1.6. The indication of the parameter 13 is confirmed by the The values of the nondimensional parameters are: behaviour of the 3 cascade controllers since a too strong correction in Y2 takes place in all cases. 11 = 1.1; 1'2 1.11; 13 = 1.17. (11)

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Fig. 5. Dynamic behaviour of the 4 control structures for a step disturbance

Fig. 6. Dynamic behaviour of the 4 control structures for a step disturbance ("(2 = 1.6)

Control structure # 4 is still the only one which manages to lead both controlled variable to their steady state in a short time. In order to change "Yl, variations in 1c12 have been considered. Figure 8 shows the behaviour of the 4 controllers when 1c12 4.55 and "Yl 2. Control structures # 2 and # 3 show in this case stronger oscillations due to the interactions between the composition loops while the performance of control structure # 4 degrades only marginally. It is relevant to point out that the improvement in performance from control structure # 2 to # 3 is due to the elimination of the cascade interactions while the further improvement of structure # 4 is due to the elimination of the residual multivariable interactions of the top and bottom loops.

On the other side, the IMC controllers has the main advantage of giving a one-shot solution to the control problem so that at any time step the explicit control law is available with no need of solving a different optimization problem each time. If the process dynamics are of the kind in section 3, the proposed controller is very attractive since it is simple to design, easy to work and effective in performance. If the dynamics are more complex or the role played by the constraints is important, the predictive controller can be a reasonable alternative to the above proposed controller. In this case, according to the interactions and to the relative speed of the two loops, one can use a conventional controller in the inner loop and a predictive controller in the outer loop or a predictive controller in both loops. While the inner controller requires the dynamic relationship between the manipulated variables and the secondary outputs, the outer predictive controller requires the dynamic relationship between the secondary outputs set-points and the primary outputs together with the dynamic relationship between the n - m manipulated variables Ul and the primary outputs. The model that links the secondary outputs set-points and the primary outputs is the less obvious extension of the predictive controller to the cascade structure and may

=

=

4. PREDICTIVE CONTROL IN CASCADE SYSTEMS The use of predictive controllers in place of modelbased controllers of the IMC family offers two main advantages: 1. no explicit inversion of the multi variable model with the consequent separation of minimum phase and nonminimum phase components is required; 2. constraints are explicitely taken into account.

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Fig. 7. Dynamic behaviour of the 4 control structures for a step disturbance bl = 1.6)

Fig. 8. Dynamic behaviour of the 4 control structures for a step disturbance bl = 2)

be accomplished in a number of different ways. The comparison of the different control structures that arise from different choices is the subject of some further research.

Filter in Cascade Control Schemes. Ind. Eng. Chem. Re!., 31, 2694. Lorenzi, G., D. Semino, C. Scali and A. Brambilla (1995). Analysis and Design of Cascade Control Schemes for Distillation. In: Proceeding' of DYCORD+ '95., pp. 255-260 . Helsingor, Denmark. Luyben, W. (1973). Parallel Cascade Control. lnd. Eng. Chem. Fund., 12, 463. Mc Avoy, T . J. and N. Ye (1995). Nonlinear Inferential Parallel Cascade Control. In: Proceeding' of DYCORD+ '95., pp. 441-446. Helsingor, Denmark. Porcari, S. (1998) . Control of Chemical Proce" through Multivariable Calcade. Thesis. Dept. of Chemical Engineering, Pisa, Italy. Semino, D. and A. Brambilla (1996). An Efficient Structure for Parallel Cascade Control. lnd. Eng. Chem. Rei., 35, 1845. Shen, S. H. and C. C. Yu (1990). Selection of Secondary Measurement for Parallel Cascade Control. AIChE J ., 36, 1267. Wolff, E. A. and S. Skogestad (1996). Temperature Cascade Control of Distillation Columns. Ind. Eng. Chem. Rei., 35, 475. Yu, C. C. (1988). Design of Parallel Cascade Control for Disurbance Rejection. AIChE J., 34, 1833.

5. CONCLUSIONS In this paper it has been shown how an advanced controller can be designed for multi variable cascade systems. The control scheme, which may seem complicated in the general case, has shown its simplicity in the application to a practical case of a distillation column. In all the tests, the performance of the control structure is better than the one of all alternative structures that have been tried. Even though, the best achievable performance is closely related to the choice of the secondary measurement 50 that a correct choice is ~eceS5ary to make the advanced structure as profItable as possible; the parameters introduced by Lorenzi et al. (1995) can be helpful in this respect. The use of cascade predictive controllers as an alternative to the proposed scheme has been analysed in the last part of the paper.

6. REFERENCES

Brambilla, A. and D. Semino (1992). Nonlinear

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