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(1981), where we have shown that the Langmuir-Hinshelwood equation describes the experimental data well. We also showed the energy of activation values for the reaction and the adsorption. Finally, as was pointed out in our work, the pyrolysis was performed first in situ, followed the gasification. Data were obtained taking this into account, so that the procedure for the determination of the reaction rates is correct.
Literature Cited Marcllio et al. "Proceedings, International Conference on Coal Science"; Dusseldorf, 1981: p 203.
COOPEIUFRG Coordenacao dos Programas de Pos-Graduacao de Engenharia Universidade Federal do Rio de Janeiro Rio de Janeiro, Brazil
Martin Schmal
Comments on "Some Results on Dynamlc Interaction Analysis of Complex Control Systems" Sir: In a recent article, McAvoy (1983) advocates the use of the relative gain array (RGA) as a tool in analysis of dynamic interactions. The present correspondence points to a danger with the use of RGA and shows that in several of the examples used by McAvoy (1983) inspection of the transfer function matrix (TFM) is a simpler and better way of analyzing dynamic interaction.
[c e]
RGA Counter Example Consider a system with the following TFM G(S) =
s+1
[a ;]
(1)
The steady-state and dynamic RGA of this system is A(S) =
(2)
which when using the normal interpretation of the RGA should indicate that there is no interaction at any frequency, and hence a simple dual loop control system should work well. However, this is far from being the case. There is considerable interaction due to similar gains in diagonal and off-diagonal elements, and furthermore the off-diagonal element (1,2) has an unstable right plane pole. With a dual loop control system the closed loop TFM is also triangular. The above result is general; that is, any system with a triangular TFM has the identity matrix as its dynamic relative gain array.
RGA and Distillation Column Control As a case for dynamic interaction analysis using the RGA, McAvoy (1983) compared conventional and material balance control of a distillation column. The analyzer dynamics are modeled by pure dead time, and dead times of 1 and 5 min are considered. A better model of the analyzer dynamics would probably be a zero-order hold. However, regardless of how the analyzer dynamics are modeled, the interaction can be analyzed from the TFM alone. On the other hand, the quality of the resultant control system depends on the sampling frequency, Le., the analyzer dead time. For conventional control the model contains a time delay in the off-diagonal elements. This delays interactions even though the dynamics between XDand L and V respectively are otherwise very similar and have gains of the same magnitude. The dynamics between XB and respectively L and V are also similar except for the time delay, and again the gains are of the same magnitude. McAvoy (1983) misleadingly states that the lower loop is decoupled from the upper loop, while the upper loop 0196-4305/85/1124-0228$01.50/0
exhibits some interaction with an analyzer dead time of 1min. A particular analyzer dead time, which influences all elements of the system transfer function matrix equivalently, does nothing to decouple loops or change the interaction in the system. However, the sampling frequencies' relation to the system time constants naturally influence the quality of control. Looking at the time constanta, it is seen that an analyzer dead time of 1 min is less than the time constants in the XD dynamics, but of the same magnitude as the fast time constants in the XB dynamics. It would therefore be expected that XDwould be controlled well by such a system, but XB would show a poor response. An analyzer dead time of 5 min is still less than the time constants of XD dynamics, but not significantly. A dead time of 5 min is between the time constants of the XD dynamics. Such a system would therefore give poor control of XD and very poor control of XB. In the analysis of McAvoy (1983),the magnitudes of the dynamic relative gains elements are used. These magnitudes for one row or one column of the array do not add up to 1 as is the case for the steady-state RGA. On the other hand, the real part of the elements in one row or column does add up to 1, and there is no evidence to suggest that the magnitude should be used in dynamic RGA analysis. For material balance control the model contains gains of identical magnitude in the second column, i.e., the influence of boil-up on both concentrations. These two terms also exhibit time constants of similar magnitude. Hence considerable interaction exists in the system. This interaction can only be removed by a nondiagonal decoupler designed by any suitable method such as INA, DNA, or CL. Looking at the system time constants, it is seen that the off-diagonal elements contain terms with time constants around 1 min, and the elements of XDdynamics have time constants around 5 min. Hence both analyzer dead times would give poor control if material balance control were used. The preceding analysis shows without any numerical calculations that with an analyzer dead time of 1min the conventional control scheme is better than material balance control. However, for an analyzer dead time of 5 min the analysis gives no hint to which scheme is the better one. The RGA is unnecessary in reaching these conclusions. Furthermore, Jensen et al. (1984) show that the RGA does not give a measure of closed loop interaction. Hence it is incorrect that for a very slow analyzer the steady-state RGA would give an accurate measure of the amount of interaction present. Also since disturbances are in general arbitrary and unknown, it is not possible to establish a priori whether a particular interaction transmittance is helpful or harmful (Jensen, 1981).
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RGA and Control System Design In a fifth case for dynamic interaction analysis using the
RGA and Paper Machine Head Box Control As a second case for dynamic interaction analysis using the RGA, McAvoy (1983) discusses control of a paper machine head box. Since only a parametric TFM is given, it is difficult to analyze how significant interaction is in the closed loop system. However, the closed loop interaction will depend on all the parameters of the TFM, not just e and d as the RGA analysis indicates. Control of a paper machine head box is also discussed by Belletrutti (1971), and his numerical model indicates that the diagonal terms of the TFM are the dominating ones.
RGA, McAvoy (1983) considers the use of simple models to predict a control problem. The use of simple approximate models to analyze a control system at the design stage is a very powerful technique. The simplified dynamic model of a distillation column given in eq 24 of McAvoy (1983) shows gains of similar size in the (1,l) and (2,l) elements of the TFM. Hence according to the simplified model, Q B influences P and T in much the same way. Likewise D influences P and T in the same way. Thus considerable interaction is present in the system. Calculation of the RGA adds nothing to this result.
RGA and Flash Tower Control The problem of pairing variables in controlling a flash tower is the third case for dynamic interaction analysis using the RGA presented by McAvoy (1983). Inspecting the dynamic model and pairing variables so the largest gains are moved to the diagonal indicates that the interface I should be controlled by M2,Fl by M3,and Pl by MI. The remaining gains show that the I-M, loop is to a large extent decoupled from the other two loops. The gains also indicate that there is considerable interaction between the F1-M3 and the Pl-Ml loops. The RGA is unnecessary in reaching these conclusions, as are any other numerical calculations.
Conclusion Several aspects of dynamic interaction analysis as treated by McAvoy (1983) have been commented upon. It has been shown that the RGA can give misleading results about closed loop interaction or variable pairing and resulting control quality in triangular systems. It has also been shown that in all the examples presented by McAvoy, a straightforward examination of the dynamic model can give valuable information about variable pairing, closed loop interaction, and control quality without any numerical calculations, and the calculation of the steady-state or dynamic RGA adds little insight into the interactions of the systems. Acknowledgment The author thanks Exxon Chemical Canada and Exxon Chemical Co. for their kind permission to publish this work. However, the views presented are those of the author and not necessarily the positions of Exxon Chemical Canada and Exxon Chemical Co. Literature Cited
RGA and Large System The fourth problem considered by McAvoy (1983) is the pairing of variables in a 5 X 5 distillation column control problem. This is a further case in which the standard RGA approach gives a variable pairing which is in conflict with current industrial practice according to McAvoy (1983). This conflict should be recognized for what it is, Le., a further indication that RGA does not give a good measure of closed loop interaction. McAvoy (1983) discussed which loops should be closed in analyzing intercations in large systems and how the RGA analysis is affected by such loop closures. It ought to be clear that any loops whose pairings are not in question or whose system dynamics are significantly faster than the dynamics of the interaction being analyzed should be closed. Of course in closing loops the appropriate controller dynamics of the closed loops should be considered in the process of reducing the order of the dynamic model.
Beiletrutti, F. F. Ph.D. Thesis, University of Manchester, Manchester, England, 1971. Jensen, N.; Fisher, D. G.: Shah, S. L. AIChEJ. 1984 (accepted for publication). Jensen, N. Ph.D. Thesis, University of Alberta, Edmonton, Alberta, Canada, 1981. McAvoy. T. J. Ind. Eng. Chem. Process Des. Dev. 1983. 22, 42-49.
Exxon Chemical Canada Sarnia, Ontario N7T 7M5 Canada
Niels Jensen
Response to Comments on “Some Results on Dynamic Interaction Analysis of Complex Control Systems” that as an indicator of closed loop difficulties the RGA is “useful”. This conclusion would seem to answer the question raised by Jensen if one is interested in the ultimate engineering application of interaction measures. When one recognizes that Jensen is using a different measure of interaction than that used in developing the RGA, most of his comments can be understood. For example, in the distillation tower example he questions whether the RGA gives an accurate measure of interaction when a slow analyser is used. For the head box he also questions the interaction results produced by an RGA analysis. In terms of indicating closed loop difficulties the RGA results are accurate in both of these examples. A number of other comments made by Jensen are difficult to justify. In discussing conventional distillation column control he concludes that a 5-min dead time would give “very poor control” of X B and only “poor control” of xD. He arrives at this conclusion based on the fact that
Sir: In a recent correspondence, Jensen (1984)questions whether or not the relative gain array (RGA) is needed. His comments are directed toward a recent paper by McAvoy (1983a). There is one fundamental problem with the arguments raised by Jensen as well as a number of specific problems. The fundamental problem rests on the question of how one defines a measure of closed loop interaction and what the purpose of the measure is. In a related article, Jensen et al. (1984) propose one definition for an interaction measure and then proceed to claim that other interaction indices are inferior measures of interaction based on their definition. As associated question is: what is the goal of measuring closed loop interaction? Traditionally, the goal has been to determine whether or not there will be difficulties in closed loop controller design and operation due to interaction as opposed to simply measuring interaction for its own sake. Indeed, Jensen et al. (1984) concluded 0196-4305/85/ 1124-0229$01.50/0
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