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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.
E x x o n 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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1984 American Chemical Society
230 Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
the xD time constants are both 7.74 while the 3tg time constants are 13.8 and 0.4. Since the sum of the time constants in both cases is about the same one would expect the quality of control of both loops to be about the same for a 5-min analyser dead time. Jensen goes through a similar analysis for material balance control, part of which includes a statement that time constants of 1.25,3.3, and 5.76 min are similar! Ultimately he concludes that for a 1-min dead time conventional control is superior while for a 5-min dead time his “analysis gives no hint to which scheme is the better”. In fact, the RGA analysis shows that the material balance scheme will become better and better relative to the conventional scheme as the loops become slower and slower. Nonlinear dynamic tower simulations, given in McAvoy (1983a), bear out the RGA predictions. In his discussion of the decanter (called the Flash Tower) Jensen pairs variables by simply inspecting the gains in the open loop process model. His methods involves pairing “so the largest gains are moved to the diagonal”. It is straightforward to show that this procedure can lead to incorrect pairings in some problems. If the F1variable in the decanter problem is replaced with p = Fl/100 then the resulting process model is
[F] -[%:;i3
1.03
-3.42 145.11s
-3.42 -4.301s
]El
The RGA would give the same, correct pairings for this modified problem as are given in McAvoy (1983a). The reasons is that the RGA pairing results are independent of scaling. Jensen’s ad hoc approach would result in the incorrent pairing of mz-I, ma-P1, and ml-p. Finally, Jensen offers an RGA counter example which involves a triangular system with an off-diagonal unstable element. I t is highly unlikely that any real industrial system would exhibit such dynamic behavior. I feel that his conclusion on triangular systems is much too broad. He states that the “RGA can give misleading results on variable pairing in triangular systemsn. This statement should be qualified by the fact that some elements of the
model are open loop unstable. If all model elements are stable, then the RGA will give the correct pairing. Lastly, it can be pointed out that McAvoy (1983b) already has discussed the limitations and interpretation of the RGA for the important, industrial case of crude tower control which has a triangular gain matrix. In his correspondence, Jensen appears to be arguing against the use of the RGA. He feels that he can get the same information that the RGA gives through examination of the open loop process model. However, as discussed above, several of his ad hoc analyses are incorrect. Insight into a process model can be invaluable if such insight is developed on a sound basis. Jensen’s approach should be judged in light of the ease with which the steady-state RGA can be calculated. An analogous situation would be to look at D , V , p, and p independently and decide if a flow were turbulent. Alternatively, one could calculate a Reynolds number, DVplp, and use it to decide on the nature of the flow. Clearly, if one has gone to the trouble of developing a dynamic process model, then they should not limit their analysis to the dynamic RGA. The advantage of a dynamic RGA analysis is that it can suggest for what types of systems steady state information alone is insufficient and one must consider dynamics. In summary, a careful examination of the points raised and arguments used by Jensen leads one to conclude that the RGA is indeed needed.
Literature Cited Jensen, N.: Fisher, D. G.: Shah, S. L. AIChE J . 1984 (accepted for publication). Jensen, N. Ind. Eng. Chem. Process Des. Dev. 1084, preceding correspondence in this issue. McAvoy, T. J. Id. Eng. Chem. Process Des. D e v . 1983a, 22, 42-49. McAvoy, T. J. “Interaction Analysis Prlnciples and Applications”, Instrument Society of America, Chapter 3, 1983b.
Department of Chemical & Nuclear Engineering University of Maryland College Park, Maryland 20742
CORRECTION Use of the Bristol Array in Designing Noninteracting Control Loops. A Limitation and Extension, J. C. Friedly, Ind. Eng. Chem. Process Des. Deu. 1984, 23, 469. Page 472. The following reference should have been included in the Literature Cited section: Hammarstrom, L. G.; Waller, K. V.; Fagervik, K. C. “Model Mismatching in Multivariable Distillation Control”; Proceedings of the 2nd World Congress of Chemical Engineering, Montreal, 1981.
Thomas J. McAvoy
