Ind. Eng. Chem. Res. 1987,26, 180-181
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As shown in Figure 2, the cascade control system comprises two loops. The primary loop measures gas temperature and manipulates the set point of the secondary top-skin-temperature controller. The top-skin-temperature controller manipulates the furnace power in order to track its set point. The cascade controller performance is shown in Figure 3 for a 5.75 “C increase in gas temperature set point. The gas temperature settled at the desired set point after only 40 min. The top skin temperature is seen to follow a complex trajectory; a simple step change in furnace power would have resulted in a much slower gas temperature response. The controller also performs well in the face of load disturbances. Figure 4 shows the gas temperature response to a 30% decrease in power to the bottom furnace zone.
Acknowledgment I gratefully acknowledge the General Electric Company for permission to publish this work. Literature Cited Berty, J. M. Chem. Eng. h o g . 1974, 70(5), 78.
James M. Silva General Electric Company Corporate Research and Development Center Schenectady, New York 12301 Received for review June 20, 1985 Accepted June 10, 1986
CORRESPONDENCE Comments on “Shortcut Operability Analysis. 1. The Relative Disturbance Gain” Sir: In a recent article, Stanley et al. (1985) proposed a new steady-state control loop interaction index, the Relative Disturbance Gain (RDG). The present correspondence addresses two problems associated with the concept of the RDG in relation to its development and its use in assessing the benefits of loop decouplers. 1. “Perfect Control” and the Transfer Function or Gain Matrix
Stanley et al. (1985) and other recent workers in the area of interaction analysis (Friedly, 1984; McAvoy, 1983;Tung and Edgar, 1981) use a transfer function matrix or the corresponding steady-state gain matrix as the basis of their analysis. This is despite the fact that such a representation may hide valuable information about the interactive properties of the system (Johnston and Barton, 1985). This is particularly true when the concept of “perfect control” enters the interaction index definition, as it does with the RDG. An example used by Stanley et al. is a case in point. The model considered had the form x = A x + Bm + D d (1) with A =
R = [:‘5
-2.0 - 5 7
:,J D
=
[:1
where x1 was paired in a control loop with m, (loop 1)and with m2 (loop 2 ) . The RDG is based on a steady-state gain matrix which for the example above is given by x2
From eq 2, it is possible to calculate the RDG elements for the two loops pi = -4.0 p2
= 0.926
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Stanley et al. claim that the large value of PI indicates unfavorable interaction from loop 2 to loop 1. However, consider the state-space model, eq 1,which is shown as a cause and effect graph in Figure 1. If x 2 is held exactly at its set point at all times through perfect control action, then m2 will have no effect whatsoever on xl. This is because m2 can only influence x1 via changes in x 2 ; the perfect control action blocks this influence and effectively decouples loop 2 from loop 1. Similarly, when x l is under perfect control, loop 1is decoupled from loop 2. If perfect control were possible at all times, the loops would be completely decoupled and there would be no interaction. Any interaction index, such as RDG, based on perfect control behavior should reveal the increased decoupling effect as control quality improves (in the limit approaching perfect control). However, this is only possible if the analysis is based on the state-space model rather than on a steady-state gain or transfer function matrix. No interaction index is perfect, and as McAvoy (1985) has pointed out, different workers define their measures of interaction in different ways and for different purposes. However, it is important to note that significant interaction information may be hidden in steady-state gain or transfer function matrices, whereas this information is revealed explicitly by a state-space model. It is appreciated that in many cases, a state-space model will not be available. However, in cases where such a model is available (as in the examples given by Stanley et al.), methods such as the steady-state and dynamic “direct gain” interaction analysis of Johnston and Barton (1985) can be used to a great advantage. 2. RDG as a Guide to the Need for Decouplers Stanley et al. claim that the RDG may be interpreted as giving a comparison of multiloop control to ideal decoupled control. They base this claim on the statement (Stanley et al., 1985, p 1184) that “if the two loops in the system under consideration were decoupled by using an ideal decoupler, then their response would be the same as the SISO response”. This is only true of the response of the controlled variables, not the response of the m a n i p 0‘ 1987 American Chemical Societv
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Literature Cited Friedly, J. C. Ind. Eng. Chem. Process Des. Dev. 1984,23,469-472. Johnston, R. D.;Barton, G. W. Int. J. Control 1985, 41(4), 1005-1013. McAvoy, T. J. Znd. Eng. Chem. Process Des. Dev. 1983,22,42-49. McAvoy, T. J. Ind. Eng. Chem. Process Des. Dev. 1985,24,229-230. Stanley, G.; Marino-Galarraga, M.; McAvoy, T. J. Ind. Eng. Chem. Process Des. Dev. 1985,24, 1181-1188. Tung, L. S.; Edgar, T. F. AIChE J. 1981,27(4),690-693.
u Figure 1. Cause and effect graph for eq 1.
ulated variables, whereas it is the latter on which the RDG is based. It is therefore incorrect to interpret the RDG in this way. A decoupler modifies the response of a manipulated variable in an attempt to compensate for the interactive effects of another manipulated variable. Finally, in relation to decouplers, a distinction should be made between the controller output signal and the value of the manipulated variable which influences the process (typically a flow rate). Stanley et al. appear not to make this distinction, treating these as one and the same, whereas the RDG is actually based on the response of the manipulated variables.
Richard D. Johnston* School of Chemical Engineering and Industrial Chemistry The University of N e w South Wales Kensington, N.S. W. 2033, Australia
Geoffrey W. Barton Department of Chemical Engineering T h e University of Sydney Sydney, N.S. W. 2006,Australia Received for review January 28, 1986 Accepted August 3, 1986
Response to Comments on “Shortcut Operability Analysis. 1. The Relative Disturbance Gain” Sir: In their comments Johnston and Barton discuss the concept of “perfect control”. It is appropriate that these terms are put in quotes since the point they make rests on their definition of “perfect control” and not the definition used in the paper by Stanley et al. (1985). In the paper the RDG is defined as ’a steady-state index, and perfect control is achieved by using reset action. Thus, “perfect control” in the paper is perfect steady-state control. For practical problems, perfect dynamic control is not realistic and it cannot be achieved. On the other hand, perfect steady-state control is achieved all the time through reset action. Thus, the RDG, which is based on perfect steady-state control, is a tool that can be used to assess the operability of practical industrial control loops. Once one is aware that Johnston and Barton are using a different definition for “perfect control” than that used in the paper, their comments can be understood. We contend that while perfect dynamic control is an interesting topic, it is of limited utility in analyzing practical interaction problems. Concerning their comments on the difference between controller output and manipulative variables, we agree that this distinction should be made. For nondecoupled systems, the controller output equals the manipulative variable. In the notation, we defined m ias the manipulated variable. For more clarity it would have been better to define mias the controller output. In defining the RDG, we were careful to state on 1183 that ”& is defined as the
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change in controller output”. Throughout the paper, the RDG is described as referring to a change in controller outputs. In fact, basing an RDG definition on anything but the controller output does not make sense. For a given disturbance, the change in manipulative variables is always the same regardless of whether decouplers are used or not. Finally, Johnston and Barton point out that the RDG should be interpreted in regard to controlled and not manipulated variables. As discussed, above, the RDG definition is based on controller output and not on manipulative variables. This distinction between controller output and manipulative variables only becomes important when decouplers are used. However, regardless of whether or not decouplers are used, the RDG does give a comparison of controlled variable response as well as controller output changes. I t can indeed be interpreted in this manner. In summary, we feel that the points raised by Johnston and Barton are based on semantics. They interpret perfect control differently than is done in the paper. Also, in the paper the RDG is defined and consistently discussed as referring to controller output. Thomas J. McAvoy University of Maryland Department of Chemical Engineering College Park, Maryland 20742
0 1987 American Chemical Society
