- Email: [email protected]
A new approach to defining a dynamic relative gain
Control Engineering Practice 11 (2003) 907–914
A new approach to defining a dynamic relative gain Thomas Mc Avoya,*, Yaman Arkunb, Rong Chena, Derek Robinsonc, P. David Schnellec a
Institute for Systems Research/Department of Chemical Engineering, University of Maryland, College Park, MD 20742, USA b College of Engineering, KOC University, Istanbul, Turkey c E.I. DuPont de Nemours & Company, 1007 Market Street (N6512), Wilmington, DE 19898, USA Accepted 19 September 2002
Abstract A new approach to defining a dynamic RGA (DRGA) is presented. The approach assumes the availability of a dynamic process model which is used to design a proportional output optimal controller. The new DRGA is defined based on the resulting controller gain matrix. Two examples in which the traditional RGA gives the wrong pairings and an inaccurate indication of the amount of interaction present are discussed. One example involves transfer function models and the other an industrial recycle/reactor system. In both cases the new DRGA indicates the best pairings to use and it accurately assesses the extent of interaction present. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic interaction; Relative gain; Loop pairing; Optimal control
1. Introduction Since its introduction in 1966 by Bristol (1966) the relative gain array (RGA) has been widely used in control system design and analysis. The primary use of the RGA is in determining control loop pairings in multi-loop SISO (MISO) control systems, and in assessing the extent of interaction present in such systems. The use of the RGA was popularized by Shinskey (1988), who applied it to numerous examples, including blending, energy conservation, and distillation. Mc Avoy (1983) published a monograph in which many of the applications of the RGA up to that date were presented. Among the advantages of the RGA are the following. It requires minimal process information, and due to its ratio nature even approximate process models can give useful results. It is independent of control system tuning and process disturbances, and it is simple to calculate. Thus, from a cost benefit point of view the RGA rates very highly. However, the RGA also has some deficiencies, and in some cases it leads to incorrect conclusions about how control loops should be *Corresponding author. Department of Chemical Engineering, University of Maryland, College Park, MD 20742. Tel.: 301-4051939; fax: 301-314-9920. E-mail address: [email protected] (T. Mc Avoy).
paired and how much loop interaction exists. In MISO systems disturbances can have a profound effect on the transient performance that is achieved (Stanley, MarinoGalarraga, & Mc Avoy, 1985). Since the RGA does not consider disturbances it does not give any insight into these cases. In other cases one way interaction, which is not measured by the RGA, can also be severe. Perhaps the most important limitation involves the fact that the RGA does not consider dynamics, and as a result it can lead to incorrect loop pairings. This paper presents a new approach to defining a dynamic RGA (DRGA) that overcomes this limitation. Any approach that includes dynamics clearly requires additional information compared to the traditional RGA approach. The first paper to present an approach to defining a DRGA was published by Witcher and Mc Avoy (1977). They used a transfer function model in place of the traditional steady-state model used for the RGA calculation. The denominator of the DRGA involved achieving perfect control at all frequencies, while the numerator was simply the open loop transfer function. Bristol also published an approach to calculating a DRGA (Bristol, 1979), as did a number of other authors (Tung & Edgar, 1981; Gagnepain & Seborg, 1982). In some of these papers (Tung & Edgar, 1981) the approach presented required that a somewhat detailed feedback controller design be carried out. Since the
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DRGA is most valuable for screening alternate control system designs, the requirement of an extensive controller design tends to defeat the utility of these methods. A better approach to defining a useful DGRA should involve relatively little user interaction in the controller design aspect of the analysis. In this paper it is assumed that a dynamic model of the process is available. If the model is a transfer function model then it is converted to a state space model. A proportional output optimal feedback controller is used, and the controller gain matrix is calculated for the expected value of the initial state being on a unit sphere (Levine & Athans, 1970). Since it is calculated for an expected value of a disturbance, the resulting controller is independent of specific disturbances, and its calculation can be carried out automatically using optimization. Numerical approaches to solving this optimization problem have been investigated by many researchers and Chen, Mc Avoy, Robinson, and Schnelle (2001) give recommendations on which algorithms are the best to use in terms of speed and convergence properties. The output optimal controller does require specification of weighting matrices for the process measurements, manipulated variables, and states. How to choose these weights is discussed in the paper. Two examples are given in which the traditional RGA gives rise to incorrect pairings because of dynamics. The first involves a 2 2 transfer function model, and the second involves a model of a 4 4 industrial reactor/recycle system.
where BS ; and CS can be calculated from B; and C using the scale factors. The optimal control problem that is solved is a linear quadratic regulator with output feedback (Levine & Athans, 1970). The objective function to be minimized is given by Z 1 N T # dt: J¼ ðy# Qy# þ u# T RuÞ ð5Þ 2 0 In our calculations Q and R are typically taken as identity matrices, since u and y have been scaled by their operating ranges. Such a choice for Q and R treats all measurements and manipulated variables equally. An output feedback matrix gives the admissible controls # u# ¼ K y:
Levine and Athans (1970) developed the solution to the expected value of J in Eq. (5) for the case where the initial state forcing lies on a unit sphere around the origin. Since K is calculated based on the dynamic model of the process, it in essence contains information about the process dynamics. It is proposed to use K to develop a new definition of a dynamic RGA. A definition of a dynamic RGA: In this section it is assumed that the number of manipulated and measured variables is equal, and therefore K is a square matrix. If K is non-square then the approach proposed by Chang and Yu (1990) can be used. The derivation of the DRGA, is based on the controller K; and the i; jth element of the DRGA is defined as lDij
1.1. New definition for a dynamic relative gain array Optimal control problem solved: In the plantwide control design methodology it is assumed that a linear state space dynamic model is available for the plant. This model is given by dx=dt ¼ Ax þ Bu;
ð1Þ
y ¼ Cx;
ð2Þ
with xð0Þ ¼ x0 : The first step in designing a plantwide architecture is to scale Eqs. (1) and (2). The manipulated variables, u; are scaled by their operating range, uR ; and the measurements, y; by the range in which it is desired to hold them, yR : The state scaling only affects the initial condition, x0 : In the examples discussed below transfer function models are used and the states are not scaled. Another approach to scaling the states would be to use their steady state values as scale factors. If the scaled # then Eqs. (1) and (2) can be variables are u# and y; written in as # dx=dt ¼ Ax þ BS u;
ð3Þ
y# ¼ CS x;
ð4Þ
ð6Þ
@ui =@yj juk a0;kai ; @ui =@yj juk ¼0; kai
ð7Þ
Both terms in Eq. (7) give the gain of ui to yj during a transient in which the process is controlled using the optimal output proportional gain matrix, K: The numerator gives the change in manipulative variable, ui ; to a change in measurement, yj ; for the case where the optimal controller is bringing the system back to the origin starting from a random initial state on the unit sphere. The denominator is calculated using the same optimal controller gain matrix, K; used in the numerator. This K is calculated by minimizing J in Eq. (5) subject to Eqs. (3), (4), and (6). However, the trajectory traversed in the denominator is such that only a single manipulated variable, ui ; changes. This trajectory is an idealized transient that cannot be achieved in practice for random x0 forcing, but it provides a useful reference point for assessing how a variable’s sensitivity changes depending on the nature of the transient response considered. Similarly, when transfer function models are used to define a dynamic RGA (Witcher & McAvoy, 1977), the denominator involves the idealized case where perfect control is achieved at all frequencies. Such idealized control cannot be achieved in practice as well, but the dynamic RGA based on it has proven to be useful. From Eq. (6) the partial derivatives in Eq. (7) can be
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calculated to give the following result for lDij : Kij lDij ¼ ¼ Kij K# ji ; 1=K# ji
ð8Þ
where K# is equal to K 1 : Before discussing the application of Eq. (8), another possible definition of a dynamic RGA based on optimal control can be given. An advantage of this definition is that it does not involve the idealized, hypothetical transients that the denominator of Eq. (7) does. To estimate manipulated variable interaction a base case optimal control problem, Eqs. (3)–(6), is solved with the R matrix equal to R0 : Then the optimal control problem is re-solved with each of the manipulated variables emphasized. First, all the diagonal entries in the R matrix are multiplied by 100, except the entry for the manipulated variable being emphasized, assumed to be ui : Its R entry remains the same. Then Eqs. (3)–(6) are solved and a relative gain is calculated as the ratio of the gains for ui from the base case divided into the gains when ui is emphasized. Mathematically this dynamic RGA is given by sij
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manipulated variable has a negative value of sij ; then its behavior switches sign depending on how aggressively the other manipulated variables are moving. Such a pairing should be avoided since interaction is likely to be high. Large values of sij also indicate large changes in the behavior of a manipulated variable, which in turn indicate that interaction is high. This pairing rule is a heuristic since at present a theoretically based pairing rule has not been developed. Eq. (8) can be interpreted in the same manner as Eq. (9). To use Eq. (8) for pairing a SISO system one would choose pairings in which the DRGA is close to one. For such pairings the coupling between ui and yj should remain the same whether or not other uk ’s are moving. While Eq. (9) generates a dynamic RGA from feasible transient paths, its elements do not sum to 1, and this is a disadvantage for their interpretation. Chen, Mc Avoy, Robinson, and Schnelle (2000) have used Eq. (9) to determine loop pairings for use in plantwide control design. In the results given below only Eq. (8) is used for illustration. 1.2. Examples of the use of the DRGA
@ui =@yj jR¼R0 ; @ui =@yj jR¼R1
ð9Þ
where R1 is the weighting Rkk ¼ 100R0kk ; kkai and Rii ¼ R0ii : Both terms in Eq. (9) give the gain of ui to yj during a transient in which the process is controlled using an optimal output proportional controller. The numerator gives the change in manipulative variable, ui ; to a change in measurement, yj ; for the case where the full base case optimal controller is bringing the system back to the origin. The denominator gives the same change for the case control is achieved primarily using only ui since the other manipulated variables are heavily penalized. The sij ratio given by Eq. (9) is scale independent, and its determination requires the solution of a number of optimal control problems. The following interpretation of sij is proposed. If sij is close to 1, then the optimal gain between ui and yj remains the same regardless of whether the remaining manipulated variables are changing aggressively or they are moving very little. If sij E1:0 then one can interpret this result to mean that the manipulated variable under consideration does not interact with the other manipulated variables in so far as yj is concerned. Thus, pairing yj with ui would be one candidate for SISO control. If a
Example 1—transfer function model: To demonstrate the utility of the DRGA definition given above the following transfer function is used: 2 3 5e40s K12 e4s 6 100s þ 1 10s þ 1 7 7 G¼6 ð10Þ 4 K e4s 5e40s 5 21 10s þ 1 100s þ 1 with values of K12 ¼ 1; and K21 ¼ 5; 1; and 5 and K12 ¼ 0; and K21 ¼ 1; 5 and 10. It is assumed that the gains in Eq. (10) are scaled. The model given by Eq. (10) involves a system that has fast off diagonal dynamics relative to its diagonal dynamics. First, Eq. (5) is solved for this model using K12 = 1 and a third order Pade’ approximation (Truxal, 1955) for the deadtimes. The Pade approximation is used since the methods for solving the output optimal control problem are based on a state space approach. To test the sensitivity of the DRGA (lD ) to the scaling used in Eq. (5) three different values for R are used and the resulting 1,1 relative gain elements are given in Table 1. In all cases in Table 1 the steady-state RGA suggests the use of a diagonal pairing which should have a small amount of interaction. The
Table 1 Steady-state (RGA) and DRGA ðlD Þ 1,1 elements for K12 ¼ 1 and Q ¼ I R¼I
R ¼ 10I
R ¼ 0:1I
K21
RGA
DRGA
Pairing
DRGA
Pairing
DRGA
Pairing
1 5 5
1.04 0.833 1.25
3.38 0.0732 0.353
Diagonal Off-diag. Off-diag.
2.24 0.122 0.250
Diagonal Off-diag. Off-diag.
6.89 0.0726 0.594
Diagonal Off-diag. Off-diag.
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DRGA suggests that interaction as the result of process dynamics will be larger than that indicated by the RGA if diagonal pairings are used. Further for the cases where the magnitude of K21 is 5, the DRGA suggests off diagonal pairings that can take advantage of the speed of the off diagonal elements relative to the diagonal elements. As Table 1 shows the suggested pairings are insensitive to large changes in the value of the weighting matrix, R: However, the magnitude of the DRGA does change with R: To test whether the pairing indications given in Table 1 are correct the following approach is taken. Two SISO PI controllers are used for the cases of diagonal and off diagonal pairing. The model for each controller is Z ui ¼ Kci ei þ KIi ei dt: ð11Þ The four controller parameters, two controller gains, Kci ; and two integral gains, KIi ; are optimized for a step change in y1 followed by a step change in y2 using the same objective function that was used for the optimal control calculation, namely Eq. (5). Equal weighting is given to the measurements and the manipulated variables. In calculating the errors for y1 and y2 the difference between these measurements and their set points is used. Since the minimization involved in tuning the PI controllers is non-convex, several different
starting points are used to determine the best values of the tuning parameters. Figs. 1 and 2 give the results of the calculation, and the tuning parameters used are given in Table 2. Fig. 1 presents results for the case where K21 ¼ 5; while Fig. 2 gives results for the case where K21 =5. For both cases the DRGA suggests that off diagonal pairings should be used. The transient results in both figures are similar. The off diagonal pairings take advantage of the fast y21 transfer function to achieve excellent responses for y2 : There is very little interaction from y1 to y2 : Figs. 1 and 2 show that the off diagonal y2 responses are significantly better than the diagonal y2 responses. In the case of y1 its response to a step in its set point is a little better when a diagonal pairing is used than when an off diagonal pairing is used for both K21 ¼ 5 and 5. For the case of K21 ¼ 5 the y1 response of the diagonal pairing is somewhat better than that produced by the off diagonal pairing when the set point for y2 is step changed. For K21 ¼ 5 the y1 response for a diagonal pairing is much better than that for an off diagonal pairing when the set point for y2 is step changed. Comparing the responses in Figs. 1 and 2 one can draw several conclusions. First, the traditional RGA does not give a correct assessment of the amount of interaction present when diagonal pairings are used. Second, the DRGA correctly predicts that when K21 ¼ 5 more interaction exists than when K21 ¼ 5: Finally, the DRGA correctly indicates that an off diagonal pairing
Fig. 1. Transient response for diagonal and off diagonal pairings for K21 ¼ 5; K12 ¼ 1:
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Fig. 2. Transient Response for Diagonal and Off Diagonal Pairings for K2;1 ¼ 5; K1;2 ¼ 1:
Table 2 Optimized tuning parameters for cases where DRGA suggests an off diagonal pairing ðK12 ¼ 1Þ Controller parameters
K21 ¼ 5 diagonal
K21 ¼ 5 off diagonal
K21 ¼ 5 diagonal
K21 ¼ 5 off diagonal
Kc1 Kc2 KI1 KI2
1:94e 01 4:90e 01 1:57e 03 2:81e 03
3:63e 01 6:41e 01 4:62e 02 1:02e 03
8:96e 02 2:94e 01 7:50e 04 1:56e 03
3:68e 01 1:98e 01 5:69e 02 5:10e 04
produces a better overall response than a diagonal pairing. The excellent y2 response more than makes up for the poorer y1 response. For K21 ¼ 5 achieving the response shown in Fig. 2 for the off diagonal pairing requires using positive feedback for the y1 u2 loop, as can be seen in Table 2. When K21 =5 the 2 2 system is paired on a negative steady-state RGA, so positive feedback must be used in one loop for stability. This example presents a control system designer with a pairing dilemma. If control system integrity is important and one desires a control system that is robust to instrument failure, then pairing on positive RGAs should be carried out. In this case the potential advantages indicated by the DRGA cannot be achieved. On the other one may be willing to pair on a negative
Table 3 Steady-state (RGA) and DRGA ðlD Þ 1,1 elements for K12 ¼ 0 and Q¼R¼I K21
RGA
DRGA
Pairing
1 5 10
1.00 1.00 1.00
1.05 0.471 0.117
Diagonal Off-diag. Off-diag.
RGA in order to achieve the dynamic benefits indicated by the DRGA. Three additional cases of Eq. (10) were also studied for which K12 ¼ 0: These three cases are given in Table 3, where it can be seen that the DRGA suggests an off diagonal pairing when K21 ¼ 5; and 10. Fig. 3 shows the response produced by optimized PI controllers for both the diagonal and off diagonal paired systems for K21 ¼ 10: The same optimization approach discussed above is used to calculate the PI parameters which are given in Table 4. The responses in Fig. 3 are similar to those in Figs. 1 and 2 and the same conclusions can be drawn about them as are given above. As can be seen the DGRA correctly predicts that the off diagonal paired system will produce a better overall response than the diagonally paired system. For the diagonally paired system there is significant one way
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Fig. 3. Transient Response for Diagonal and Off Diagonal Pairings for K21 ¼ 10; K12 ¼ 0:
Table 4 Tuning parameters for cases where K12 ¼ 0 Controller parameters
K21 ¼ 5 diagonal
K21 ¼ 5 off diagonal
Kc1 Kc2 KI1 KI2
6:51e 02 4:02e 01 5:51e 04 2:79e 03
1:96e 01 5:21e 01 2:60e 02 6:02e 04
interaction from y1 to y2 ; and only in the direction from y2 to y1 is there no interaction. Previous dynamic RGA’s based on transfer functions would incorrectly call for a diagonal pairing for the second and third cases shown in Table 3. The fact that the dynamic RGA presented here correctly calls for an off-diagonal pairing is a clear advantage of this new definition. Example 2—an industrial reactor/recycle system: In an earlier paper (Robinson, Chen, Mc Avoy, & Schnelle, 2001) an optimal control approach was used to synthesize a plantwide control system for an industrial reactor recycle system, shown in Fig. 4. In the earlier paper a detailed dynamic model of the process, the expected disturbances and set point changes are presented. The process has 3 main units, a reactor, flasher, and recycle tank. The reaction that takes place is A þ B-C: The goal of the plant is to produce as much product C as possible. It is desirable for economic
reasons to sell as much A in the product as possible, up to 2.1 wt% A. If the product exceeds 2:1 wt% A it cannot be sold and must be diverted to waste. The setpoint for the product is normally kept at 2:0 wt% A. There is an impurity, D which enters with the fresh B stream. This impurity needs to be regulated since a build up of D in the system impedes the reaction and allows unreacted A to exit the reactor. The reactor is a CSTR and it is assumed that the temperature is established by the enthalpy of the feed and that the reactor jacket medium is able to maintain the reactor temperature at a constant value, and that the liquid level in the reactor is controlled by its exit flow. Thus, reactor temperature and level control are not addressed for this process. The product flasher separates all of the A and C into a bottoms stream, and the B and D are separated into an overhead stream. There are four variables to be controlled, the production rate, the wt% A in the product, the wt% D in the recycle tank, and the recycle tank level. There are four manipulated variables that can be used, the setpoints of the A and B feed flow controllers, the recycle valve, and the purge valve. The original model was nonlinear, and identification used on it to develop a transfer function approximation, which is given in Robinson et al. (2001), together with the scale factors used for u and y: The state variables, x; are not scaled. In optimizing Eq. (5) the transfer function model was converted to state space form, and the numerical
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Recycle Level wt % D Recycle Valve Recycle Tank
Fresh B Feed
FC
Purge Valve
Production Rate wt % A in Product
LC
Reactor Fresh A Feed
Flasher
FC
Fig. 4. Schematic of reactor recycle process.
approach discussed by Chen et al. (2001) was used to calculate K: The resulting K is: Prod
%DRecl
0:023
0:042 0:057
6 1:218 6 K ¼6 4 0:300
0:043 0:230 0:758 0:089
2
0:340
0:024
Pairing Purge Valve Recycle Valve
%AProd
0:973
Level 3 1:287 pur 0:026 7 7 Recl 7 1:564 5 B 0:046 A
ð12Þ
From the model, which contains integrating variables, the following steady-state RGA can be calculated using the method proposed by Arkun and Downs (1990): 2
Pur
0 6 0:072 RGA¼ 6 6 40 1:072
Recl
B
0:107 0:196
0:148 0:829
0:723
0:067
0:027 0:044
Table 5 Feedback pairing candidates
A
3 0:745 Prod 0:046 7 7 %D Recl : 7 0:210 5 %AProd Level 0 ð13Þ
If one rules out negative and zero RGA pairings, then the only possible choice for level control is to use the purge flow. Further, if one rules out pairings that occur on RGA values less than 0.1, then there are three possible pairings and these are shown in Table 5. By far the best choice from an RGA point of view is pairing 1 which involves using the A flow to control the product ðRGA ¼ 0:745Þ; the B flow to control the %D in the product ðRGA ¼ 0:829Þ; and the recycle to control the %A in the product ðRGA ¼ 0:723Þ: Another choice, pairing 2, involves controlling the product flow with the recycle ðRGA ¼ 0:107Þ; the %D in the recycle with the B flow ðRGA ¼ 0:829Þ; and the %A in the product with the A flow ðRGA ¼ 0:210Þ: The final pairing 3 involves controlling the product flow with the B feed ðRGA ¼ 0:148Þ; the %D in the recycle with the recycle flow ðRGA ¼ 0:196Þ; and the %A in the product with the A flow ðRGA ¼ 0:210Þ: From a purely RGA
1 2 3
B Flow
A Flow
Recyl. Level wt% A Prod. wt% D Recyl. Prod. Rate Recyl. Level Prod. Rate wt% D Recyl. wt% A Prod. Recyl. Level wt% D Recyl. Prod. Rate wt% A Prod.
point of view one would not choose either pairing 2 or 3 because of the small RGA pairing values that are involved. Now consider the DRGA calculated from Eq. (8) and given below
Prod %DRecl %AProd Level 2 3 0:001 0:062 0:005 0:932 Pur 6 0:925 0:015 0:059 0:000 7 Recl 6 7 LD ¼ 6 : 7 4 0:010 0:920 0:004 0:066 5 B 0:063 0:003 0:933 0:002 A
ð14Þ
Eq. (14) indicates that the best pairing from a DGRA point of view is pairing 2, and since all its DRGA values are close to 1, this pairing should exhibit little dynamic interaction. The results given by Robinson et al. (2001) support this conclusion and they show that the transients produced by pairing 2 are significantly superior to those produced by pairing 1. This superiority results even though additional feedforward controllers are used to improve the transient performance of scheme 1 (Robinson et al., 2001). Thus, this industrial example helps to confirm the utility of the DRGA defined in this paper for application to practical problems.
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2. Conclusions A new approach to defining a dynamic RGA has been presented. The approach assumes that a dynamic process model is available. A proportional output optimal controller is designed using a state space approach. The resulting controller gain matrix is used to define a dynamic RGA. Several examples in which the traditional RGA gives the wrong pairings and an inaccurate indication of the amount of interaction present are discussed. One set of examples involves a transfer function model and another example involves an industrial recycle/reactor system. In all cases the new DRGA indicates the best pairings to use and it accurately assesses the extent of interaction present.
References Arkun, Y., & Downs, J. (1990). A general method to calculate input– output gains and the RGA for integrating processes. Computers and Chemical Engineering, 14, pp. 1101–1110. Bristol, E. (1966). On a new measure of interaction for multivariable process control. IEEE Transactions on Automatic Control, AC-11, 133. Bristol, E. (1979). Recent results on interactions in multivariable process control. Presented at 71st Annual AIChE Meeting, Houston, TX. Chang, J., & Yu, C. (1990). The relative gain for non-square multivariable systems. Chemical Engineering Science, 45, 1309–1323.
Chen, R., Mc Avoy, T., Robinson, D., & Schnelle, P. D. (2000). An optimal control approach to designing plantwide control system architectures. Presented at annual AIChE meeting, Los Angeles, CA. Chen, R., Mc Avoy, T., Robinson, D., & Schnelle, P. (2001). Designing plantwide control system architectures by using optimal control. Proceedings of 6th World congress of chemical engineering, Melbourne, Australia. Gagnepain, J. P., & Seborg, D. E. (1982). Analysis of process interactions with application to multiloop control system design. Industrial and Engineering Chemistry, Process Design and Development, 21, 5–11. Levine, W., & Athans, M. (1970). On the determination of the optimal constant feedback gains for linear multivariable systems. IEEE Transactions on Automatic Control, AC-15, 44–48. Mc Avoy, T. (1983). Interaction analysis. Research Triangle Park, NC: Instrument Society of America. Robinson, D., Chen, R., Mc Avoy, T., & Schnelle, D. (2001). An optimal control based approach to designing plantwide control system architectures. Journal of Process Control, 11, 223–236. Shinskey, F. G. (1988). Process control systems (3rd ed.). New York: McGraw-Hill. Stanley, G., Marino-Galarraga, M., & Mc Avoy, T. (1985). Short-cut operability analysis: 1. The relative disturbance gain. Industrial and Engineering Chemistry, Process Design and Development, 24, 1181–1188. Truxal, J. (1955). Automatic feedback control system synthesis. New York, NY: McGraw-Hill, p. 550. Tung, L., & Edgar, T. (1981). Analysis of control-output interactions in dynamic systems. A.I.Ch.E. Journal, 27, 690–693. Witcher, M., & McAvoy, T. J. (1977). Interacting control systems: Steady state and dynamic measurement of interaction. ISA Transactions, 16, 35–41.
A new approach to defining a dynamic relative gain Thomas Mc Avoya,*, Yaman Arkunb, Rong Chena, Derek Robinsonc, P. David Schnellec a
Institute for Systems Research/Department of Chemical Engineering, University of Maryland, College Park, MD 20742, USA b College of Engineering, KOC University, Istanbul, Turkey c E.I. DuPont de Nemours & Company, 1007 Market Street (N6512), Wilmington, DE 19898, USA Accepted 19 September 2002
Abstract A new approach to defining a dynamic RGA (DRGA) is presented. The approach assumes the availability of a dynamic process model which is used to design a proportional output optimal controller. The new DRGA is defined based on the resulting controller gain matrix. Two examples in which the traditional RGA gives the wrong pairings and an inaccurate indication of the amount of interaction present are discussed. One example involves transfer function models and the other an industrial recycle/reactor system. In both cases the new DRGA indicates the best pairings to use and it accurately assesses the extent of interaction present. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic interaction; Relative gain; Loop pairing; Optimal control
1. Introduction Since its introduction in 1966 by Bristol (1966) the relative gain array (RGA) has been widely used in control system design and analysis. The primary use of the RGA is in determining control loop pairings in multi-loop SISO (MISO) control systems, and in assessing the extent of interaction present in such systems. The use of the RGA was popularized by Shinskey (1988), who applied it to numerous examples, including blending, energy conservation, and distillation. Mc Avoy (1983) published a monograph in which many of the applications of the RGA up to that date were presented. Among the advantages of the RGA are the following. It requires minimal process information, and due to its ratio nature even approximate process models can give useful results. It is independent of control system tuning and process disturbances, and it is simple to calculate. Thus, from a cost benefit point of view the RGA rates very highly. However, the RGA also has some deficiencies, and in some cases it leads to incorrect conclusions about how control loops should be *Corresponding author. Department of Chemical Engineering, University of Maryland, College Park, MD 20742. Tel.: 301-4051939; fax: 301-314-9920. E-mail address: [email protected] (T. Mc Avoy).
paired and how much loop interaction exists. In MISO systems disturbances can have a profound effect on the transient performance that is achieved (Stanley, MarinoGalarraga, & Mc Avoy, 1985). Since the RGA does not consider disturbances it does not give any insight into these cases. In other cases one way interaction, which is not measured by the RGA, can also be severe. Perhaps the most important limitation involves the fact that the RGA does not consider dynamics, and as a result it can lead to incorrect loop pairings. This paper presents a new approach to defining a dynamic RGA (DRGA) that overcomes this limitation. Any approach that includes dynamics clearly requires additional information compared to the traditional RGA approach. The first paper to present an approach to defining a DRGA was published by Witcher and Mc Avoy (1977). They used a transfer function model in place of the traditional steady-state model used for the RGA calculation. The denominator of the DRGA involved achieving perfect control at all frequencies, while the numerator was simply the open loop transfer function. Bristol also published an approach to calculating a DRGA (Bristol, 1979), as did a number of other authors (Tung & Edgar, 1981; Gagnepain & Seborg, 1982). In some of these papers (Tung & Edgar, 1981) the approach presented required that a somewhat detailed feedback controller design be carried out. Since the
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DRGA is most valuable for screening alternate control system designs, the requirement of an extensive controller design tends to defeat the utility of these methods. A better approach to defining a useful DGRA should involve relatively little user interaction in the controller design aspect of the analysis. In this paper it is assumed that a dynamic model of the process is available. If the model is a transfer function model then it is converted to a state space model. A proportional output optimal feedback controller is used, and the controller gain matrix is calculated for the expected value of the initial state being on a unit sphere (Levine & Athans, 1970). Since it is calculated for an expected value of a disturbance, the resulting controller is independent of specific disturbances, and its calculation can be carried out automatically using optimization. Numerical approaches to solving this optimization problem have been investigated by many researchers and Chen, Mc Avoy, Robinson, and Schnelle (2001) give recommendations on which algorithms are the best to use in terms of speed and convergence properties. The output optimal controller does require specification of weighting matrices for the process measurements, manipulated variables, and states. How to choose these weights is discussed in the paper. Two examples are given in which the traditional RGA gives rise to incorrect pairings because of dynamics. The first involves a 2 2 transfer function model, and the second involves a model of a 4 4 industrial reactor/recycle system.
where BS ; and CS can be calculated from B; and C using the scale factors. The optimal control problem that is solved is a linear quadratic regulator with output feedback (Levine & Athans, 1970). The objective function to be minimized is given by Z 1 N T # dt: J¼ ðy# Qy# þ u# T RuÞ ð5Þ 2 0 In our calculations Q and R are typically taken as identity matrices, since u and y have been scaled by their operating ranges. Such a choice for Q and R treats all measurements and manipulated variables equally. An output feedback matrix gives the admissible controls # u# ¼ K y:
Levine and Athans (1970) developed the solution to the expected value of J in Eq. (5) for the case where the initial state forcing lies on a unit sphere around the origin. Since K is calculated based on the dynamic model of the process, it in essence contains information about the process dynamics. It is proposed to use K to develop a new definition of a dynamic RGA. A definition of a dynamic RGA: In this section it is assumed that the number of manipulated and measured variables is equal, and therefore K is a square matrix. If K is non-square then the approach proposed by Chang and Yu (1990) can be used. The derivation of the DRGA, is based on the controller K; and the i; jth element of the DRGA is defined as lDij
1.1. New definition for a dynamic relative gain array Optimal control problem solved: In the plantwide control design methodology it is assumed that a linear state space dynamic model is available for the plant. This model is given by dx=dt ¼ Ax þ Bu;
ð1Þ
y ¼ Cx;
ð2Þ
with xð0Þ ¼ x0 : The first step in designing a plantwide architecture is to scale Eqs. (1) and (2). The manipulated variables, u; are scaled by their operating range, uR ; and the measurements, y; by the range in which it is desired to hold them, yR : The state scaling only affects the initial condition, x0 : In the examples discussed below transfer function models are used and the states are not scaled. Another approach to scaling the states would be to use their steady state values as scale factors. If the scaled # then Eqs. (1) and (2) can be variables are u# and y; written in as # dx=dt ¼ Ax þ BS u;
ð3Þ
y# ¼ CS x;
ð4Þ
ð6Þ
@ui =@yj juk a0;kai ; @ui =@yj juk ¼0; kai
ð7Þ
Both terms in Eq. (7) give the gain of ui to yj during a transient in which the process is controlled using the optimal output proportional gain matrix, K: The numerator gives the change in manipulative variable, ui ; to a change in measurement, yj ; for the case where the optimal controller is bringing the system back to the origin starting from a random initial state on the unit sphere. The denominator is calculated using the same optimal controller gain matrix, K; used in the numerator. This K is calculated by minimizing J in Eq. (5) subject to Eqs. (3), (4), and (6). However, the trajectory traversed in the denominator is such that only a single manipulated variable, ui ; changes. This trajectory is an idealized transient that cannot be achieved in practice for random x0 forcing, but it provides a useful reference point for assessing how a variable’s sensitivity changes depending on the nature of the transient response considered. Similarly, when transfer function models are used to define a dynamic RGA (Witcher & McAvoy, 1977), the denominator involves the idealized case where perfect control is achieved at all frequencies. Such idealized control cannot be achieved in practice as well, but the dynamic RGA based on it has proven to be useful. From Eq. (6) the partial derivatives in Eq. (7) can be
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calculated to give the following result for lDij : Kij lDij ¼ ¼ Kij K# ji ; 1=K# ji
ð8Þ
where K# is equal to K 1 : Before discussing the application of Eq. (8), another possible definition of a dynamic RGA based on optimal control can be given. An advantage of this definition is that it does not involve the idealized, hypothetical transients that the denominator of Eq. (7) does. To estimate manipulated variable interaction a base case optimal control problem, Eqs. (3)–(6), is solved with the R matrix equal to R0 : Then the optimal control problem is re-solved with each of the manipulated variables emphasized. First, all the diagonal entries in the R matrix are multiplied by 100, except the entry for the manipulated variable being emphasized, assumed to be ui : Its R entry remains the same. Then Eqs. (3)–(6) are solved and a relative gain is calculated as the ratio of the gains for ui from the base case divided into the gains when ui is emphasized. Mathematically this dynamic RGA is given by sij
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manipulated variable has a negative value of sij ; then its behavior switches sign depending on how aggressively the other manipulated variables are moving. Such a pairing should be avoided since interaction is likely to be high. Large values of sij also indicate large changes in the behavior of a manipulated variable, which in turn indicate that interaction is high. This pairing rule is a heuristic since at present a theoretically based pairing rule has not been developed. Eq. (8) can be interpreted in the same manner as Eq. (9). To use Eq. (8) for pairing a SISO system one would choose pairings in which the DRGA is close to one. For such pairings the coupling between ui and yj should remain the same whether or not other uk ’s are moving. While Eq. (9) generates a dynamic RGA from feasible transient paths, its elements do not sum to 1, and this is a disadvantage for their interpretation. Chen, Mc Avoy, Robinson, and Schnelle (2000) have used Eq. (9) to determine loop pairings for use in plantwide control design. In the results given below only Eq. (8) is used for illustration. 1.2. Examples of the use of the DRGA
@ui =@yj jR¼R0 ; @ui =@yj jR¼R1
ð9Þ
where R1 is the weighting Rkk ¼ 100R0kk ; kkai and Rii ¼ R0ii : Both terms in Eq. (9) give the gain of ui to yj during a transient in which the process is controlled using an optimal output proportional controller. The numerator gives the change in manipulative variable, ui ; to a change in measurement, yj ; for the case where the full base case optimal controller is bringing the system back to the origin. The denominator gives the same change for the case control is achieved primarily using only ui since the other manipulated variables are heavily penalized. The sij ratio given by Eq. (9) is scale independent, and its determination requires the solution of a number of optimal control problems. The following interpretation of sij is proposed. If sij is close to 1, then the optimal gain between ui and yj remains the same regardless of whether the remaining manipulated variables are changing aggressively or they are moving very little. If sij E1:0 then one can interpret this result to mean that the manipulated variable under consideration does not interact with the other manipulated variables in so far as yj is concerned. Thus, pairing yj with ui would be one candidate for SISO control. If a
Example 1—transfer function model: To demonstrate the utility of the DRGA definition given above the following transfer function is used: 2 3 5e40s K12 e4s 6 100s þ 1 10s þ 1 7 7 G¼6 ð10Þ 4 K e4s 5e40s 5 21 10s þ 1 100s þ 1 with values of K12 ¼ 1; and K21 ¼ 5; 1; and 5 and K12 ¼ 0; and K21 ¼ 1; 5 and 10. It is assumed that the gains in Eq. (10) are scaled. The model given by Eq. (10) involves a system that has fast off diagonal dynamics relative to its diagonal dynamics. First, Eq. (5) is solved for this model using K12 = 1 and a third order Pade’ approximation (Truxal, 1955) for the deadtimes. The Pade approximation is used since the methods for solving the output optimal control problem are based on a state space approach. To test the sensitivity of the DRGA (lD ) to the scaling used in Eq. (5) three different values for R are used and the resulting 1,1 relative gain elements are given in Table 1. In all cases in Table 1 the steady-state RGA suggests the use of a diagonal pairing which should have a small amount of interaction. The
Table 1 Steady-state (RGA) and DRGA ðlD Þ 1,1 elements for K12 ¼ 1 and Q ¼ I R¼I
R ¼ 10I
R ¼ 0:1I
K21
RGA
DRGA
Pairing
DRGA
Pairing
DRGA
Pairing
1 5 5
1.04 0.833 1.25
3.38 0.0732 0.353
Diagonal Off-diag. Off-diag.
2.24 0.122 0.250
Diagonal Off-diag. Off-diag.
6.89 0.0726 0.594
Diagonal Off-diag. Off-diag.
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DRGA suggests that interaction as the result of process dynamics will be larger than that indicated by the RGA if diagonal pairings are used. Further for the cases where the magnitude of K21 is 5, the DRGA suggests off diagonal pairings that can take advantage of the speed of the off diagonal elements relative to the diagonal elements. As Table 1 shows the suggested pairings are insensitive to large changes in the value of the weighting matrix, R: However, the magnitude of the DRGA does change with R: To test whether the pairing indications given in Table 1 are correct the following approach is taken. Two SISO PI controllers are used for the cases of diagonal and off diagonal pairing. The model for each controller is Z ui ¼ Kci ei þ KIi ei dt: ð11Þ The four controller parameters, two controller gains, Kci ; and two integral gains, KIi ; are optimized for a step change in y1 followed by a step change in y2 using the same objective function that was used for the optimal control calculation, namely Eq. (5). Equal weighting is given to the measurements and the manipulated variables. In calculating the errors for y1 and y2 the difference between these measurements and their set points is used. Since the minimization involved in tuning the PI controllers is non-convex, several different
starting points are used to determine the best values of the tuning parameters. Figs. 1 and 2 give the results of the calculation, and the tuning parameters used are given in Table 2. Fig. 1 presents results for the case where K21 ¼ 5; while Fig. 2 gives results for the case where K21 =5. For both cases the DRGA suggests that off diagonal pairings should be used. The transient results in both figures are similar. The off diagonal pairings take advantage of the fast y21 transfer function to achieve excellent responses for y2 : There is very little interaction from y1 to y2 : Figs. 1 and 2 show that the off diagonal y2 responses are significantly better than the diagonal y2 responses. In the case of y1 its response to a step in its set point is a little better when a diagonal pairing is used than when an off diagonal pairing is used for both K21 ¼ 5 and 5. For the case of K21 ¼ 5 the y1 response of the diagonal pairing is somewhat better than that produced by the off diagonal pairing when the set point for y2 is step changed. For K21 ¼ 5 the y1 response for a diagonal pairing is much better than that for an off diagonal pairing when the set point for y2 is step changed. Comparing the responses in Figs. 1 and 2 one can draw several conclusions. First, the traditional RGA does not give a correct assessment of the amount of interaction present when diagonal pairings are used. Second, the DRGA correctly predicts that when K21 ¼ 5 more interaction exists than when K21 ¼ 5: Finally, the DRGA correctly indicates that an off diagonal pairing
Fig. 1. Transient response for diagonal and off diagonal pairings for K21 ¼ 5; K12 ¼ 1:
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Fig. 2. Transient Response for Diagonal and Off Diagonal Pairings for K2;1 ¼ 5; K1;2 ¼ 1:
Table 2 Optimized tuning parameters for cases where DRGA suggests an off diagonal pairing ðK12 ¼ 1Þ Controller parameters
K21 ¼ 5 diagonal
K21 ¼ 5 off diagonal
K21 ¼ 5 diagonal
K21 ¼ 5 off diagonal
Kc1 Kc2 KI1 KI2
1:94e 01 4:90e 01 1:57e 03 2:81e 03
3:63e 01 6:41e 01 4:62e 02 1:02e 03
8:96e 02 2:94e 01 7:50e 04 1:56e 03
3:68e 01 1:98e 01 5:69e 02 5:10e 04
produces a better overall response than a diagonal pairing. The excellent y2 response more than makes up for the poorer y1 response. For K21 ¼ 5 achieving the response shown in Fig. 2 for the off diagonal pairing requires using positive feedback for the y1 u2 loop, as can be seen in Table 2. When K21 =5 the 2 2 system is paired on a negative steady-state RGA, so positive feedback must be used in one loop for stability. This example presents a control system designer with a pairing dilemma. If control system integrity is important and one desires a control system that is robust to instrument failure, then pairing on positive RGAs should be carried out. In this case the potential advantages indicated by the DRGA cannot be achieved. On the other one may be willing to pair on a negative
Table 3 Steady-state (RGA) and DRGA ðlD Þ 1,1 elements for K12 ¼ 0 and Q¼R¼I K21
RGA
DRGA
Pairing
1 5 10
1.00 1.00 1.00
1.05 0.471 0.117
Diagonal Off-diag. Off-diag.
RGA in order to achieve the dynamic benefits indicated by the DRGA. Three additional cases of Eq. (10) were also studied for which K12 ¼ 0: These three cases are given in Table 3, where it can be seen that the DRGA suggests an off diagonal pairing when K21 ¼ 5; and 10. Fig. 3 shows the response produced by optimized PI controllers for both the diagonal and off diagonal paired systems for K21 ¼ 10: The same optimization approach discussed above is used to calculate the PI parameters which are given in Table 4. The responses in Fig. 3 are similar to those in Figs. 1 and 2 and the same conclusions can be drawn about them as are given above. As can be seen the DGRA correctly predicts that the off diagonal paired system will produce a better overall response than the diagonally paired system. For the diagonally paired system there is significant one way
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Fig. 3. Transient Response for Diagonal and Off Diagonal Pairings for K21 ¼ 10; K12 ¼ 0:
Table 4 Tuning parameters for cases where K12 ¼ 0 Controller parameters
K21 ¼ 5 diagonal
K21 ¼ 5 off diagonal
Kc1 Kc2 KI1 KI2
6:51e 02 4:02e 01 5:51e 04 2:79e 03
1:96e 01 5:21e 01 2:60e 02 6:02e 04
interaction from y1 to y2 ; and only in the direction from y2 to y1 is there no interaction. Previous dynamic RGA’s based on transfer functions would incorrectly call for a diagonal pairing for the second and third cases shown in Table 3. The fact that the dynamic RGA presented here correctly calls for an off-diagonal pairing is a clear advantage of this new definition. Example 2—an industrial reactor/recycle system: In an earlier paper (Robinson, Chen, Mc Avoy, & Schnelle, 2001) an optimal control approach was used to synthesize a plantwide control system for an industrial reactor recycle system, shown in Fig. 4. In the earlier paper a detailed dynamic model of the process, the expected disturbances and set point changes are presented. The process has 3 main units, a reactor, flasher, and recycle tank. The reaction that takes place is A þ B-C: The goal of the plant is to produce as much product C as possible. It is desirable for economic
reasons to sell as much A in the product as possible, up to 2.1 wt% A. If the product exceeds 2:1 wt% A it cannot be sold and must be diverted to waste. The setpoint for the product is normally kept at 2:0 wt% A. There is an impurity, D which enters with the fresh B stream. This impurity needs to be regulated since a build up of D in the system impedes the reaction and allows unreacted A to exit the reactor. The reactor is a CSTR and it is assumed that the temperature is established by the enthalpy of the feed and that the reactor jacket medium is able to maintain the reactor temperature at a constant value, and that the liquid level in the reactor is controlled by its exit flow. Thus, reactor temperature and level control are not addressed for this process. The product flasher separates all of the A and C into a bottoms stream, and the B and D are separated into an overhead stream. There are four variables to be controlled, the production rate, the wt% A in the product, the wt% D in the recycle tank, and the recycle tank level. There are four manipulated variables that can be used, the setpoints of the A and B feed flow controllers, the recycle valve, and the purge valve. The original model was nonlinear, and identification used on it to develop a transfer function approximation, which is given in Robinson et al. (2001), together with the scale factors used for u and y: The state variables, x; are not scaled. In optimizing Eq. (5) the transfer function model was converted to state space form, and the numerical
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Recycle Level wt % D Recycle Valve Recycle Tank
Fresh B Feed
FC
Purge Valve
Production Rate wt % A in Product
LC
Reactor Fresh A Feed
Flasher
FC
Fig. 4. Schematic of reactor recycle process.
approach discussed by Chen et al. (2001) was used to calculate K: The resulting K is: Prod
%DRecl
0:023
0:042 0:057
6 1:218 6 K ¼6 4 0:300
0:043 0:230 0:758 0:089
2
0:340
0:024
Pairing Purge Valve Recycle Valve
%AProd
0:973
Level 3 1:287 pur 0:026 7 7 Recl 7 1:564 5 B 0:046 A
ð12Þ
From the model, which contains integrating variables, the following steady-state RGA can be calculated using the method proposed by Arkun and Downs (1990): 2
Pur
0 6 0:072 RGA¼ 6 6 40 1:072
Recl
B
0:107 0:196
0:148 0:829
0:723
0:067
0:027 0:044
Table 5 Feedback pairing candidates
A
3 0:745 Prod 0:046 7 7 %D Recl : 7 0:210 5 %AProd Level 0 ð13Þ
If one rules out negative and zero RGA pairings, then the only possible choice for level control is to use the purge flow. Further, if one rules out pairings that occur on RGA values less than 0.1, then there are three possible pairings and these are shown in Table 5. By far the best choice from an RGA point of view is pairing 1 which involves using the A flow to control the product ðRGA ¼ 0:745Þ; the B flow to control the %D in the product ðRGA ¼ 0:829Þ; and the recycle to control the %A in the product ðRGA ¼ 0:723Þ: Another choice, pairing 2, involves controlling the product flow with the recycle ðRGA ¼ 0:107Þ; the %D in the recycle with the B flow ðRGA ¼ 0:829Þ; and the %A in the product with the A flow ðRGA ¼ 0:210Þ: The final pairing 3 involves controlling the product flow with the B feed ðRGA ¼ 0:148Þ; the %D in the recycle with the recycle flow ðRGA ¼ 0:196Þ; and the %A in the product with the A flow ðRGA ¼ 0:210Þ: From a purely RGA
1 2 3
B Flow
A Flow
Recyl. Level wt% A Prod. wt% D Recyl. Prod. Rate Recyl. Level Prod. Rate wt% D Recyl. wt% A Prod. Recyl. Level wt% D Recyl. Prod. Rate wt% A Prod.
point of view one would not choose either pairing 2 or 3 because of the small RGA pairing values that are involved. Now consider the DRGA calculated from Eq. (8) and given below
Prod %DRecl %AProd Level 2 3 0:001 0:062 0:005 0:932 Pur 6 0:925 0:015 0:059 0:000 7 Recl 6 7 LD ¼ 6 : 7 4 0:010 0:920 0:004 0:066 5 B 0:063 0:003 0:933 0:002 A
ð14Þ
Eq. (14) indicates that the best pairing from a DGRA point of view is pairing 2, and since all its DRGA values are close to 1, this pairing should exhibit little dynamic interaction. The results given by Robinson et al. (2001) support this conclusion and they show that the transients produced by pairing 2 are significantly superior to those produced by pairing 1. This superiority results even though additional feedforward controllers are used to improve the transient performance of scheme 1 (Robinson et al., 2001). Thus, this industrial example helps to confirm the utility of the DRGA defined in this paper for application to practical problems.
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2. Conclusions A new approach to defining a dynamic RGA has been presented. The approach assumes that a dynamic process model is available. A proportional output optimal controller is designed using a state space approach. The resulting controller gain matrix is used to define a dynamic RGA. Several examples in which the traditional RGA gives the wrong pairings and an inaccurate indication of the amount of interaction present are discussed. One set of examples involves a transfer function model and another example involves an industrial recycle/reactor system. In all cases the new DRGA indicates the best pairings to use and it accurately assesses the extent of interaction present.
References Arkun, Y., & Downs, J. (1990). A general method to calculate input– output gains and the RGA for integrating processes. Computers and Chemical Engineering, 14, pp. 1101–1110. Bristol, E. (1966). On a new measure of interaction for multivariable process control. IEEE Transactions on Automatic Control, AC-11, 133. Bristol, E. (1979). Recent results on interactions in multivariable process control. Presented at 71st Annual AIChE Meeting, Houston, TX. Chang, J., & Yu, C. (1990). The relative gain for non-square multivariable systems. Chemical Engineering Science, 45, 1309–1323.
Chen, R., Mc Avoy, T., Robinson, D., & Schnelle, P. D. (2000). An optimal control approach to designing plantwide control system architectures. Presented at annual AIChE meeting, Los Angeles, CA. Chen, R., Mc Avoy, T., Robinson, D., & Schnelle, P. (2001). Designing plantwide control system architectures by using optimal control. Proceedings of 6th World congress of chemical engineering, Melbourne, Australia. Gagnepain, J. P., & Seborg, D. E. (1982). Analysis of process interactions with application to multiloop control system design. Industrial and Engineering Chemistry, Process Design and Development, 21, 5–11. Levine, W., & Athans, M. (1970). On the determination of the optimal constant feedback gains for linear multivariable systems. IEEE Transactions on Automatic Control, AC-15, 44–48. Mc Avoy, T. (1983). Interaction analysis. Research Triangle Park, NC: Instrument Society of America. Robinson, D., Chen, R., Mc Avoy, T., & Schnelle, D. (2001). An optimal control based approach to designing plantwide control system architectures. Journal of Process Control, 11, 223–236. Shinskey, F. G. (1988). Process control systems (3rd ed.). New York: McGraw-Hill. Stanley, G., Marino-Galarraga, M., & Mc Avoy, T. (1985). Short-cut operability analysis: 1. The relative disturbance gain. Industrial and Engineering Chemistry, Process Design and Development, 24, 1181–1188. Truxal, J. (1955). Automatic feedback control system synthesis. New York, NY: McGraw-Hill, p. 550. Tung, L., & Edgar, T. (1981). Analysis of control-output interactions in dynamic systems. A.I.Ch.E. Journal, 27, 690–693. Witcher, M., & McAvoy, T. J. (1977). Interacting control systems: Steady state and dynamic measurement of interaction. ISA Transactions, 16, 35–41.