Ind. Eng. Chem. Process Des. Dev. 1982, 21, 5-1 1
5
‘Receivedfor review May 30, 1979 Revised manuscript received May 22,1981 Accepted August 12,1981
Ross, S. L. “Differential Equations”; Bialsdel, Ginn and Co.: London, 1964; p 282. Schwarzenbach, 0.; Flaschka, H. ”Complexometric Titrations”; Methuen & Co.,Ltd.: London, 1969a; p 132. Schwarzenbach, 0.; Flaschka, H. “Complexometric Titrations”; Methuen & Co.,Ltd.: London, 1969b; pp 257-61. V W , A. I.“Quantttative Inorganlc Analysis”; ELBS-Longmans. 1968; p 434. Vogel, A. I.“Vogel’s Text Book of Quantitative Inorganic Analysis Including Elementary Instrumental Analysls”; ELBS-Longmans. 1978a; p 533. Vogel, A. I. “Vogel’s Text Book of Quantitative Inorganic Analysis”; ELBSLongmans, 1978b; p 322. Vogel. A. I. “Vogel’s Text Book of Quantitative Inorganic Analysis”; ELBSLongmans, 1978c; p 354.
Supplementary Material Available: The mathematical details of calculation of ytheorfrom eq 5 have been given. A typical calculation for the system magnesium ammonium sulfate-copper ammonium sulfate-water (at an a value of 0.333,K = 1.1,and m = 1)has also been described (5 pages). Ordering information is given on any current masthead page.
Analysis of Process Interactions with Applications to Multiloop Control System Design‘ Jean-Plerre Gagnepaln’ and Dale E. Seborg” Department of Chemical and Nuclear Engineering, University of Callfornia, Santa Barbara, Santa Barbara, California 93106
The traditional industrial control strategy for multivariable control problems is to use a multiloop control scheme consisting of several conventional PI or PID controllers. In this approach, the key design decision Is to determine the best pairing of controlled and manipulated variables. This paper presents a systematic method for determining the best pairing using a new measure of process interactions. The new measure is based on the concept of average dynamic gains which are easily calculated from open-loop step responses. The usefulness of the new measure is demonstrated in simulation examples.
large number of possible pairings. To illustrate why some multiloop control configurations do not provide satisfactory control, consider the block diagram of a 1-1/2-2 control c o n f i a t i o n in Figure 2. In general each manipulated variable affects both controlled variables, so there are four process transfer functions: Gll(s), G12(s),G2,(s),and G&). These process interactions may cause severe control loop interactions, as illustrated by the following sequence of events. Suppose that y1deviates from its set point R1.Then the feedback controller for loop 1, Gcll, will take corrective action by adjusting ul. But u1 affects y z as well as y1 and consequently the controller for the second loop, Gm,adjusts u2accordingly. But u2 affects y1 as well as y2. Thus, we see that the initial corrective action in loop 1 (Le., a change in ul) propagates around the two control loops and results in a subsequent change in the other manipulated variable u p Under certain conditions, these control loop interactions can pose serious problems or even prevent the control system from performing properly. For example, if process transfer functions, G12(s)and G21(~),represent the dominant process interaction (Le., large gains, small time delays and small time constants) while Gll(s) and G&) exhibit weak interactions (characterized by small gains, large time constants, and large time delays), then the control system in Figure 2 will not perform very well. In this situation better results would be obtained by using the 1-2/2-1 control configuration. The above example has indicated that process interactions between manipulated and controlled variables should be taken into account when determining the proper control configuration. In the next section we review the existing methods for characterizing process interactions. Measures of Process Interactions The most prominent and widely used approach for characterizing process interactions is the Relative Gain
Introduction In recent years there has been considerable interest in developing process control strategies for multivariable control problems, that is, problems where several process variables are to be controlled and several variables can be manipulated. Although a variety of advanced control techniques are available (Edgar, 1976), only a few industrial applications have been reported (Rijnsdorp and Seborg, 1976). The conventional industrial approach to multivariable control problems is to use a “multi-loop control system” consisting of n conventional PI or PID controllers where n is the number of controlled variables. To illustrate the multiloop control strategy, consider the process in Figure 1 which has two controlled variables, y1 and y2, and two manipulated variables, u1 and u2. The disturbances are considered to be unmeasurable and/or unknown. There are two possible multiloop control configurations. One approach would be to control y1 by adjusting u1 and to control y2 by adjusting u2;this will be referred to as a 1-1/2-2 control configuration. The alternative approach is to control y1 by adjusting u2 and y2 by adjusting ul, i.e., a 1-2/2-1 canfiguration. In designing a multiloop control system, a key design decision is to determine the proper pairing of controlled and manipulated variables. If an incorrect pairing is used, the resulting control system may perform very poorly or even be inoperable (Shinskey, 1979). The process in Figure 1 has only two possible pairings of controlled and manipulated variables; however, a process with n controlled variables and n manipulated variables has n! possible pairings. Thus it would be useful to have a simple, systematic method for determining the best controller pairing from among the A preliminary version of this paper was presented at the 72nd Annual AIChE Meeting, San Francisco, Nov 1979. Esso SAF, Bordeaux Refinery. 0196-4305/82/1121-0005$01.25/0
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1981 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982
The RGA method can be used to categorize process interactions and to provide a recommendation concerning the proper controller configuration (Bristol, 1966;Shinskey, 1979). If Ai, is zero, or close to zero, then uj has little effect on yi. On the other hand, if lAijl is large, then this interaction is significant. In the RGA approach, the recommended controller pairing is the one corresponding to the relative gains which have the largest positive values. For example
Disturbances
u1._ Uj
-+
U,
4
Process Yn
Figure 1. Multivariable process with n manipulated variables and n controlled variables. u1
L
I
L
Figure 2. Block diagram of a multiloop control system (1-1/2-2 configuration).
Array (RGA) method proposed by Bristol (1966). The chief advantages of the RGA approach are that it is easy to use and only requires a crude process model, namely, the process gains which can be determined from steadystate information. To illustrate the RGA method, consider the steady-state process model in eq 1 Y58 = KU8, (1) where yasand u, represent steady-state changes in the n controlled variables and n manipulated variables, respectively, and K is the nXn steady-state process gain matrix. The relative gain between the ith controlled variable, yi, and the j t h manipulated variable, uj, will be denoted by Aip This dimensionless quantity is defined as open-loop gain x.. = closed-loop gain (2) Mathematically, the relative gain can be expressed as (3)
In eq 3 the partial derivative in the numerator is evaluated with all of the manipulated variables except uj held constant. Similarly, for the partial derivative in the denominator, all of the controlled variables except yi are held constant. This latter condition can be achieved (in principle) by adjusting the other n - 1manipulated variables. It is apparent from eq 1that the open-loop gain between yi and u is K j j ,an element of the process gain matrix, K. Bristol(1966) has demonstrated that the closed-loop gain is l / K i jwhere K i .is an element of the matrix inverse of the transpose of k
K
=
[&,I
= (KT’
-1
Y1
(4)
Thus the expressions for hi,in eq 2 and 3 are equivalent to eq 5 A,, 11 = K..K.. 51 11 (5) The Relative Gain Array (RGA) is defined to be the n X n matrix, A = [A,]. A useful property of the RGA is that the sum of the elements in each row and each column is equal to one (Bristol, 1966).
Although the RGA method provides useful information, it neglects the process dynamics. In some situations dynamic characteristics may be the key factor in determining the correct controller configuration. Consequently, other measures of process interaction have been proposed which include both static and dynamic behavior of the process (Rijnsdorp, 1965; Davison, 1969, 1970; Nisenfeld and Schultz, 1971; Witcher and McAvoy, 1977; Tung and Edgar, 1981; Bristol, 1978; Jensen et al., 1980). In a recent paper, Bristol (1978) has generalized his original RGA analysis by replacing steady-state gains in eq 5 with the corresponding process transfer functions. His analysis provides useful insight into dynamic behavior but leaves the controller pairing question still unanswered. Davison (1969,1970) and Tung and Edgar (1981) have examined process interactions in linear, state-space models. The analysis of Tung and Edgar (1981) is especially useful since it provides a measure of dynamic interactions as well as a vigorous derivation of Bristol’s RGA. The Relative Dynamic Array Method Witcher (1977) and Witcher and McAvoy (1977) have proposed an interaction measure based on open-loop step responses. This type of information is easily obtained from experimental data or from the transient response of a process model, for example, a transfer function or statespace model. Consequently,this approach has widespread applicability since it does not require that a particular type of process model be available. The interaction measure proposed by Witcher and McAvoy (1977) is based on a “dynamic potential”, 4 1 , which is the integral of the open-loop step response, y,(tj, to a unit step change in uI at time, t = 0.
4Jt9) = J o y l ( t ) dt (i, j = 1, 2 , ..., n)
(6)
The time period, 8, over which the integration is performed is somewhat arbitrary; Witcher (1977) suggests that 6 be specified as 20% to 100% of the dominant time constant of the process. By substituting the dynamic potential, 4, (e), for the steady-state gain, K,, in eq 5, Witcher and dcAvoy (1977) constructed a “Relative Dynamic Array (RDA)” with element AJO) given by h1(t9)= hl(MIl(fl) (7) where ?,,(t9)
is an element of &t9) and = [4(e)q-1 (8) The RDA provides the same type of normalization as the RGA since the sum of the elements in each row and each column is one. Consequently, the recommended controller pairing is determined in the same manner, namely, the pairing which corresponds to large, positive elements in the RDA is selected. As t9 m, the RDA asymptotically approaches the RGA. Witcher (1977) has applied the RDA
-
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982 7 Table I. Relative Dynamic Arrays
e
h
1 2 5 10 20 30 35 40 50
l(
e ), example 1
h
,( e ), example 2
undefined -0.03
undefined 0.03 0.09 0.21 0.38 0.46 0.49 0.51 0.52
-0.11 -0.35 -1.53 -6.50 - 28.98 t 21.04 t 6.62
t 2.29
m
0.64
analysis to transfer function models of two distillation columns and to a state-space model of a fluid catalytic cracker. Although the RDA analysis does provide considerable insight into the static and dynamic interactions that occur in multivariable processes, it has two shortcomings. (i) Since Xij(6) is a function of the time period 6,the degree of interaction and the recommended pairing determined by the RDA analysis will also depend on 6. Consequently, the RDA may be difficult to interpret. It would be very desirable to have this information reduced to a single interaction measure so as to avoid the problem of considering (ii) The RDA elea time-varying matrix, A(6) = [A,(@]. ment, A,@), may “blow up” due to passing through a discontinuity. To illustrate these difficulties with the RDA approach, consider two hypothetical examples which have the following transfer function matrices and RGA values. example 1
G ( s )=
r-1 02e-S stl
1LX
1.5e-S
1.5e-S 7 s t l 2e-S
10s+
I IJ
= 2 .29
(9)
example 2
less than zero since the corresponding transfer function in eq 10 has a negative gain. Thus R(6) < 0 for all 0 and no discontinuity occurs. Examples 1 and 2 are models of highly interacting processes as indicated by their RGA values. In each case steady-state considerations favor the 1-1/2-2 controller pairing while dynamic considerations favor the alternate 1-2/2-1 pairing. (The process dynamics suggest the 1-2/2-1 pairing due to the small time constants in the off-diagonal transfer functions of eq 9 and 10). Although the RDA analysis provides useful information, it would be better to have a simpler interaction measure that is easier to interpret than the time-varying matrix, A(@. A New Measure of Process Interactions: the Average Relative Gain Matrix In this section a new interaction measure, the average relative gain, pij*, is introduced. This measure is an extension of the RDA concept since it is also based on open-loop step responses. However, the resulting response data are used in a different manner in order to avoid the disadvantages of the RDA method. To illustrate the new measure, consider the following process model As) = G(s) 4 s ) (13) where each process transfer function is expressed as a first-order plus time delay model. K,.;e-dijs
Gij(s)=
Tijs + 1
Suppose that the process is initially at steady state ( u = y = 0 ) and that a unit step change in uj occurs at t = 0. During the time interval [0, dij],yi is not affected by u, and we can say that the “dynamic gain” of the process for this time period is zero. For the time interval [dij, 61, the “average dynamic gain”, Dij*, between yi and uj can be calculated as Dij* = (average change in yi)/(change in uj) (15) But since a unit step change in u .was made with the process initially at steady state, it foiows that Dij* is equal to the average value of y over the time interval [dij,e]
; h , , = 0.64
G(s) = 10s
(10)
+ IJ
The RDA values for these two examples are shown in Table I. For example 2 and 6 < 40, the Aii(6) values are small, which suggests that the 1-2/2-1 pairing should be used. On the other hand, if one considers 6 > 40, then the 1-1/2-2 pairing is favored. Note that for example 1, Aii(6) passes through a discontinuity during the time period, 35 < 6 < 40, since its value goes from -a to +m. To understand why this occurs, consider a process with two inputs and two outputs. Then eq 7 can be written in the equivalent form (Gagnepain, 1979)
where
From eq 11 and 12, it is clear that if R(0) passes through a value of one, then Aii(6) becomes discontinuous as in example 1. For example 2, three of the four dynamic potentials are positive while the fourth, (P11(6), is always
Next, it is convenient to define a piecewise constant function, Dij(t) Di,(t) = 0 for t < dij = Dij*
for dij I t I 6
(17)
In analogy with the RGA and RDA, a new relative gain, pij(t), is defined in terms of Dij(t) F i j ( t ) = Dij(t)Dji(t) (18) where dji is an element of D = [DT]-l. The relative gain pij(t) is a piecewise constant function which is defined for t > el, where dl is the smallest value of t for which the matrix inverse of D ( t ) exists. Gagnepain (1979) has shown that 61 = min b” (dll, d22), max ( 4 2 , d2J (19) To provide a simple measure of process interactions, we consider the average value of p&) over the time interval [61, 61, where 6 is yet to be specified.
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Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982
Finally, in order to use this new interaction measure, we must specify 8. Since most of the step response (63.2%) occurs in a period of time equal to one time constant, we choose 8 as 8 = 81 TM (21)
+
where TM is the largest time constant in the process transfer function matrix, G(s). Stability Considerations One important attribute of the best controller configuration is that the resulting closed-loop system must be stable and remain stable after a transmitter or actuator fails; i.e., the closed-loop system should exhibit a high degree of “integrity” (MacFarlane, 1972). In determining the proper controller pairing it would be desirable to be able to eliminate all unstable configurations by means of an easily applied stability criterion. Fortunately, a useful theorem due to Niederlinski (1971) provides such a result. Niederlinski’s theorem was developed for multivariable systems which can be represented by the following transfer function model As) = G(s) 4 s ) (22) In eq 22, G(s) is an nXn process transfer function matrix. The theorem is derived under the following assumptions: (i) A multiloop control system is employed consisting of n feedback controllers in a 1-1/2-2 .../n-n configuration. Each controller is tuned so that a stable system results when the other n - 1 feedback control loops are opened. (ii) The elements, (Gi.(s)],of G(s) are stable, rational functions. (iii) The feedback controllers all contain integral control action (e.g., I, PI, or PID controllers). Niederlinski’s theorem can be stated as follows: The closed-loop system consisting of the multivariable system in eq 22 and a multiloop control system is structurally monotonic unstable if and only if
In eq 23, IG(0)l denotes the determinant of the steady-state gain matrix G(0). The term “structurally monotonic unstable” means that the closed-loop system is unstable for all possible values of the controller constants. For 2 x 2 systems, it can easily be shown (Gagnepain, 1979) that the stability criterion in eq 23 is equivalent to the condition that All < 0. Thus it follows that for a 2x2 system, a controller pairing which corresponds to negative relative gain elements should not be selected since it results in an unstable system is integral control action is used. This result provides a firm theoretical basis for the previous awertions by Bristol(1966) and Shinskey (1979) that controller pairings which correspond to negative relative gains should be avoided. Gagnepain (1979) has shown that the stability criterion in eq 23 also provides insight into the controller pairing problem for 3x3 systems. If the steady-state process gain matrix, G(O),is denoted by K as in eq 1, then eq 23 can be expressed as
IK‘
KllK22K33 A11
=
lKl
1
*-
(x11)2,3
where (X11)2,3 denotes the (1,l) element of the RGA for the 2x2 minor of K which is obtained by deleting the first row is a relative gain for the 2x2 and first column. Thus subsystem consisting of u2,u3,y2, and y3. (In general, All # (A11)2,3.) Similar expressions can be derived for Xz2 and A33 KllKZ2K33 A22
=
IKI
1
*-
(x11)1,3
By combining eq 24,26,27, and 28, necessary and sufficient conditions for a structurally monotonic unstable system are obtained xll(x11)2,3 (29) &2(h11)1,3
