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Interaction in two-variable control systems for distillation columns—I
Automatlca, Vol, 1, pp.
15-28. Pergamon Pre~, 1965. Printed in Great Britain.
INTERACTION IN TWO-VARIABLE CONTROL SYSTEMS FOR DISTILLATION COLUMNS--I THEORY J. E. Rm~SDORP Koninklijkc/Shell-Laboratorium, Amsterdam SmnmarymIn general, multivariable control systems are too complex for even a qualitative description of their behaviour. However, for linear two-variable control systems with two control lOOps, the situation is more favourable. In this article, the influence of one loop on the other is deduced from a polar diagram. The interaction quotient of the process K=(G12 G21)/(GII G22) in which the G's represent response functions of the process is used as a convenient measure for the degree of mutual influence. This study is focussed on control systems for continuous distillation columns and gives the theoretical background. The application to distillation columns will be treated in Part II. 1. I N T R O D U C T I O N IN THE application o f control to complicated systems it is quite c o m m o n that the feedback loops for the various controlled conditions show mutual interaction. It is then incorrect to consider the feedback loops separately; the system should rather be studied as a whole, as a multivadable control system. Some cases where mutual interaction can occur are: the the the the the
translational and rotational modes o f airplanes, rockets, etc., azimuth and elevation positioning o f radar antennae [1], r o t o r speed and turbine temperature o f jet engines [2], conditions in boilers [3], conditions in distillation processes [4-6].
The last-mentioned ease is the focal point o f this study. 'More specifically, we want to analyse the interaction in the following two-variable control systems for distillation columns: (a) temperature and pressure control, (b) dual quality control. [1] A. A. KnASOVSKn: Avtomat. Telemekh. 18, 126-36 (1957); Automn remote Control 139--49 (1957). [2] A. ]]OI¢~ENnOMand R. HOOD: NACA Report 980, Washington, April (1949). [3] H. K. CHA11rJO~: In Automatic and Remote Control, VoL I, pp. 132-7. Proc. 1st IFAC Congress, Moscow (1960). [4] H. H. ROSmq~OCK: Trans. lnstn, chem. Engnrs, Lond. 40, 35-53 (1962). [5] O. R ~ v . ~ o ¢ ~ and J. E. RIJNSDORP: Third World Petroleum Congress, Section VII, Paper 5. New York (1959). [6] J. E. Rt~SDORP and A. M~Rt~,cet.o: In Joint Sympon. Instrumentation and Computation, edited by P. A. RoT~,mtnto, pp. 135-43. Inst. of Chem. Engrs, London (1959). 15
16
J.E. RLINSDORP
In Part I of this study the theoretical background will be developed. Simple twovariable control systems in general are considered, keeping in mind the specific requirements of process control. The application to distillation columns will be the subject of Part II. In the following sections, we shall give a critical review of the literature on the theory of multivariable control systems and see how far it can be used for our purpose. 2. T H E
STRUCTURE
OF .THE
PROCESS
In multivariable control systems, the process (controlled system) has several inputs and outputs. Consequently, there are many possible configurations of the responses in the process. MESAROVI~ [7] has investigated the choice of the most suitable configuration for a given problem. For this, he introduces so-called canonical structures. Figures 1 and 2 show the P- and V-canonical structure, respectively, for the case of a two-variable control system. On pages 10 and 32 of his book, MESAROVI6compares the conditions to be satisfied for non-interaction. He finds different values for the transfer functions of the controllers for the P- and for the V-canonical structure.
lYl
l+.,j.~I
-
PROCESS+
+
\
+
/
/Glz
\
4-
i ,r FIG. I. Two-variable control system, the process having P-canonical structure.
PROCESS o'~
C~ Csl
C,z~
1
F21"/
2<
+
/
F1z
FIG. 2. Two-variable control system, the process having V-canonical structure.
The cause of this difference is simply that he has adopted different requirements for the overall responses of the systems. For the P-canonical structure (see Fig. 1) these are: /~1=G110"1 ,
/']1 --~-Gl1~'1 ,
or ~]2 = G220"2 , [7] M. D. M~AROVI~: The Control of Multivariable
(1) /72 = G22~2 ,
Systems.
M . I . T . and Wiley (1960).
17
Interaction in two-variablecontrol systems for distillation columns
for non-interaction with respect to trl, a2, or with respect to (1, (2 respectively, where 111, 2 are the controlled conditions, a l, 2 are the set values, (1, 2 are disturbances in the correcting conditions, GI i and G22 are P-canonical process responses. On the other hand, for the V-canonical structure the requirements are respectively: 111=G1(1 ,
111= GLO'1 ,
(2)
or 112=
112= G2o" 2 ,
G2~2,
where Gx, 2 are V-canonical process responses. As G1 # G11, and Gz # G22, (1) is not the same as (2), and one can expect different values for the controller responses C11, Cz2, CIz and C21. We prefer to use requirements that are independent of an assumption about the structure of the process. More specifically, if non-interaction is desirable, logical requirements are: 112 not dependent on al ~ (for non-interaction with respect to if1, t72, 111 not dependent on a 2 3
or
112 not dependent on ~1 ~ (for non-interaction with respect to 111 not dependent on (2
(3)
(I, (2
J
Then we find for the P-canonical structure: CI IG21 q- C2tG22 = 0, C22G12 q" C12G11=0,
(4)
or G2t
C21
G I 1 G 2 2 - - GI2G21,
C12 =
G12 GIIG22- GI2G2t.
(5)
(See also References 2 and 8, respectively.) For the V-canonical structure the results are:
CllG1F21+C21=O, C22G2FIz+ Cl2 =0,
(6)
or
Czl =F21, Ca2 =F12By using the relationships between the structures [8] R. J. KAVANAOrI: Trans. A.LE.E. 77, (2) 425-9 (1958).
(7)
18
J.E. RLINSDORP
Gl t = GIJ, G22 = G2J,
Gx2 = G2Fx2GtJ,
(8)
G2t = GIF21G2J, where J = ( I - G t F 2 t G 2 F 1 2 ) -1, one can show that (4) is equivalent to (6) and (5) to (7). Hence the choice of the structure does not influence the result, when (3) is used as a requirement for non-interaction. Now we are free to choose a structure corresponding to a physical interpretation of the process. This is desirable for acquiring good insight in the problem. Generally, distillation processes are inherently stable, and the controlled conditions are principally influenced by the correcting conditions, and not vice versa. This makes the V-canonical structure less suitable, because it contains the loop Gt F2tG2Ft2, which could be unstable. Moreover the direction of F2t and F 12 is opposite to the physical signal flow. Consequently, we choose the P-canonical structure for our discussion. As long as there is no need for four controllers, we shall only use two of them (see Fig. 3). A source of disturbance usually affects both controlled conditions. In Fig. 3 this is indicated by the transfer functions F I a n d / ' 2 . _
,..
/
o,( crz
-&
\ "-+
/ 1
/
o,, PROCESS
Fla. 3. Simple two-variable control system. 3. RESPONSES OF THE SIMPLE TWO-VARIABLE SYSTEM In the literature there are many discussions of two-variable systems. Some will be mentioned here. KRASOVSKII [1] considers a conical-sweep-tracking-angle servo system. This has an inherent interaction of V-canonical structure. The internal loop gain has the form (see also Fig. 2): l 1 -a 2 GIF2IG2FI 2 = s(1 + sT--'--"--)x as x ~ x ( - as) = (1 + sT) 2' (9) where s is the operator of Laplace. Evidently, the internal loop has negative feedback and is always stable. When a is large, the loop gain is large too, which decreases the sensitivity of the process to its inputs. This is an explanation for the high controller gains found by KRASOVSKII.
Interaction in two-variable control systems for distillation columns
19
S T ~ [9] discusses two-variable control systems with V-canonical structure too. But here the internal loop has positive feedback, which results in an unstable process when the static loop gain is larger than one. Both studies refer to particular cases which cannot be fitted to our examples of distillation control systems. The same can be said about an article by N~WMAr~ [10], who considers symmetrical systems of V-canonical structure, and about a study by BOHN [I 1], who uses controllers with equal dynamic behaviour (apart from a constant of proportionality). MITCHELL and WmaB [12] consider the response of 71 (see Fig. 3) to variations of the correcting condition ~'1. They conclude that the deviations become smaller when x approaches 1 ; G12G21 x= - (10) GIIG22 However, this is only true for the chosen location of the disturbances, corresponding to F I = G l l and F2--G21 in Fig. 3. Here the response o f e 1 to ( equals:
-ex 1 +(1 - ~ ) A 2 "T- = G111 +Ax +A2 + ( 1 - r ) A 1 A 2
(11)
where A1-CtGII,
(12a)
A2 - C2G22.
(12b)
[A2I~>
A1 and A 2 are the subloop gains. If [A1], 1, the numerator of (11) decreases more than the denominator when x ~ l . Thus the remaining deviations become smaller too. In contrast with (11), the general expression for the response of e~ to ( is: - el _ Fx(1 +A2)-F2(G12/G22)A2 - 1+AI+A2+AIA2(I_r ) (13) It can be seen that generally the numerator does not become small here when x-~ 1. As the denominator still becomes small for low frequencies, control even becomes poor. This can be illustrated by substituting IAI[, IA21 l, and x = 1 into (13): - ~l~l.f. ~ F x - F2(G12/G22) ] 1 +A1/A2 •
(14)
There is no guarantee that this expression has a small value, contrary to what one desires for low frequencies. Apparently x ~ I is favourable when there are disturbances in the correcting conditions, but it is unfavourable for all other disturbance locations. For distillationcolumns there are disturbances from many sources, each having its own values of F1 and F2. A m o n g these may be disturbances in the correcting conditions. The calculation of the effect of all disturbance sources on the controlled conditions is therefore a very complex problem, requiring more data than are usually available. Evidently, if we want to say something in general, we have to make simplifying assumptions. It seems logical to pay more attention to unfavourable cases than to favourable cases. [9] R. ST.~K~m~^~n~: Regelungstechnik 7 (9),301-306 (1959). Ibid. 8, (8) 257-61 (1960). Ibid. 10, (I0) 433 (1962). [10] D. B. NEWMAN: IRE Transactions AC-5, No. 4 314-20 (1960). [11] E. V. BonN: IRE Transactions AC-5, No. 4 321-7 (1960). [12] D. S. MITCHELLand C. R. W~Im: in Automatic and Remote Control, Vol I, pp. 142-51. Proc. 1st IFAC Congress, Mo-~-.ow(1960).
20
J . E . RIJNSDORP
Therefore we shall in the following leave disturbances in the correcting conditions out of consideration. Useful information about their effect on control can be found in Mn'Ca-mLL and Wmm's paper. Further we shall write the general expression (13) in the form: ( =LFI-F2v22 l+A2
1 +A1 + A 2 + A 1 A 2 ( 1 - r ) ] "
(15)
Here the first factor indicates how the disturbance ( would affect the controlled condition r/I if controller C1 should become inoperative (see Fig. 3). It contains the expression A2 1 +A2
(16)
which is equal to the follow-up response of the second subloop. For low frequencies, (16) is about equal to one. For higher frequencies, it can show a resonance peak. However, when controller C2 has been adjusted in a conservative way (and our study will show that the interaction makes this unavoidable) the peak is either small or absent. For very high frequencies, (16) becomes small. But in this region control is ineffective, thus it need not be considered. Therefore it seems reasonable to assume that (16) is equal to one. Then we can introduce the disturbance effect:
which is independent of the controller responses. The response of the deviation 81 to ~* is now equal to the second factor of (15): - el _
(*
1 +A2 1 +AI +A2 +AIA2(I - x)
(18)
This expression, and the corresponding one for the response of 62, will be further analysed in the next section. 4. FURTHER ANALYSIS OF THE SIMPLE TWO-VARIABLE SYSTEM Formula (18) can be written in a number of ways (19a) (*
I+A,
1-K\I+AI/\1+A2]_ I
. I-l+(1-x)A21 +A 1 . . . .
I+AI[1
rA2 1 I+A2J
(19b)
(19c)
Interaction in two-variablecontrol systems for distillation columns
21
Formula (19a) indicates that the response without interaction (1 + A I ) - 1 is multiplied by a factor. This factor corresponds to the response of a positive feedback loop, consisting of the interaction quotient r, and the follow-up responses of the subloops. Formula (19b) indicates that the loop gain without interaction has to be multiplied by a factor. Formula (19c) gives a different expression for this factor. In all expressions, r plays an important role. It is a convenient measure for the interrelation in the process, because it is dimensionless, and independent of the addition of series transfer functions. The latter can be seen from Fig. 4 where the introduction of F¢,, F¢,, F~, and F,, has no effect on the value of r.
1~o. 4. Intrc~luetionof ~l"ies tra~er f~etio~. A further feature is that r often is less dependent on the frequency than its component responses G11,Gzz, Glz and Gz,. For instance, in the ease of distillation columns the denominators of the G's are two by two identical, hence they cancel in the expression for r. On the other hand, the expression (GI1 Gzz-GlzGz~)is less convenient, because it does not have the above-mentioned features. The same applies to the "interrelation strengths" introduced by ~ o x q ~ (see Reference 7, pages 55-6). We shaU first assume constant and real values for r. There are now three cases: 0 < r < l , r < 0 , and r > l . (a) 0 < r < 1. The interaction factor in formula (19b) can be transformed into:
_l-I ~+A2
\l-rJ (20a) - ( - 1)+A 2 This can easily be interpreted as the ratio of two vectors in the complex diagram (see Fig. 5a), multiplied by (1 - r). Loci for constant modulus of/1 have been derived in Appendix I. They are circles with centres on the negative real axis, the phase loci are circles with centres on the perpendicular bisector of the line joining - 1 and - (1 - r ) - x and going through these points. 1 I 0 o ~ _ . 4t-K ~ m -* 180 o - ~ 180°~1 -/, =(l-r)
\ +90 ° \ J
1~o. 5a. Intecaction when 0 < tc< 1. I11 : I z d > 1 - : lift
Aa
22
J.E. RUlqSI~m, I I
-,
O"
_ ,.ool
oo
FIG. 5b. Interaction when r < 0. III : 1121>1-,~-
: I1~1<1 1 K-I
I0 - t
_~ t 8 0 o
_~ t 8 0 o
J
j
/
J
/ / /
N
/ /'
,
\ / t
/
/ I
1~o. 5¢. Interaction when x > l ; for loop with positive feedback. Ill : 1121>1 -
: lI~l<(K--1)
I
1 ~-1
oo
\
~
-t
oo
'0
~
o°
--....
\
''-......, \
X
^,
/
--~
290°
/
r.
1
/
Flo. 5d. Interaction when r > l ; for loop with negative feedback. Ill : II~1>1 -
: 1131<(K--I)
In Fig. 5a a few important loci have been shown in the third quadrant (which is the most important for control). When K is small, the points - 1 and - ( 1 - x ) - 1 are close together, and the effect of the interaction is only felt in a region close to these points. However, when
Interact/on in two-variablecontrol systems for distillation cob~mn.q
23
is almost equal to one, the point - ( I -so)-z is far away, and the interaction is sio~ificant over a large part of the quadrant. Figure 5a shows that the interaction gives phase advance; attenuation over most of the quadrant (in particular around - ( 1 - ~ ) - t ) , but amplification around the point - 1. When the loop A 2 has a low resonance frequency compared to At, its effect on A t is small. This can also be seen from (20a), where putting A 2 ---0 gives I t = 1. When A 2 has a high resonance frequency compared to At, its effect on At is not negligible. Substituting JA2J>>1 into (20a) gives I 1~ (1 - K). Hence the loop gain A t should be increased by (1 - ~:)- x, in order to compensate for the interaction. However, when A 2 has the character of a double integration, its polar curve passes close to the point - (1 - i¢)- t which causes a dip in Ix. The effect on At is a deterioration of control over the frequency region corresponding to the dip. When At and A 2 have equal resonance frequencies, the effect of the interaction is worst. Both loops increase each other's loop gain in the resonance region, and decrease each other's loop gain at low frequencies. In particular, when At and A 2 pass close to - ( 1 - ~ : ) - t control will be strongly deteriorated over the corresponding frequency region. It is even possible that the latter behaves as a second resonance region. By way of illustration, Fig. 6 shows the response of one of the controlled conditions of a symmetric two-variable system for different values of ~:. The disturbance is a step change of the corresponding set value. This corresponds to F I = 1 and F2=O in formula (15), hence (18) and (19) are exact expressions here.
1-"
oIK= 0
" --
bt K=0-9
i
I
~
~
i
¢ ; K =- 0-9 ,
.
. . . . . . .
(,~JBLOOPWITHPOSITIVEFEEDBACK)
FIo. 6. Response of controlled condition to step change of set value.
24
J . E . RtmSOORP
The loop gains are: A
1 1 =A2 = [(0.1 + s)(~+0.1s, t6 ] × IK,(1 +~'~)1
(21)
where Kp is the proportional gain of the controller. Ti is the integral action time of the controller. (1 +0.1s) 1° is an approximation for the distance velocity lag e -s. Table 1 gives the settings of Kp and Ti for the different curves. TAet~ 1. CONa'ROLLr.RSe~lx~r~OSn~ FIG.6 x
K~
T~
Curve
0 0"9 --0.9 +I.I
0"93 0"41 0-34 0.51
4.4 3-9 11.5 43
a b c d, e
Curve a shows the case without interaction. Curve b shows a case where t¢ is near to one. Here Kp had to be reduced by about a factor 2 in order to maintain the same relative stability. The resonance frequency is about the same as without interaction. The slow return to equilibrium is the result of poor control in a low frequency region. If the process response contains an integration (for instance if the term 0. I is lacking), Am and A2 pass close to - ( l - x ) - t , and the slow return to equilibrium is oseiUatory. (b) x < 0 . Here (20a) can be used too. Figure 5b shows the vector diagram. The interaction gives phase lag here, and some attenuation in the region around - ( 1 - x ) -1. This usually leads to a reduction of the resonance frequency. Here too, the effect is most pronounced when A t ,~A2, and when their polar curves pass dose to the points - 1 and - ( 1 - x)-1. Then the additional phase lag even approaches 180°. Figure 6, curve c shows the case r = - 0 . 9 . The response is slower than that without interaction, but there is no slow return to equilibrium here. (c) x > 1. The interaction factor in formula (19b) is now negative for Ih l >>1. This means that the sign of A1 should be reversed in order to maintain overall negative feedback. Alternatively, there is the possibility of changing the sign of A2. In any ease, the system is only stable if one of the subloops has positive feedback (At or A2 with negative sign). From the practical point of view, it is not attractive to have a subloop with positive feedback: If the other subloop becomes inoperative, the former subloop becomes exponentially unstable [5, 13]. The interaction factor between brackets in formula (19b), after replacing At by - A t , can be written in the form: 1 x - 1 ~-A2 (20b) 12 ----(/¢ -- 1) -(-1)+A 2 The corresponding vector diagram is shown in Fig. 5c, for the third and the fourth quadrant. Apparently, just as in Fig. 5a there is attenuation over most of the quadrants [I 3] E. M. GRAn~: Handbook of Automation, Computation and Control, Part IIl, pp. 10-36. Wiley,New York (1961).
Interaction in two-variablecontrol systemsfor distillation columns
25
(in particular around (~:-1)-1), but amplification around - 1 . However, in contrast to Fig. 5a, the interaction now gives phase lag instead of phase advance. In particular when ~¢ is close to one, this phase lag is usually large, and the resonance frequency of the control system is much lowered. Now we should also pay attention to the interaction for the other control loop, as the two-variable control system is qualitatively asymmetrical. The interaction factor becomes: i3 =(ic_ 1) - - ( - - ~ - - 1) +AI +l-A t
(20c)
The corresponding vector diagram is shown in Fig. 5d for the third and the fourth quadrant. Because I3 has a positive real pole, the phase of I3 is 180° for lad,> 1. For stability, the polar curve for A213 should consequently encircle the point - 1 once in anticlockwise direction. Just as in Fig. 5e, in Fig. 5d there usually is a strong phase lag, in particular when x~, 1. Hence control is made slow, and it might even become impossible if A2 should have too much phase lag of its own. In view of the latter circumstance it is desirable to put the positive feedback in the slowest loop. Figure 6, curves d and e show the ease x = 1.1 for both loops. Apparently, the ease K> 1 means slow and poor control. Consequently, both from the practical and the theoretical point of view, it is advisable to connect C1 and C2 to ~1 and ~2 in such a way that x becomes smaller than one. (d) r depends on o~. When x is dependent on co, it is more convenient to write the interaction faetor of formula 20a in the following form 1
14 --
- [- 1 +
1 ~-~2- ( -1 )
(20d)
Here it has been assumed that the static value of +¢is smaller than 1, hence both subloops have negative feedback. When K(0) > 1, one of the subloops should have positive feedback. As control is then unsatisfactory (see Section 4e), we shall not discuss it further. Formula (20d) can easily be interpreted as the ratio of two vectors in the complex diagram. One vector connects the curve for - 1 +~¢(o)) to the curve for A2 - t , the other connects the latter curve to the point - 1. As an illustration we select the following case from the multitude of possibilities: ~co) = K(0)e-10' •
(22)
This is of interest for the application to distillation columns (see Part II'). Figure 7 shows the polar diagram with A 2 given by formula (21), ~c(0)=0.9, z=0.3, and Kp and T~ given by Table 1 (curve b in Fig. 6). Apparently the deterioration of control at low frequencies is here less pronounced than it is for constant x.
26
J . E . Rmcsr~RP
Finally it should be remarked that the polar diagram determined by formula (20d) and illustratexl by Fig. 7 is also very useful when ~¢is independent of o~. It only requires some experience with inverse Nyquist diagrams, which are used less frequently than ordinary ones. "5
--1 -oE
""5
l
t,O
-I ÷K(w)
FIG. 7. Interaction when ~(¢0)=0.9 exp (-jart). 5. C O N C L U S I O N S
We have tried to get some insight into the behaviour of, two-variable control systems with two controllers. For application to the control of distillation columns, the P-canonical structure is more suitable than the V-canonical structure. A convenient measure for the interaction is r=(GI2G21)/(GIIG22). It is dimensionless, independent of the addition of series transfer functions, and often not strongly dependent on co. When r is constant and almost equal to one, the interaction causes poor control in a region of low frequencies. This can even cause an additional resonance effect, in particular when the other subloop contains two integrations. When the static value of r is almost equal to one, and r shows increasing phase lag for increasing frequencies, the deterioration of control for low frequencies is less pronounced. When ~¢is constant and negative, the interaction decreases the resonance frequencies of the subloops.
Interaction in two-variablecontrol systemsfor distillation column~
27
When sc is constant and larger than one, the subloop with the lowest resonance frequency should have positive feedback. The responses are much slower than for the case where the connections between controllers and control valves (correcting conditions) have been reversed. In the latter case ~c has the reciprocal value, hence it is smaller than one.
LIST OF SYMBOLS a
G1, a2
C, F, G,
I, J, ;7,
K,, M, $,
T, Tt, X, Y, ~l, K,
A, O',
03,
constant; constants (see Appendix I); controller response; process response; process response; interaction factor;
J(-1);
see formula (S); proportional gain of controller; modulus (see Appendix I); operator of Laplace; process time constant; integral action time of controller; Cartesian co-ordinate (see Appendix I); Cartesian co-ordinate (see Appendix I); deviation of controlled condition from set value; disturbance; controlled condition, interaction quotient; loop gain; controller output and process input; set value; time constant; correcting condition; frequency.
APPENDIX
I
Modulus Loci in Fig. 5 Here the formulae for the loci in Fig. 5 will be given. They pertain to the expressions for the interaction defined by formula 20. (a) ~ real, ~ < 1 (formula 20a). The modulus loci are given by:
(x.i M2at'-at'~ 2
2_[Ma,(al-1)~ 2 '
) + Y - ~ M2at 2 - 1 /
(I-1)
where al ffi 1/(1 - g ) . M is the value of the modulus. When Mal ffi 1, this circle degenerates into the straight line: X= --~a I + 1). (I-2)
28
J.E. ~ R p
(b) ~ real, x > 1, f o r the l o o p with positive f e e d b a c k (formula 20b)
X'4 M2a22+a2"~2 Y2=~Ma~(a~+!)~~ ) ÷ / ' where
a2= 1/(x-1).
When
Ma2 =
(I-3)
1, this circle degenerates into the straight line: X = ½(a2 - 1).
(I-4)
(c) 1¢ real, x > 1, for the l o o p with negative f e e d b a c k ( f o r m u l a 20c):
X M2a22+a2~2 Y2 ( Ma~(az+l)~z ) + / When
Ma2= 1, this
circle degenerates into the straight line: X = - ½ ( a ~ - 1).
(I-6)
R/~mnt---En general, les syst~mes de r~glage ~ variables multiples sont trop complexes pour une description, m~me qualitative, de leur comportement. La situation est. toutefors, plus favorable pour les syst~mes de r~glage line,ire/t deux variables avec denx boucles de r~glage. Dam cet article, rinfluence d'une boucle sur rautre est d~luite d'un diagramme polalre. Le quotient d'interaction du processus = (GI 2G21)/(G 11G22)
dan~ lequel les G representent les fonctions de reponse du processus, est employ~ en tant que mesure convenable du degr6 d'influence mutueUe. Cette etude est adaptbe vers les sysN:mes de r~glage pour colonnes de distillationcontinues et donne les bases thb)riques. L'application aux colonnes de distillationscra traitde dans la partie II.
Zusammeafumng--Im allgemeinen sind vermaschte Regelungssysteme sonar for eine qualitative Beschriebung ihres Verhaltens zu komplex. Jedoch ist die Situation ffir lineare Zweifachregelnngssysteme mit zwei Regelkreisen gfiustiger. In diesem Artikel wird der ~influss des einen Regelkreises auf den anderen yon einem Vektordiagramm abgeleitet. Der Kopplungsquotient des Prozesses K = (G 12G2 I)/(G 11G22),
wobei die Gv~ Antwortfunktionen des Prozesses darsteUen, ist ein brauchbares Mass for den Grad des gegenseitigen Einflusses. Die vorliegende Arbeit kon~entriert sich ~uf Regelungssysteme for kontinuierliche Destillatiouskolonnen und liefert die theoretische Grundlage. Die Anwendung auf DestiLlationskolonnen wird im Teil II behandelt. AtcrlPmcr--O~mo. c~creM~x yapanaemcq c MHOrO.mcaemua~m n e p e M e ~
c0aamxoM
CJIO~bI R.rLq XOMSl-'lbbIKaq~"TBeHHOrO OIIH~wgm[ ilX I10"be~eHIDL OjIBa.KO, noJ"io]KeliilefflJDqec~i" b o n ~ NIaHonpRffrlIMM ~
JlgHeib~
CllffreM yIipaBJ'Iellg£
c osynmx nepeMemn.-vm H O~yMS y n p a m m i o n m M K xo~rypaMa. B Imc'ramnelt craTbe, ~mmme o~moro xowrypa Ha ~pyrot~ m~momrrcx w3 noJmprog ~ u r p a ~ . tlacr~oe
naa~o~IcttCTB~ npoHecca
K=(G12G21)/(GI1G22) rJIe G n p e ~ C T ~ T ¢O~Ott nepeRaTo~oe ~pymo~m~npot~ecca, xCnOmb3yeTc~gax no~xo~m e e a3 Mepemm crenem~ B3amdo~eltc~m~. I-lacTosmce xccne~omume HapamIeno Ra cgcreM~a y n p a e a e m m a e n p e p ~ , m H U ~ neperoHKOJIOHHaMH H ~aeT TeOpeTgqeCICg¢ OT~DCHOBaHH~. I']pHMCHCHHe K n c p e r o ~ ~OJIKHHaM IbyjIeTpaccMapmmTcg B qqaCTH If.
15-28. Pergamon Pre~, 1965. Printed in Great Britain.
INTERACTION IN TWO-VARIABLE CONTROL SYSTEMS FOR DISTILLATION COLUMNS--I THEORY J. E. Rm~SDORP Koninklijkc/Shell-Laboratorium, Amsterdam SmnmarymIn general, multivariable control systems are too complex for even a qualitative description of their behaviour. However, for linear two-variable control systems with two control lOOps, the situation is more favourable. In this article, the influence of one loop on the other is deduced from a polar diagram. The interaction quotient of the process K=(G12 G21)/(GII G22) in which the G's represent response functions of the process is used as a convenient measure for the degree of mutual influence. This study is focussed on control systems for continuous distillation columns and gives the theoretical background. The application to distillation columns will be treated in Part II. 1. I N T R O D U C T I O N IN THE application o f control to complicated systems it is quite c o m m o n that the feedback loops for the various controlled conditions show mutual interaction. It is then incorrect to consider the feedback loops separately; the system should rather be studied as a whole, as a multivadable control system. Some cases where mutual interaction can occur are: the the the the the
translational and rotational modes o f airplanes, rockets, etc., azimuth and elevation positioning o f radar antennae [1], r o t o r speed and turbine temperature o f jet engines [2], conditions in boilers [3], conditions in distillation processes [4-6].
The last-mentioned ease is the focal point o f this study. 'More specifically, we want to analyse the interaction in the following two-variable control systems for distillation columns: (a) temperature and pressure control, (b) dual quality control. [1] A. A. KnASOVSKn: Avtomat. Telemekh. 18, 126-36 (1957); Automn remote Control 139--49 (1957). [2] A. ]]OI¢~ENnOMand R. HOOD: NACA Report 980, Washington, April (1949). [3] H. K. CHA11rJO~: In Automatic and Remote Control, VoL I, pp. 132-7. Proc. 1st IFAC Congress, Moscow (1960). [4] H. H. ROSmq~OCK: Trans. lnstn, chem. Engnrs, Lond. 40, 35-53 (1962). [5] O. R ~ v . ~ o ¢ ~ and J. E. RIJNSDORP: Third World Petroleum Congress, Section VII, Paper 5. New York (1959). [6] J. E. Rt~SDORP and A. M~Rt~,cet.o: In Joint Sympon. Instrumentation and Computation, edited by P. A. RoT~,mtnto, pp. 135-43. Inst. of Chem. Engrs, London (1959). 15
16
J.E. RLINSDORP
In Part I of this study the theoretical background will be developed. Simple twovariable control systems in general are considered, keeping in mind the specific requirements of process control. The application to distillation columns will be the subject of Part II. In the following sections, we shall give a critical review of the literature on the theory of multivariable control systems and see how far it can be used for our purpose. 2. T H E
STRUCTURE
OF .THE
PROCESS
In multivariable control systems, the process (controlled system) has several inputs and outputs. Consequently, there are many possible configurations of the responses in the process. MESAROVI~ [7] has investigated the choice of the most suitable configuration for a given problem. For this, he introduces so-called canonical structures. Figures 1 and 2 show the P- and V-canonical structure, respectively, for the case of a two-variable control system. On pages 10 and 32 of his book, MESAROVI6compares the conditions to be satisfied for non-interaction. He finds different values for the transfer functions of the controllers for the P- and for the V-canonical structure.
lYl
l+.,j.~I
-
PROCESS+
+
\
+
/
/Glz
\
4-
i ,r FIG. I. Two-variable control system, the process having P-canonical structure.
PROCESS o'~
C~ Csl
C,z~
1
F21"/
2<
+
/
F1z
FIG. 2. Two-variable control system, the process having V-canonical structure.
The cause of this difference is simply that he has adopted different requirements for the overall responses of the systems. For the P-canonical structure (see Fig. 1) these are: /~1=G110"1 ,
/']1 --~-Gl1~'1 ,
or ~]2 = G220"2 , [7] M. D. M~AROVI~: The Control of Multivariable
(1) /72 = G22~2 ,
Systems.
M . I . T . and Wiley (1960).
17
Interaction in two-variablecontrol systems for distillation columns
for non-interaction with respect to trl, a2, or with respect to (1, (2 respectively, where 111, 2 are the controlled conditions, a l, 2 are the set values, (1, 2 are disturbances in the correcting conditions, GI i and G22 are P-canonical process responses. On the other hand, for the V-canonical structure the requirements are respectively: 111=G1(1 ,
111= GLO'1 ,
(2)
or 112=
112= G2o" 2 ,
G2~2,
where Gx, 2 are V-canonical process responses. As G1 # G11, and Gz # G22, (1) is not the same as (2), and one can expect different values for the controller responses C11, Cz2, CIz and C21. We prefer to use requirements that are independent of an assumption about the structure of the process. More specifically, if non-interaction is desirable, logical requirements are: 112 not dependent on al ~ (for non-interaction with respect to if1, t72, 111 not dependent on a 2 3
or
112 not dependent on ~1 ~ (for non-interaction with respect to 111 not dependent on (2
(3)
(I, (2
J
Then we find for the P-canonical structure: CI IG21 q- C2tG22 = 0, C22G12 q" C12G11=0,
(4)
or G2t
C21
G I 1 G 2 2 - - GI2G21,
C12 =
G12 GIIG22- GI2G2t.
(5)
(See also References 2 and 8, respectively.) For the V-canonical structure the results are:
CllG1F21+C21=O, C22G2FIz+ Cl2 =0,
(6)
or
Czl =F21, Ca2 =F12By using the relationships between the structures [8] R. J. KAVANAOrI: Trans. A.LE.E. 77, (2) 425-9 (1958).
(7)
18
J.E. RLINSDORP
Gl t = GIJ, G22 = G2J,
Gx2 = G2Fx2GtJ,
(8)
G2t = GIF21G2J, where J = ( I - G t F 2 t G 2 F 1 2 ) -1, one can show that (4) is equivalent to (6) and (5) to (7). Hence the choice of the structure does not influence the result, when (3) is used as a requirement for non-interaction. Now we are free to choose a structure corresponding to a physical interpretation of the process. This is desirable for acquiring good insight in the problem. Generally, distillation processes are inherently stable, and the controlled conditions are principally influenced by the correcting conditions, and not vice versa. This makes the V-canonical structure less suitable, because it contains the loop Gt F2tG2Ft2, which could be unstable. Moreover the direction of F2t and F 12 is opposite to the physical signal flow. Consequently, we choose the P-canonical structure for our discussion. As long as there is no need for four controllers, we shall only use two of them (see Fig. 3). A source of disturbance usually affects both controlled conditions. In Fig. 3 this is indicated by the transfer functions F I a n d / ' 2 . _
,..
/
o,( crz
-&
\ "-+
/ 1
/
o,, PROCESS
Fla. 3. Simple two-variable control system. 3. RESPONSES OF THE SIMPLE TWO-VARIABLE SYSTEM In the literature there are many discussions of two-variable systems. Some will be mentioned here. KRASOVSKII [1] considers a conical-sweep-tracking-angle servo system. This has an inherent interaction of V-canonical structure. The internal loop gain has the form (see also Fig. 2): l 1 -a 2 GIF2IG2FI 2 = s(1 + sT--'--"--)x as x ~ x ( - as) = (1 + sT) 2' (9) where s is the operator of Laplace. Evidently, the internal loop has negative feedback and is always stable. When a is large, the loop gain is large too, which decreases the sensitivity of the process to its inputs. This is an explanation for the high controller gains found by KRASOVSKII.
Interaction in two-variable control systems for distillation columns
19
S T ~ [9] discusses two-variable control systems with V-canonical structure too. But here the internal loop has positive feedback, which results in an unstable process when the static loop gain is larger than one. Both studies refer to particular cases which cannot be fitted to our examples of distillation control systems. The same can be said about an article by N~WMAr~ [10], who considers symmetrical systems of V-canonical structure, and about a study by BOHN [I 1], who uses controllers with equal dynamic behaviour (apart from a constant of proportionality). MITCHELL and WmaB [12] consider the response of 71 (see Fig. 3) to variations of the correcting condition ~'1. They conclude that the deviations become smaller when x approaches 1 ; G12G21 x= - (10) GIIG22 However, this is only true for the chosen location of the disturbances, corresponding to F I = G l l and F2--G21 in Fig. 3. Here the response o f e 1 to ( equals:
-ex 1 +(1 - ~ ) A 2 "T- = G111 +Ax +A2 + ( 1 - r ) A 1 A 2
(11)
where A1-CtGII,
(12a)
A2 - C2G22.
(12b)
[A2I~>
A1 and A 2 are the subloop gains. If [A1], 1, the numerator of (11) decreases more than the denominator when x ~ l . Thus the remaining deviations become smaller too. In contrast with (11), the general expression for the response of e~ to ( is: - el _ Fx(1 +A2)-F2(G12/G22)A2 - 1+AI+A2+AIA2(I_r ) (13) It can be seen that generally the numerator does not become small here when x-~ 1. As the denominator still becomes small for low frequencies, control even becomes poor. This can be illustrated by substituting IAI[, IA21 l, and x = 1 into (13): - ~l~l.f. ~ F x - F2(G12/G22) ] 1 +A1/A2 •
(14)
There is no guarantee that this expression has a small value, contrary to what one desires for low frequencies. Apparently x ~ I is favourable when there are disturbances in the correcting conditions, but it is unfavourable for all other disturbance locations. For distillationcolumns there are disturbances from many sources, each having its own values of F1 and F2. A m o n g these may be disturbances in the correcting conditions. The calculation of the effect of all disturbance sources on the controlled conditions is therefore a very complex problem, requiring more data than are usually available. Evidently, if we want to say something in general, we have to make simplifying assumptions. It seems logical to pay more attention to unfavourable cases than to favourable cases. [9] R. ST.~K~m~^~n~: Regelungstechnik 7 (9),301-306 (1959). Ibid. 8, (8) 257-61 (1960). Ibid. 10, (I0) 433 (1962). [10] D. B. NEWMAN: IRE Transactions AC-5, No. 4 314-20 (1960). [11] E. V. BonN: IRE Transactions AC-5, No. 4 321-7 (1960). [12] D. S. MITCHELLand C. R. W~Im: in Automatic and Remote Control, Vol I, pp. 142-51. Proc. 1st IFAC Congress, Mo-~-.ow(1960).
20
J . E . RIJNSDORP
Therefore we shall in the following leave disturbances in the correcting conditions out of consideration. Useful information about their effect on control can be found in Mn'Ca-mLL and Wmm's paper. Further we shall write the general expression (13) in the form: ( =LFI-F2v22 l+A2
1 +A1 + A 2 + A 1 A 2 ( 1 - r ) ] "
(15)
Here the first factor indicates how the disturbance ( would affect the controlled condition r/I if controller C1 should become inoperative (see Fig. 3). It contains the expression A2 1 +A2
(16)
which is equal to the follow-up response of the second subloop. For low frequencies, (16) is about equal to one. For higher frequencies, it can show a resonance peak. However, when controller C2 has been adjusted in a conservative way (and our study will show that the interaction makes this unavoidable) the peak is either small or absent. For very high frequencies, (16) becomes small. But in this region control is ineffective, thus it need not be considered. Therefore it seems reasonable to assume that (16) is equal to one. Then we can introduce the disturbance effect:
which is independent of the controller responses. The response of the deviation 81 to ~* is now equal to the second factor of (15): - el _
(*
1 +A2 1 +AI +A2 +AIA2(I - x)
(18)
This expression, and the corresponding one for the response of 62, will be further analysed in the next section. 4. FURTHER ANALYSIS OF THE SIMPLE TWO-VARIABLE SYSTEM Formula (18) can be written in a number of ways (19a) (*
I+A,
1-K\I+AI/\1+A2]_ I
. I-l+(1-x)A21 +A 1 . . . .
I+AI[1
rA2 1 I+A2J
(19b)
(19c)
Interaction in two-variablecontrol systems for distillation columns
21
Formula (19a) indicates that the response without interaction (1 + A I ) - 1 is multiplied by a factor. This factor corresponds to the response of a positive feedback loop, consisting of the interaction quotient r, and the follow-up responses of the subloops. Formula (19b) indicates that the loop gain without interaction has to be multiplied by a factor. Formula (19c) gives a different expression for this factor. In all expressions, r plays an important role. It is a convenient measure for the interrelation in the process, because it is dimensionless, and independent of the addition of series transfer functions. The latter can be seen from Fig. 4 where the introduction of F¢,, F¢,, F~, and F,, has no effect on the value of r.
1~o. 4. Intrc~luetionof ~l"ies tra~er f~etio~. A further feature is that r often is less dependent on the frequency than its component responses G11,Gzz, Glz and Gz,. For instance, in the ease of distillation columns the denominators of the G's are two by two identical, hence they cancel in the expression for r. On the other hand, the expression (GI1 Gzz-GlzGz~)is less convenient, because it does not have the above-mentioned features. The same applies to the "interrelation strengths" introduced by ~ o x q ~ (see Reference 7, pages 55-6). We shaU first assume constant and real values for r. There are now three cases: 0 < r < l , r < 0 , and r > l . (a) 0 < r < 1. The interaction factor in formula (19b) can be transformed into:
_l-I ~+A2
\l-rJ (20a) - ( - 1)+A 2 This can easily be interpreted as the ratio of two vectors in the complex diagram (see Fig. 5a), multiplied by (1 - r). Loci for constant modulus of/1 have been derived in Appendix I. They are circles with centres on the negative real axis, the phase loci are circles with centres on the perpendicular bisector of the line joining - 1 and - (1 - r ) - x and going through these points. 1 I 0 o ~ _ . 4t-K ~ m -* 180 o - ~ 180°~1 -/, =(l-r)
\ +90 ° \ J
1~o. 5a. Intecaction when 0 < tc< 1. I11 : I z d > 1 - : lift
Aa
22
J.E. RUlqSI~m, I I
-,
O"
_ ,.ool
oo
FIG. 5b. Interaction when r < 0. III : 1121>1-,~-
: I1~1<1 1 K-I
I0 - t
_~ t 8 0 o
_~ t 8 0 o
J
j
/
J
/ / /
N
/ /'
,
\ / t
/
/ I
1~o. 5¢. Interaction when x > l ; for loop with positive feedback. Ill : 1121>1 -
: lI~l<(K--1)
I
1 ~-1
oo
\
~
-t
oo
'0
~
o°
--....
\
''-......, \
X
^,
/
--~
290°
/
r.
1
/
Flo. 5d. Interaction when r > l ; for loop with negative feedback. Ill : II~1>1 -
: 1131<(K--I)
In Fig. 5a a few important loci have been shown in the third quadrant (which is the most important for control). When K is small, the points - 1 and - ( 1 - x ) - 1 are close together, and the effect of the interaction is only felt in a region close to these points. However, when
Interact/on in two-variablecontrol systems for distillation cob~mn.q
23
is almost equal to one, the point - ( I -so)-z is far away, and the interaction is sio~ificant over a large part of the quadrant. Figure 5a shows that the interaction gives phase advance; attenuation over most of the quadrant (in particular around - ( 1 - ~ ) - t ) , but amplification around the point - 1. When the loop A 2 has a low resonance frequency compared to At, its effect on A t is small. This can also be seen from (20a), where putting A 2 ---0 gives I t = 1. When A 2 has a high resonance frequency compared to At, its effect on At is not negligible. Substituting JA2J>>1 into (20a) gives I 1~ (1 - K). Hence the loop gain A t should be increased by (1 - ~:)- x, in order to compensate for the interaction. However, when A 2 has the character of a double integration, its polar curve passes close to the point - (1 - i¢)- t which causes a dip in Ix. The effect on At is a deterioration of control over the frequency region corresponding to the dip. When At and A 2 have equal resonance frequencies, the effect of the interaction is worst. Both loops increase each other's loop gain in the resonance region, and decrease each other's loop gain at low frequencies. In particular, when At and A 2 pass close to - ( 1 - ~ : ) - t control will be strongly deteriorated over the corresponding frequency region. It is even possible that the latter behaves as a second resonance region. By way of illustration, Fig. 6 shows the response of one of the controlled conditions of a symmetric two-variable system for different values of ~:. The disturbance is a step change of the corresponding set value. This corresponds to F I = 1 and F2=O in formula (15), hence (18) and (19) are exact expressions here.
1-"
oIK= 0
" --
bt K=0-9
i
I
~
~
i
¢ ; K =- 0-9 ,
.
. . . . . . .
(,~JBLOOPWITHPOSITIVEFEEDBACK)
FIo. 6. Response of controlled condition to step change of set value.
24
J . E . RtmSOORP
The loop gains are: A
1 1 =A2 = [(0.1 + s)(~+0.1s, t6 ] × IK,(1 +~'~)1
(21)
where Kp is the proportional gain of the controller. Ti is the integral action time of the controller. (1 +0.1s) 1° is an approximation for the distance velocity lag e -s. Table 1 gives the settings of Kp and Ti for the different curves. TAet~ 1. CONa'ROLLr.RSe~lx~r~OSn~ FIG.6 x
K~
T~
Curve
0 0"9 --0.9 +I.I
0"93 0"41 0-34 0.51
4.4 3-9 11.5 43
a b c d, e
Curve a shows the case without interaction. Curve b shows a case where t¢ is near to one. Here Kp had to be reduced by about a factor 2 in order to maintain the same relative stability. The resonance frequency is about the same as without interaction. The slow return to equilibrium is the result of poor control in a low frequency region. If the process response contains an integration (for instance if the term 0. I is lacking), Am and A2 pass close to - ( l - x ) - t , and the slow return to equilibrium is oseiUatory. (b) x < 0 . Here (20a) can be used too. Figure 5b shows the vector diagram. The interaction gives phase lag here, and some attenuation in the region around - ( 1 - x ) -1. This usually leads to a reduction of the resonance frequency. Here too, the effect is most pronounced when A t ,~A2, and when their polar curves pass dose to the points - 1 and - ( 1 - x)-1. Then the additional phase lag even approaches 180°. Figure 6, curve c shows the case r = - 0 . 9 . The response is slower than that without interaction, but there is no slow return to equilibrium here. (c) x > 1. The interaction factor in formula (19b) is now negative for Ih l >>1. This means that the sign of A1 should be reversed in order to maintain overall negative feedback. Alternatively, there is the possibility of changing the sign of A2. In any ease, the system is only stable if one of the subloops has positive feedback (At or A2 with negative sign). From the practical point of view, it is not attractive to have a subloop with positive feedback: If the other subloop becomes inoperative, the former subloop becomes exponentially unstable [5, 13]. The interaction factor between brackets in formula (19b), after replacing At by - A t , can be written in the form: 1 x - 1 ~-A2 (20b) 12 ----(/¢ -- 1) -(-1)+A 2 The corresponding vector diagram is shown in Fig. 5c, for the third and the fourth quadrant. Apparently, just as in Fig. 5a there is attenuation over most of the quadrants [I 3] E. M. GRAn~: Handbook of Automation, Computation and Control, Part IIl, pp. 10-36. Wiley,New York (1961).
Interaction in two-variablecontrol systemsfor distillation columns
25
(in particular around (~:-1)-1), but amplification around - 1 . However, in contrast to Fig. 5a, the interaction now gives phase lag instead of phase advance. In particular when ~¢ is close to one, this phase lag is usually large, and the resonance frequency of the control system is much lowered. Now we should also pay attention to the interaction for the other control loop, as the two-variable control system is qualitatively asymmetrical. The interaction factor becomes: i3 =(ic_ 1) - - ( - - ~ - - 1) +AI +l-A t
(20c)
The corresponding vector diagram is shown in Fig. 5d for the third and the fourth quadrant. Because I3 has a positive real pole, the phase of I3 is 180° for lad,> 1. For stability, the polar curve for A213 should consequently encircle the point - 1 once in anticlockwise direction. Just as in Fig. 5e, in Fig. 5d there usually is a strong phase lag, in particular when x~, 1. Hence control is made slow, and it might even become impossible if A2 should have too much phase lag of its own. In view of the latter circumstance it is desirable to put the positive feedback in the slowest loop. Figure 6, curves d and e show the ease x = 1.1 for both loops. Apparently, the ease K> 1 means slow and poor control. Consequently, both from the practical and the theoretical point of view, it is advisable to connect C1 and C2 to ~1 and ~2 in such a way that x becomes smaller than one. (d) r depends on o~. When x is dependent on co, it is more convenient to write the interaction faetor of formula 20a in the following form 1
14 --
- [- 1 +
1 ~-~2- ( -1 )
(20d)
Here it has been assumed that the static value of +¢is smaller than 1, hence both subloops have negative feedback. When K(0) > 1, one of the subloops should have positive feedback. As control is then unsatisfactory (see Section 4e), we shall not discuss it further. Formula (20d) can easily be interpreted as the ratio of two vectors in the complex diagram. One vector connects the curve for - 1 +~¢(o)) to the curve for A2 - t , the other connects the latter curve to the point - 1. As an illustration we select the following case from the multitude of possibilities: ~co) = K(0)e-10' •
(22)
This is of interest for the application to distillation columns (see Part II'). Figure 7 shows the polar diagram with A 2 given by formula (21), ~c(0)=0.9, z=0.3, and Kp and T~ given by Table 1 (curve b in Fig. 6). Apparently the deterioration of control at low frequencies is here less pronounced than it is for constant x.
26
J . E . Rmcsr~RP
Finally it should be remarked that the polar diagram determined by formula (20d) and illustratexl by Fig. 7 is also very useful when ~¢is independent of o~. It only requires some experience with inverse Nyquist diagrams, which are used less frequently than ordinary ones. "5
--1 -oE
""5
l
t,O
-I ÷K(w)
FIG. 7. Interaction when ~(¢0)=0.9 exp (-jart). 5. C O N C L U S I O N S
We have tried to get some insight into the behaviour of, two-variable control systems with two controllers. For application to the control of distillation columns, the P-canonical structure is more suitable than the V-canonical structure. A convenient measure for the interaction is r=(GI2G21)/(GIIG22). It is dimensionless, independent of the addition of series transfer functions, and often not strongly dependent on co. When r is constant and almost equal to one, the interaction causes poor control in a region of low frequencies. This can even cause an additional resonance effect, in particular when the other subloop contains two integrations. When the static value of r is almost equal to one, and r shows increasing phase lag for increasing frequencies, the deterioration of control for low frequencies is less pronounced. When ~¢is constant and negative, the interaction decreases the resonance frequencies of the subloops.
Interaction in two-variablecontrol systemsfor distillation column~
27
When sc is constant and larger than one, the subloop with the lowest resonance frequency should have positive feedback. The responses are much slower than for the case where the connections between controllers and control valves (correcting conditions) have been reversed. In the latter case ~c has the reciprocal value, hence it is smaller than one.
LIST OF SYMBOLS a
G1, a2
C, F, G,
I, J, ;7,
K,, M, $,
T, Tt, X, Y, ~l, K,
A, O',
03,
constant; constants (see Appendix I); controller response; process response; process response; interaction factor;
J(-1);
see formula (S); proportional gain of controller; modulus (see Appendix I); operator of Laplace; process time constant; integral action time of controller; Cartesian co-ordinate (see Appendix I); Cartesian co-ordinate (see Appendix I); deviation of controlled condition from set value; disturbance; controlled condition, interaction quotient; loop gain; controller output and process input; set value; time constant; correcting condition; frequency.
APPENDIX
I
Modulus Loci in Fig. 5 Here the formulae for the loci in Fig. 5 will be given. They pertain to the expressions for the interaction defined by formula 20. (a) ~ real, ~ < 1 (formula 20a). The modulus loci are given by:
(x.i M2at'-at'~ 2
2_[Ma,(al-1)~ 2 '
) + Y - ~ M2at 2 - 1 /
(I-1)
where al ffi 1/(1 - g ) . M is the value of the modulus. When Mal ffi 1, this circle degenerates into the straight line: X= --~a I + 1). (I-2)
28
J.E. ~ R p
(b) ~ real, x > 1, f o r the l o o p with positive f e e d b a c k (formula 20b)
X'4 M2a22+a2"~2 Y2=~Ma~(a~+!)~~ ) ÷ / ' where
a2= 1/(x-1).
When
Ma2 =
(I-3)
1, this circle degenerates into the straight line: X = ½(a2 - 1).
(I-4)
(c) 1¢ real, x > 1, for the l o o p with negative f e e d b a c k ( f o r m u l a 20c):
X M2a22+a2~2 Y2 ( Ma~(az+l)~z ) + / When
Ma2= 1, this
circle degenerates into the straight line: X = - ½ ( a ~ - 1).
(I-6)
R/~mnt---En general, les syst~mes de r~glage ~ variables multiples sont trop complexes pour une description, m~me qualitative, de leur comportement. La situation est. toutefors, plus favorable pour les syst~mes de r~glage line,ire/t deux variables avec denx boucles de r~glage. Dam cet article, rinfluence d'une boucle sur rautre est d~luite d'un diagramme polalre. Le quotient d'interaction du processus = (GI 2G21)/(G 11G22)
dan~ lequel les G representent les fonctions de reponse du processus, est employ~ en tant que mesure convenable du degr6 d'influence mutueUe. Cette etude est adaptbe vers les sysN:mes de r~glage pour colonnes de distillationcontinues et donne les bases thb)riques. L'application aux colonnes de distillationscra traitde dans la partie II.
Zusammeafumng--Im allgemeinen sind vermaschte Regelungssysteme sonar for eine qualitative Beschriebung ihres Verhaltens zu komplex. Jedoch ist die Situation ffir lineare Zweifachregelnngssysteme mit zwei Regelkreisen gfiustiger. In diesem Artikel wird der ~influss des einen Regelkreises auf den anderen yon einem Vektordiagramm abgeleitet. Der Kopplungsquotient des Prozesses K = (G 12G2 I)/(G 11G22),
wobei die Gv~ Antwortfunktionen des Prozesses darsteUen, ist ein brauchbares Mass for den Grad des gegenseitigen Einflusses. Die vorliegende Arbeit kon~entriert sich ~uf Regelungssysteme for kontinuierliche Destillatiouskolonnen und liefert die theoretische Grundlage. Die Anwendung auf DestiLlationskolonnen wird im Teil II behandelt. AtcrlPmcr--O~mo. c~creM~x yapanaemcq c MHOrO.mcaemua~m n e p e M e ~
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