Znd. Eng. Chem. Res. 1991,30,1187-1193
1187
Disturbance Rejection Properties of Control Structures at One-Point Control of a Two-Product Distillation Column Peter M. Sandelin,' Kurt E. Haggblom, and Kurt V. Waller* Process Control Laboratory, Department of Chemical Engineering, Abo Akademi, SF-20500 Abo, Finland
In multiloop process control the choice of control structure, that is, the pairing of controlled and manipulated variables, can have a profound effect on the disturbance rejection properties of a control system: different structures are "self-controlling" to different degrees. This paper investigates the disturbance rejection properties of four common distillation control structures [(L,V), (0, V), ( D / ( L + D),V), and ( D / ( L+ D),V / B ) ]and one structure (DRD) especially designed to eliminate certain disturbances. The main question studied is how well the control structures reject the effect of disturbances on both products under one-point control of a two-product distillation column. I t is shown through both simulations and experiments that indirect two-point control can be obtained through one-point control with some of the structures, whereas other structures are not suitable for that purpose. These properties can be predicted by a simple analysis of the process and disturbance gains for the control structures. The gains for arbitrary structures can be calculated if the gains for one structure are known.
Introduction In industrial practice, the most common approach to multivariable process control is to rely on multiloop control, that is, to decompose the control system into SISO loops. It has long been known that the way the system is decomposed can have a profound effect on the interaction between the control loops, and thus on the obtainable control quality. Less recognized is the fact that different decompositions can result in very different disturbance sensitivities: different control structures are "selfcontrolling" to different degrees (Waller et al., 1988b). For distillation columns, a number of "standard" control structures exist. These structures have different properties for different separations, with respect to both interaction and disturbance sensitivity. It is also possible to synthesize model-based control structures with given (nominal) properties. Such a control structure is the DRD (disturbance rejection and decoupling) structure (Haggblom and Waller, 1990),which has the property that, in steady state, the control loops are completely decoupled and the primary outputs are insensitive to certain disturbances. However, as models are prone to be inexact and there usually are unmodeled disturbances, feedback control cannot usually be disposed of even for the DRD structure. It may, however, be sufficient to apply one-point control to obtain acceptable distwbance rejection of both products in a two-product distillation, especially if the uncontrolled product is an intermediate product in the plant, that is, not an end product with stringent specifications. In this paper, the disturbance rejection properties of some well-known distillation control structures and the DRD structure are investigated under one-point control of a distillation column. This is done through a steadystate analysis of the models for the control structures as well as through simulations. One of the standard control structures, namely, the (L,V) structure, and the DRD structure are also tested experimentally on a pilot distillation column. The study shows that some standard control structures can be used to achieve approximate two-point control through one-point control, whereas other structures are unsuitable for that purpose. The performance of the DRD structure is superior to that of the standard structures, 'Present address: Neste Oil,Porvoo Refinery, SF-06101Porvoo,
Finland.
although it is dependent on the accuracy of the model used in the synthesis.
Steady-State Analysis of One-Point Control Consider a two-product distillation column. The process can be described by the steady-state model Ay = KyuAu + KyuAv+ KywAw (la)
+
Ax = KxuAu Kx,Av
+ KxwAw
(1b) where y and x are primary output variables (to be controlled), u and u are manipulated variables in a chosen control structure, w is a disturbance variable, and K is the gain between the two variables implied by the subscript. The model thus describes how the manipulated variables in the control structure and a given disturbance variable affect the outputs in steady state. Suppose x is controlled by u. Equation l b then gives the control action AV = Kx{' (AX- KX& - KXwAw) (2) where Ax is a setpoint change of x. Elimination of Au from eq l a by eq 2 now gives Ay = KyuxAu+ Kyw:'Aw+ KyxxAx (3) where (44 KyuX = Kyu - KyuKxu-lKxu
Kyw* = Kyw - KyuKxu-lKxw
(4b)
KyxX= KyvKxu-' (44 Equations 3 and 4 describe how y is affected by a change in u,w, and x when x is controlled by v. More specifically, Kyuxshows the effect of u,Kywxthe effect of w, and Kyxx the effect of a setpoint change of x . Kywxand Kyxxthus indicate the disturbance sensitivity of y at one-point control of x . Similarly, the behavior of x when y is controlled by u is given by Ax = KxuYAv+ KxwYAw+ KxyYAy (5) where (64 KXVY = Kxu - KxuKyu-lKyu KXWY = Kxw - KxuKyu-lKyw KxyY = KxuKy,,-l
0888-588519112630-1187$02.50/0 0 1991 American Chemical Society
(6b) (64
1188 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 Table I. Nominal Steady-State Data feed flow rate F distillate flow rate D bottoms flow rate B feed composition z distillate composition bottoms composition reflux flow rate L rate of steam flow to reboiler V feed temp reflux temp
Table 11. Transfer Functions of Four Distillation Control Structures"
-
200 kg/h 60 kg/h 140 kg/h 30wt % 87 wt % 5wt% 60 kg/ h 72 kg/h 65 O C 62 O C
struct
( L , v)
tiyF
ti,
GX"
GX"
GZF
GXZ
-0.045e4'" 8.1s 1
0.048e4.% -O.OO1e-l,Os 11s 1 10s 1
0.004e-1.b 8.5s + 1
-0.23e-',% 8.1s 1
0.55e4,% -0.16e-1.b 10s + 1 5.5s + 1
-0.65e-1,b 9.2s 1
0.074e4.% 14s 1
-0.052e-1.b -0.005e-1.08 -0.076e-3.b 23s+1 15s+1 20s 1
0.38e-l.% 15s 1
0.03e4.b 10s 1
+
+
+
(D'v,
used to assess the feasibility of some distillation control structures for achieving indirect two-point control by one-point control. The results are also illustrated by simulations.
(74
AU = -0.60AL + 1.36AV - 0.73AD - 0.13AB (7b)
For given controller outputs Au and Au and measured flow
+
+
-0.18e-1~0s -1.06e-1,b 7.5s 1 15s 1
+
+
-0.026e-2.b 23s 1
+
12s
+ 0.043AV + 0.0025AD - 0.0012AB
+
+
+
Control Structures Studied The distillation process used in this study is a 15-plate pilot column separating a mixture of ethanol and water. The column was operated at the steady state given in Table I. The control structures studied are (see Table I for notation) as follows: the energy balance structure (L, V) with inventory controlled by D and B; the material balance structure (D, V) with inventory controlled by L and B; Ryskamp's (1980) control structure ( D / ( L + D), V) with inventory controlled by L + D and B the two-ratio structure ( D / ( L+ D), V/B) with inventory controlled by L + D and B (Takamatsu et al., 1982; Shinskey, 1984);the DRD structure with inventory controlled by D and B (Haggblom and Waller, 1990). The first four control structures are labeled after the two manipulators by which the primary outputa are controlled in the respective structures. The controlled variables are the temperatures on plates 4 and 14 (counting from the top), the first manipulator in the control structure notation being paired with the temperature on plate 4 and the second with the temperature on plate 14. In the DRD structure the primary manipulators are linear combinations of L , V, D, and B such that the desired nominal properties are obtained, that is, perfect steady-state disturbance rejection and decoupling. The four standard control structures have previously been studied in Waller et al. (1988a), where experimentally determined transfer function models for the control structures are given. As the gains of these models were determined from experimental data, they did not exactly satisfy certain consistency relationships existing between them (Haggblom and Waller, 1988). The gains were therefore reconciled subject to these relationships (Haggblom, 1989). The resulting dynamic models, which in this paper are used to simulate the behavior of the control structures, are given in Table I1 (Waller et al., 1988b). The DRD structure is simulated by means of the model for the ( L , V) structure. The part of the model containing the primary outputs 0,and x ) is given in Table 11, and the part containing the inventory control manipulators (Dand B ) is given in Table 111. The manipulators used for the DRD structure in this study are (Haggblom and Waller, 1990)
-
-
ti,"
+
In the following application, the gains in eqs 3 and 5 are
AU = -0.043AL
-
G,,
+1
0.35e4.% 10s 1
-0.17e-1.08 4.5s 1
-0.81e-1,b 13s 1
I.Oe4.b
O.OO1e-l.Oa 6.0s 1
-0.029e-7.58 18s 1
+
25s 23e-',&
+1
34e4.% 15s 1
26s + 1
+
+ +
+
+
-0.05e-1~08 -0.89e-1.b 7.5s 1 7.5s 1
+
+
"Units: flow rate, kg/h; composition, wt %; temperature, " C ; time, min.
Table 111. Transfer Functions for Inventory Control Manipulators in the (L,V ) Structurea
GDV GBV
GDL GBL
GDZ GBZ
GDF GBF
-0.61e4.% 1.0s 1
1.35e4.% 1.0s 1
0.056e-1.b 4.0s 1
1.08e-l.O" 2.0s 1
0.61e-l.% 1.5s 1
1.35e-'.&
0.944e-'.0S 1.5s 1
1.08e-1,b 3.5s 1
+
+
+
1.0s
+1
+
+
+
+
"Units: flow rate, kg/h; composition, wt %; time, min.
Table IV. Open-Loop Steady-State Gainsa
( L , V)
(D, V) ( D I G + D ) , V)
( D / ( L+ D ) , V I B )
DRD
-0.045 -0.23 0.074 0.38 6.7 34 6.4 23 1.02 0.51
0.048 0.55 -0.052 0.03 0.010 0.35 1.0 34 -0.002 0.96
-0.001 -0.16 -0.005 -0.18 -0.003 -0.17 0.001 -0.05 -0.001 -0.01
0.004 -0.65 -0.076 -1.06 -0.026 -0.81 -0.029 -0.89 -0.003 -0.04
"Units: flow rate, kg/h; composition, wt %; temperature,
OC.
rates ALI and AB, the needed adjustments of L and V are calculated as AL = -42Au + 1.3Au + l.lAD + 0.12AB (8a) AV = -18Au
+ 1.3A~+ l.0AD + 0.14AB
(8b)
Open-Loop Responses Table IV shows the open-loop steady-state gains of the five control structures. Here "open-loop" means that the primary control loops are open, but the inventory control loops are closed. The gains for the four standard control structures are taken from Table 11, and the gains for the DRD structure are calculated from the complete steadystate model for the (L,V) structure (with the gains given in Table I1 and 111)by elimination of AL and AV by means
Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991 1189 0.5
"C
AT
Table V. Steady-State Gains at K struct -. (L, v) -0,025 0.30 (D,v) 0.73 0.30 ( D / ( L + D),v) 5.73 0.30 ( D / ( L+ D),V / B ) 5.72 30 DRD 1.02 0.96
.........
xy
0.0
-0.5 0.0
O C -2.0
AT14
T
One-Point Control" KyxX KYFI KY2 KxVy K,rY Kzzy -. 0.061 0.087 0.013 5.11 -0.15 -0.67 -1.73 -0.32 -1.91 5.14 -0.15 -0.67 0.029 0.002 -0.003 5.07 -0.15 -0.68 0.029 0.002 -0.003 3.59 -0.05 -0.79 -0.002 -0.001 -0.003 -0.04 0.50 -0.01 -
