Control structure selection for energy integrated distillation column

ELSEVIER

J. Proc. Cont. Vol. 8, No. 3, pp. 185 195, 1998 ',C' 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0959-1524/98 $19.00 + 0.00

PII: S0959-1524(97)00039- 5

Control structure selection for energy integrated distillation column Jens Erik Hansen, t Sten Bay Jorgensen, t* Jonathan Heath s and John D. Perkins~ tDepartment of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark ~Centrefor ProcessSystemEngineering,Imperial College, London, UK, SW7 2BY This paper treats a case study on control structure selection for an almost binary distillation column. The column is energy integrated with a heat pump in order to transfer heat from the condenser to the reboiler. This integrated plant configuration renders the possible control structures somewhat different from what is usual for binary distillation columns. Further the heat pump enables disturbances to propagate faster through the system. The plant has six possible actuators of which three must be used to stabilize the system. Hereby three actuators are left for product purity control. An MILP screening method based on a linear state space model is used to determine economically optimal sets of controlled and manipulated variables. The generated sets of inputs and outputs are analysed with frequency dependent relative gain array (RGA), relative disturbance gain (RDG) and condition number (CN) to determine the best structure in terms disturbance rejection and setpoint tracking. The pairing and controller design are implemented and evaluated through nonlinear simulation. The suggested control structure is also qualitatively compared to a control structure applied experimentally. ~£) 1998 Elsevier Science Ltd. All rights reserved

Keywords: distillation; control structure selection; mixed integer linear programming (MILP)

used for control structuring, these include the open loop static and dynamic relative gain array (RGA), e.g. Skogestad and Postlethwaite 5, the Morari resiliency index, e.g. Morari and Zafiriou 6, or the relative disturbance gain (RDG), which is a closed loop measure. In this application study the former type of methods are used to generate candidate controller structuring for decentralized control and subsequently these candidates are investigated using the interaction type of control structuring measures. The present case study is on control structure selection for an energy integrated binary distillation column, equipped with a heat p u m p to transfer heat from the condenser to the reboiler. The distillation benchmark problem of Koggersbol and Jorgensen 7 is used as a point of reference. The lower stabilising control layer of the system is assumed fixed, and the challenge is to design an upper control layer to minimize variations in the product purities under oscillatory disturbances in feed flow rate and composition. At the upper control layer there are three potentially manipulated variables and 15 possible measurements. Based on a linear model mapping all inputs to outputs a mixed integer linear programming (MILP) screening method is used to determine a set of optimal control structures. The method, (Heath et al.8), searches structures that minimize an

Process designers need to be aware of latent control problems during the late phases of process design, as operational problems might seriously jeopardize economical benefits of otherwise cunningly designed plants. Control structuring forms the basis for providing desired operational functionality of a processing plant. For integrated process plants satisfying the goals related to operational functionality often is a much more demanding task than for non-integrated plants. Thus the control structure design problem is becoming increasingly important as environmental considerations are incorporated in to plant design. Control structuring has been attracting significant attention after the Foss, 19731 paper. Morari et al. 2 suggested solving the closed loop perfect control structuring problem as a generic rank problem. Georgiou and Floudas 3 solved this type of problem using a mixed integer linear p r o g r a m m i n g (MILP) formulation. Narraway and Perkins 4 considered backing off from optimal operating conditions, to evaluate the achievable performance of a given structure by formulating the control structure selection problem as a hybrid MILP. Other indicators for control loop interactions have also been

*To w h o m c o r r e s p o n d e n c e s h o u l d be addressed; e-mail:sbj/a k t . d t u . d k

185

186

Control structure selection: J. E. Hansen e t a I.

objective function under the assumption of perfect control and with specified constraints on all inputs and outputs. Also an extended version which simultaneously determines optimal pairings and tuning of decentralized PI-controllers has been applied. Previous work by Koggersbel 9 has addressed control configuration of this particular distillation plant. Several structures were selected in a heuristic manner and compared for stability and dynamic resiliency. All structures included both product compositions under direct control. In the present work potential structures are established in a completely automated way, where indirect control of the product purities is also considered. In distillation control it is common to control the column pressure individually and then perform control configuration analysis on the resulting system. In this work column pressure control is included with composition control configuration. The automatically selected structures are compared on the basis of (1) relative gain array (RGA) (Bristol l°, Hovd and Skogestad 11 and Skogestad and Moran 12) for interactions and robustness, (2) condition number and disturbance condition number for directionalities and alignment of disturbances with the plant, (e.g. Morari and Zafiriou6). Further right half plane zeros and relative disturbance gain ( R D G ) (Stanley et al. 13) are used to differentiate between the structures. The most promising structures are implemented in the non-linear simulation with decentralized PI-controllers which are tuned individually by loop shaping. The column is an industrial size pilot plant located at The Technical University of Denmark. The analyses in this paper are based exclusively on a simulation program, which contains a detailed model of every process unit, and thermodynamic data for the components. The modelling assumptions are described in detail in Koggersbel 9. To facilitate a later experimental verification of the obtained results it is chosen not to change the configuration of the lower stabilizing control layer in simulation based analyses. Thus focus is here only on control structure selection for the upper product purity control layer. The purpose of this paper is to demonstrate how a suitable decentralised control configuration can be determined through application of a fully automated screening method and subsequent differentiation between selected structures by means of conventional controllability measures.

Plant description A process flow sheet is shown in Figure 1. The column consists of 19 sieve trays, a reboiler, a total condenser and a reflux drum. The system separates a mixture of iso-propanol and methanol with a small contamination of water. The feed tray is no. 10. The column is equipped with a refrigerant (Rl14) based heat pump to transfer energy from the condenser to the reboiler. In a binary distillation column without energy integration,

there is a fast interaction from the boilup to the top of the column, and a slow interaction from the reflux to the bottom. With the heat pump an additional fast interaction from the column top to the bottom through the heat pump is introduced. All available actuators and measurements are listed in Tables 1 and 2, respectively. No direct measurements of product compositions are available, but instead the product purities are estimated from temperature and pressure measurements throughout the column. However in the following (simulation based) analyses it is assumed that the exact measurements are accessible with no time delay. Only the variables B, L, D, and K (Table 1) can be manipulated. Obviously the flows cannot directly be manipulated but rather the setpoints to PI-flow controllers using the valve positions (orB, otL, aD, o~K) are the real manipulated variables. Further note that the possible side stream outlet is closed and not used for control in this paper. Without any control at all the plant is unstable (Koggersbel et all4). To stabilize the system one needs to control the levels and break the positive energy feedback in the heat pump. The reboiler level is controlled by the bottom product flow rate, the condenser level is controlled by the flow to the accumulator and the accumulator level is controlled by the reflux flow rate. All these loops are implemented by IMC-tuned PIDs. Using the reflux flow rate for level control leaves the distillate product flow for product purity control, this will result in a (D,V)-type control configuration. In Koggersbol et al. 15 it has been experimentally verified that an (L,V)-type configuration may render the system unstable. The operation of the compressor is assumed fixed and cannot be used for control, so the only place to remove energy from the system is in the secondary condenser. Since this energy removal does not happen in a selfregulating manner, the heat pump will exhibit a positive energy feedback. One way to break this rather undesirable feature is to control the pressure somewhere in the heat pump by manipulating air-cooler bypass control valve CV8, that is to adjust the amount of heat withdrawn from the system. The (high) pressure in the reboiler PH on the heat pump side has been chosen as measurement for this purpose, and a PI controller is employed. The boilup rate may now indirectly be manipulated by the setpoint Pn..~ to this loop. The fast interaction from the column top to the column bottom introduced by the heat pump is very much dependent on the performance of this loop. With this lower level control the plant has been stabilized. The refrigerant flow rate control valve CV9 should not be used directly for product purity control either. Instead the (low) pressure on the heat pump side of the condenser, PL, is controlled by manipulating CV9, and the setpoint PL,~ is left for control of purities. The dynamic response from CV9 to PL is very fast, thus CV9 would have been a bad choice as manipulated variable in the upper control layer, since this layer is implemented with discrete controllers and using CV9 would require a very short sampling period.

Control structure selection." J. E. Hansen et al.

187

~CV9

Condenser ..~. i -

. . . . . .

K

D Super heatin~

Compressor

Secondary Condenser~, ),, Reboiler .-

Figure I

Table ! B L D K O~('l"8 ff('l'9

t i

I I i.

Of CV8

i I .

.

.

.

.

.

.

.

.

.

.

.

.

Distillation column, flow sheet. Note that the decanter in this paper simply functions as an accumulator

Manipulable variables

Table 2

Bottom product, volumetric flow Reflux, volumetric flow Distillate, top product volumetric flow Volumetric flow rate from total condensor to accumulator Position of control value CV8 Position of control value CV9

Ti

Ti o Ti9 Treboiler P]o t919

Prehoiler HC P~ = P5

•

The resulting control problem is now firstly to choose from the three actuators, PH,, P c : and D~, and 15 potential outputs a set of actuators and controlled variables to be included into the product purity control structure. And secondly to determine which decentralized control configurations is the best for that particular choice. Figure 2 illustrates schematically the lower level control configuration. The lower level control layer is already assumed implemented to stabilize the experimental distillation column. For the herein applied control structure selection methods a stable plant is needed.

Benchmark problem Based on the benchmark problem formulation of Koggersbool and Jorgensen v, the operating point in Table 3 was found to be in the middle of the operating window regarding column pressure P I 9 and boil-up flow rate (Figure 3). F, ZMeOH and 2iPrO H are the feed flow rate and compositions. The proposed sine wave disturbances are:

F(t) -- Fo + AFsin(mf, s,t)

2MeOH(I) = 2MeOH,O q- A :

sin(ws/owt)

T~ T7 TI 72_ 7"3 • HD XD,MeOH -¥B,iPrOtt • HB •

PH

PL •K oL •

Measured variables Temperature on tray 1 Temperature on tray 10 Temperature on tray 19 Temperature in reboiler Pressure on tray 10 Pressure on tray 19 Pressure in reboiler Level in condenser Pressure after control valve CV9 Temperature at compressor inlet Temperature after compressor Temperature in second condenser Temperature in receiver Temperature before expansion valve Relative level in decanter Top product purity (methanol) Bottom product purity (i-Propanol) Level in condenser Pressure in reboiler Pressure on condenser Volumetric flow rate to decanter Volumetric reflux flow rate

The variables marked with • are not potential controlled outputs for composition control. They are included as output variables in order to be able to specify bounds on the m a x i m u m deviation from the operating point. This bounding is particularly important for the reflux flow rate, which else might move outside the operating window and cause either flooding or weeping in the column.

2iPrOH(1) =: 2iPrOH,O -- A z

sin(w~/owt)

with coj,,t = 2rr/15 min - 0.70 10 e s and ~Ost,,,,= 2rr/120 min = 0.86 10 -3 s. A nominal situation and two enhanced stress levels should be considered (Table 4). Low and high amplitudes are defined relative to the steady state value of the particular inputs. Low is 5%, and high is 25%. Further, hard constraints are given for the product purity by a maximum deviation of 0.5mole% from the stationary value. The control objective is to fulfil this requirement, and keep the

Controlstructure selection:J. E. Hansen et al.

1 88

PH,s, PL,~ I

15 potential output

D~

¢ I PI(I

Cascade I

VID's

I

K8

P~

L~ Bs

I PID

l[

PID's

[

~K

9~CV9 O~CV8

]

:~O

D

K ~L L ~B B

HC HB HD

Distillation Column & Heat P u m p F ZMeOH ZiPrOH

Figure2 Table 3

Lowercontrol layer

Nominal operating point

F

3.0 litre m i n t

ZMeOH,O zip,on.o XD

xn P19

0.4950 0.4950 0.9750 0.9644 75.0 kPa

Vapour flow ( mS/h, condensed ) 2.25-

1.75

1.25

purities as close to their setpoints as possible. The two purities are assumed equally important. 0.75

Preliminary controllability analysis 0.25

First it is investigated how hard the plant actually is to control. A linearized state space model of the plant, mapping the three manipulable variables and the disturbance to the measured variables, was found by linearisation of the non-linear model at the operating point. M a x i m u m allowed deviations have been specified on all inputs and outputs, whereafter the linear model was scaled that these deviations correspond to a value of 1. The m a x i m u m allowed deviations of the manipulable variables have been set to the actuator limits specified in the problem definition. The m a x i m u m deviations of the two disturbances correspond to the nominal case level. Regarding the outputs the deviations are set to 10°C on the column temperatures and 25°C on heat p u m p temperatures. Column pressures are allowed to deviate by 10 kPa. And the deviation of the product compositions reflects the hard limit of 0 . 5 m o l e % . The m a x i m u m deviation on the reflux flow rate ensures no violation of the operating window. The scaled linear model has the following form: .~ = A x + Bu + C d y = D x + Eu + Fd

. . . . . . . . . . . . . . . . . . . . . . 30 50 70 90 110 Column presure ( kPa )

, 130

Figure3

Operating window. The attainable column pressure (PI9) is limited by the constraints on PL imposed by the compressor operating range. The boil-up vapour flow is limited by flooding and weeping limits. The nominal operating point is indicated by a cross

where y is 22 dimensional, describing all measured variables of Table 2, including the ones that are not potential controlled variables. Vector u describes the actuators PH,s, PL,s and Po.s. The disturbance vector d is two dimensional reflecting the feed flow rate disturbance and the composition disturbance. The corresponding input-output relation is denoted by y = Gu + Gdd. Figure 4 shows the frequency response of the feed

flow rate and composition disturbances. Since the gains Table 4 Amplitudes in oscillatory disturbances. Three different situations Nominal case Stress level 1 Stress level 2

AF is low AF is low AF is high

A= is low A: is high A: is high

Control structure selection: J. E. Hansen et al. i i iiii1[]

lO

0

189

i

I

ITIIIII

i i iiii1[ ]

i i[1111

100

10 .2

10 -2

(.9

,

10 .4

10 .4

10 -6

10 6 __

\ ~ I~lHIl]

I I I1'1111

10 5

10 -4

10 3 10 .2 Frequency ( rad/sec )

10 -1

10 °

1 0 .5

10 4

I llljtljI

.i llllmt

10 -3 10 -2 Frequency ( rad/sec )

ll~

10 1

10 a

Figure 4 Open loop frequency response of disturbances. Solid lines: product composition (xD.u¢on and xB.mmn): dashed lines: the rest of the variables in the output list. Right: feed flow disturbance: left: feed composition disturbance

of all outputs are less than 1 at the given frequencies of the oscillating disturbances, it may be concluded that system is self regulating for the nominal case of sine wave disturbances, and no control is needed to keep the product purities within the hard limits. When considering the high amplitudes at stress levels 1 and 2 the frequency gains must be multiplied by a factor of 5. This causes the curves on Figure 4 to be shifted upwards by 5. It is seen that the fast flow rate disturbance will not give any problems at all, but control is needed to remove the effect of the slow composition disturbance. Additionally the figure indicates that above the frequency ~ 10 2 rad s -~ control is unnecessary, thus the required control system bandwidth is COd= 10 2 rad s t. Another simple test to perform, is to check whether lack of input power might be a limiting factor in control of the system. For this purpose the gains of the three actuators necessary to perfectly control the three hardest controllable outputs, under the worst case disturbance, are calculated for all frequencies up to the bandwidth. Intuitively the three outputs, a m o n g the list in Table 2, that will be affected the most by the disturbances, are the two product purities (XD.MeOH, XB,iPrOn) and the column pressure at the top tray (Pl9). Figure 5 shows the input gains corresponding to the nominal case disturbance amplitude. Since the curves do not exceed i over the frequency range below the bandwidth, neither in the nominal case, nor when shifted upwards by a factor of 5 according to the high amplitude disturbances, there is a strong indication that actuator constraints will not exhibit control limitations. Actually the input gains are upper bounds corresponding to the worst case disturbances. If the square transfer function from the three actuators to the three selected outputs is denoted by G*, and the transfer function from the (nominal amplitude) disturbances is G*d~ then the curves are the row sums of (G*) IG,~evaluated at every frequency. After having established an overview of the control problem, the structure screening method will be discussed.

[

I

I i

• -03

\

(.9 0.01 \

'\,

'\

1.e~4e_ 5

I

[ i I ,~L ~_ILIIIIIi 0.001 0.01 0.1 F r e q u e n c y ( rad/sec )

I [ I II]~

Figure 5 Input power to remove worst case disturbance, under perfect control assumption. Solid line: Pn.,: dashed: PL,.,.;dot-dash: D~

Control structure screening method Two structure selection methods have been applied. They are briefly described in the following. Both are based on a M I L P formulation of the control objective. The first method assumes perfect control of the selected controlled variables up to a specified frequency COd.The algorithm minimizes a linear objective function formulated in terms of an approximation to the largest time domain deviation (from the worst case disturbance) of the uncontrolled outputs and manipulated inputs, subject to the constraints given by the maximum allowed deviation of all variables. The objective function is written as m i n ( ~-~ otivi + Z l

fljh/)

j

where )~g and hj are estimates of the largest (time domain) deviations. X and Y are vectors the dimension

Control structure selection: J. E. Hansen et al.

190

of which equals the number of measurements and actuators respectively. If the flh output is included into the control structure then Yj is one (else zero) and ~j = 0. Similarly for X and the potential actuators, if the ith input is not included then Xi = 0 and /-li = 0. The algorithm enforces the same number of controlled as manipulated variables. In the present work all the weights, oti,/~i, are set equal to zero, except o/16 and a17 which are both one, reflecting the control objective of the benchmark problem to minimize deviation of the product purities, disregarding all other outputs and the actuator consumption, and further that the two purities are of equal importance. Combinations with outputs that are not potential controlled variables, marked with • in Table 2, are excluded from being selected. As a measure of the worst case time domain deviation of the outputs the following frequency approximations are used: fii =

Z k

max

to<_wa

~v,cl Amax V ik t~ k

u cl

may

max IG.~ Idk to
~li = Z k

-

where Gy,d is the closed loop transfer function from the disturbances to all measurements based on the perfect control of the outputs selected to be included into the structures. Similarly G ",d is the closed loop transfer function from disturbances to the actuators that are included in the structure. A formalized derivation of Gy'c! and G u'cl from G, Gd and the vectors X and Y may be found in Appendix A. For the scaled distillation column model, the magnitude of the largest disturbances, a~kax is set to one for k = 1, 2, which corresponds to the normal case, low amplitude disturbances. The variables fii and fii are frequency approximations to time domain deviations. The frequency domain up to the bandwidth Wd is discretized into a set of distinct frequencies, and the upper bound on the worst amplification of any disturbance is calculated at all these frequencies. For the distillation column Wd is chosen to be 10-2 rad s -1" The algorithm is implemented such that a specified number of most feasible solutions may be ranked according to the lowest value of the objective function. The second screening method is based on realistic control rather than the assumption of perfect control. The method searches all structures and simultaneously finds the optimal pairing of inputs and outputs based on Ziegler-Nichols (ZN) tuned PI controllers. For a given pairing, the ZN-parameters are determined, along with a set of detuning factors to formulate the tuning optimization within the M I L P framework. The transfer functions to determine the upper bound to the worst case deviations are now the closed loop based on decentralized PI-controllers.

Results and analysis First the structures selected by perfect control screening will be discussed and compared based on conventional controllability measures, whereafter the results of the realistic control screening are presented. Perfect control screening

The perfect control screening method selected the five structures (A,B,C,D,E) listed in Table 5. Structure F is the existing one. Structures A and B (and F) show the optimal cost of the objective function, due to perfect control of both product purities. Since no other structures with both purities controlled were selected, it can be concluded that such structures all are rated infeasible by the screening method. Which means that either actuators or uncontrolled outputs are forced beyond the acceptable maximum deviations. This must also be the case for the existing structure F. It must be emphasized that this statement only is true for perfect control and when the linear model applies. The remaining structures C, D and E all have just one purity on control along with a column temperature in the opposite section, and also a column pressure. Also it is interesting to note that all selected structures use all three potential actuators. In the following the new and the existing structures are compared based on the controllability measures, RGA, CN, RDG, and RHP-zeros. Relative gain array. The R G A matrix measures the interaction between control loops in a decentralized control configuration of a given structure. The (i,j)th element of the matrix may be interpreted as the gain of the actuator j to perfectly control the output i when no other loops are closed, relative to this gain when all other outputs are perfectly controlled as well. Table 6 shows a comparison of the static R G A based on the static gain from all inputs to the outputs of a given structure. It is seen that structure C is very interactive and structure D and E have almost diagonal RGA matrices, indicating only little static interaction. Figure 6 presents the frequency dependent RGA-elements of the transfer function from actuators to controlled variables of a given structure. Only the magnitude of the four upper left most elements of each transfer function matrix are plotted. At higher frequencies all structures show enhanced interactions. Again structures D and E exhibit the least interactions,

Table 5 Selected structures by the perfect control selection method (A, B, C. D, and E), and the existing structure (F) Output A B C D E F

T19, XB, XD T1, XB, XD TI9, P19, Xs 7"1, PIo, XD T1, PI9, XD XB, el9, XD

Actuator PH, s, Pt.,~, D~ PH, s, PL..~, D., PH.s, PL.s, D~ P,..~, PL.~, D.~ Pn,s, PL,,~,D.~ PH..~, PL..,-, D,.

Cost 0 0 3 10-3 3 10 -3 3 10 3 0

Controlstructure

Table 6

A:

StaticRCA

matrices

0.7501

0.2498

0.0000

0.0807 0.1691

0.2534 0.4967

0.6658 0.3342

for structures

14.490

-14.70

1.2078

0.1663 -13.66

0.4879 15.209

0.3458 -0.554

1.0223

0.0146

0.0328 -0.055

0.9608 0.0246

0.9822

0.0178

0.0000

0.0099 0.0079

0.3248 0.6574

0.6653 0.3347 I

1 [I D:

-0.073

1.0227

0.0499

0.0070 .0656

0.0854 -0.108

0.9076 0.0425

1 [ F:

J. E. Hansen et a I.

191

and C the worst, but the RGA-measure regarding structure A, B, and F.

A. B, C, D, E. and F

1 [ B:

selection:

1

0.7545

0.2455

0.0001

0.0765 0.1690

0.2577 0.4969

0.6658 I 0.3342

is inconclusive

Relative disturbance gain. The RDG is a measure of interactions on decentralized control configurations, that takes into account the directions of the disturbances (Stanley et al.‘s). The RDG matrix is defined by

The (i.j)th element may be interpreted as the gain of actuator i to perfectly reject the effect of disturbance .i

~~*,~~

ou

,&-_

j_~.I<_l-j

L

1O-4

(C) 100 r

1o-3 lo-* Frequency(rad/sec)

10-l

1o-4

(D)5~

1O-3

lo-*

10-l

Frequency( radkec)

/ 41

84

3

21

011111/LUi 1o-4

1o-3 lo'* Frequency(rad/sec)

10-l Frequency( radlsec) (F)

5r :

Frequency(rad/sec) Figure 6

Absolute value of the four upper

left most elements

Frequency in the frequency dependent RCA.

( radkec )

The structures

are A, B, C. D. E. and F. row-wise

Controlstructure selection. J. E. Hansen et a l.

1 92

on output i when all other loops are closed, relative to that gain when all other loops are open. This means that a value less than one indicates that closing the other loops actually has a positive impact on the control loop in question. The R D G will here be a (3 × 2) matrix. The loop pairings are the ones suggested by the steady state RGA, which is the direct 1-2-3 pairing as listed in Table 5. Figure 7 shows the results. It is seen that all structures possess some extent of unfortunate high frequency interactions. Structures A, C and F have high interactions even for the feed flow rate disturbance. When the composition disturbance is considered

structures D and E must be favoured over the others, due to very small interactions below the desired bandwidth.

Right half plane zeros. In perfect control the open loop transfer function is inverted, and hereby right half plane zeros are transformed into unstable poles of the closed loop. Then R H P zeros, with a real part less than the bandwidth will seriously affect the stability margin of the closed loop. If the R H P zero is greater than the bandwidth but near to this value, the performance of the closed loop will be reduced.

(A) 102

(B) 102

1

10

10

0

.

10

10"1

100

.

1 0 -2 -4

10

-

.........

101

-3

.........

-2

10 10 Frequency ( rad/sec )

.........

-1

10

(C) 102 I

101 I

10 -4 10

-2

-1

10

(D) 102

"

101

10~ E~_~__._~,

100

101

10"1

10 -4 10

-3

10 10 Frequency ( rad/sec )

-3

-2

10 10 Frequency(rad/sec)

-1

10

10 -4 10

/ /

-3

-2

-1

10 10 Frequency ( rad/sec )

10

2 (F) 10

(E)IO 1

1

10

10

t /

100

100

101

10-1

10 -4 10

-2

10-3 10 Frequency ( rad/sec )

-1

10

10 -4 10

-3

/j //~

j

-2

10 10 Frequency ( rad/sec )

-1

10

Figure 7 R D G elements of the six structures. Solid lines: feed flow rate disturbance; dotted lines: composition disturbance; the structures are A, B, C, D, E, and F, row-wise

Control structure selection.'J. E. Hansen

7

Table

A

Zeros

B

1.45x 105

Structure Zeros

Table

C

2.73x 105

0.0016

D

E

F

0.70±i 21.82

0 . 1 6 ± i 21.82

1.45x105

All structures show one R H P transition zero, but except for C they are all of minor importance since control is only required to be effective below 10-2rads i. The RHP zeros of structure A,B and F are so large that they have no implication on the closed loop performance, and these structures must be favoured based on R H P zeros.

Condition number. The condition number based on the transfer function from actuators to controlled ouputs in a given structure is plotted as functions of frequency in Figure 8. Since all inputs and outputs are scaled according to importance, the conditions numbers are the actual ones and not bounds on the minimum numbers over all scalings. Based on the condition numbers structures C and F should be excluded. According to the condition number it is not possible to differentiate between the other structures. Summary. Table 8 summarizes the controllability measures for the six structures. The table suggests the overall qualitative ordering: 1

2 3 4

D,E A,B F C

It is interesting to note that the R G A and R D G measures are the most conclusive indicators for the favoured structures, and that these measures all favour structures D and E. Thus structures D and E will most likely give the best control performance of the controlled variables.

2000~ 1500 "'\.\ \

1000~

\,

\\

C

\

A B C D E F

Static RGA

Dynamic RGA

Dynamic R D G

ic ic

-

6

ic

-

+

ic ic

+ +

~+

ic ic

ic ic

ic

-

+

When controlling the plant through structure D, with integral action in all loops, the uncontrolled product purity, particularly the bottom purity, will exhibit a steady state offset if the plant is exposed to step disturbances. An estimate of the resulting static error is given by Yo = GY'd(O)do. This of course would emphasize the structure A or B, since they have both purities controlled. But the objective is to regulate against oscillating disturbances, a situation where the achievability of zero steady state error is of lesser importance.

Nonlinear simulation. The structures A, D and F have been implemented in the nonlinear simulation program to verify the above result. Discrete time PI-controllers are used with a sampling time of 15 s. This sampling interval is presently the sampling time of the experimental pilot plant distillation column. The distributed controllers are paired according to the static RGA matrices (direct 1-2-3 pairing as listed in Table 5), and the tuning parameters are determined in a uniform way by adjusting the gain and phase margin of the individual transfer functions, with addition of a 15 s time delay for conservative control design. Table 9 shows the maximum and minimum time domain deviations from the steady state of the product purities when the column is subjected to the stress level 2 oscillatory disturbances. The simulations confirm the above ordering of the structures for best performance. The new structures A and D show improved performance compared to the existing structure F. Further it is seen that actually only structure D is able to meet the control objective. Also responses to feed flow rate step disturbances have been simulated to demonstrate the control performance. The product compositions responses are shown in Figure 9. The lack of integral action on the product purities for structure D isobvious, but this structure clearly shows the least overshoot and the fastest settling time. Table 9 Minimum and maximum deviation on product purities, with stress level 2 oscillatory disturbances. A, D, and F are structures of the perfect control screening, and D is of the realistic control screening

\ I/

YD

Structure

10-4 Figure

8

+ + ic

RHP CN

\\

5ooi 0L

193

8 Summary of controllability measures for the structures selected by the perfect control screening. Symbols: - indicates structure is not favoured by the measure; + indicates structure favoured: and ic that the particular measure is inconclusive

Multivariable zeros of structures A, B, C, D, E, and F

Structure

et al.

10-3

Condition number

10

-2

10

-1

A D F G

XB

Min

Max

Min

Max

-0.6 -0.4 -1.7 0.22

6 0.6 60.5 6 1.0 +0.17

-1.5 - 1.0

+(I.7 + 0.6 + 1.3 + 0.5

- 1.4

-0.8

Controlstructure selection.'J. E. Hansen et al.

1 94 0.12 0.1 t I / /I\

O.OE O.OE

it

/

/

r / x\

\

\ x \ \

\

\

It

\

II

0.84

product purity responses to the feed flow rate step disturbance are seen in Figure 9. It is interesting to see that this structure gives far the best performance under the oscillatory disturbances and it has an overshoot and settling time similar to those of structure D. From the fact that the realistic control structure screening method have rejected all other structures it may be concluded that it is not possible to tune the structures of the perfect control screening method with Z N parameters and obtain a feasible solution, one that does not force some variables beyond their maximum allowable deviations. That ZN tuning parameters are infeasible for this particular distillation column is consistent with previous analysis of the column (Koggersbo19). That structure G was not among the selected structures of the perfect control screening indicates that this structure will not work with very tight control.

\

II

\

ill

\\

II

--

\

0.02 C -0.02 -0.04

1.1

1:2

1'.3

1:,

0.05

// //

1 / -0.05

1'.5

Time (hour)

1:8

1'.,

1'.5

11,

x \

Conclusions

~\\

/

t

-0.1 \

111

/

112

113

114

115 116 Time (hour)

117

118

119

2

Figure

9 Responses to step disturbance in feed flow rate of 5%. Upper: xv; lower: xs. Solid: structure A; dashed: structure D; dotted: structure G; dot-dash: structure F

Structure D may be viewed as an indirect control of the bottom product purity. Integral action could be achieved by a cascade using, for example, the setpoint of T1 as manipulable variable. Whether this will improve performance compared to structure A or F in the step disturbance situation is still to be investigated.

The control objectives of the benchmark problem were met by three new structure, D, E, and G. The two first have the bottom product purity indirectly controlled. Far the best performance was achieved with structure D, that has both purities indirectly controlled. Based on the five selected structures of the perfect control screening, two feasible structures could be selected by analysis with conventional controllability measures prior to implementation in a rigorous nonlinear dynamic simulation. Also it has been indicated that Ziegle~Nichols tuning with a single detuning factor for each loop is inadequate for this particular energy integrated distillation column. The screening methods have several limitations. The perfect control screening concept has the very nice feature of being independent of any configuration, in terms of tuning and pairing, which means that the results are applicable for both M I M O controllers and decentralized controllers. But the effect of e.g. time delay introduced by sampling cannot be incorporated into the analysis. In contrast this incorporation of a delay is possible for the realistic control screening, which however is limited by Ziegler-Nichols tuning of the decentralized PI-controllers.

Real&tic control screening The second screening method was run with a 15 s delay on all measurements in order to attempt an automated tuning of the distributed PI-controllers. Just one feasible structure, G, was obtained. The structure and the pairing is given in Table 10. The automatic Ziegler-Nichols based tuning did work, but it was far from optimal in the sense that is produced very oscillatory responses even to step disturbances. For this reason the loops were retuned in a similar way as the structure of the perfect control screening. The maximum and minimum deviation of the product purities when the system is subjected to the high amplitude disturbances are also given in Table 9 and the

Acknowledgement The work has been partially supported by the EECContract CHRX-CT94-0672.

Table 10

G

Result of the realistic control screening Output

Input

Tj TI9 PI9

PH,.~ PL,~ Ds

Control structure selection. J. E. Hansen e t al.

1 95

References

Appendix A: Closed loop transfer functions

I. Foss, A. S., Critique of chemical process control theory. American Institute of Chemical Engineers Journal, 1973, 19(2), 209. 2. Morari, M., Arkun, Y. and Stephanopoulos, G., Studies in the synthesis of control structure for chemical processes. American Institute ol" Chemical Engineers Journal, 1980, 26(2), 220. 3. Georgiou, A. and FIoudas, C. A., Structural analysis and synthesis of feasible control systems: theory and applications. Chemical Engineering Research and Design, 1989, 67(11), 60(~618. 4. Narrawya, L. T. and Perkins, J. D., Selection of process control structure based on linear dynamic economics. Industrial & Engineering Chemist O" Research, 1993, 32(11), 2681 2692. 5. Skogestad, S. and Postlethwaite, I., Muhivariable Feedback Control Analysis and Design. John Wiley & Sons, New York, 1996. 6. Morari, M. and Zafiriou, E., Robust Process Control. Prentice Hall, Englewood Cliffs, N J, 1988. 7. Koggersbol, A. and Jorgensen, S. B., Distillation column control benchmarks: lbur typical problems. In DYCORD'95, 4th Symposium on Dynamics and Control of Chemical Reactors, Distillation Column and Batch Processes, ed. J. B. Rawlings. Danish Automation Society, Copenhagen, June 1995, pp. 323 325. 8. Heath, J., Perkins, J. D. and Walsh, S., Control structure selection based on linear dynamic economics multiloop P1 structures for multiple disturbances. Technical report, Centre for Process Systems Engineering, Imperial College, London, 1996. 9. Koggersbol. A., Distillation column dynamics, operability and control. P.h.D. thesis, Technical University of Denmark, 1995. 10. Bristol, J. G., On a new measure of interaction for multivariable process control. IEEE Transactions on Automatic Control, 1996, AC-Ii, 133 134. II. Hovd, M. and Skogestad, S., Simple frequency-dependent tools for control system analysis, structure selection and design. Automatica, 1992, 28(5), 989 996. 12. Skogestad, S. and Morari, M., Implication of large RGA-elements on control performance. Industrial & Engineering Chemistry Research, 1987, 26, 2323 2330. 13. Stanles. G., Marion-Galarrage, M. and McAvoy, T. J., Shortcut operability analysis. 1. The relative disturbance gain. Industrial & Enginecrin~ Chemistry Process Design and Development, 1985, 24, 1181 1188. 14. Koggersbol, A., Andersen, B. R., Nielsen, J. S. and Jorgensen, S. B.. Control configuration for an energy integrated distillation. Computers in Chemical Engineering, 1996, 20(B), $835. 15. Koggersbol, A., Andersen, T. R., Bagterp, J. and Jorgensen, S. B., An oulput multiplicity in binary distillation: experimental verification. Computers in Chemical Engineering, 1996, 20(B), $253.

T h e c l o s e d l o o p t r a n s f e r f u n c t i o n s f r o m the dist u r b a n c e s to all m e a s u r e m e n t s , G v,ct, a n d to the selected a c t u a t o r s , G ",Ct, m a y easily be d e r i v e d . Let m a t r i c e s C r a n d C x be d e f i n e d by: C r = diag{ Y } C x = d i a g { X }

w i t h all r o w s o f p u r e z e r o s r e m o v e d . T h e n C r y a n d C x u a r e v e c t o r s o f selected o u t p u t s a n d i n p u t s i n c l u d e d in the structure. W h e n the selected o u t p u t s are p e r f e c t l y c o n t r o l l e d , the f o l l o w i n g applies:

t' = GC~xlCxu + Gdd Cy)'

C y G C T C x u + CyG~Id = 0

C x u = - ( C r G C T) - 1C rG~ld ,, =

C,,G,,a +

Gv,cl = - G C x (TC r G C x ) T a n d G "''/ = - C x (YC r G C ~ , )Y

I C r G a + G,/ I CrUel

T h e c l o s e d t r a n s f e r f u n c t i o n for the realistic c o n t r o l m a y be f o u n d in a s i m i l a r way. But the r e l a t i o n b e t w e e n C r y a n d C r u is n o w given by: Cxu

- K C rY

W h e r e K is the t r a n s f e r f u n c t i o n r e p r e s e n t i n g the dist r i b u t e d PI c o n t r o l l e r s .