Design of multiloop siso systems

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Reprinted from I&EC PROCESS DESIGN8 DEVELOPMENT, 1986, 25, 498.

Copyright @ 1986 by the American Chemical Society and reprinted by permission of the copyright owner.

Design of Multiloop S I S O Controllers in Multivariable Processes

Cheng-Chlng Yu and William L. Luyben*

Process Modeling and Control Center, Department of Chemical Engineering, Lehlgh University, Bethlehem, Pennsylvania 18015

A practical design procedure is proposed for determining the structure, variable pairing, and tuning of multiloopSISO controllers in a multivariable-process environment. The selection of controlled variables relies primarily on

engineering judgment. The choice of manipulated variables is based on Morari's Resiliency Index (minimum singularvalue of the process transfer function matrix). Variable pairing is decided by first eliminating those pairings thatgive negative RGA's or negative Niederlinksi or Morari Indexes of integral controllability. The final variable-pairing

selection is based on the Tyreus load rejection criterion. The simple BLT controller tuning method, recently proposed

by Luyben, is used to give consistent, logical comparisons of alternatives. The method is illustrated by applying

it to a multivariable ternary distillation column example.

A recent paper (Luyben, 1985) proposed a practical,

easy-to-use method (BLT tuning) for tuning SISO (sin-

gle-input-single-output) controllers in a multivariable

environment. The technique is based on detuning all

controllers equally from Ziegler-Nichols settings until the

desired degree of closed-loop stability of the system is

achieved as indicated by a multivariable Nyquist plot. The

structure of the system must be specified to use this tuning

method: what variables are controlled, what variables are

manipulated, and how they are paired in the SISO

structure.This paper addresses these structural questions. A

logical, systematic design procedure is proposed which

produces a stable, workable multiloop SISO system.

I t should be em~has izedhat multivariable controllers

are not consideredin this paper. There are no claims that

the multiloop SISO structure gives the best control system.

It is possible that a multivariable controller could improve

control performance. Multivariable controllers (such as

Internal Model Control [IMC], Dynamic Matrix Control

[DMC], Linear Quadratic Control [LQ], etc.) will be com-

pared to multiloop SISO systems in a future paper.

What is claimed about the proposed procedure is that

it willproduce a workable, stable, simple SISO system with

only a modest amount of engineering effort. This con-ventional SISO system can then serve as a realistic

benchmark, against which more complex multivariable

controller structures can be compared. This procedure has

O 1986 American Chemical Society



Ind. Eng. Chem. Process Des. Dev., Vol. 25 , NO. 2, 1986 49 9

been tested on open-loop stable processes only. The ap-

plication to open-loop unstable processeswill be the subject

of future research.

Outline of Procedure

Before the proposed design procedure is discussed in

detail, a brief outline of the steps is given below.

1. Select Controlled Variables: Use primarily engi-neering judgment based on process understanding.

2. Select Manipulated Variables: Find the se t of ma-

nipulated variables that gives the largest Morari ResiliencyIndex (MRI).

3. Eliminate Unworkable Variable Pairings: Eliminate

', pairings with negative RGA's or negative measures of in-

tegral controllability (Niederlinski and Morari Indexes).

4. Find the Best Pairing from the Remaining Sets: (a)

Tune all combinations using BLT tuning and check for

stability (characteristic loci) and robustness (Doyle-Stein

Index). (b) Select th e pairing that gives the lowest mag-

nitude closed-loop load transfer functions: Tyreus Load-

Rejection Criterion (TLC).

Discussion of ProcedureThe starting poin t is plant transfer functions relating

all controlled variables to all manipulated variables. I t is

assumed that these I@ transfer functions are available ( N

is the number of SISO loops in the multivariable process).

The N transfer functions relating the controlled variables

to the load disturbance must also be known.

This aspect of the problem is by no means a trivial job,

particularly for the many chemical engineering systems

that exhibit strong nonlinearities (reactors, high-purity

distillation columns, etc.).

Plant test dat a can sometimes be used to develop these

transfer functions, but noise and changing conditions make

these transfer functions difficult to obtain reliably. Pulse

testing of a nonlinear mathematical model of the processis reasonably successful in most systems, but analytical or

numerical linearization methods must be used in some

nonlinear processes.

A. Choice of Controlled Variables. Usually the

controlled variables are fairly easy to select. Engineering

judgment, based on a good understanding of the process,

leads in most cases to a logical choice of what needs to be

controlled. Considerations of economics, safety, con-

straints, availability and reliability of sensors, etc., must

be factored into this decision. We offer no new guidance

on this question in this paper.

I t should be remembered that these controlled variables

can be directly measured (temperature, pressure, compo-

sition, etc.) or computed from other directly measuredvariables (ratios. heat removal rates. inferred ~ r o d u c t

compositions, mass flow rates, extensive variables, etc.).

Occasionally some guidance can'be obtained from singu-

lar-value decomposition (Downs, 1984).

B. Choice of Manipulated Variables. Once the

controlled variables have been specified, the "control

structuren depends only on the choice of manipulated

variables. For a given process, selecting different manip-

ulated variables will produce different "control structuren

alternatives. These "control structuresn are independent

of the "controller structuren, i.e., multiloop SISO con-

trollers or one multivariable controller.

For example, in a distillation column, the controlled

variables might be chosen as distillate and bottoms com-positions (XDand X,). Now we could choose the ma-

nipulated variables reflux and vapor boilup ( R and V).

This choice will give one possible control structure.

Had we chosen to manipulate distillate flow and vapor

boilup (D and V), we would have a different control

structure for the same basic distillation process.

Morari has studied methods for assessing the inherent

resilience of alternative control structures (Morari, 1983).

Based on this work, we propose to use the "Morari Resi-

liency Indexn (MRI) to guide the selection of manipulative

variables.

MRI = g[G(iw)] (1)The MRI is the minimum singular value (g) f the plant

transfer function matrix G(iw). The set of manipulated

variables that gives the largest minimum singular value

over the frequency range of interest is the best.

This selection of control structure (manipulated varia-

bles) is independent of variable pairing (controller struc-

ture) and controller tuning. The MRI is a measure of the

inherent ability of the process (control structure) to handle

disturbances, model plant mismatches, changes in oper-

ating conditions, etc. The larger the value of MRI, the

more resilient the control structure.

The problem of the effect of scaling on singular values

is handled by expressing the gains of all the plant transfer

functions in dimensionless form. The gains with engi-neering units a re divided by transmitter spans and mul-

tiplied by valve gains. This yields the dimensionless gain

that the control system has to deal with.

C. Eliminate Unworkable Variable Pairings.1. Eliminate Pairings with Negative RGA's. Pairing

on negative RGA's is undesirable because it produces ei-

ther an unstable system or a system tha t lacks integrity

(will go unstable with some loops on manual: Grosdidier

et al., 1985).

2. Eliminate Pairings with Negative NiederlinskiIndexes. Th e Niederlinski Index (NI) is defined

where G(o)= steady-state gain matrix of plant transfer

functions and gii(,)= diagonal elements of G(o). If all the

SISO controllers contain integral action and have positive

loop gains, a negative value of the Niederlinski Index shows

that the system will be closed-loop unstable with this

variable pairing for any controller tuning (Niederlinski,

19711.

3. Eliminate Pairings with Negative Morari In-

dexes of Integral Controllability (MIC)

MIC = X [ G , ]

where X[G]=

eigenvalues of the matrix G and Gfo)=

plantsteady-state gain matrix with the signs adjusted so tha t

all diagonal elements have positive signs.

If all the SISO controllers contain integral action and

have positive loop gains, a negative value of any of the

eigenvalues of Gfo) shows that th e variable pairing will

produce an unstable closed-loop system. Note tha t the

application of MIC is limited to multiloop SISO controllers

and strictly proper process transfer function matrices. MIC

is a special case of Theorem 7 presented by Grosdidier et

al. (1985).

D.- Final Variable-Pairing Selection. All the re-

maining pairing possibilities must be examined. For each

case, controllers are tuned by using the BLT procedure.

Then the closed-loop load-rejection performances are

compared in order to find the pairing which does the best

job of keeping controlled variables constant in the face of

load disturbances.

1. BLT Tuning (Luyben, 1985). This simple proce-

dure uses a multivariable Nyquist plot and equal detuning



500 Ind. Eng. Chem. Process Des. D ev., Vol. 25 , No. 2, 1986

of all loops from the single-loop Ziegler-Nichols settings.

(a) Calculate the Ziegler-Nichols settings for each PIcontroller by using the diagonal elements of G.

(b) Assume a detuning factor "F,nd calculate con-

troller settings for all loops.

(c) Define the function W(iu)

W(,,, = -1 + de t [I + G(L,B(i,)] (6)

where B = diagonal matrix of SISO feedback controllers.

(d) Calculate the closed-loop function

= 20 log-1 Ywl

(e) Adjust the detuning factor F until the peak in the

LC og modulus curve (L,) is equal to 2N

LC, = 2N (8)

where LC, is the biggest log modulus in the LC urve and

N is the number of SISO loops in the multivariable system.

Luyben (1985) demonstrated tha t this method gives

reasonable, conservative estimates for the controller set-

tings on 10 case studies. I t provides a method of obtaining

settings for all possible pairings on a consistent basis sothat realistic comparisons can be made.

2. Tyreus Load-Rejection Criterion (TLC). Tyreus

(1984) proposed the use of the load-rejection capability of

the closed-loop system as a rational method for selecting

the best variable pairing.

The closed-loop relationship between the load variable

L and the controlled variables X i s

where X = vector of N controlled variables, GL vectorof open-loop plant transfer functions relating the controlled

variables to the load disturbance, and L = a single-load

disturbance.

The at tenuation of the load disturbance by the processand the control system can be shown in a plot of the

magnitude of each element of X as a function of frequency.

A perfect control system would keep all the Xi's a t zero.

The best variable pairing is the one that gives the

smallest magnitudes of the Xi's. This is the TLC criterion.We have formulated the procedure above for a single-

load disturbance. Obviously the method can be appliedto systems with several load disturbances provided ap-

propriate weighting factors can be defined for th e effectsof each load input.

E. Robustness of Closed-Loop System. The ability

of the closed-loop system to remain stable despite changes

in process parameters is called robustness. Doyle and Stein

(1981) proposed two useful measures of robustness.

DSI = g[I + (BG)-l] : (11)

The final variable pairing and controller tuning should

be checked for robustness by plotting DSO and DSI as

functions of frequency. Singular values below 0.3-0.2 in-

dicate a lack of stability robustness. Further controller

detuning may be required.

Distillation Example

The ternary distillation column, studied by Yu and

Luyben (1984) for single-loop control, is presented as an

w e i r length

6.35 crn

1.61 rn

Figure I . Steady-state design of ternary deisobutanizer column.

R -V - I D-V- I R R - V

Figure 2. Alternative control schemes.

example. Several other multivariable distillation systems

have been studied by using this procedure and will be

reported in later papers dealing with these process ap-

plications: heat-integrated columns (Chiang, 1985) andcomplex distillat ion configurations (Alatiqi, 1985).

The steady-state design of the system is given in Figure

1. The ternary propane/isobutane/normal butane system

is studied a t 120 psia. Propane is the lighter-than-light

key component, and its concentration in the feed is 6 mol

%. The column has 32 trays, a total condenser, a partial

reboiler, saturated liquid reflux, and theoretical trays.A. Controlled Variables. Since we wish to control the

distillate and bottoms product compositions, XDN nd XBNwill be chosen as controlled variables where XDN= nor-

malized mole fraction of iC4 n the distillate = mol of iC4

in distillate/(mol of iC4+ mol of nC4 in distil late), XBN= normalized mole fraction of nC4 in bottoms = mol of nC4

in bottoms/(mol of iC4+mol of nC4 in bottoms). These

normalized compositions are used so as to avoid the

problems t ha t occur when trying to control the absolute

composition in the presence of non-key components.

Composition analyzers are used with two different sam-

pling times (T, 1 and 4 min).

B. Manipulated Variables. Many possible choices

of manipulative variables exist. The conventional alter-

natives are reflux flow and vapor boilup (R and V), dis-

tillate flow and vapor boilup (D and V), and reflux ratioand vapor boilup (RR and V). Figure 2 shows some pos-sible control schemes. Table I gives the open-loop process

transfer functions G and GL or this process with severalchoices of manipulated variables. The load disturbance



Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 501

Table I. Open-Loop Transfer Function Matr ix an d Load Trans fer Functions

R- Va D- Vb RR- Vc

"Transmitter span of 0.2 mol fraction for XDN nd XBN,valve gain of 1457 g-molls for V, and valve gain of 1420 g-moll s for R. bValvegain of 160 g-molls for D. CValvegain of 17.75 for RR. With units of mole fraction/mole fraction.

Table 11. Steady-State Analysis of Alternative Control - - V--- - -

Struc tu resD -V

"..... .. RR-Vcontrol variable MRI MIC .

structure pairing a@,GA NI X ~ ~ $ , ,

R-V R-V-1 0.114 -86.1 -1186.1 45.1, -0.16

(XDN-R, XBN-V)R-V-2 0.114 87.1 1187.1 44.1, 0.24

(X~N-v ,XBN-R)D-V D-V-1 1.86 0.56 110.56 2.04 * 1.89i

(XDN-D, XBN-V)

D-V-2 1.86 0.44 110.44 2.06*

1.75i(XDN-V, X n r D )

RR-V (XDrRR, ' 0.89 1.48 111.48 5.33, 0.9

XBN-V)

is considered to be the propane concentration in the feedZF (LLK)

X = GM+ &ZF(LLK) (12)

Table I1 gives the MRI values, the RGA values, theNiederlinski Indexes (NI), and Morari Indexes of IntegralControllability (MIC).

The D-V structure has a bigger MRI value, and there-fore this choice of manipulated variables is recommended(Table I1 and Figure 3) .

The R-V structure has a very small MRI (0.114 atw =

0). Therefore, this structure is inherently sensitive. Forexample, a 1% change in steady-state gains could makeG(,,singular. Also notice that the R-V scheme with theconventional pairing XDrR and XBrV (R-V-1) showsa negative RGA. This unusual result is due to the mul-ticomponent nature of the system. As can be seen in thegains of the open-loop transfer functions (Table I), V hasmore of an effect on XDNhan does R.

At thispoint, the manipulated variables would be chosenas D and V. However, for purposes of illustration, the R-Vand RR-V structures will be studied further in order todemonstrate that t he MRI is an effective criterion.C. Eliminate Unworkable Variables Pairings.

(1) Values of Values of RGA indicate both pairings (D-V-l and D-V-2) for the D-V structure give stable systems(Table 11). As mentioned earlier, the conventional R-V-1pairing gives a negative RGA. Therefore, only the R-V-2

B ..-.0. I I I I I,001 .a I I 10 100

WFigure 3. Morari Resiliency Indexes MRI for R-V, D-V, and RR-Vcontrol structures.

pairing can be used. The RR-V structure also has onestable pairing as indicated by RGA.

(2) For a 2 x 2 system, NI contains essentially the sameinformation as RGA (Table 11). However, this is not truefor higher-order systems.

(3) Values of MIC also show tha t both D-V-1 and D-V-2are integral-controllable, and there is only one possiblepairing for R-V and RR-V structures.

For a 2 X 2 system, RGA, NI, and MIC give essentiallythe same results. But for higher-order systems, there arecases where MIC gives more information than RGA andNI.

Before getting into the final variable pairing selection,the R-V structure will be investigated briefly. Table I11

Table 111. Dynamic Analyses of Alternative Control Stru ctu res

control tuning stabilityb sta bil ityscheme TS, min method P actor Kc1/K, 7i1/7i2 DSO DSI gaina margin (linear) (nonlinear)

R-V-1 0 Buckle9 1 1.8410.39 4.2130.0 Oe no Yes(SISO)

R-V-2 0 Buckley 2 -1.951-0.46 4.3119.8 0.50 0.70 5.5 Yes no(SISO)

D- -1 1 BLT-4 3.35 -2.4510.54 13.1118.4 0.63 0.26 4.1 Yes Yes4 BLT-4 2.02 -1.17/0.31 26.9128.7 0.67 0.36 2.2 yes yes

D-V-2 1 BLT-4 3.88 0.2510.60 138.2/100.2 0.49 0.30 1.7 yes no4 BLT-4 3.21 0.3010.35 130.21124.4 0.42 0.38 1.6 Yes no

RR- V 1 BLT-4 2.34 5.3211.01 24.3111.9 0.45 0.10 3.0 Yes no1 BLT-2 3.98 3.1210.59 41.4120.3 0.87 0.22 5.4 Yes Yes4 BLT-4 1.73 2.010.42 39.2124.6 0.68 0.20 2.4 Yes no4 BLT-1 2.48 1.3910.29 56.0135.2 0.89 0.30 3.5 Yes Yes

a Smallest gain margin of characteristic loci. Stability on linear transfer function model. Stability on nonlinear rigorous column model.dBuckley's single-loop tuning (Luyben, 1973). System with positive feedback.



502 Ind. Eng. Chem. Process Des. Dev., Vol. 25 , No. 2, 1986

R - V - I ( T , a O )I

R - V - 2 ( T s a O )

Figure 4. Characterist ic loci plots for R-V-1 and R-V-2 pairings.

R - V ( T c O )

I

- - V - I

Figure 5. Nonlinear step responses for L-V-1 and L-V-2 schemes

for a step change in ZF(LLK) (ZF(LLK) = 0.10).

gives the controller parameters for both R-V-1 and R-V-2pairings with no analyzer dead time (T, 0). Charac-teristic loci plots (Figure 4) show that the R-V-1 schemeis closed-loop unstable as indicated by MIC. Characteristicloci plots also indicate that R-V-2 is stable and has a gainmargin of 5.5. However, the nonlinear simulations, Figure5, show that the R-V-1 is stable and the R-V-2 is unstable.This structure is very sensitive to nonlinearities and waseliminated from further study.

D. Fina l Variable-Pair ing Selection. (1)There arethree possible control schemes left: D-V-1, D-V-2, andRR-V. All the controllers were tuned with BL T tuningfor L , = 4 (BLT-4). Table I11 summarizes the controllerparameters. Reasonable stability margins were obtainedas shown in characteristic loci plots (Figure 6) and gainmargins in Table 111.

(2) The Tyreus Load-Rejection Criterion (TLC) wasused to determine the final variable pairing. Figure 7

Figure 6. Characteristic loci plots for D-V-1 and D-V-2 schemes

with BLT-4 tuning.

D-V-I (T,=41 -O-V-ZCr<41 ----RR -V (Tsa4) ..........

tnol .lol 1 I I iI

W

10 Do

Figure 7. Load-rejection plot (TLC) for D-V-1, D-V-2, and RR-V

schemes.

shows that the D-V-1 scheme attenuates the load dis-turbance (ZF(LLK)) much better than the D-V-2 scheme.Therefore, D-V-2 is eliminated according to TLC. Thereis no clear preference between the D-V-1 and RR-Vschemes (Figure 7) in terms of TLC.

E. Robustness of Closed-Loop System. Doyle andStein Indexes (DSO and DSI) give good measures of ro-bustness. Figure 8 shows the lack of robustness for theRR-V scheme (DSI = 0.2 at w = 0.2). To get the samedegree of robustness, the RR-V scheme had to be detunedto a +1 LC,. Therefore, D-V-1 is the control schemerecommended. Figure 9 shows the responses of the columnfor step changes in feed composition. The D-V-1 schemegives good response. The RR-V scheme is unstable forthe +4 LC, tuning.

It should be emphasized that we can determine the bestcontrol structure at the very beginning (from MRI).



Ind. Eng. Chern. Process Des. Dev., Vol. 25, No. 2, 1986 503

1 D-V- I (Ts=41-RR-V (Tsm4) ----

.o a .a .I I K) KmW

Figure 8. Doyle-Stein Indexes (DSO and DSI) for D-V-1 andRR-V schemes.

RR-V (BLT-41 ----

$4R R- V I BLT-I)-

T IM E , r n ln

Figure 9. Nonlinear step responses of D-V-1 an d RR-V schemesfor a step change in ZF(LLK) (ZF(LLK) = 0.10).

However, for purposes of illustration, several alternatives

were included throu ghout this example.

Conclusion

A practical design procedure'is presented for deter-

mining the control structure, variable pairing, and tuning

of multiloop SISO controllers in a multivariable process.

Thi s procedure gives a workable, stable, simple control

system with a modest amount of engineering manpower.

It has been successfully tested on a number of multivar-

iable distillation examples.

Nomencla ture

B = bottoms flow rateB = controller transfer function matrixD = distillate flow rateF = feed flow rateF = detuning factor in BLTG = open-loop process transfer function matrixg,. ijt h element of G&L = load transfer function vectorI = identity matrixK,. = controller gain of i th controllerKZNi Ziegler-Nichols gain of ith controllerL = load variableLC= closed-loop log modulusM = vector of manipulated variablesL C , = maximum closed-loop log modulusN = order of system (number of SISO controllers)

R = reflux flow rateRR = reflux ratio = RIDT, analyzer dead timeV = vapor boilup rateW = -1 + det (I + GB)X = vector of controlled variablesXB= bottoms compositionXBN= normalized bottoms composition (mole fraction nC4)XD= distillate compositionXD N= normalized distillate composition (mole fraction iC4)ZF(LLK) = feed composition of lighter-than-light key

Greek Letters

X = eigenvalueu = minimum singular value

a = maximum singular value

rI i= reset time of i th controller, minT Z N ~= Ziegler-Nichols reset time of ith controller, mino = frequency, rad/min

Li tera ture Ci ted

Alatiqi, I. Ph.D. Thesis, Lehigh University, 1985.Chiang, T. P. Ph.D. Thesis, Lehigh University, 1985.Doyle, J. C.; Stein, G. IEEE Trans. 1981, AG2 6, 4 .Downs, J., paper presented at the Lehigh University Distillation Control Short

Course, Bethlehem, PA. May 1984.Grosdidier, P.; Ho k, 8 . R.; Morari, M. Ind. Eng. Chem. Fundam. 1985, 24,

221.Luyben, W. L. "Process Modeling, Simulation, and Control for Chemical

Engineers"; McGraw-HIII: New York. 19 73; p 420.Luyben, W. L. Ind . Eng. Chem . Process Des. Dev. 1986, 25, 326.Morari, M. Chem. Eng. Scl . 1983, 38 , 1881.Niederlinski. A. Automatics 1971, 7, 691.Tyreus, 8. D., paper presented at the Lehigh University Distillation Control

Short Course, Bethlehem, PA, May 1984.Yu, C. C.; Luyben, W . L. Ind. Eng. Chem. Process Des. Dev. 1984, 23 ,

590.

Received for review April 8, 1985Revised manuscript received August 30, 1985

Accepted September 18, 1985

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Design of multiloop siso systems