Reprin ted fro m I&EC PROCESS DESIGN 8 DEVELOPMENT, 1986, 25 , 498. Copyright @ 1986 by the American Chemical Society and reprinted by permission of the copyright owner. Design of Multiloop SISO 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 pra ctic al design procedure is proposed for determining the struc ture, variable pairin g, and tuning of multiloop SISO controllers in a multivariable-process environment. The selection of controlled variables relies primarily on engineering judgment . Th e choice of manipulated variables is based on Morari's Resilie ncy Index (minimum singular value of the proc ess transfer function matrix). Variable pairing is decided b y first eliminating those pai rings that give negative RGA's or negative Niederli nksi or Morari Indexes of integral controllabil ity. The final variable-pairing select ion is based on the Tyr eus load rejection criter ion. The simple BL T controller t uni ng meth od, recently proposed by Luybe n, is used to give consistent, logical comparisons of alternatives . The method is illustrated by applyi ng it to a multivariable ternary distillation column example . A recent paper (Luyben, 1985) proposed a practical, easy-to -use met hod (B LT tuning ) for tuning SISO (sin- gle-in put-single-output) controllers in a multivariable environment. Th e technique is based on detuni ng al l controllers equally from Ziegl er- Nich ols settin gs until t he desired degree of closed-loop stability of the system is achieved as indicated by a multivariable Nyquist plot. The struc ture o f th e system must be specifie d 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. It should be em~ has izedha t multivariable controllers are not consideredin this pape r. There are no claims th at th e multiloo p SISO structure giv es the best control system. I t is possible th at a multivar iable controller could improve control performance. Multiva riable controllers (such as Intern al Model Control [IMC], Dynamic Matrix Control [DMC], Linear Quadratic Control [LQ], etc .) will be com- pared to multil oop SISO sys tems in a future paper. What is claimed about the proposed procedure is that it will produce a work able, stable , simple SISO system with only a modest amount of engineering effort. Th is con- ventional SISO system can then serve as a realistic benchmark, against which more complex multivariable control ler structur es c an b e compar ed. Thi s procedure has O 1986 American Chemical Societ y
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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
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
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--- - -
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)
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.
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
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).
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