Implications of Large RGA Elements on Control Performance

8/4/2019 Implications of Large RGA Elements on Control Performance

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I n d . E n g . C h e m . R e s . 1 9 8 7 , 2 6 , 2323-2330 2323Greek Symbolsa = dimensionless parameter = u i h C / r i A A PT = osmotic pressure of solute in the solution of concentrationy = dimensionless osmotic pressure = bCX,,,/APA = fractional solvent recovery = 1- V0 = dimensionless solute permeability = ( D h / K 6 ) / ( A A F / C )7 = dimensionless mass-transfer coefficient = k C / A h P6 = effective membrane thickness of the hollow fiber, cm

C,, atm

Subs c r ip t sf = high-pressure-side feed a t r = rij = 1= feed stream, 2 = high-pressure side of the hollow fiber,0 = values for high rejection membranes, i.e., X 2 3= 01, 2 = conditions at r = ri and at r = rb , respectively3 = permeate stream

Supe r s c r ip t s- = average value over the moduleLiterature CitedHandbook of Mathematical Functions; Abramowitz, H., Stegun, I.A., Eds.; Applied Mathematical Series 55 , Bureau of Standards:Washington,D.C., 968; p 238.Gupta, S. K. Ind. Eng. Chem. Process Des. Dev. 1985, 24 , 1240.Gupta, S. K. Ind. Eng. Chem . Process Des. Dev. 1986,25(4), 1022.Ohya, H.; Nakajima, H.; Takagi,K.; Kagawa,S.; egishi, Y. Desa-Sirkar, K. K.; Rao, G . H. In d. Eng. Ch em. Process Des. Deu. 1982,Soltanieh, M.; Gill W. N. Chem. Eng. Commun. 1982, 18, 311.Sourirajan,S. Reverse Osmosis; Logos: London, 1970.

lination 1977, 21, 257.21, 517.

Received f or review October 21, 1986Accepted June 9, 1987

Implications of Large RGA Elements on ControlPerformance

Sigurd Skogestadt and Manfred Morar i*Chemical Engineering, 206-41, California Institute of Technology, Pasadena, California 91125Large elements in th e RG A imply a plant which is fundamentally difficult to control. (1)The plantis very sensitive to uncorrelated uncertainty in the transfer matrix elements. (2) The closed-loopsystem with an inverse-based controller is very sensitive t o diagonal input uncertainty. With adiagonal controller, t he system is not sensitive to diagonal input uncertainty, bu t the controller doesnot correct for t he strong directionality of the plant and may therefore give poor performance evenwithout uncertainty.

1. IntroductionEach element in the Relative Gain Array (RGA) is de-fined as the open-loop gain divided by the gain betweenthe same two variables when all other loops are underperfect control (Bristol, 1966)($Yi/auj)uk+j( ~ 3 y ~ / d u ~ ) y ~ , ~

gain all other loops opengain all other loops closed (1)= -The elements X i , form the RGA A. Definition 1 s com-pactly written in terms of the transfer matrix G ( s )= (gi,)= ( ( d y i / du j ) u kJnd its inverse G-(s )= {&) = {(dui/dyj)yk)as

A ( G ) = {Xi,)= {g ig i j ]= G ( s ) X G - ( s ) ~ (2a)where X denotes element-by-element multiplication. For2 x 2 plants,

Although definition 1 is limited to steady state (s = 0),formulas 2 may be used to compute the RGA as a functionof frequency (s = jw).Since ita introduction more than 20 years ago (Bristol,1966), he RGA has found widespread use in industry. Inhis original paper, Bristol makes a number of interstingclaims about the RGA, but few of these were actuallyproved. Therefore, for a number of years the RGA re-mained an empirical tool with little rigorous theoreticalPresent address: Chemical Engineering, Norwegian Instituteof Technology (NTH),N-7034 Trondheim, Norway.

0888-5885/87/ 2626-2323$01.50/0

basis. The RGA was originally defined as a measure ofinteractions (eq 1)when using single-loop controllers ona multivariable plant (decentralized control), and Gros-didier e t al. (1985) have proved a number of results whichdemonstrate the usefulness of the RGA in this respect. Inparticular, pairings corresponding to negative RGA ele-menta should be avoided whenever possible. Shinskey(1967, 1984) uses the RGA extensively as a tool for se-lecting control confi ia ti on s for distillation columns. Alsohe interprets the RGA primarily as an interaction measure.However,if this were the case, then an RGA analysis wouldbe of no interest if multivariable rather than decentralizedcontrollers were chosen. Experience indicates that this isnot the case, and that the RGA is a measure of achievablecontrol quality in a much wider sense than just as a toolfor choosing pairings for decentralized control. In fact,Bristol indicates in his original paper th at large RGA el-ements imply a plant which is fundamentally difficult tocontrol. This claim has also been made more recently(McAvoy, 1983; Grosdidier et al., 1985). The followingidentity, which is easily derived from the results ofGrosdidier e t al. (1985), gives a good intuitive feeling forwhy large RGA elements may cause problems:

d g i j (3)./-..-A, . -i gii t i g i jThis identity shows that t he elements i j i f the inverse(G- l )are extremely sensitive to small changes in th e ele-menta g i j of G if the RGA elements are large. This seemsto indicate tha t plants with large RGA elements are verysensitive to modeling errors, and this is indeed true as weshow in this paper.More specifically, he objective of this paper is to answerthe following two questions. (A ) Is a plant with large

0 1987 American Chemical Society


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2324 Ind. Eng. Chem. Res., Vol. 26, No . 11, 1987elements in the RGA always difficult to control? (B) Isa plant with small elements in the RGA always easy tocontrol (in the absence of other limitations on controlperformance, such as constraints and RHP zeros)? We w i l llook at the questions in the context of model uncertainty.The results are illustrated with a simplified distillationcolumn as an example. All results below involving theRGA, singular values, and condition numbers apply to anyfrequency (s = jw) unless otherwise stated.2. Relat ionships between the RGA and theCondit ion Num berLet G(s) denote the linear transfer function model ofthe plant. The plant outputs y (s)are then related to theplant inputs by y ( s )= G(s)u (s). Let the magnitude ofthe vectors y and u be defined in terms of the 2-norm,which is obtained by taking the root mean square of thecomponents at any frequency (s = jw)

IlYjl* = llullz = (cui2)12A multivariable plant has the property that the magnitudeof the output vector (y) epends on the direction of theinput vector ( u ) . We define the maximum gain of theplant at any frequency as

llG4lzU # O llull2 U # O l l ~ l l 2max-ax gain: a,,,(G) = max-l Y l l 2That is, a,,(G) is obtained by choosing the direction ofu such that Ilyllz/llu is maximized. SimilarlyIlGullzu+o llullzin gain: ami,(G) = min-

a,(G) and umh(G)are the maximum and minimum sin-gular values of G (Klema and Laub, 1980). The conditionnumber of the plant is the ratio between the maximum andminimum gain at any frequencyr(G) = am,(G) /amin(G)

We say that a plant has a strong directionality and isill-conditioned if y(G) is large.Note the property umh(G)= l/u-(G-l) which yieldsr(G) = C T ~ ~ ( G ) U ~ , ~ ( G - ~ ) .ristol (1966) himself pointedout the resemblance between the definition of the con-dition number and expression 2 for the RGA. However,Grosdidieret al. (1985) were the first to establish a rigorousrelationship between the condition number and the RGA,and these results have later been extended by Skogestadand Morari (1987a) and Net t and Manousiouthakis (1987).The usefulness of these results in the control context issomewhat limited, since the results which link the con-dition number and control quality are somewhat weak.These results include (1) the bound on the matrix additiveuncertainty by Grosdidier et al. (1985) (theorem 14 in theirpaper) and the slightly more powerful extension to elementuncertainty by Skogestad and Morari (1987a) (also pres-ented as result 1 in section 3 below) (these results dem-onstrate that ill-conditioned plants are sensitive to additivekinds of uncertainty) and (2) the result presented byMorari and Doyle (1986) (eq 63 in their paper) whichindicates that the control performance of ill-conditionedplants may be very sensitive to input uncertainty.Next we will briefly review some of the relationshipsbetween the RGA and the condition number. First notethat the RGA is scaling independent:

A(G) = A(S1GS2)Here SI nd S 2 are diagonal *scaling matrices with real

positive entries and SlGS2corresponds physically to an-other choice of units for the inputs and outputs of the plantG. On the other hand, the condition number r (G) isscaling dependent. One way of making it scaling inde-pendent is to minimize r(SIGSz)over all possible scalingsr*(G) = min r(S1GSz) (4)SI3SZ

Not surprisingly, the tightest relationships between theRGA and the condition number are in terms of r*(G). Thefollowing inequalities show that plants with large elementsin the RGA are always ill-conditioned (Nett and Ma-nousiouthakis, 1987)r(G) 2 r*(G) 2 IlAllm - l / r*(G) 2 11AIlm - 1 (5)

Here the m-norm is defined as (also see Nomenclaturesection)I I 4 l m = 2 max (IIAIlil, Il4Ii-1 (6)

From (5) we see that large elements in the RGA alwaysimply a large value of -y*(G) and r (G). Since ill-condi-tioned plants as ndicated above are generally believed tocause control problems, (5) gives some justification to theclaim that plants with large elements in the RGA arefundamentally difficult to control. Note that r (G) can besignificantly larger than r*(G),nd the plant may there-fore be ill-conditioned (r(G) large) even if all the elementsin the RGA are small. In particular, 2 X 2 plants with anodd number of negative elements in G always have r*(G)= 1and IlAll, = 2, but r( G) may be arbitrary large. Forexample,

From (5) we know that r*(G) is always large when thereare large elements in the RGA. And, similarly, a largevalue of r*(G) always implies large elements in the RGA.This is seen from the following bound in terms of llAlll(sum of element magnitudes) which applies to 2 X 2 plants(Grosdidier et al., 1985)2 x 2: r*(G) 5 Il4ll (7 )and from the followingconjecture for n X n plants (Sko-gestad and Morari, 1987a; Nett and Manousiouthakis,1987)conjecture: r * W 5 l l A l l l + k ( n ) (8)with k(2) = 0, k ( 3 ) = 1, and k(4) = 2. Note that for 2 X2 plants, the 1-and the m-norm of the RGA are identical2 x 2: l l4 l l = IlAllm = 2llAlli- = 2llAllilCombining ( 5 ) and (7), one shows that llAlll and r*(G) arealways close in magnitude (in particular when they arelarge):

(9)Consequently, for 2 X 2 plants, the difference between llAnland r*(G) is at most equal to l/y*(G) and r*(G)- lAlllas llAlll - . Numerical evidence suggests that this alsoholds for n X n plants.3. T h e RGA and Model UncertaintyIn this section we present two results which directlyrelate the RGA to control stability and performance.Result 1 has been derived previously (Skogestad andMorari, 1987a) and is presented here to more clearly showthe relationship to the RGA. Result 1 (for example, eq 12 )

12 x 2: l l4 l l - r*(c> 5 r*W l l4ll


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Ind. Eng. Chem. Res., Vol. 26, No . 11,1987 2325quently, a large value of Aij means tha t only small relativeerrors on the corresponding plant element gij are tolerated.The main restriction inherent in these results is theassumption of i nde pe nde n t element uncertainty. Con-ditions 12 and 13 and the result of Yu and Luyben (1987)may be very conservative if the element uncertainties arecorrelated. For example, for distillation columns, eventhough llAlll is large and the elements in G may vary widelywith operating conditions ( r may be close to 11, it can beshown that the elements are correlated such that the plantnever becomes singular (Skogestad and Morari, 1986).Therefore, for distillation column control, these results donot explain why plants with large values of the RGA ingeneral are difficult to control.3.2. Uncertainty on Each Manipulated Input. Re-sult 2 below introduces the RGA as a measure of howperformance is affected by uncertainty on each manipu-lated input. This result is of more general interest thanresult 1, because uncertainty on the manipulated inputsis always present: We never know the exact value of theinputs u which are applied to the plant.Let uc idenote the desired value of the ith manipulatedinput as computed by the controller, and let A, representthe relative uncertainty on this input. Then the actualplant input is ui = uci(l+ Ai) or in vector form u = u,(I+ AI) where AI = diag ( A i ] epresents the diagonal inputuncertainty. Alternatively, we may define the perturbedplant as

G, = G( I + AI) AI = diag ( A i ] (14)Any real plant has diagonal input uncertainty, and forprocess control applications, it seems unlikely that oneshould be able to get the uncertainty on each input lowerthan about !+lo%. This is probably the case even whencascaded loops based on flow measurements are used tocorrect for the nonlinear valve characteristics. Note thatthese errors (uncertainty) are no t on the absolute valueof the flows, but rather on their change. Fo r example,assume we want to increase a flow rate from 100 to 110kmol/min. However, due to errors in the flow measure-ment, the actual change is from 100 to 111 kmol/min.Although the absolute measurement error in this case isonly 1% ,he corresponding error in the change is lo%,i.e., IAil = 0.1, for this input. Even though it seems unlikelyin a real plant to reduce the magnitude of IAil below, say,0.05, it is nevertheless clear that the effect of input un-certainty on the closed-loop system can be strongly reducedin some cases by using cascaded loops based on flowmeasurements instead of manipulating the valves directly.In this paper we consider each manipulated input (valve)as the source of the input uncertainty. Since there is noreason to assume tha t these manipulated inputs influenceeach other, this results in a diagonal input uncertaintymatrix, AI . The case of unstruc tured input uncertainty(A I is a full matrix) is often considered in the literature(e.g., Morari and Doyle, 1986). This may be convenientfor mathematical purposes, but for the reasons mentionedabove, this is often not a proper description of the actualuncertainty, and the resulting conditions for robustnessand performance may be unnnecessary conservative. Forexample, Morari and Doyle (1986) show that robust per-formance may be poor for plants with a large value of r(G)when there is unstruc tured input uncertainty. However,below we show that if we consider diagonal input uncer-tainty, then large RGA elements (or equivalently a largevalue of r*(G) (eq 9)), rather than a large value of r(G),indicate control problems. For example, we present anill-conditioned plant (r(G) = 142) with small RGA ele-ments (llA(G)lll= 2) and which therefore is easy to control

shows that plants with large RGA elements will easilybecome singular if small relative errors ( r )on each transfermatrix element (gij)occur. Tight control of a plant whichmay become singular is not possible. The uncertainties(errors) on each element have to be i nde pe nde n t forresult 1 (eq 12 ) to be nonconservative. However, the ele-ment uncertainties are usually correlated, and result 1 snot very useful in this case.The most important result in this paper is thereforeresult 2, which directly shows tha t inverse-based controllershould never be used for plants with large RGA elementsbecause of the presence of input uncertainty. This resultis subsequently used to argue tha t plants with large RGAelements are fundamentally difficult to control.3.1. Inde pende nt Relative Element Uncertainty.Thi s result introduces llA(G)lll as a sensitivity measurewith respect to independent uncertainty on the plant el-ements.Result 1 (2 X 2) (Skogestad and Morari, 1987a).Assum e each t rans fer m atr ix e lement has a re la t ive un-certainty of mag nitude r; tha t is, th e actual (perturbed)p l an t i s

T h e unc e r t a in t i e s on e ac h e l e m e n t a re as sum e d t o beindependent; tha t is , there is no correlation between t heAijs. T h e n t h e p l a nt G, remains nons ingular a t s tea dys t a t e (w = 0 ) or an y real per turbat ions , -r I ijI ifand on l y if1r C -r*(G)

which is sat isf ied if

Comment. Condition 1 2 also holds for complex per-turbations lAijl I and w > 0 (Skogestad and Morari,1987a). Condition 11 does no t hold in these cases.Condition 11is necessary and sufficient. Condition 1 2is only sufficient, but it is also tight because of the closerelationship between r*( G) and llA1ll shown in (9). Con-ditions similar to (12) are derived for n X n p l a n t s usingconjecture8 above and theorem 6 in Skogestad and Morari(1987a).Conjecture 1 (nX n ). Th e plant remains nonsingularat a ny frequen cy w for complex relat ive errors of mag-n i t ude r (w ) on e ach e l e m e n t i f

r ( w ) < l / ( l lGw)l l l + Nn ) ) (13)wi t h k(2)= 0 (in his case the conjecture is proven t o becorrect), k(3)= I , a nd k(4)= 2.The control implications of conditions 11-13 follow fromthe fact tha t if a plant is singular at a certain frequency( w ) , hen the plant has a zero on the j w axis. The presenceof this RHP zero limits the achievable control quality(Morari, 1983). In particular, i t is impossbile to haveintegral control for a plant which may become singular atsteady state ( w = 0) (Skogestad and Morari, 1987a).Consequently, if there are large elements in the RGA andllAlll is large, we can allow only very small uncertaintiesin the elements without having control problems.A very similar result has recently been published by Yuand Luyben (1987). It applies to n X n plants a t steadystate when only one element varies at the time: Let theij th element be gJl + Aij). Then the plant becomes sin-gular with a relative pertubation Ai j = - l / X i j . Conse-


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2326 Ind. Eng. Chem. Res., Vol. 26 , No. 11, 1987

Figure 1. Classical feedback structure. d represente the effect ofthe disturbance on he output.also in the presence of diagonal input uncertainty (Figure5).The loop transfer matrix, GpC, is closely related toperformance because of the identity (Figure l ) , = (I +GpC)-'d. GpCmay be written in termsof the nominal GCand an "error term", C-'AIC or GArG-l, as

G,C = GC(1 + C-'AIC) (15 4GpC = (I + GArG-')GC (15b)

For SISO plants, a relative input error of magnitude A1on g results in the same relative change in g,c = g c ( 1 +Al), but for multivariable plants the effect of the inputuncertainty on GpCmay be amplified significantyas shownbelow.R e s u l t 2. For 2 X 2 plants the error term C-'AIC in(15a) may be expressed in terms of the RGA o f the con-troller C asab/ac

c12c11 ,) (16)

Fo r n X n plants i t is easily shown that the diagonalelements of the error term C-lAIC ma y be writ ten as astraightforward generalization of the 2 X 2 case

Ali(C)A1 + h ( c ) A 2 xii(C)-(Ai - A,)-All(C)-(Al - A,) A12(C)A1 + b2(C)A2i ::

GAblG-'=( g2 1

C-'AIC =

n;=l

(C-lAIC)ii = C ji(C)Aj (17)Similarly, for 2 X 2p lan ts, the error term GAIG-' in (15b)may be expressed in terms of the RGA of the plant

ad/ae g12 ) (18)AIIAI+ A 1 2 4 -A -(Ai - A dA -(AI - A,) &iAi + &A2

l'g22

"g1,

(Here Xij = Xij(G) denotes the RGA elements of the plan t.)For n X n plan ts, the diagonal elements o f the error termGA1G-l aren

;= l(GAIG-')ii = CXij(G)Aj (19)Comment . Similar results, bu t with, for example, gllreplaced by -gZ and g , replaced by -gll in the off-diagonal

elements in (18), may be derived for the case of outputuncertainty and performance measured at the input of theplant. This case is generally of less interest.The RGA is independent of scaling, but the off-diagonalelements in (16) and (18) will depend on the scaling of theplant outputs. For the correct interpretation of theseelements, the plant outputs should be scaled such that anoutput deviation of magnitude 1 has equal significance forall outputs.Controllers with large RGA elements will lead to largeelements in the matrix C I A & , and plants with large RGAelementswill lead to large elements in the matrix GAIG-'.Equations 15a and 15b seem to imply tha t either of thesecases will lead to large elements in GpC and therefore poor

performance when there is input uncertainty (A I # 0).However, this interpretation is generallynot correct sincethe directionality of GC may be such that the elementsin GpC remain small even though C-lAIC or GAIG-l havelarge elements. This should be clear from the followingtwo extreme cases.(1) Assume the controller has small RGA elements(small elements in h(C)). In this case, the elements in theerror term C-'AIC are similar to AI in magnitude (eq 16and 17). Consequently,GpCis not particularly influencedby input uncertainty, even though the plant itself may bestrongly ill-conditioned with large RGA elements (largeelements in h(G)).(2 ) Assume the plant has small RGA elements (smallelements in h(G)). In this case, the elements in the errorterm GAIG-' are similar to AI in magnitude (eq 18 and 19).Consequently,GpC is not particularly influenced by inputuncertainty, even though the controller itself may havelarge RGA elements. (Comment: From a practical pointof view, one might argue tha t i t is unlikely that anyonewould design a controller with large RGA elements for aplant with small RGA elements.)From (1 ) and (2),we conclude that for a system to besensitive to input uncertainty, both the controller and theplant must have large RGA elements. These results agreewith Doyle's conditions for robust performance (RP) aspresented in a paper by Skogestad and Morari (1986) (eq34 in their paper): R P in the presence of unstructuredinput uncertainty (A , is a full matrix) is automaticallyimplied by nominal performance (NP) and robust stability(RS) provided the condition number of either the con-troller, r ( C ) , or the plant, r ( G ) , is close to 1. Note thatour results (eq 15-19 above) are in terms of diagonal inputuncertainty (A I diagonal) and involve the RGA rather thanthe condition number.3.3. Inverse-Based C ontroller. For "tight" control,it is desirable to use an inverse-based controller, C(s) =G-l(s)K(s), where K(s) is a diagonal matrix. A special caseof such an inverse-based controller is a decoupler. WithC(s) = G-l(s)K(s), we find A(C) = A(G-'K) = h(G-') =hr(G). Thus, if the elements of h(G) are large, so will bethe elements of h (C) , and from the discussion above, weexpect high sensitivity to input uncertainty. We also seedirectly from

G,C = K(s)(I + GA1G-l) = K(s)(I + C-lAIC) (20)that large elements in GA1G-l (o r equivalently large ele-ments in C-'AIC) imply that the loop transfer matrix GpCis very different from the nominal one GC = K(s ) , andpoor response or even instability is expected with AI #0 (in this case GC = K has no "directionality" that maymake GpC remain small).Decouplers have been discussed extensively in thechemical engineering literature, in particular in the contextof distillation columns (e.g., Luyben (1970) and Arkun etal. (1984)). The idea of using a decoupler (D ) is that themultivariable aspects are taken care of by the decouplerand tuning of the control system is reduced to a series ofsingle-loop problems. Let the diagonal matrix K(s) denotethese "single-loop" controllers. The overall controller, C,including the decoupler is

C(s) = DK(s) (21)A steady-state decoupler is obtained with D = G(O)-'. Thesensitivity of decouplers to decoupler errors has beendiscussed in the literature (e.g., Toijala (Waller) andFagervik (1972)), and the observed sensitivity for sucherrors is in fact easily explained from result 1 (eq 12).However, the most important reason for the robustness


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Ind. Eng. Chem. Res., Vol. 26, No. 11,1987 2327C diagonal: r(GC) 1 *(G) (25)follows since a diagonal controller merely corresponds toa scaling of the input to the plant. Applying (5) yieldsC diagonal: r(GC)1 lA(G)ll, - 1 (26)

problems encountered with decouplers is probably notdecoupler errors but rather inp ut uncertainty. Recall from(20) that a n y controller of the form C(s) = G-(s)K(s) issensitive to input uncertainty if the plant has large RGAelements. Decouplers are generally of this form and shouldtherefore not.be used for plants with large RGA elements.Let Gdiagdenote the matrix consisting of the diagonalelements in G. Then for the decouplers most commonlystudied in the literature, we findideal decoupling: D = G-lGdiag ( 2 2 4simplified decoupling: D = G-( (G-)diag)- (22b)In both of these cases, the controller is of the form C(s)= G-K(s) and will lead to serious robustness problems ifthe plant has large RGA elements. On the other hand, ifone-way decoupling is used, then D is triangular andA(C) = A ( D K ) = I . A one-way decoupler is thereforemuch less sensitive to input uncertainty (recall (16) and(17)).Control Implications of (17), (19), a n d (20). (i) Aninverse-based controller (and in particular a decoupler)should never be used for a plant with large elements in theRGA. (ii) One-way decouplers are much less sensitive toinput uncertainty. (iii) Inverse-based controllers may givepoor response even if the elements in the RGA are small.This may happen if g12/g2, r g2,/g11 s large (eq 18 and20). One example is a triangular plant which always hasAl l = 1, but where the response obtained with an in-verse-based controller may display large interactions inthe presence of uncertainty.It should be added that i t is the behavior of G,C aroundcrossover (llG CII 1)which is primary importance for thestability a n 8 performance of the closed-loop system.Therefore, control problems are expected if the RGA haslarge elements in this frequency range.3.4. Diagonal C ontroller. A diagonal controller alwayshas Xll(C) = 1, and the error term in (15a) becomes

C-AIC = AI (23)Therefore, the response is only weakly influenced by thepresence of input uncertainty. However, it may be difficultto achieve a good nominal response when the controlleris restricted to being diagonal (this may be the case evenif All is close to one as for a nearly triangular plant): Thediagonal controller gives limited correction for thedirectionality of the plant and r(GC) may be large. Inthis case, the response depends strongly on thedisturbance direction: Let d represent the effect of thedisturbance on the output. The response is poor for adisturbance ( d )with a large disturbance condition number(Skogestad and Morari, 1987b):

yd(GC) ranges in value between 1 and r(GC). A valueclose to 1 ndicates that the disturbance is in the gooddirection, corresponding to the high loop gain, a,,(GC).A value close to r (GC) indicates that the disturbance isin the bad direction, corresponding to the low loop gain,a,,(GC). d may also represent the effect of a se t -pointchange. If arbitrary set-point changes are allowed, thenthere exists a set-point change y s uch that r,,(GC) =r(GC).Diagonal controllers do not generally correct for thedirectionality of the plant and r(GC) s large wheneverA(G) has large elements (see (26) below). The inequality

and we see tha t a plant with large RGA values always willhave $GC) arge and will yield poor performance (at leastif arbitrary set-point changes are considered).One special case when a diagonal controller may yieldacceptable performance for an ill-conditioned plant (y(G)large) is when the plant is naturally decoupled a t theinput (V= I ) . This plant has all RGA elements less than1as shown below. Write the singular value decomposition(SVD) of G as

For the case V = I (or, more generally, when V has onlyone nonzero element in each row and column, which giveV = I by rearranging the inputs), a diagonal controller canbe found which removes most of the directionality in theplant: Choose C(s) = c ( s ) Z to get GC = c(s)Uwhich hasyd(GC)= 1 or all disturbances. Note, however, that theresponse is not decoupled (unlessU is diagonal). Also notethat r*(G) = y*(UZ) = 1 n this case ( 2 s diagonal andrW) l) , and it follows from (5) that the elements in A(G)are less than 1 in magnitude.3.5. General Controller Structu re. (1)From (20) weconcluded that the system is always sensitive to inputuncertainty if an inverse-based controller is used for aplant with large RGA elements (in this case, A(C) has largeelements). (2) On the other hand, we know from (23) thatthe system is never sensitive to this uncertainty if a di-agonal controller is used. (In this case A(C) = I ) .What can be said in other cases? Is the RGA of thecontroller, A(C), a useful indicator of a systems sensitivity

to input uncertainty? In general, the answer is no (thisis clear from (18) and (19) above). However, for practicalpurposes, where C(s) is designed based on G(s), he answeris yes. The reason is that one property of any well-de-signed multivariable controller is to remove some of thedirectionality in G by making GC more diagonal thanG. This excludes, fo r example, using a controller withlarge RGA elements for a diagonal plant.) Therefore, largeelements in the error term C-AIC will lead to some degreeto large elements in G,C (eq 15a). Consequently, a plotof the magnitude of the elements of A(C) as a function offrequency may be useful for evaluating the systems sen-sitivity to input uncertainty: A controller with small RGAelements at all frequencies is generally insensitive to inputuncertainty. On the other hand, a controller with largeRGA elements is likely to result in a system which issensitive to input uncertainty.

3.6. Finding W orst-case Conditions from the RGA.It is of interest to know th e worst case combination ofAjs (input uncertainty) to use in simulation studies.Consider the error term GAIG-which for an inverse-basedcontroller is directly related to the change in GC (eq 20).If all Ajs have the same magnitude (lA,l < rI), then from(19) the largest possible magnitude (worst case) of anydiagonal element in GAIG- is given by rIIIA(G)llim(maximum row sum). To obtain th is worst case value,the signs of the Ajs should be the same as those in the rowof A(G) with the largest elements.


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2328 Ind. Eng. Chem. Res., Vol. 26 , No . 11, 1987Table 11. S teady-S tate Data for Disti l lation Columnable I . Guidelines for Choice of Best MultivariableController Structu re (Large Implies a Comparison w i thO n e. T v o i c a l h >IO)

laree smallIIA(G)lll large (diagonal) diagonalllA(G)lll small inverse-based (V = I: diagonal) inverse-based

Example. Consider a plant with steady-state gain(diagonal)

matrix 0.1 -2G(0)= ( 2 -:)-0.1 -1

The RGA is-1.89 -0.13 3.02 )A(G(0)) = 3.59 3.02 -5.61( 0.70 -1.89 3.59

Assume that the relative uncertainties AI, A2, and A3 oneach manipulated input have the same magnitude. Thenthe second row of A(G) has the largest row sum (IIA(G)lli,= 12.2), and the worst combination of input uncertaintyfor an inverse-based controller isA1 = A2 = -A3 = A

We find-5.0 7.5 14.3( 3.0 -3.7 -6.2 )A1G-l = -9.1 12.2 21.5 A

Note that in this specific example, we would arrive at thesame worst case diagonal elements by considering row 1or row 3. Therefore, the worst case will always be obtainedwith A, and A2 of the same sign and A3 with a differentsign even if their magnitudes are different. In some caseswe may arrive at a different conclusion by consideringother frequencies. Also note that, unless an inverse-basedcontroller is used, i t is not guaranteed that the worse caseuncertainties are deduced by using this approach.4. Choice of Control ler Structure

An important decision facing the engineer is the choiceof the controller struc ture. Two extremes are consideredhere: diagonal controller and inversebased controller. Thediagonal controller has advantages: it has fewer tuningparameters, is easier to understand and retune, and canbe made failure tolerant more easily. These issues are notconsidered here. We want to decide which of the twochoices above may result in the best multivariable con-troller. On the basis of the discussion above, Table I wasprepared to assist the engineer in making this choice. Thetable should be used only as a rough guideline, since di-agonal input uncertainty is the only source of uncertaintyconsidered.5. Large RGA Elements Are Bad News

Let us now answer the two questions presented in sec-tion 1.(A) Is a plant with large elements in the RGA alwaysdifficult to control? Yes. This follows from results 1and2. However, if the following conditions are satisfied,control may still be acceptable. (1)The transfer matrixelements are correlated, and despite the large values in theRGA, the plant is not likely to become singular. (2) Thereexists a controller with small RGA elements (e.g., a diag-onal controller) which gives an acceptable response for all

Binary Sep aration, Constant Molar Flows, Feed Liquidre1 volatility C = 1.5no. of theoretical trays N = 40feed tray (1= reboiler) NF = 21feed com position Z F = 0.5product compositions y$ = 0.99, x; = 0.01product ratesreflux rate L I F = 2.706D f F = BfF = 0.5Steady-State Gain M atrices

important disturbances. This is the case if all importantdisturbances are in the good direction (i.e.,-yd(G)is smalldespite the fact that -y(G) s large).Note that condition 2 implies tha t the plant is actuallynot ill-conditioned for the expected disturbances. We willgive an example of such a case below (response to y s 2 nFigure 4) .(B) Is a plant with small elements in the RGA alwayseasy to control? No . As seen from (18),an inversebasedcontroller results in serious interactions f there is inputuncertainty and some of the off-diagonal elements in theplant are large. A diagonal controller gives large inter-actions even in the absence of uncertainty, if the plant isnearly triangular. (Consider, for example, the plant

G = (i too)which has A = I ) .Let us also answer the following additional question.(C ) Is a plant with a large condition number alwaysdifficult to control? No. On the basis of the uncertaintydescriptions investigated in this paper, the RGA ratherthan r (G ) gives a measure of the plants sensitivity todiagonal input uncertainty. We will show in an examplebelow that an inverse-based controller gives very goodcontrol for a plant with y (G ) = 71 even in the presenceof uncertainty (Figure 5).6. Examples

The distillation column described in Table I1 is used asan example. The product compositions y D and xB re tobe controlled by manipulating the reflux ( L )and eitherthe boilup (V ) or the distillate flow (D). The column isassumed to have no dynamics. (This is, of course, not true.However, we make the crude assumption that the dynam-ics are given in terms of a single first-order lag, which isexactly cancelled by a zero in the controller.) We stressthat the objective of the examples is to demonstrate theusefulness of the RGA as a tool for screening design al-ternatives and to support the results presented in TableI, rather than to provide a realistic study of distillationcolumn control.We show simulations for two different configurationsof manipulated inputs: LV configuration, y(GLV) 142,Xll(GLV) 35 , llAlll = 142;DV configuration, -y(GDv) 71 ,X,,(GDv) = 0.45, llAlll = 2. We also consider two controllersfor each of these: inverse-based controller (GC 1(0.7/s));diagonal controller.The controllers are given in the figure texts, and theirgains were adjusted to guarantee robust stability for rel-


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Ind. Eng. Chem. Res., Vol. 26,No. 11, 1987 2329A , = ( : :) (; a 2 )

Figure 2. Controllers satisfy the robust stability condition (29).p(CG(1 + CG)-') is shown as a function of frequency for (1 ) in -verse-based controllers for LV an d DV configurations, (2 ) diagonalLV controller, an d (3 ) diagonal DV controller.Table 111. RGA, Condit ion Numbers, and SV D forDist i l la t ion Column

ConfigurationLV DV

RGA; A i1 35.1 0.45condition no., y(G) 141.7 70.8dist. condition no., yI(G)d = F (feed rate) 11.8 4.3I lA l l l 138.3 2

d = zF feed composition) 1.5 1.4d = ( (set point in yD) 110.7 54.9d = c ) ( s e t p oi nt in xB ) 88.5 44.6

S V decomp., G = U Z V HUI:V 0.707 0.708 1.000 -0.001

-0.630 0.777($% 4::;;)-0.777 -0.630):.0139) ( '393

(-0.708 0.707) (0.001 1.000)ative uncertainty on each manipulated input with amagnitude bound

5s + 10.5s + 1WI(S) = 0.2-

This implies an input error of up to 20 % at low frequen-cies, as is used in the simulations. The uncertainty in-creases at high frequency, reaching 100% at about w = 1min-l. This increase at high frequency may take care ofneglected flow dynamics. Robust stability is guaranteedfor this uncertainty if and only if (Skogestad and Morari,1987a)p(CG(1 + CG)-') 5 l/lwII v w (29)

where the structured singular value p is computed withrespect to a diagonal matrix. Condition 29 is satisfied forthe controllers used as shown graphically in Figure 2.For each of these four systems, the responses to twoset-point changes are shown:YSl = (;) Ysz = (::$The set-point change y s lhas a large component in the"bad" direction corresponding to the low plant gain (Yd(G)= 110.7 or the LV configuration and yd(G)= 54.9 for theD V configuration). ys2 as the same direction as a feedflow disturbance and has yd(G) = 11.8 nd 4.3 or the twoconfigurations (Table 111).The responses are shown both for the nominal case (AI= 0) and with 20% relative uncertainty on each manipu-lated input

Figure 3. LV configuration . Closed-loop responses y 1 nd y z orinverse-based controller. C(s) = (0 . 7 / s )G & = (0.7/s)(;::; :E$).Responses are shown for two different set-point changes, ya l nd y s z ,both for the nom inal case with no uncertain ty (left) and with 20 %error on the manipulated inputs AL an d AV (right). Th e simulationsillustrate th at a n inverse-based controller (e.g.,a decoupler) shouldnever be used for plants with large RG A elements because of thesensitivity to input uncertainty.

0.82E IB 20 30

T I M E c a i n )0 !E 20 30T I ME I m i n l ~ ( 6 )142A,l(C) = 35

______._____.6 ____._._____._____.--..

, , , , , , , , , , , , / , , , 1, / , ,

8 . 1 : 8 . 1 :U* r = ( 4)

8. 2

BE IB 20 30 0 I B 20 30

T l n E I m i n ) \ l n E ( n i n )

Figure 4. LV configuration. Closed-loo p responses, y 1 nd y z , ordiagonal controller C(s) = (l/s)(A '?J. The plant has large RGAelements, and a diagonal controller yields responses which arestrongly dependent on the disturbance (or set-point) direction. Theresponses to y e z re acceptable, bu t th e response to t he set-pointchange y s l s extremely sluggish.which give the following error terms (eq 18) or GC whenan inverse-based controller is used:

Conclusion. The simulations illustrate the followingpoints.An inverse-based controller gives poor response whenthe plant has large RGA elements (Al1 is large) and thereis input uncertainty (Figure 3).A diagonal controller cannot correct for the strong di-rectionality of a plant with large RGA elements (recall eq25). This results in responses which are strongly de-pendent on the disturbance (or set-point) direction (Figure4). The response to ysz disturbance in F) which has Yd(G)= 11.8 is acceptable, but the response to the set-pointchange y s l s extremely sluggish. This system may beacceptable, despite the large value of All, provided set-pointchanges are not important.


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2330 Ind. Eng. Chem. Res., Vol. 26, No. 11,1987

Figure 5. DV configuration. Closed-loop response, y an d y 2 fo rinverse-based controller C(s)= (0.7/s)G& = (0.7/s)($:$ 2i3):ninverse-based controller may give very good response for an ill-con-ditioned pla nt, even in the presence of input uncertainty, providedthe RGA elements are small.. --:- /

,., ( q ; ; ,, , , , , , ,a :a .__-____......_.....-a 2

a :0 20 30T!rE : m , .

Figure 6 . DV configuration. Closed-loop responses, y1 nd y 2 , ordiagonal controller C(s)= (-0.2Js)A-1 = ( O . ~ / S ) ( ? ~ ~ ~ A diago-nal controller performs satisfactory ill-conditioned plants w ith V =I (which implies small RGA elements). However, interactions arestill present because U = (2:;: $&) is not diagonal.An inverse-based controller may give very good responsefor an ill-conditioned plant with diagonal input uncer-tainty, provided Al l is small (Figure 5).A diagonal controller may remove most of the direc-tionality in an ill-conditioned plant if V = I. However,interactions are still present because

= -0.78 -0.63.78)is not diagonal (Figure 6) .NomenclatureC ( s )= transfer matrix of controllerG =UZVH,singular value decomposition (Klema and Laub,1980)

G(s) = {ggi j ) ,ransfer matrix of the plantG&) = perturbed plant (with uncertainty)K(s) = diagonal transfer matrix of single-loop controllers

Greek SymbolsAI, = relative element uncertainty, g,,, = g,(l + AcJ),AIl1 < rA, = relative uncertainty on input iAI = diag (A,},matrix of relative input uncertainties, G, = G (IA(M) = M(s) X M- ( s ) ~ ,RGA of the transfer matrix M (XA, A(G) = RGA of plantA(C) = RGA of controllerllAllrl ; max,~ ~ A ~ ~ l mmax,llAll1 = C I JX l , l , 1-norm (sum of element magnitudes)Z(G) = diag (ulpax(G),..,umln(G)I- singular valuesumm(G)= maximum singular valueum,(G)= minimum singular valuer(G) = um=(G)/umIn(G),ondition numberr*(G) = mins ,s r(S ,GS2) ,minimized scaled condition num-ber (S, nd b, are diagonal matrices with real, positiveentries)

+ AI )denotes element by element multiplication)

IX,,I, induced 1-norm (maximum columnIXVI, induced --norm (maximum rowsum )sum)

11~11m= 2 max ( l l 4 l l 1 1 ll4l1-1

Literature CitedArkun, Y.; Manousiouthakis, B.; Palozoglu, A. Znd. Eng. Chem.Bristol, E. H. ZEEE Trans. Autom. Control 1966, AC-11, 133-134.Grosdidier, P.; Morari, M.; Holt, B. R. Znd. Eng. Chem. Fundam.Klema, V. C.; Laub , A. J. ZEEE Trans. Autom. Control 1980, AC-25,Luyben, W. L. AZChE J. 1970, 16 , 198-203.McAvoy, T. J. Interaction Analysis; Instrumen tal Society of Am -erica; Research Triangle Park, NC, 1983.Morari, M. Chem. Eng. Sci. 1983, 38, 1881-1891.Mor ari, M.; Doyle,J. C. A Unifying Framew ork for Control SystemDesign Under Uncertainty and its Implications for Chemical

Process Control, In Chemical Process Control-CPC ZZC Morari,M., McAvoy, T. J., Eds.; CACH E-Elsevier: Amsterdam , 1986.Nett, C. N.; Manousiouthakis, V. ZEEE Trans. Autom. Control 1987,Shinskey, F. G . Process Control Syst ems; McGraw-Hill: New York,1967.Shinskey, F. G. Distillation Control, 2nd ed.; McGraw-Hill: NewYork, 1984.Skogestad, S.; Morari, M. Control of Ill-Conditioned Plants:High-Purity D istillation, AIChE A nnual M eeting, Miami B each,1986; Paper 74a.Skogestad, S.; Morari, M. Design of Resilient Processing Plants:Effect of Model Uncertainty on Dynamic Resilience, Chem. Eng.Sci. 1987a, in press.Skogestad, S.; Morari, M. Effect of Disturbance Directions onClosed-Loop Performance, Znd. Eng. Chem. Res . 1987b, in press.Toijala (Waller), K.; Fagervik, K. A Digital Simulation Study ofTwo-Point Feedback Control of Distillation Columns, KemianTeollisuus 1972, 29, 1-12.Yu, C.-C.; Luyben , W. L. Znd. Eng. Chem. Res. 1987,26, 1043-1045.

Process. Des. Dev. 1984,23, 93-101.1985,24, 221-235.164-176.

AC-32, 405-407.

Received fo r reuiew October 13, 1986Revised manuscript received J u n e 23 , 1987Accepted J u l y 23, 1987

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Implications of Large RGA Elements on Control Performance