Optimal Operation by Controlling the Gradient to Zero

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1 Optimal Operation by Controlling the Graient to Zero Johannes Jäschke Sigr Skogesta Department of Chemical Engineering, Norwegian University of Science an Technology, NTNU, Tronheim, Norway ( Abstract: From an optimization point of view, the graient is the key variable which gives information abot the optimality of a process. In this paper we present how the graient is relate to the loss from optimality, an show how etermining a goo set of controlle variables can be consiere as weighte approximation of the graient. We show that even if there are setpoint changes for the controlle variables, this can still be consiere as approximating the graient. Keywors: Self-optimizing control, Graient control, Optimizing control, Control strctre selection 1. INTRODUCTION The overall objective of process operation is to minimize thecostj (oreqivalentlytomaximizetheprofitp = J) sbject to given constraints. However, when sing control, the objective is to keep selecte controlle variables c at their optimal setpoints, c(y) = c s. (1) With respect to these two goals, Morari et al. 198] state...or main objective is to translate the economic objectives into process control objectives. In other wors we want to fin a fnction coftheprocessvariables...]whichwhenhel constant leas atomatically to the optimal ajstment of the maniplate variables, an with it, the optimal operating conitions. However, they o not give a systematic metho for fining the controlle variables, nor o they mention that for the nconstraine case, the obvios approach to get consistency between economic an process control objectives is to select the graient as the controlle variable. That is, to select c = J (,), () an keep the setpoint constant at zero, c s =. Here are the nconstraine egrees of freeom, are nmeasre istrbances, an J (,) = J(,)/ is the graient. Irrespective of the istrbance, the optimal vale of J is zero, (Figre 1). This was propose by Halvorsen an Skogesta 1997a], who write that the ieal controlle variable wol be c = c 1 J c, (3) where c an c 1 are constants. The iea has also been propose by Halvorsen an Skogesta 1997b, 1999], Bonvin et al. 1], Cao 3, 5], Srinivasan et al. 8], an intitively it seems to be an excellent iea. The elements Cost J(,) J(, 1 ) J(, ) J ( 1 ) = J ( 1 ) J ( ) J ( ) = opt ( 1 ) opt ( ) J J = Inpt Fig. 1. Cost an graients for ifferent istrbances Cost J(,) J J < Fig.. Cost an graient vales J = opt J > Inpt of the graient change sign when moving from one sie of the optimm to another sie (Figre ), ths, it is well site for feeback control. However, in practice, we rarely have a measrement of the graient an it is often not clearly efine what it means to control the graient to zero. The graient is a vector, an in many practical cases is not possible to control all elements exactly to zero. What shol we o in these cases? A first attempt to answer this qestion will be to fin a control strctre, which minimizes the norm of the graient. This is a goo start, however, it is important to keep in min that or ltimate goal is to minimize the cost J, so this original criterion has to be applie to evalate the possible control strctres. The starting point is to write the controlle variables as a fnction of measrements y, c = Hy, (4) Copyright by the International Feeration of Atomatic Control (IFAC) 673

2 which is controlle tozero,c = Hy =. Sincethe graient is optimally at zero, we can consier c = Hy as an approximation of the graient. If the approximation is exact, Hy = J then we will have optimal operation whenever c =, provie convexity. If it is not possible to control the graient (becase of e.g. nmeasre istrbances, noise an missing measrements), there will be some loss associate to the chosen control strctre. To evalate the performance of the chosen control policy, we se the original cost fnction an efine the loss from optimality: L = J(,) J( opt,) (5) Consiering the problem of selecting the best control strctres, there are two important qestions, which we wol like to aress in this paper: Q1. Does a H which minimizes J (Hy = ) also minimize L(Hy = )? Q. If not, is the ifference significant? Intermsof Q1.weshowinTheorem1thatminimizingthe norm of the graient is not qite the same as minimizing the loss L. In terms of Q. we show that it is important in the case, when we have strctral constraints on H. That is, we have control strctres involving ifferent measrements, Another contribtion of this paper is an extremely simple erivationofthenllspacemethoalstaanskogesta, 7]. Frthermore, we show we show how setpoint changes of the controlle variables can be seen in the context of minimizing the loss or approximating the graient. This paper is strctre sch that the next section presents or main reslt, a erivation of the expression for the economic loss base on the graient. In Section 3 we escribe how this interpretation is connecte to existing methos, an its importance. Section 4 iscsses how the case of varying setpoints can be treate in this framework. After presenting a istillation case sty in Section 5, we close the paper with a iscssion an conclsions.. DERIVATION OF THE LOSS EXPRESSION USING THE GRADIENT.1 Preliminaries Consier the feeback system in Figre 3, where the variables c an c s enote the vector vale controlle variable an its setpoint, an where the variables n y,n c enote the noise an the steay state control error, respectively. The noisy measrements are y m = y n y, an we assme that the controllers have integral action so that there is no steay state error, n c = ; then at steay state c s = cn, where n = Hn y. After all active constraints are satisfie (controlle), the remaining nconstraine problem can be formlate as minj(,) (6) which in the neighborhoo of the optimal point can be approximate by a qaratic problem 1 min T T ] J J J J ] ] (7) c s Controller Plant cn n c = y Combination c = H(y m) y m Fig. 3. Control strctre (with integral action, c s = cn at steay state, n c = ) where we assme that J >. For small eviations aron the nominal optimm, the plant can be escribe by the linear moel ] y = G y G y = G y, (8) where G y an G y are the steay state gain matrices from an to the otpts y. Or goal is to fin controlle variables of the form c = Hy m, (9) which, when controlle to zero, yiel optimal or near optimal operation.. Approximating the graient The first orer accrate expression for graient of (6) is J (,) = J J ] = J ] J () Assming that (7) matches the real plant, the necessary conition for optimality is J (,) = J J ] =. (11) ] As mentione above, the ieal controlle variable is the graient, c = J (,). When it is known exactly, sing it as a controlle variable is the best choice an works fine. In practice, however, the graient has to be somehow estimate sing measrement information. Then the controlle variable becomes c = J ˆ. (1) Obtaining the graient estimate Ĵ can be one in several ways, sch as e.g. black box moelling or estimating the graient sing statistical methos. In the case of zeromean noise, the effects may cancel ot, bt if there is a constant non-zero offset, the noise can eteriorate performance severely, ths we have to incle the noise in the analysis, too. A first approach wol be to fin a controlle variable c = Hy m which minimizes the worst case graient norm, e.g. to select H as ( min n y ). (13) H = arg max J Hy m H In the non-ieal case, when Hy m J, controlling Hy m to zero will reslt in a graient which has nonzero elements, an therefore has nonzero norm, J (Hy m = ). (14) The norm of the graient may seem a goo criterion to evalate sboptimality, however it oes not trly reflect 674

3 J J() J( 1 ) J( opt) J ( 1 ) L( 1 ) = J J opt = 1 J( 1)J 1 J ( 1 ) J ( opt) constraints will generally be more important than tight control of the nconstraine variable c. 3. MINIMIZING THE GRADIENT VS. MINIMIZING THE LOSS Theorem 1 shows that a controlle variable which minimizes J,oesnotnecessarilyminimizethelossL.One case, where J 1/ has no effect is, when it is orthogonal, J 1/ = J 1/ T, or scalar. In the next sections, we examine in which frther cases an H which minimizes J is the same that minimizes the loss L. opt 1 Fig. 4. Loss L impose by non-optimal operation the performance in terms of the original cost fnction. To qantify the sboptimality, we consier the loss L, which is efine as the ifference between the actal cost an the optimal cost for a given istrbance, L = J(,) J( opt,). (15) Note that we are consiering the loss with respect to the trly optimal instea of the cost. The loss has the properties of a weighte norm. Theorem 1. The local economic loss can be expresse to first orer in terms of the crrent graient vale as L = 1 J (16) Proof From Halvorsen et al. 3] it is known that the loss can be written as L = 1 ( opt ) T J ( opt ). (17) Solving J = (11) for opt = J 1 J an inserting into (17) yiels (note that J is symmetric): L = 1 ( J 1 J ) T J ( J 1 J ) = 1 (T T J T J T )J ( J 1 J ) (18) = 1 (T J T T J T )J 1 (J J ) = 1 JT J 1 J = 1 J. At the optimm, opt, the graient J =, an the loss L =. Aron the optimm J, the loss L is eqal to the norm the weighte graient, where the weight factor is J 1/, Figre 4, Remark 1. (Effect of constraints). The above analysis is locally vali for a system where all active constraints are known an have been satisfie, g(,) =. If an active constraint is not satisfie exactly, g(,) = ǫ, then the effect on the objective fnction will be given by the corresponinglagrangianmltipliernocealanwright, 6]: λ = J/ ǫ (19) A pertrbation of the constraints ǫ has therefore a first orer effect on the cost fnction, while from (16), a small change in J has a secon orer effect on the cost. From an economic point of view, tight control of the active 3.1 Enogh measrements, no noise, fll H: same H If it is possible to have zero loss (no noise an sfficient measrements), optimal operation correspons to J =. Then, J 1/ has no effect. Assme that y contains all available information, then we reqire that J = Hy. () Theorem. (Nll space metho, no noise). Givenalinear moel as in (8), with a sfficient nmber of inepenent measrements (n y n n ) an no noise (n y = ), selecting H = J J ] Gy ] 1 (1) an controlling Hy = gives zero loss from optimal operation. Here, Gy is the gain matrix of any sbset of n n measrements. Proof The graient from () is: J = J J ] ] () We want to eliminate the variables, ] T sing the available measrements, ] y = G y. (3) Solving for T, T ] T, 1y, = Gy] (4) ] an inserting into () gives: J = J J ] Gy ] 1 y = Hy Controlling c = Hy = reslts in zero loss. (5) This is a new erivation of the nll space metho reporte in Alsta an Skogesta, 7]. It shows that the optimal controlle variable fon by selfoptimizing control is ientical to the graient, c = J = Hy. (6) 3. Enogh measrements, noise, fll H: same H The case of fining a controlle variable combination, which minimimizes the loss in presence of sfficient (noisy) measrements an a fll H matrix is aresse in the exact local metho Alsta et al. 9]. First, we scale the istrbances an the noise, sch that = W an 675

4 n y = W n yn y ] where T n yt 1 an W an W n y are iagonal scaling matrices of appropriate sizes. Then we: (1) Express L as a fnction of H, an n y (assming c = H(yn y ) = ) () Then fin an expression for the worst-case loss L(H) (worst-case w.r.t. an n y ); which is the maximm singlar vale σ(m). Here an where M = J 1/ (HG y ) 1 HY (7) Y = FW W n y], (8) F = yopt (9) is the optimal measrement sensitivity matrix, see Halvorsen et al. 3]. (Kariwala et al. 8] have shown that the average loss is given by M F, where F enotes the Frobenis norm) (3) Fin a convex problem formlation for fining H (see Alsta et al. 9]). The convex problem for fining an H which minimizes the average an worst case loss for a given set of istrbances is Alsta et al., 9]: min HFW W n y] H F (3) sbject to HG y = Q, where Q is any nonsinglar n n matrix, an F as efine in (9). Here, too, J is not neee for fining the best measrement combination. However, if we want to know the actal worst case loss, we nee J in (7). In the case of no strctral constraints on H, it is fon that J is not neee for fining the minimm. That is, a controlle variable which minimizes J minimizes also the loss L = 1 J 3.3 Strctral constraints on H: not the same H Intheabovecases,weseallmeasrementsytogenerate the controlle variables as linear combinations of all measrements. In practice however, there are often strctral constraintsonthecontrollevariables.examplesforstrctral constraints incle controlling single measrements, or sing only two measrements from the rectifier section an two measrements from the stripping section of a istillation colmn. When we have to ecie between two or more controlle strctres, the norm of the graient (if it is nonzero) oes no longer give accrate information abot what controlle variable is best. To be able to make a goo ecision in these cases, we nee to consier norm of the weighte graient L = 1 J. Consier a process with ] 44 J = an J = ]. (31) an assme that we have the choice between the two controlle variables ] ] ] ] c 1 = an c.78 =..186 (3) Setpoint Calclation c s Controller n c cn Prices p Combination y n y Plant Fig. 5. Feeback control strctre with setpoint calclation For a istrbance = 1, the corresponing graients are J (c 1 = ) = 1 1] T an J (c = ) = 1 1] T. (33) The norm of the graient is in both cases J (c 1 = ) = J (c = ) =. This inicates that the two controlle variables give eqivalent performance. However, if we consier the loss impose by the two ifferent control strctres, we have that L(c 1 ) = 1 J 1/ J (c 1 ) =.5 (34) an L(c ) = 1 J 1/ J (c ) = (35) When we want to compare sets of controlle variables with each other, we nee to examine the loss, as the graient oes not give sfficient information. When searching for the best linear combination for a given set of measrements, it is sfficient to consier the graient. 4. VARYING SETPOINTS FOR THE CONTROLLED VARIABLES Many processes are operate sch that the setpoints of the controlle variables are change, when for example proct prices p an specifications change, Figre 5. To hanle this in the framework above, we consier the reason for the setpoint change as a measre istrbance. The relationship between the measrements y an the controlle variables is c = Hy, (36) an the relationship between the setpoint change an the prices p is c set = H set p. (37) We efine Ĵ = c c set y Ĵ = Hy H set p = H H set ] p] (38) = H ag y ag The gain matrices are agmente accoring to ] G y G y ] n ag = y n, G y np n, ag, = G y np np, (39) np n I np np an the scaling matrices accoring to ] W W,ag = n n p, (4) np n W pnp n p ] Wn W n,ag = ny np. (41) np ny W npnp n p Here, W p an W np are iagonal matrices with the expecte price variations an ncertainties, respectively. 676

5 Stage 4 F Stage V LC L Stage N F =1 LC B x B D x D Fee istrbance 5 F time min] Fee composition istrbance 5 z time min] Fee liqi fraction 5 q time min] Price change 5 q time min] Fig. 6. Distillation colmn If the prices are known exactly, W np,ag =. The sensitivity matrix F ag may be fon by reoptimization or by evalating F ag = G y ag, Gy agj 1 J,ag, Alsta an Skogesta, 7]. After the problem has been formlate, the optimal H ag, which minimizes the loss is fon by solving min H ag F ag W,ag W n,ag ] H F ag (4) sbject to H ag G y = Q De-partitioning H ag = H H set ], the controlle variables an the setpoint pates are c = Hy an c set = H set p. 5. DISTILLATION CASE STUDY 5.1 Problem escription an setp A binary istillation colmn is se to emonstrate the reslts. The colmn moel is taken from Skogesta 1997]. It is controlle in the LV configration an has 41 stages, Figre 6. We assme that the temperatres on stage 9, 16, 4, an 33 are measre an that they can be se for control,i.e.y = T 9, T 16, T 4, T 33 ] T.Thetemperatresare calclate as alinear fnction of the liqi composition for the respective stages i, T i = (1 x i ) (43) Thiscorresponstoapreproctboilingpointifference of C.Inorertobeabletosellthetopproct,aprity of 99% is reqire for the istillate D. This is consiere an active constraint, an is controlle to its setpoint sing the liqi reflx L. The remaining egree of freeom (the boilp V) can be se to maximize the profit which is the same as minimizing the ifference between the costs for the fee an evaporation, an the profit from selling the prifie procts: J = (p D D p B B p V V p F F) (44) Assming the price for the fee is eqal to the price of the bottom proct, p F = p B, an introcing the overall mass balance, the cost fnction can be simplifie to ( ) pf p D J = p V D V = p V (p D V). (45) p V The only parameter which affects the minimm is the relative price ifference of the fee an the istillate, p = (p F p D )/p V, We assme p = 64 crrency nits. As istrbances, we consier the flow rate F, composition z an liqi fraction q of the fee. These istrbances are Fig. 7. Distrbance trajectories etectable throgh the measrement moel in eviation variables: y = G y G y (46) In aition, we assme that the proct prices change an are known. We se the prices to pate the setpoint of c The self-optimizing controlle variable is selecte as a linear combination of the for tray measrements. The agmente gain matrices an the agmente optimal sensitivity matrix are G y ag = 1.36 Gy,ag = (47) F ag = (48) The weighting matrix W n,ag is chosen sch that all temperatre measrements have an ncertainty of.5 C, an the price ncertainty is zero. The expecte variation in the istrbances is captre in W,ag = iag(.1,.1,.1, 6.4]), which correspons to % variation in every istrbance variable. The corresponing secon erivatives are J = 4.85 (49) J = 15.64, 3.68,.87,.]. This gives a controlle variable combination c = Hy with H =.3,.69,.8,.4] (5) an the setpoint is pate sing c set = H set p with H set =.71. (51) The first orer loss from optimality estimate is calclate accoring to L = J or alternatively accoring to Halvorsen et al. 3] as J 1/ (H ag G y ag) 1 H ag F ag W,ag W n,ag ], (5) an eqals crrency nits. 5. Simlations We consier istrbances in the flow rate, F = %, the fee concentration, z = %, the fee liqi fraction, q = %, an the price, p = %. The istrbance scenario is given in Figre 7, an the reslting profit is plotte together with the inpts in Figre

6 Profit, ( J/pV) Inpts L, V Profit time min] L V time min] Fig. 8. Profit an inpts for the istillation colmn Top composition x SOC variable Controlle variable Setpoint time min] Top composition Setpoint time min] Fig. 9. Controlle variables: Self-optimizing controlle variable an top composition In Figre 9, the controlle variables are given together with their setpoints. The self-optimizing controlle variable is nicely controlle back to the setpoint after a istrbance enters the process. As long as the prices are constant, the setpoint is zero. When the price ratio p changes, the setpoint is aapte to the new vale. The top composition is controlle well at its specification, as can be note from the plotting scale. 6. DISCUSSION AND CONCLUSIONS We have given a first orer approximation of the loss as the weighte norm of the graient, an we have shown that if all measrements are se, the weighting is not reqire. However, when selecting between ifferent sets of controlle variables, we nee to consier the weighte graient, becase neglecting the weighting can be seriosly misleaing. The previosly pblishe exact local metho is basically inicating how close the norm of the weighte graient is to zero when a particlar set of controlle variables c = Hy is se. The key points are to weight the graient when approximating it an to incle noise in the analysis. Otherwise we may approximate the graient well while still sffering from nnecessary economic loss. The controlle variables obtaine by this metho have a robstness against measrement noise. However, the nerlying linear moel an the cost fnction parameters are assme to be locally exact, that is all the ncertainty is assme to be taken care of in the measrement noise an istrbances. Or analysis is base on the assmption that the active constraints o not to change. If the active constraints change, it is necessary to ajst the control strctre to satisfy the new active set. However, if there are nconstraineegreesoffreeominthenewactiveset,theabove analysis can be reapplie. The secon part of this paper eals with istrbances which o not enter throgh the moel. By consiering them as aitional measrements, this can be formlate in terms of minimizing the weighte graient, an the techniqes from self-optimizing control can be se to pate the setpoints to ensre optimal operation for all consiere process an price istrbances REFERENCES Viar Alsta an Sigr Skogesta. Nll space metho for selecting optimal measrement combinations as controlle variables. Instrial & Engineering Chemistry Research, 46: , 7. Viar Alsta, Sigr Skogesta, an Earo Shigeo Hori. Optimal measrement combinations as controlle variables. Jornal of Process Control, 19(1): , 9. Dominiqe Bonvin, Bala Srinivasan, an Davi Rppen. Dynamic Optimization in the Batch Chemical Instry. in: Chemical Process Control-VI, 1. Yi Cao. Self-optimizing control strctre selection via ifferentiation. In Proceeings of the Eropean Control Conference, pages , 3. Yi Cao. Direct an inirect graient control for static optimisation. International Jornal of Atomation an Compting, :6 66, 5. Ivar J. Halvorsen an Sigr Skogesta. Inirect on-line optimization throgh setpoint control. AIChE 1997 Annal Meeting, Los Angeles; paper 194h, 1997a. Ivar J. Halvorsen an Sigr Skogesta. Optimizing control of petlyk istillation: Unerstaning the steaystate behavior. Compters & Chemical Engineering, 1 (Spplement 1):S49 S54, 1997b. Ivar J. Halvorsen an Sigr Skogesta. Optimal operation of petlyk istillation: steay-state behavior. Jornal of Process Control, 9(5):47 44, Ivar J. Halvorsen, Sigr Skogesta, John C. Mor, an Viar Alsta. Optimal selection of controlle variables. Instrial & Engineering Chemistry Research, 4(14): , 3. Vinay Kariwala, Yi Cao, an S. Janarhanan. Local self-optimizing control with average loss minimization. Instrial & Engineering Chemistry Research, 47: , 8. Manfre Morari, George Stephanopolos, an Yaman Arkn. Sties in the synthesis of control strctres for chemical processes. part i: formlation of the problem. process ecomposition an the classification of the control task. analsysis of the optimizing control strctres. AIChE Jornal, 6(): 3, 198. Jorge Noceal an Stephen Wright. Nmerical Optimization. Springer, 6. Sigr Skogesta. Dynamics an control of istillation colmns: A ttorial introction. Chemical Engineering Research an Design, 75(6):539 56, Bala Srinivasan, Lorenz T. Biegler, an Dominiqe Bonvin. Tracking the necessary conitions of optimality with changing set of active constraints sing a barrierpenalty fnction. Compters & Chemical Engineering, 3:57 79,

G y 0nx (n. G y d = I nu N (5) (6) J uu 2 = and. J ud H1 H

G y 0nx (n. G y d = I nu N (5) (6) J uu 2 = and. J ud H1 H 29 American Control Conference Hatt Regenc Riverfront, St Lois, MO, USA Jne 1-12, 29 WeC112 Conve initialization of the H 2 -optimal static otpt feeback problem Henrik Manm, Sigr Skogesta, an Johannes