Optimal measurement combinations as controlled variables

Transcription

1 Available online at Journal of Process Control 9 (009) Optimal measurement combinations as controlled variables Vidar Alstad, Sigurd Skogestad *, Eduardo S. Hori Department of Chemical Engineering, Norwegian Universit of Science and Technolog, Trondheim, Norwa Received 5 March 007; received in revised form 9 November 007; accepted 3 Januar 008 Abstract This paper deals with the optimal selection of linear measurement combinations as controlled variables, c ¼ H. The objective is to achieve self-optimizing control, which is when fixing the controlled variables c indirectl gives near-optimal stead-state operation with a small loss. The nullspace method of Alstad and Skogestad [V. Alstad, S. Skogestad, Null space method for selecting optimal measurement combinations as controlled variables, Ind. Eng. Chem. Res. 46 (3) (007) ] focuses on minimizing the loss caused b disturbances. We here provide an explicit expression for H for the case where the objective is to minimize the combined loss for disturbances and measurement errors. In addition, we extend the nullspace method to cases with extra measurements b using the extra degrees of freedom to minimize the loss caused b measurement errors. inall, the results are interpreted more generall as deriving linear invariants for quadratic optimization problems. Ó 008 Elsevier Ltd. All rights reserved. Kewords: Control structure selection; Self-optimizing control; Measurement selection; Process control; Quadratic optimization; Plantwide control. Introduction Optimizing control is an old research topic, and one method for ensuring optimal operation in chemical processes is real-time optimization (RTO)[0]. Using RTO, the optimal values (setpoints) for the controlled variables c are recomputed online based on online measurements and a model of the process, see ig.. In most RTO-applications, a stead-state model is used for the reconciliation (parameter/disturbance estimation) and optimization steps [0,9], however dnamic versions of the RTO-framework have also been reported [7]. However, the cost of installing and maintaining RTO sstems can be large. In addition, the sstem can be sensitive to uncertaint. A completel different approach to optimizing control is to focus on selecting the right variables c to control, which is the idea of self-optimizing control [3]. The objective is * Corresponding author. Tel.: ; fax: address: skoge@chemeng.ntnu.no (S. Skogestad). Present address: StatoilHdro, Oil and Energ Research Centre, N Porsgrunn, Norwa. to search for combinations of measurements (), for example, linear combinations c ¼ H, which when controlled, will (indirectl) keep the process close to the optimum operating conditions despite disturbances and measurement errors. The need for a RTO laer to compute new optimal setpoints c s can then be reduced, or in man cases even eliminated. Thus, the implementation is trivial and the maintenance requirements are minimized. The idea in this paper is extend this approach, b providing explicit formulas for the optimal matrix H. The issue of selecting H can also be viewed as a squaring down problem, as illustrated in ig.. The number of output variables that can be independentl controlled (n c ) is equal to the number of independent inputs (n u ), but in most cases the number of available measurements (n )is larger, that is, n > n u. The issue is then to select the nonsquare matrix H such that the map (transfer function) G ¼ HG from u to c is square, see ig.. However, selecting H such that G is square is not the onl issue. More importantl, as mentioned above, control of c should (directl or indirectl) result in acceptable operation of the sstem /$ - see front matter Ó 008 Elsevier Ltd. All rights reserved. doi:0.06/j.jprocont

2 V. Alstad et al. / Journal of Process Control 9 (009) Optimizer (RTO) eedback Controller Measurement combination Process ig.. eedback implementation of optimal operation with separate laers for optimization (RTO) and control. ig.. Combining measurements to get controlled variables c (linear case). To quantif acceptable operation we introduce a scalar cost function J which should be minimized for optimal operation. In this paper, we assume that the (economic) cost mainl depends on the (quasi) stead-state behavior, which is a good assumption for most continuous plants in the process industr. Self-optimizing control [3] can now be defined. It is when a constant setpoint polic (c s constant) ields acceptable loss, L ¼ Jðu; dþ J opt ðdþ, in spite of the presence of uncertaint, which is here assumed to be represented b () external disturbances d and () implementation errors n,c s c, see ig.. The implementation error n has two sources: () the stead-state control error n c and () the measurement error n ; and for linear measurement combinations n ¼ n c þ Hn. In ig., the control error n c is shown as an exogenous signal, although in realit it is determined b the controller. In an case, we assume here that all controllers have integral action, so we can neglect the stead-state control error, i.e. n c ¼ 0. The implementation error n is then given b the measurement error, i.e. n ¼ Hn. Ideas related to self-optimizing control have been presented repeatedl in the process control literature, but the first quantitative treatment was that of Morari et al. []. Skogestad [3] defined the problem more carefull, linked it to previous work, and also was the first to include the implementation error. He mainl considered the case where single measurements are used as controlled variables, that is, H is a selection matrix where each row has a single and the rest 0 s. Halvorsen at al. [3] considered the approximate maximum gain method and also proposed an exact local method for the optimal measurement combination H. The proposed to obtain H numericall b solving min H r½mðhþš, but did not sa anthing about the properties of this optimization problem. Kariwala [8] proposed an iterative method involving singular value and eigenvalue decomposition. Hori et al.[5] considered indirect control, which can be formulated as a subproblem of the extended null space method presented in this paper. Additional related work is presented in [5 7] on measurement-based optimization to enforce the necessar condition of optimalit under uncertaint, with application to batch processes. Bonvin et al. [] extend these ideas and focus on stead-state optimal sstems, where a clear distinction is made between enforcing active constraints and requiring the sensitivit of the objective to be zero. This paper is an extension of the nullspace method of Alstad and Skogestad [], where it was found that, in the absence of implementation errors (i.e., n ¼ 0), it is possible to have zero loss with respect to disturbances, provided the number of (independent) measurements (n ) is at least equal to the number of (independent) inputs (n u ) and disturbances (n d ), i.e., n P n u þ n d.itis then optimal to select H such that H ¼ 0, where ¼ d opt =dd T is the optimal sensitivit with respect to disturbances d []. Note that it is not possible to have zero loss with respect to implementation errors, because each new measurement adds a disturbance through its associated measurement error, n. In this paper, we include the implementation error and provide the following new results: () Optimal H for combined disturbances and implementation errors (Section 3). () Optimal H for disturbances using possible extra measurements to minimize the effect of implementation error (extended null space method, Section 4).

3 40 V. Alstad et al. / Journal of Process Control 9 (009) inall, in the discussion section, the results are interpreted in terms of adding linear constraints to quadratic optimization problems. L ¼ ðu uopt Þ T J ðu u opt Þ¼ zt z where ð4þ. Background The material in this section is based on [3], unless otherwise stated. The most important notation is given in Table. The objective is to achieve optimal stead-state operation, where the degrees of freedom u are selected such that the scalar cost function Jðu; dþ is minimized for an given disturbance d. Parameter variations ma also be included as disturbances. We assume that an optimall active constraints have been implemented, so that u includes onl the remaining unconstrained stead-state degrees of freedom. The reduced space optimization problem then becomes min Jðu; dþ ðþ u The objective of this work is to find a set of controlled variables c, or more specificall an optimal measurement combination c ¼ H, such that a constant setpoint polic (where u is adjusted to keep c constant; see ig. ) ields optimal operation (), at least locall. With a given d, solving Eq. () for u gives J opt ðdþ, u opt ðdþ and opt ðdþ. In practice, it is not possible to have u ¼ u opt ðdþ, for example, because of implementations errors and changing disturbances. The resulting loss (L) is defined as the difference between the cost J, when using a non-optimal input u, and J opt ðdþ [4]: L ¼ Jðu; dþ J opt ðdþ The local second-order accurate Talor series expansion of the cost function around the nominal point (u ; d ) can be written as Jðu; dþ ¼Jðu ; d Þþ½J u J d Š T Du þ Du T J J ud Du ð3þ J T ud J dd where Du ¼ðu u Þ and ¼ðd d Þ. or a given disturbance ( ¼ 0), the second-order accurate expansion of the loss function around the optimum (J u ¼ 0) then becomes Table Notation u Vector of n u unconstrained inputs (degrees of freedom) d Vector of n d disturbances Vector of n selected measurements used in forming c c Vector of selected controlled variables (to be identified) with dimension n c ¼ n u n Measurement error associated with ðþ z,j = ðu uopt Þ In this paper, we consider a constant setpoint polic where the controlled variables are linear combinations of the measurements Dc ¼ HD We assume that n c ¼ n u, that is, the number of (independent) controlled variables c is equal to the number of (independent) stead-state degrees of freedom ( inputs ) u. The constant setpoint polic implies that u is adjusted to give c s ¼ c þ n where n is the implementation error for c (see ig. ). As mentioned in Section, we assume that the implementation error is caused be the measurement error, i.e. n ¼ Hn. We now want to express the loss variables z in terms of d and n when we use a constant setpoint polic, but first some additional notation is needed. The linearized (local) model in terms of deviation variables is written as D ¼ G Du þ G d ¼ G e Du ð7þ Dc ¼ GDu þ G d ð8þ where ð5þ ð6þ eg ¼½G G d Š ð9þ is the augmented plant. rom Eqs. (6) (8) we get G ¼ HG and G d ¼ HG d ð0þ The magnitudes of the disturbances d and measurement errors n are quantified b the diagonal scaling matrices W d and W n, respectivel. More precisel, we write ¼ W d d 0 n ¼ W n n 0 ðþ ðþ where we assume that d 0 and n 0 are an vectors satisfing d 0 6 ð3þ n 0 A justification for using the combined vector -norm in Eq. (3) is given in the discussion section of Halvorsen et al. [3]. The nonlinear functions u opt ðdþ and opt ðdþ are also linearized, and it can be shown that [3] Du opt ¼ J J ud D opt ¼ ðg J J ud G d fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Þ n c Control error associated with c (this paper: n c ¼ 0) n Implementation error associated with c; n ¼ n c þ Hn We use D to denote deviation variables. Often, the D is omitted and we write, for example, c ¼ H. ð4þ ð5þ

4 V. Alstad et al. / Journal of Process Control 9 (009) where we have introduced the optimal sensitivit matrix for the measurements. In terms of the controlled variables c we then have ðu u opt Þ¼G ðc c opt Þ¼G ðdc Dc opt Þ Dc opt ¼ HD opt ¼ H Dc ¼ Dc s n ¼ n ¼ Hn ð6þ ð7þ ð8þ where we in the last equation have assumed a constant setpoint polic (Dc s ¼ 0). Upon introducing the magnitudes of and n from Eqs. () and () we then get for the loss variables z in (5) for the constant setpoint polic: z ¼ M d d 0 þ M n n 0 where ð9þ M d ¼ J = ðhg Þ HW d ð0þ M n ¼ J = ðhg Þ HW n ðþ Introducing M, ½M d M n Š ðþ gives z ¼ M d0, which is the desired expression for the n 0 loss variables. A non-zero value for z gives a loss L ¼ kzk (4), and the worst-case loss for the expected disturbances and noise in (3) is then[3] L wc ¼ max d 0 n 0 6 L ¼ ðr½mšþ ð3þ where the last equalit follows from the definition of the singular value r and the assumption about the normalized disturbances and measurement errors being -norm bounded, see Eq. (3). Thus, to minimize the worst-case loss we need to minimize rðmþ with respect to H. This is the exact local method in Halvorsen et al.[3], and note that we have expressed M d in (0) in terms of the easil available optimal sensitivit matrix. 3. Explicit formula for optimal H for combined disturbances and measurement errors rom (3), the optimal measurement combination is obtained b solving the problem ( exact local method ) H ¼ arg min H rðmþ ð4þ It ma seem that this optimization problem is non-trivial as M depends nonlinearl on H, as shown in (0) (). Halvorsen et al. [3] proposed a numerical solution and Kariwala [8] provides an iterative solution for the optimal H involving the singular value and eigenvalue decompositions. However, (4) is in fact eas to solve, as shown in the following. We start b introducing M n,j = ðhg Þ ¼ J = G ð5þ which ma be viewed as the effect of n on the loss variables z. We then have M ¼ ½M d M n Š ¼ M n HW ½ d W n Š ð6þ Next, we use the fact that the solution of Eq. (4) is not unique, so that if H is an optimal solution, then another optimal solution is H ¼ DH, where D is a non-singular matrix of dimension n u n u. or example, this follows because M d and M n in (0) and () are unaffected b the choice of D. One implication is that G ¼ HG ma be chosen freel (which also is clear from ig. since we ma add an output block after H which allows G to be selected freel). Alternativel, and this is used here, it follows from (5) that M n ma be selected freel. However, the fact that M n ma be selected freel, does not mean that one can, for example, simpl set M n ¼ I in (6) and then minimize rðmþ with M ¼ HW ½ d W n Š. Rather, one needs to minimize rðmþ subject to the constraint M n ¼ I. Introducing e, ½W d W n Š ð7þ the optimization problem (4) can then be stated as H ¼ arg min rðhþ e subject to HG ¼ J = ð8þ H This is fairl eas to solve numericall because of the linearit in H in both the matrix H e and in the equalit constraints. In fact, an explicit a solution ma be found, as shown below. Choice of norm. The optimization problems (4) and (8) involve the singular value (induced -norm) of M, rðmþ, which represents the worst-case effect of combined -norm bounded disturbances and measurement errors on the loss. A closel related problem is to minimize the robenius norm (Euclidean or -norm) of M, kmk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pi;j jm ijj, which represents some average effect of combined disturbances and measurement errors on the loss. Actuall, which norm to use is more a matter of preference or mathematical convenience than of correctness. irst, the difference in minimizing the two norms is generall minor; the main difference is that minimizing rðmþ usuall puts more focus on minimizing the largest elements. Second, as discussed below, it appears that for this particular problem, we have a kind of super-optimalit, where the choice of H that minimizes kmk, also minimizes rðmþ [9]. Scalar case. or the scalar case (c is a scalar and n u ¼ n c ¼ ), M and H T are (column) vectors, rðmþ ¼kMk, and an analtic solution to (8) is easil derived. The optimization problem (8) becomes min k et H T k subject to G T H T ¼ J = H T ð9þ and from standard results for constrained quadratic optimization, the optimal solution is (see proof in Appendix) H T ¼ð e et Þ G ðg T ð e et Þ G Þ J = where it is assumed that e et has full rank. ð30þ

5 4 V. Alstad et al. / Journal of Process Control 9 (009) General case. The explicit expression for H in (3) holds also for the general case, that is, for minimizing kmk for the case when H is a matrix. This can be proved b rewriting the general optimization problem (8) for the matrix case, into a vector problem b stacking the columns of H T into a long vector (see Appendix A.). In addition, Kariwala et al. [9] have shown, as alread mentioned, that the matrix H that minimizes the robenius-norm of M also minimizes the singular value of M [9]. However, the reverse does not necessaril hold, that is, a solution that minimizes rðmþ does not necessaril minimize kmk [9], which is because the solution to the problem of minimizing rðmþ is not unique. Our findings can be summarized in the following Theorem (see Appendix A. for proof). Theorem. or combined disturbances and measurement errors, the optimal measurement combination problem in terms of the robenius-norm, min H kmk with M given b (0) (), can be reformulated as min H khk e subject to HG ¼ J =, where e ¼½W d W n Š. is the optimal measurement sensitivit with respect to disturbances, and W d and W n are diagonal weighting matrices, giving the magnitudes of the disturbances and measurement noise, respectivel. Assuming e et is full rank, we have the following explicit solution for the combination matrix H, H T ¼ð e et Þ G ðg T ð e et Þ G Þ J = ð3þ This solution also minimizes the singular value of M, rðmþ, that is, provides the solution to the exact local method in (4). Note that e e T ¼ ½W d W n Š½W d W n Š T in (3) needs to be full rank. This implies that (3) does not generall appl to the case with no measurement error, W n ¼ 0, but otherwise the expression for H applies generall for an number n of measurements. One special case, when the expression for H in (3) applies also for W ¼ 0, is when n 6 n d, because e et then remains full rank. 4. Extended nullspace method The solution for H in (3) minimizes the loss with respect to combined disturbances and measurements errors. An alternative approach is to first minimize the loss with respect to disturbances, and then, if there are remaining degrees of freedom, minimize the loss with respect to measurement errors. One justification is that disturbances are the reason for introducing optimization and feedback in the first place. Another justification is that it ma be easier later to reduce measurements errors than to reduce disturbances. If we neglect the implementation error (M n ¼ 0), then we see from (0) that M d ¼ 0 (zero loss) is obtained b selecting H such that H ¼ 0 ð3þ This provides an alternative derivation of the nullspace method of []. It is alwas possible to find a non-trivial solution (i.e. H 0) H satisfing H ¼ 0 provided the number of independent measurements (n ) is greater than the number of independent inputs (n u ) and disturbances (n d ), i.e. n P n u þ n d []. One solution is to select H as the nullspace of T []: H ¼ Nð T Þ ð33þ The main disadvantage with the nullspace method is that we have no control of the loss caused b measurement errors as given b the matrix M n ¼ M n HW n. In this section, we stud this in more detail, b deriving an explicit expression for H,see(37) and (4), that allows us to compute the resulting M n,see(4) and (44). The explicit expression for H allows us to extend the nullspace method to cases with extra or too few measurements, i.e., to cases when n n u þ n d. 4.. Explicit expression for H for original null space method rom the expansion of the loss function we have, see eqs. (5) and (4) e J zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ z ¼½J = J = J J Du ud Š ð34þ We assume that H is selected to have zero disturbance loss, which is possible if n P n u þ n d. Then from (9) and (6), z ¼ M n Hn. With the controlled variables c ¼ H fixed at constant setpoints (Dc ¼ Dc s ¼ 0) we then have D ¼ n, and get z ¼ M n Hn ¼ M n HD ¼ M n HG e Du ð35þ where G e ¼½G G d Š is the augmented plant. Comparing Eqs. (34) and (35) ields M n H e G ¼ e J ð36þ where e J is defined in (34). We then have the following explicit expression for H for the case where n ¼ n u þ n d such that e G is invertible H ¼ M n e J½ e G Š ð37þ This explicit expression gives H for a case with zero disturbance sensitivit (M d ¼ 0), and thus gives the same result as (33). Note that M n can be regarded as a free parameter (e.g. we ma set M n ¼ I, see Remark below). 4.. Extended nullspace method The explicit solution for H in (37) forms the basis for extending the nullspace method to cases where we have extra measurements (n > n u þ n d ) or too few measurements (n < n u þ n d ). Assume that we have n u independent unconstrained free variables u, n d disturbances d, n measurements, and we want to obtain n c ¼ n u independent controlled variables c that are linear combinations of the measurements, c ¼ H. rom the results in Section, the loss imposed b a constant

6 V. Alstad et al. / Journal of Process Control 9 (009) setpoint polic is L ¼ zt z where z ¼ M d d 0 þ M n n 0. Define E as the error in satisfing Eq. (36): E ¼ M n H e G e J ð38þ We want to derive a relationship between E and M d. rom (5) and (9) the optimal sensitivit can be written as " # ¼ G e J J ud ð39þ I which combined with (6) gives " # " # M d ¼ M n HG e J J ud W d ¼ðEþ e JÞ J J ud I I Here e J J J ud ¼ 0 which gives " I # M d ¼ E J J ud W d I W d ð40þ Note that the disturbance sensitivit is zero (M d ¼ 0) if and onl if E ¼ 0. qffiffiffiffiffiffiffiffiffiffiffiffiffi Let kek ¼ denote the robenius (Euclidean) P i;j e ij norm of a matrix, and let denote the pseudo-inverse of a matrix. Then we have the following theorem: Theorem (Explicit expression for H in extended nullspace method). Selecting H ¼ M n e JðW n G e Þ W n ð4þ minimizes kek, and in addition minimizes the noise sensitivit km n k among all solutions that minimize kek. Proof. Rewrite the definition (38) for E as E ¼ M n HW n W fflfflfflfflffl{zfflfflfflfflffl} n G e e J ð4þ M n rom the theor of linear algebra [8], the solution for M n that minimizes kek, and in addition minimizes km n k among all solutions that minimize kek, is given b M n ¼ e JðW n G e Þ, which gives (4). To see this, note that minimizing kek is equivalent to finding the leastsquare solution X ¼ BA to XA ¼ B, where X ¼ M n, A ¼ W n G e and B ¼ e J. h Remark. If we have enough measurements (n P n u þ n d ) then the choice for H in Eq. (4) gives E ¼ 0 and thus M d ¼ 0. However, for the case with too few measurements the above choice for H minimizes kek, whereas we reall want to minimize km d k. Nevertheless, since km d k 6 kek k J J ud I W d k, we see that minimizing kek will result in a small value of km d k. Remark. The matrix H is non-unique and the matrix M n in (4) can be viewed as a parameter that can be selected freel. or example, one ma select M n ¼ I, or one ma select M n to get a decoupled response from u to c, i.e. G ¼ HG ¼ I. However, note that M n H, and the measurement noise sensitivit M n ¼ M n HW n, are not affected as M n H is given b (36) and (4). Remark 3. It is appropriate at this point to make a comment about the pseudo-inverse A of a matrix. In general, we can write the least-square solution of XA ¼ B as X ¼ BA where the following are true: () A ¼ðA T AÞ A T is the left inverse for the case when A has full column rank (we have extra measurements). In this case, there are an infinite number of solutions and we seek the solution that minimizes kxk. () A ¼ A T ðaa T Þ is the right inverse for the case when A has row column rank (we have too few measurements). In this case there is no solution and we seek the solution that minimizes the robenius norm of E ¼ B XA (regular least squares). (3) In the general case with extra measurements, but where some are dependent, A has neither full column or row rank, and the singular value decomposition ma be used to compute the pseudo-inverse A Special cases of Theorem We have some important special cases of Theorem : Just-enough measurements (original nullspace method) When n ¼ n u þ n d, the measurements and disturbances are independent, so e G is invertible and (4) becomes H ¼ M n e Jð e G Þ ð43þ as derived earlier in (37). This choice gives M d ¼ 0 (zero disturbance loss) and from (6) the resulting effect of the measurement noise is M n ¼ e J½ e G Š W n ð44þ Note that we in this case have no degrees of freedom left for affecting the matrix M n Extra measurements: select just-enough subset If we have extra measurements (n > n u þ n d ), then one possibilit is to select a just-enough subset (such that we get n ¼ n u þ n d ) before forming c and then obtain H from (43) to achieve zero disturbance loss (M d ¼ 0). The degrees of freedom in selecting the measurement subset can then be used to minimize the loss with respect to the measurement noise, that is, to minimize the norm of M n in (44). The worst-case loss caused b measurement noise is L wc ¼ max L ¼ kn 0 k 6 rðm n Þ ¼ rðe JðG f Þ W n Þ 6 ðrðe JÞrð f G ÞrðW n ÞÞ ð45þ

7 44 V. Alstad et al. / Journal of Process Control 9 (009) The selection of measurements does not affect the matrix e J, since it from (33) depends onl on the Hessian matrices J and J ud. However, the selection of measurements affects the matrix e G. Thus, in order to minimize the effect of the implementation error, we propose the following two rules: () Optimal: In order to minimize the worst-case loss, select measurements such that rðm n Þ¼ rð e J½G f Š W n Þ is minimized. () Sub-optimal: Assume that the measurements have been scaled with respect the measurement error such that W n ¼ I. rom the inequalit in Eq. (45), it then follows that the effect of the measurement error n will be small when rðg e Þ (the minimum singular value of G e ) is large. Thus, it is reasonable to select measurements such that rðg f Þ is maximized. the optimal rule requires evaluation of all possible measurement combination, which ma be impractical. On the other hand, for the sub-optimal selection rule of maximizing rð f G Þ there exists efficient branch and bound algorithms [9]. The sub-optimal rule was used successfull in [] to select measurements from 60 candidates for a Petluk distillation case stud Extra measurements: use all or the case with extra measurements (n > n u þ n d ), we ma alternativel use all the measurements when forming c and obtain H from (4) in Theorem. This gives the solution that minimizes the implementation (measurement error) loss subject to having zero disturbance loss (M d ¼ 0). More precisel, when n > n u þ n d and the measurements and disturbances are independent, the choice for H in (4), where denotes the left inverse, minimizes km n k (robenius norm) among all solutions with M d ¼ 0. Note that we need to include the noise weight before taking the pseudo-inverse in (4) Too few measurements If there are man disturbances, then we ma have too few measurements to get M d ¼ 0. or the case when both the measurements and disturbances are independent, we have too few measurements when n < n u þ n d. In this case, the optimal H given in (4) in Theorem is not affected b the noise weight, and (4) becomes H ¼ M n e Jð e G Þ ð46þ where denotes the right inverse and M n is, as before, free to choose. However, this explicit expression for H minimizes kek, whereas, as noted in Remark, we reall want to minimize km d k. Minimizing km d k is equivalent to solving the following optimization problem H ¼ arg min khw d k subject to HG ¼ J = ð47þ H or this case, with few measurements and no consideration of measurement error, we have not been able to derive an explicit expression for H, similar to (3) in Theorem. However, for practical applications, Eq. (46) is most likel acceptable, at least provided we scale the sstem (i.e. G d ) such that W d ¼ I. There ma also be cases where we do have enough measurements, but we nevertheless want to use too few measurements to simplif implementation. In this case, we have that M n ¼ e Jð e G Þ W n and to minimize rðm n Þ (the effect of measurement error), we ma first select the set of measurements that maximizes rðm n Þ, and then select H according to (46). Also, note that if we have sufficientl few measurements, i.e. n < n d, then (3) applies with W n ¼ 0 (see comment following Theorem ). 5. Example As a simple example, consider a scalar problem with n u ¼ and n d ¼ [3]. The cost function to be minimized is J ¼ðu dþ ð48þ where the nominal disturbance is d ¼ 0. Assume that the following four measurements are available: ¼ 0:ðu dþ; ¼ 0u; 3 ¼ 0u 5d; 4 ¼ u We assume that the sstem is scaled such that jdj 6 and jn i j 6, i.e., W d ¼ ; W n ¼ I ð49þ and we want to find the optimal measurements or combinations to control at constant setpoints. Solution. rom (48), it is clear that J opt ðdþ ¼0 8d and the optimal input is u opt ðdþ ¼d. We find J ¼ and J ud ¼ and G T ¼½0: 0 0 Š and G T d ¼½ 0: 0 5 0Š ð50þ The optimal sensitivit matrix is obtained from (5) or (39). This gives ¼ ½0 0 5 Š T. 5.. Single measurement candidates Let us first consider the use of individual measurements as controlled variables (c ¼ i, i ¼ ; ; 3; 4). The losses L wc ¼ rðmþ are L wc ¼ 00; L wc ¼ :005; L3 wc ¼ 0:6; L4 wc ¼ ð5þ Measurement has D opt ¼ 0, so it happens to have zero disturbance loss (M d ¼ 0). However, this measurement is sensitive to noise (as can be seen from the small gain in G ) and we see that this choice actuall has the largest loss L wc ¼ is the best single measurement candidate. This illustrates the importance of taking into account the implementation error (measurement noise). 5.. Measurement combinations: use two of the four measurements Consider combining two measurements, c ¼ H ¼ h i þ h j. Let us first consider combinations that give

8 V. Alstad et al. / Journal of Process Control 9 (009) zero disturbance loss M d ¼ 0, which is possible since n u þ n d ¼ n ¼. The null space combination (H ¼ðh h Þ) is most easil obtained using (33). or example, for measurements (; 3), ¼ ½0 5 Š T and H ¼½h h Š¼Nð½0 5ŠÞ ¼ ½ 0:45 0:970Š ð5þ The controlled variable is then c ¼ 0:45 þ 0: The same result is obtained from (4). The results with the nullspace method for all six possible combinations are given in Table. The table gives the worst-case loss L wc caused b the measurement error. We have L wc ¼ rðmþ, where, since M d ¼ 0, M ¼ M n ¼ ejð G e Þ W n. To compare, we also show in Table rðg e Þ, which according to the sub-optimal rule for selecting measurements should be maximized in order to minimize the implementation error. We note that for this example that maximizing rðg e Þ gives the same (correct) ranking as minimizing L wc. rom Table, we see that combinations involving measurement are all sensitive to noise. Combination ði; jþ ¼ð; 3Þ is the best, followed b (3, 4), while (, ), (, 4) and (, 3) have the same noise sensitivit when the are combined using the nullspace method. The reason is that Nð T Þ¼½ 0Š, so that onl measurement is used. Combination (; 4) ields infinite noise sensitivit to noise with the nullspace method, since G e is singular. Next, consider the optimal combination of two measurements for disturbances and measurements error ( exact local method ). The results are summarized in Table 3. Again, we find that the best combination is ð; 3Þ with a Table Combinations of two measurements, c ¼ h i þ h j, with zero disturbance loss (M d ¼ 0) and resulting loss L wc caused b measurement error i j H from (4) rðg e Þ h h L wc Table 3 Combinations of two measurements, c ¼ h i þ h j, with minimum loss L wc for combined disturbances and measurement error i j H from (3) h h L wc loss L 3 wc ¼ 0:0406. This gives M d ¼ 0:0635 so, as expected, the disturbance loss is non-zero. or this combination, the result is ver similar to the extended nullspace method which gave L 3 wc ¼ 0:045 and M d ¼ 0. However, for the other 5 two-measurement combinations, the differences are much larger as the use of (3) gives a significantl lower sensitivit to measurements error, see Table 3. However, note that L wc for those five cases is onl slightl better than using a single measurement, see (5) Measurement combinations: use all four measurements Consider again first the case when we want zero disturbance loss (M d ¼ 0), and Eq. (4) in the extended nullspace method gives (after normalizing (scaling) the elements in H to get khk ¼ ): H ¼ ½0:006 0:49 0:9700 0:0 Š ð53þ which gives G ¼ 4:85 and M n ¼ 0:95. The loss contributions from the disturbance and the noise are M d ¼ 0 and M n ¼ ½ 0:0060 0:0705 0:87 0:0035 Š, respectivel. The corresponding loss is L wc ¼ r ½M d M n Š= ¼ 0:0448. To compare, the optimal combination ( exact local method ) with respect to combined disturbances and measurement noise, obtained from (3) is (after normalizing to get khk ¼ ) [3] H opt ¼½0:008 0:37 0:975 0:06 Š ð54þ which gives G ¼ 5:08 and M n ¼ 0:783. The loss contribution from the disturbance and the noise are M d ¼ 0:0606 and M n ¼½ 0:0057 0:0645 0:706 0:003Š, respectivel. The resulting loss is L wc ¼ 0:0405, which is ver similar to the extended null space method, and onl marginall improved compared to using onl two measurements (L 3 wc ¼ 0:0406). The reduction in loss is small compared to using onl two measurements (L 3 wc ¼ 0:0406). In summar, the simple two-step nullspace method, where one first selects a just-enough set of measurements b maximizing rðg e Þ, and then obtains H from the null space method, using either Eq. (43) or (33), works well for the example. 6. Example : control of refrigeration ccle or a more phsicall motivated example, we consider the optimal operation of a CO refrigeration ccle [6], for example, it could be the air condition (AC) unit for a house, see ig. 3. The ccle has one unconstrained degree of freedom (n u ¼ ), which ma be viewed as the high pressure ( ¼ p h ) in the ccle. Ideall, p h should be kept at its optimal value b varing the free input (u); which is the choke valve position. However, simpl keeping a constant setpoint p h;s is far from optimal because of disturbances. Three disturbances (n d ¼ 3) are considered: the outside temperature T H, the inside temperature T C (e.g. because of a setpoint change) and the heat transfer rate UA. As a start, control of single variables is considered, because this is the simplest and is the preferred choice if such a variable

9 46 V. Alstad et al. / Journal of Process Control 9 (009) P h successfull to a distillation case stud where the issue is to select temperature combinations [4]. k 7. Discussion T T H T 7.. Local method P s h,combine PC z Q C V l Q H The above derivations are local, since we assume a linear process and a second-order objective function in the inputs and the disturbances. Thus, the proposed controlled variables are onl globall optimal for the case with a linear model and a quadratic objective. In general, we should alwas, for a final validation, check the losses for the proposed structures using a nonlinear model of the process. T H Q loss T3 T 4 Pl W s can be found. The best single controlled variable (with a large scaled gain and a small loss) was found to be the mass holdup (c ¼ M c ) in the condenser [6], but it is ver difficult to measure in practice. Therefore, a combination of measurements needs to be controlled. Theoreticall, we need to combine at least four measurements (n u þ n d ¼ 4) to get zero loss, independent of the disturbances (i.e., to get M d ¼ 0). However, to simplif the implementation we prefer to use fewer measurements. Two measurements which are eas to measure and have a reasonabl large scaled gain [6], are the high pressure ( ¼ p h ) and the temperature before the choke valve ( ¼ T h ). It is not possible to get M d ¼ 0 with onl two measurement, so instead H ¼½h h Š was obtained numericall b minimizing km d k ; see Eq. (46) (actuall, the matrix H can be obtained explicitl from (3) with W n ¼ 0, since we have n ¼ < n d ¼ 3; see comment following Theorem ). This gives a controlled variable c ¼ h þ h ¼ h p h þ h T h. To get a more phsical variable, we select h ¼, which gives a controlled variable in the units of pressure. We find [6] c ¼ p h;combined ¼ p h þ kðt h 5:5 CÞ where k ¼ h =h ¼ 8:53 bar= C and the (constant) setpoint for c ¼ p h;combined is 97.6 bar, which is the nominall optimal value for the high pressure. Controlling c ma be viewed as controlling the high pressure, but with a temperature-corrected setpoint. A more detailed analsis using a nonlinear model shows that this combination gives ver small losses for all disturbances and measurement errors [6]. Other case studies. The results of this paper, and in particular the use of (3) in Theorem have also been applied T 4 T C TC ig. 3. Proposed control structure for the refrigeration ccle. s T C 7.. Quadratic optimization problem The following reformulations of the results in this paper ma be useful for extending them to other applications and comparing them with other results. irst, we give a reformulation of the original nullspace method in []. Theorem 3 (Linear invariants for quadratic optimization problem). Consider an unconstrained quadratic optimization problem in the variables u (input vector of length n u )andd (disturbance vector of length n d ) J J ud u min Jðu; dþ ¼½u dš ð55þ u d J T ud J dd In addition, there are measurement variables ¼ G uþ G d d. If there exists n P n u þ n d independent measurements (where independent means that the matrix G e ¼½G G d Š has full rank), then the optimal solution to (55) has the propert that there exists n c ¼ n u linear variable combinations (constraints) c ¼ H that are invariant to the disturbances d. Here, H ma be obtained from the nullspace method using (33) (where the optimal sensitivit ma be obtained from (5)) or from the explicit expression (37). Next, the result on the worst-case loss [3] and the explicit expression for the exact local method in Theorem can be reformulated as follows. Theorem 4. (Loss b introducing linear constraint for nois quadratic optimization problem). Consider the unconstrained quadratic optimization problem in Theorem 3, J J ud u min Jðu; dþ ¼½u dš u d J T ud J dd and a set of nois measurements m ¼ þ n, where Y ¼ G u þ G d d. Assume that n c ¼ n u constraints c ¼ H m ¼ c s are added to the problem, which will result in a non-optimal solution with a loss L ¼ Jðu; dþ J opt ðdþ. Consider disturbances d and noise n with magnitudes d ¼ W d d 0 d 0 ; n ¼ W n n 0 ; 6 n 0

10 V. Alstad et al. / Journal of Process Control 9 (009) Then for a given H, the worst-case loss is L wc ¼ rðmþ =, where M is given in (0) (), and the optimal H that minimizes rðmþ is given b (3) in Theorem. This optimal H also minimizes kmk. Note from Theorem 3 that if there are a sufficient number (n P n u þ n d ) of noise-free measurements, then we can obtain zero loss in Theorem Relationship to indirect control Indirect control is when we want to find a set of controlled variables c ¼ H such that the primar variables are indirectl kept at constant setpoints. The case of indirect control is discussed in more detail b Hori et al. [5] and their results are a special case of the results for the nullspace method presented in this paper if we select J ¼ k s k ¼ ½ s ŠT ½ s Š To show this, write D ¼ G Du þ G d ¼ G f Du ð56þ ð57þ and assume n ¼ n u,sog is a square matrix. We find that J ¼ G T G J ud ¼ G T G d ð58þ ð59þ Consider the case with n ¼ n u þ n d, where we can achieve perfect indirect control with respect to disturbances. Substituting (58) and (59) into the explicit expression (37) for the nullspace method gives H ¼ P c;0 e G ð e G Þ ð60þ where we have introduced the new free parameter P c;0 ¼ G J = M n ¼ G G. This is identical to the results of Hori et al. [5]. 8. Conclusion Explicit expressions have been derived for the optimal linear measurement combination c ¼ H. The null space method [] for selecting linear measurement combinations c ¼ H has been extended to the general case with extra measurements, n > n u þ n d, see Eq. (4) in Theorem. The idea of the extended nullspace method is to first focus on minimizing the stead-state loss caused b disturbances, and then, if there are remaining degrees of freedom, minimize the effect of measurement errors. Alternativel, one ma minimize the effect of combined disturbances and measurements errors, which is the exact local method of Halvorsen et al. [3]. In this paper, we have derived an explicit solution for H for this problem, see Eq. (3) in Theorem. This expression applies to an number of measurements, including n < n u þ n d. To simplif, one often uses onl a subset of the available measurements when obtaining the combination c ¼ H. A simple rule, which can aims at minimizing the effect of measurement errors, is to select measurements to maximize rð e G Þ or even better, to minimize rð e Jð eg Þ W n Þ. Here, eg is the stead-state gain matrix from the inputs and disturbances to the selected measurements. inall, the results can be interpreted as adding linear constraints that minimize the effect on the solution to a quadratic optimization problem; see Theorems 3 and 4. Appendix A. Analtical solution for the exact local method A.. Scalar case The minimization problem for the scalar case in (9) can be rewritten as: min k et xk ¼ min x T e e T x subject to G T x ¼ J = x x ða:þ where we have introduced x,h T, which is a column-vector in the scalar case. The solution to this problem must satisf the following KKT-conditions (e.g. [, p. 444]): " e e # T G x ¼ 0 G T 0 k J = ða:þ To find the optimal x, we must invert the KKT-matrix and from the Schur complement of the inverse of a partitioned matrix (e.g. [4, p. 56]), we obtain that the optimal x is x ¼ H T ¼ð e e T Þ G ðg T ð e e T Þ G Þ J = ða:3þ Comment: In the scalar case, G is a (column) vector and J = is a scalar. However, the expression and proof also applies if G were a matrix and J = were a vector. This fact is important for the extension to the multivariable case. A.. Extension to multivariable case To show that the solution for the scalar case also applies to the multivariable case, we first transform the multivariable case into a scalar problem. In this proof, we consider a sstem with two controlled variables (n u ¼ ), but it can easil be extended to an dimension. The optimization problem is min H TkHk e subject to HG ¼ J =, where we introduce X ¼ HT. The matrices X and J = are split into vectors X,½ x x Š; J = ¼½J J Š ða:4þ We further introduce the long vectors x n ¼ x ; J n ¼ J x J and the large matrices " # " G T n ¼ GT 0 ; n e ¼ e # 0 0 G T 0 e ða:5þ ða:6þ

11 48 V. Alstad et al. / Journal of Process Control 9 (009) " Then, H e ¼ X T e x T ¼ e ¼ xte # x Te and for the -norm, x T the following applies " khk e x T ¼ # x T e ¼k½x T e x T e Šk ¼kx Te n n k ða:7þ where it is noted that kk ¼kk for a vector. The constraints G T X ¼ J = become ½ G T x G T x Š¼½J J Š ða:8þ or G T x ¼ J and G T x ¼ J, which can be rewritten as " # G T x ¼ J or G T G T n x J x n ¼ J n ða:9þ (A.7) and (A.9) is a vector optimization problem of the form in (A.) and from (A.3) the solution is x n ¼ð f n f T n Þ G n ðgt n ðf n f T n Þ G n Þ J n ða:0þ We now need to unpack this to find the optimal H T ¼ X. Substituting the values and rearranging (A.0) 0 " #" # T x e 0 e 0 G x 0 e 0 e A 0 0 G 0 0 G T " #" # T 0 e 0 e 0 0 G 0 e 0 e A A 0 G J J we see that " x ¼ ðe et Þ G ðg T ð e e # T Þ G Þ J x ð e et Þ G ðg T ð e e ða:þ T Þ G Þ J and finall, H T ¼ X ¼ ½x x Š ¼ð e et Þ G ðg T ð e et Þ G Þ ½J J Š ¼ ¼ð e et Þ G ðg T ð e et Þ G Þ J = ða:þ This proves that the solution for the scalar case also applies for the multivariable. References [] V. Alstad, S. Skogestad, Null space method for selecting optimal measurement combinations as controlled variables, Ind. Eng. Chem. Res. 46 (3) (007) [] D. Bonvin, G. rancois, B. Srinivasan, Use of measurements for enforcing the necessar conditions of optimalit in the presence of constraints and uncertaint, J. Proc. Control 5 (6) (005) [3] I.J. Halvorsen, S. Skogestad, J.C. Morud, V. Alstad, Optimal selection of controlled variables, Ind. Eng. Chem. Res. 4 (4) (003) [4] E. Hori, S. Skogestad, Selection of controlled variables: maximum gain rule and combination of measurements, Ind. Eng. Chem. Res., submitted for publication. [5] E. Hori, S. Skogestad, V. Alstad, Perfect stead state indirect control, Ind. Eng. Chem. Res. 44 (4) (005) [6] J.B. Jensen, S. Skogestad, Optimal operation of simple refrigeration ccles. Part II: selection of controlled variables, Comp. Chem. Eng. 3 (007) [7] J.V. Kadam, W. Marquardt, M. Schlegel, T. Backx, O.H. Bosgra, P.- J. Brouwer, G. Dünnebier, D.V. Hessem, A. Tiagounovand, S.D. Wolf, Towards integrated dnamic real-time optimization and control of industrial processes, in: OCAPO 003, 4th International Conference of Computer-Aided Process Operations, Proceedings of the Conference held at Coral Springs, lorida, Januar 5, 003, pp [8] V. Kariwala, Optimal measurement combination for local selfoptimizing control, Ind. Eng. Chem. Res. 46 (007) [9] V. Kariwala, Y. Cao, S. Janardhanan, Local self-optimizing control with average loss minimization, Ind. Eng. Chem. Res. (008), in press, doi:0.0/ie [0] T.E. Marlin, A.N. Hrmak, Real-time optimization of continuous processes, in: American Institute of Chemical Engineering Smposium Series ifth International Conference on Chemical Process Control, vol. 93, 997, pp. 85. [] M. Morari, G. Stephanopoulos, Y. Arkun, Studies in the snthesis of control structures for chemical processes. Part I: formulation of the problem, process decomposition and the classification of the controller task. Analsis of the optimizing control structures, AIChE J. 6 () (980) 0 3. [] J. Nocedal, S.J. Wright, Numerical Optimization, Springer, 999. [3] S. Skogestad, Plantwide control: the search for the selfoptimizing control structure, J. Proc. Control 0 (000) [4] S. Skogestad, I. Postlethwaite, Multivariable eedback Control, second ed., John Wile & Sons, 005. [5] B. Srinivasan, CJ. Primus, D. Bonvin, N.L. Ricker, Run-to-run optimization via control of generalized constraints, Control Eng. Pract. 9 (8) (00) [6] B. Srinivasan, D. Bonvin, E. Visser, Dnamic optimization of batch processes I. Characterization of the nominal solution, Comp. Chem. Eng. 7 () (003) 6. [7] B. Srinivasan, D. Bonvin, E. Visser, Dnamic optimization of batch processes III. Role of measurements in handling uncertaint, Comp. Chem. Eng. 7 () (003) [8] G. Strang, Linear Algebra and its Applications, third ed., Harcourt Brace & Compan, 988. [9] Y. Zhang, D. Nader, J.. orbes, Results analsis for trust constrained real-lime optimization, J. Proc. Control (00) [0] Y. Zhang, D. Monder, J.. orbes, Real-time optimization under parametric uncertaint: a probabilit constrained approach, J. Proc. Control (3) (00)