1654 Ind. Eng. Chem. Res. 1988,27, 1654-1658

PROCESS ENGINEERING AND DESIGN

A Modified Heuristic for an Initial Sequence in Flowshop Scheduling

Iftekhar Karimi" and Hong-ming Ku Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60208

Most superior scheduling heuristics generate an initial sequence and then recursively improve upon it. The RA (rapid access) heuristic has been used by most workers to generate such an initial sequence for a variety of flowshops. In most multiproduct chemical plants, batch transfer times and set-up times are significant. The present form of RA assumes these times to be zero. In this paper, we present a modification of RA, which accounts for transfer and set-up times and gives better results both in the cases where these times are zero and where they are not.

A survey (Parakrama, 1985) on batch processes identi- fied scheduling as the most important area for the ap- plication of computer aids. It is appropriate that sched- uling of batch processes is receiving increasing attention in chemical engineering (Ku et al., 1987). Most batch chemical plants are multiproduct plants or flowshops; i.e., they produce multiple products using a series of processing stages, and all products follow essentially the same pro- cessing path. A common scheduling problem in such plants is to determine the order in which products should be produced to optimize a system performance criterion. The criterion of maximizing the productivity or equiva- lently minimizing the total time (called makespan) re- quired to produce all the products has been studied most in the literature.

In this paper, we consider the scheduling of serial flow- shops with minimization of the makespan as the objective. Serial flowshops are those in which every processing stage has a single batch/semicontinuous unit (see Figure 1). Although our investigation is primarily limited to serial flowshops in which there is practically an unlimited num- ber of storage units (called the unlimited intermediate storage policy or UIS) between the consecutive stages of every pair, the results for this system are used in the scheduling of systems with zero or finite storage (Ku and Karimi, 1986a,b; Rajagopalan and Karimi, 1987a,b) and also for flowshops with multiple parallel units in stages (Kuriyan and Reklaitis, 1985). Since most scheduling problems are known to be NP-complete problems, optimal and efficient (requiring computation time which is poly- nomially bounded in terms of the size of the problem) algorithms are not likely to exist. Thus, most scheduling research has been directed toward the development of suboptimal, heuristic algorithms.

A number of workers (Dannenbring, 1977; Ku and Karimi, 1986a,b; Rajagopalan and Karimi, 1987a; Kuriyan and Reklaitis, 1985) have demonstrated that the superior scheduling algorithms require some form of recursive im- provement strategy. This means that the algorithms start with an initial sequence and then improve it by making changes in it. Our experience (Ku and Karimi, 1986a,b; Rajagopalan and Karimi, 1987c) has also shown that generally it is prudent to start with as good an initial

* Author to whom correspondence should be addressed.

0888-588518812627-1654$01.50/0

sequence as possible with the least effort. A simple heu- ristic, called RA (rapid access), which was first developed and used by Dannenbring (1977) has been used by most of the aforementioned works to generate an initial se- quence. However, as we see now, there are drawbacks in the original form of RA.

The RA Heuristic The RA heuristic was developed by Dannenbring (1977)

for the scheduling of multiunit serial flowshops with the UIS policy. Let N be the number of products to be scheduled and M be the number of batch units. For the two-unit ( M = 2) UIS flowshop, a classic algorithm by Johnson (1954) is available to minimize the makespan. This is an optimal algorithm, and its computational effort is bounded by a polynomial in N . The algorithm may be stated as followed:

1. Divide the products into two groups, P and Q, such that all products in P have first-unit processing times that are strictly less than their second-unit processing times, while the remainder are placed in Q.

2. Arrange all products in P in increasing order of their first-unit processing .times and those in Q in decreasing order of their second-unit processing times.

3. Concatenate P and Q to obtain an optimal UIS product sequence.

The main idea behind RA is to devise a two-unit ap- proximation to the M-unit flowshop and solve it to get an initial sequence. The two-unit approximation is con- structed as

M

j=l t*il = X ( A 4 - j + 1)ti; i = 1, N (1)

t*iz = Cjt, i =1, N (2) M

j = l

where ti, is the processing time of product i on unit j and t*i, is the processing time of product i on unit j ( j = 1, 2) of the two-unit approximation. Then, the pseudo-two-unit problem with t*U, i = 1, N , j = 1,2, as the processing times is solved optimally by Johnson's rule (1954).

The first drawback of the RA heuristic is that it does not reduce to Johnson's algorithm for M = 2. This is obvious from eq 1 and 2 which give t*il = 2ti1 + ti, and t* iz = til + 2 4 , and not t*il = til and t*i, = tiz. In fact, as seen

0 1988 American Chemical Society

Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1655

Table I. Relative Errorsn for UIS Systems with Zero Transfer Times and Set-Up Times M = 3 M = 3 M = 5 M = 8 M = 10 ..- - -

N RA RA MRA RA MRA RA MRA RA MRA 5 0.6 1.4 0.8 1.3 1.1 1.0 0.8 1.0 0.8

10 0.6 1.5 0.9 1.4 1.1 1.0 0.7 0.9 0.7 20 0.3 1.1 0.6 1.2 0.9 0.9 0.6 0.8 0.6 30 0.2 0.8 0.4 1.1 0.7 0.8 0.5 0.7 0.6 40 0.2 0.6 0.2 1.0 0.6 0.8 0.4 0.7 0.5 50 0.1 0.5 0.1 0.9 0.5 0.7 0.4 0.6 0.4

" Relative error = [(heuristic makespan - best makespan)/best makespan1100.

Table 11. Proportion of Best Solutions for UIS Systems" with Zero Transfer Times and Set-Up Times M=2 M - 3 M = 5 M = 8 M = 10

N RA RA MRA RA MRA RA MRA RA MRA 5 77 74 80 71 76 74 77 72 74

10 55 53 69 59 63 65 71 67 72 20 43 44 64 45 56 54 64 56 63 30 34 42 68 41 57 48 60 50 58 40 33 41 69 38 56 44 60 48 55 50 32 39 71 37 59 42 60 48 54

"Proportion of best solutions = (no. of best makespans/no. of test prob1ems)lOO.

4

0 !t:FH 0 SEMICONTINUOUS 0 STORAGE U N I T U N I T

Figure 1. Serial multiproduct batch process.

from Table 11, the proportion of RA solutions which are optimal for M = 2 decreases considerably as N increases. It is natural to expect that a superior heuristic should reduce to Johnson's algorithm for M = 2.

The second drawback and a more important one is that RA assumes that times required to transfer batches of products from one batch unit to another and the times required to cleanlprepare the units for processing are negligible and hence can be lumped together in the pro- cessing times. This is rarely true in a chemical plant where transfer times and set-up times can be significant.

Clearly there is a need to modify the RA heuristic so that the presence of semicontinuous units or equivalently the transfer times and that of significant set-up times can be accounted for. We first modify the RA heuristic to im- prove its performance even when the transfer and set-up times are zero and then to account for transfer and se- quence-independent set-up times. We term the first modification as simple MRA (modified RA) and the second as general MRA.

Simple MRA A crude justification for this modification lies in the

concept of idle times on units. Idle time is the time that units spend unproductively, i.e., neither processing nor transferring a batch. To minimize the makespan, we should try to minimize the idle time on units. Let product i be the first product in a given sequence of production. Now, when product i is processed on unit 1, each of the rest of the (M - 1) units remains idle for time til. This results in a total idle time of ( M - l ) t i l on the units. Similarly, it is easy to see that, when product i is processed on unit j , a minimum total idle time of ( M - j)til results. Thus, during the processing of product i, the total amount of idle time, 11, that results is given by

(3) M

j = l I1 = c (M - j ) t i j

Now, consider the situation when product i is the last in a sequence. When product i is processed on unit M , each of the f i s t (M - 1) units remains idle for tiM, resulting in a total idle time of ( M - l ) t iM. Following a similar argument as for the case in which product i was the first, the total idle time resulting from processing of the last product, I,, is given by

(4)

The above analysis suggests that products which have smaller values of Il should be earlier in the sequence and those with the smaller values of I , should be later in the sequence. This is exactly the logic behind Johnson's rule. Thus, simple MRA uses the following weighted processing times in the pseudo-two-unit problem:

M

j = l I , = C ( j - 1)tij

M

j=l M

j = l

t*il = C ( M - j ) t i j i = l , N ( 5 )

t*i2 = C ( j - 1)tij i = 1, N (6)

Note that simple MRA is remarkably similar to RA except for a shift in weights. To see the effect of this modification, we evaluated both of the heuristics by using 1000 randomly simulated test problems for each of the N X M combinations in Table I. Processing times tij were generated from a uniform distribution over the range [O.O, 19.91 h. We use relative error and proportion of best so- lutions as the criteria for evaluation. Relative error is calculated by taking the best of the solutions from RA and MRA and then calculating the deviation of RA and MRA from that best solution. Proportion of best solutions is obtained as the percentage of problems for which each heuristic gives the best solution.

Note that, for M = 2, simple MRA gives t*il = til and t*iz = ti,. Thus, in contrast to RA, simple MRA reduces to Johnson's rule for M = 2 and gives optimal solutions. That is why a column for MRA for M = 2 is not shown in Tables I and 11. Since both heuristics are very similar, they tend to give similar results. However, simple MRA has relative errors (Table I) 50%-75% as much as those of RA. More importantly, it has much better proportions (Table 11) of best solutions. Thus, for a randomly chosen problem, simple MRA has a much higher probability of giving a better solution than RA. And since it needs the same effort as RA, it is superior to RA.

1656 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988

Table 111. Relative Errors" for UIS Systems with Nonzero Transfer Times and Set-UD Times M = 2 M = 3 M = 5 M = 8 M = l O

N RA MRAlb RA MRAl MRA2b RA MRAl MRA2 RA MRAl MRA2 RA MRAl MRA2 5 1.4 0.9 1.5 1.2 0.6 1.5 1.3 0.9 1.4 1.3 0.8 1.3 1.3 0.6

10 1.0 0.6 1.3 1.0 0.6 1.7 1.4 0.8 1.6 1.4 1.0 1.6 1.5 0.9 20 0.6 0.4 1.0 0.7 0.3 1.3 1.0 0.7 1.4 1.3 0.8 1.5 1.3 0.8 30 0.4 0.3 0.7 0.5 0.2 1.2 0.9 0.5 1.2 1.1 0.7 1.2 1.1 0.7 40 0.3 0.2 0.6 0.4 0.2 1.0 0.8 0.5 1.1 1.0 0.6 1.1 1.0 0.7 50 0.2 0.1 0.5 0.3 0.1 0.9 0.6 0.4 1.0 0.9 0.6 1.1 0.9 0.6

"Relative error 2: [(heuristic makespan - best makespan)/best makespan1100. bMRAl = general MRA with lumping; MRA2 = general MRA without lumping.

Table IV. Proportion of Best Solutions for UIS Systems' with Nonzero Transfer Times and Set-Up Times M = 2 M - 3 M = 5 M = 8 M = l O

N RA MRAlb RA MRAl MRA2' RA MRAl MRA2 RA MRAl MRA2 RA MRAl MRA2 ~

5 51 62 58 63 75 56 59 71 58 59 72 56 57 74 10 38 50 40 51 67 37 42 60 37 43 54 39 40 56 20 25 40 34 45 66 27 37 54 30 37 52 29 31 56 30 25 38 31 45 67 22 30 60 25 31 52 25 31 52 40 25 36 32 45 70 23 33 57 2 1 29 56 23 31 51 50 23 37 24 41 72 21 37 53 2 1 31 51 20 28 54

OProportion of best solutions = (no. of best makespans/no. of test prob1ems)lOO. bMRAl = general MRA with lumping; MRA2 = general MRA without lumping.

General MRA Simple RA, although outperforming RA, still assumes

the transfer and set-up times to be zero. In order to generalize simple RA to include the effect of transfer and sequence-independent set-up times, we make use of the following result (Rajagopalan and Karimi, 1987b).

Lemma I. For the two-unit UIS system with transfer and sequence-independent set-up times, the optimal se- quence is obtained by applying Johnson's rule to a pseu- do-two-unit problem with the following processing times:

t*il = ti, + ai0 + si1 - si2 (7) t*i2 = t i 2 + ai2 (8)

where sij is the set-up time for product i on unit j , aio is the time to fill unit 1, and ai2 is the time to empty unit 2. It is assumed that the time to empty unit 1 is the same as the time to fill unit 2.

If we assume the set-up times to be zero and argue ex- actly as we did for simple RA, we would obtain the fol- lowing pseudoprocessing times for general RA:

M

j = l tdil = C ( M - j ) [ti j + ai( j - l ) ]

t*i2 = C ( j - l ) [ t i j + ai,]

i = 1, N (9)

(10)

where aij denotes the time required to transfer product i from batch unit j . Here again, it is assumed that the time required to empty unit ( j - 1) is the same as the time required to fill unit j , j = 1, M. Note that, as in the case of simple MRA, general MRA with sij = 0 gives optimal solutions for M = 2. To account for the nonzero sii, we add the terms involving the set-up times as suggested by Lemma I and make sure that general MRA reduces to the optimal for M = 2. Thus, general MRA solves a two-unit UIS problem with the processing times:

M

j = 1 i = 1, N

M

;=l t*il = C ( M - j ) [ t i j + + s i j - si(,-l)] i = 1, N

(11) M

j=l = C ( j - I) [tij + aij] i = 1, N (12)

We term the above modification as general MRA without lumping, since transfer and set-up times are not lumped

into the processing times as conventionally done. Now, the problem is how to evaluate the performance

of general MRA, as RA is just not designed to handle transfer and set-up times. The lumping of transfer and set-up times into the processing times seemed to be the only reasonable option, so we used the following processing times in RA to generate a pseudo-two-unit problem.

M

j=l t*il = C ( M - j + 1) [ui(j-l) + t i j + aij + S i j ]

t*. L2 = Cj[a;(j-1) + t i j + aij + si;]

i = 1, N (13) M

j = l i = 1, N (14)

By simply comparing the performance of general MRA without lumping with this RA, it is not clear whether the improvement in makespan, if any, is due to the shift in weights or the effect of no lumping in transfer and set-up times. To distinguish between the effects of the two modifications on the makespans, we also included in the evaluation a two-unit UIS problem with the following processing times, which we term general MRA with lumping:

M

j = l t*i, = C ( M - j ) [ ~ ~ ( j - ~ ) + tij + ai, + sij] i = 1, N

(15) M

t*i2 = j=l C ( j - 1) [ ~ i [ j - i ) + t i j + aij + ~ i j ] i = 1, N

(16) Results of the comparison among the three heuristics

RA (with lumping), MRAl (MRA with lumping), and MRA2 (MRA without lumping) are shown in Tables I11 and IV. In this comparison, transfer times and set-up times were generated using a uniform distribution over the range [O.O, 4.91 h. Again, for each N X M combination, we used 1000 test problems. From Tables I11 and IV, we see that the relative errors of MRA2 are almost half those of RA and the proportions of the best solutions are con- siderably higher than those of RA. Although MRAl con- sistently gives better solutions than RA in every combi- nation, the improvements are marginal. This clearly in- dicates that even though both the weights and the proper

Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1657

Table V. Data for Example batch processing time (h)

at M of batch transfer times (h) at M of set-up times (h) at M of N 1 2 3 4 0 1 2 3 4 1 2 3 4 A B C D E F G H I J

10 15 12 8

17 6

10 6

20 9

8 20 16 9

19 8

10 14 12 15

5 14 19 15 12 15 14 6

10 6

16 18 7

13 11 18 7

19 7

20

2 3 2 4 2 5 4 1 4 6

1 5 6 1 4 5 3 6 2 4

1 m-, L .. . - 1 CENTRIFUGE

F*”M I STORAGE I- -

PUMP PUMP

HEAT REACTOR EXCHANGER FEED TANK

T R A Y D R Y E R

Figure 2. Example of a serial multiproduct batch process.

incorporation of transfer and set-up times are important, the latter is more important than the former. Thus, for initial sequence generation, MRA2 seems to be a superior way of incorporating the effects of transfer and set-up times than their simple lumping.

Example To illustrate the key features of general MRA, consider

a serial batch process shown in Figure 2, in which 10 different products, A-J, are to be produced. As indicated in Figure 2, all products undergo a fixed-time reaction in a batch reactor and then pass through a heat exchanger to a feed tank. From this feed tank, each product un- dergoes liquid removal by centrifuge, and it is finally dried by a batch tray dryer before being sent to packaging. From Figure 2, it is obvious that there are four batch units and three semicontinuous subtrains present in the process. A semicontinuous subtrain is one or more semicontinuous units used in series, in which all equipment in the subtrain must operate for the same length of time. The semicon- tinuous units are used to transfer products from one batch unit to another and hence are specified by the processing rate. Unlimited intermediate storage is assumed to be available between every two batch units.

Since there are 3 628 800 possible product sequences, determining an optimal sequence that has the minimum makespan is virtually impossible by complete enumeration. Hence, we wish to use the general MRA heuristic without lumping to determine a good, but not necessarily optimal, product sequence. Processing times, transfer times, and set-up times for the various products are given in Table V.

Using eq 11 and 12, we obtain the following pseudo- two-unit processing times for each product:

H I J

t*i1

69 128 117 78

124 87 90 81

121 72

t*i2

72 106 78 79 79 97 61 a8 56 90

6 0 0 1 1 2 2 4 0 0 2 3 1 2 3 0 0 3 4 2 3 1 0 0 4 1 1 1 3 0 0 3 2 1 2 5 0 0 2 3 3 3 2 0 0 3 2 1 2 5 0 0 3 3 3 1 3 0 0 1 3 4 3 3 0 0 2 4 4 3

Applying Johnson’s algorithm to the above two-unit UIS problem, we obtain an ordered P = (A, J , D, H, F) and an ordered Q = (B, E, C, G, I). Hence, general MRA gives a product sequence of A-J-D-H-F-B-E-C-GI. Having determined the sequence, we can now use the completion time algorithm proposed by Rajagopalan and Karimi (1987) to compute the makespan. The total production time of the above sequence is 267 h or 11.1 days.

On the other hand, if we use RA and general MRA with lumping to determine production sequences for the above example , we ob ta in t h e sequences

respectively. The total production times of the above two sequences are 273 h (11.4 days) and 272 h (11.3 days), respectively, which are longer than that obtained from MRA without lumping. Thus, these results are consistent with our conclusion that MRA without lumping is a su- perior way of incorporating the effects of transfer and set-up times than simple lumping.

Conclusion A modification was proposed to a commonly used heu-

ristic for generating an initial sequence for flowshop scheduling. In contrast to the original heuristic (RA), the modified heuristic (MRA) gives optimal solutions for the two-unit flowshops. MRA is more general as it incorpo- rates the nonzero transfer and set-up times in a superior fashion. MRA requires exactly the same amount of work as RA and still has a higher probability of giving a better solution than RA with almost half the relative error of RA. Therefore, MRA is superior to RA for generating an initial sequence for flowshop sequencing.

Acknowledgment

western University and IBM Corporation.

Nomenclature a, = transfer time of product i from batch unit j I , = total idle time defined by eq 3 I 2 = total idle time defined by eq 4 P = group of products defined in Johnson’s rule Q = group of products defined in Johnson’s rule sl, = set-up time for prouct i on unit j t , = processing time of product i on unit j t*, = processing time of product i on unit j in the pseudo-

Literature Cited Dannenbring, D. G. “An Evaluation of Flowshop Sequencing

Heuristics” Manag. Sci 1977, 23, 1174. Johnson, S. M. “Optimal Two and Three Stage Production Sched-

ules with Set-up Times Included” Nuual Res. Logist. Q. 1954,1, 61.

A-F-B-J-H-C-E-I-D-G and A-F-B-J-H-C-E-D-I-G,

We acknowledge the financial support from North-

two-unit problem

1658 Ind . Eng. Chem. Res. 1988,27, 1658-1664

Ku, H. M.; Karimi, I. A. "Scheduling in Multistage Serial Batch Processes with Finite Intermediate Storage, Part 1: MILP Formulation" Presented a t the AIChE Annual Meeting, Miami, - 1986a, Paper 72e.

Ku. H. M.: Karimi. I. A. "Scheduling in Multistage Serial Batch Processes with Finite Intermediate storage, Part 12: Approximate Algorithms" Presented at the AIChE Annual Meeting, Miami, 198613, Paper 72e.

Ku, H. M.; Rajagopalan, D.; Karimi, I. A. "Scheduling in Batch Processes" Chem. Eng. Prog. 1987, 83(8), 35.

Kuriyan, K.; Reklaitis, G . V. "Approximate Scheduling Algorithms for Network Flowshops" Znd. Chem. Eng., Symp. Ser. 1985, 92, 79.

Parakrama, R. "Improving Batch Chemical Processes" Chem. Eng. 1985, Sept, 24.

Rajagopalan, D.; Karimi, I. A. "Scheduling in Serial Mixed-Storage Multiproduct Processes with Transfer and Set-up Times" Pro- ceedings of the 1987 Foundation of Computer-Aided Operations Conference, Park City, UT, July 5-10, 1987a.

Rajagopalan, D.; Karimi, I. A. "Completion Times in Serial Mixed- Storage Multiproduct Processes with Transfer and Set-up Times". Comp. Chem. Eng. 1987b, in press.

Rajagopalan, D.; Karimi, I. A. "Scheduling in Serial Mixed-Storage Multiproduct Processes with Transfer and Set-up Times" Tech- nical Report 8703,1987~; Center for Manufacturing Engineering, Northwestern University, Evanston, IL.

Received for review September 14, 1987 Revised manuscript received March 30, 1988

Accepted April 22, 1988

An Adaptive Estimation Algorithm for Inferential Control

Mohammad T. Guilandoust, A. Julian Morris, and Ming T. Tham* Department of Chemical and Process Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, U.K.

An estimator relationship for inferring infrequently measured process outputs, from other more rapidly sampled secondary outputs, is derived. The structure of the estimator is not application dependent, and its parameters can be continuously estimated and updated. As a result, slow variations in plant or disturbance characteristics can be tracked. Closed-loop control schemes using the estimated output data are shown to exhibit superior performance over those using the slowly measured output data. In contrast to some other inferential control strategies suggested in the literature, the method proposed requires minimal effort for estimator design and secondary output selection.

In digital control systems, the infrequent measurement of some process outputs, determined by sampling limita- tions, prevents the early detection of load disturbances. This results in large deviations from the desired output and consequently long disturbance recovery times. Often, these adverse effects cannot be acceptably overcome by the use of existing advanced control algorithms and can lead to unsatisfactory control system performance. Ex- amples of industrial situations where this can occur are in product composition control of distillation columns (Patke et al., 1982) and chemical reactors (Wright et al., 1977). In these cases, the sampling delay is a direct result of the long cycle time of on-line composition analyzers. Because of potential problems due to infrequent sampling, the control of product quality of many industrial multi- component columns is commonly achieved by maintaining an a priori chosen tray temperature near to its set-point value. However, this type of single-temperature feedback control is not always effective since maintaining a constant tray temperature does not necessarily result in constant product composition (Patke et al., 1982).

The problem of controlling infrequently measured pro- cess outputs has long been studied, and publications in this area date back to the early 1970s. There have been es- sentially two approaches to the problem. One is to design special controllers for the infrequently sampled outputs. For example, Soderstrom (1980) formulated a number of minimum variance controllers enabling the manipulated (control) input to be changed between the sampling in- tervals of the primary process output. These control al- gorithms were only developed for first-order plant models, and no comparative results were presented. Parrish and Brosilow (1985) also proposed a controller design method based upon the philosophy of reconstructing the effects of disturbance inputs. The controller parameters are de-

0888-5885/88/2627-1658$01.50/0

termined on-line by heuristic tuning rules. Their simu- lation results indicated superior performance of their control strategy over that achieved by using conventional PID control.

A second approach to deal with the problem of con- trolling infrequently sampled process outputs is to use the information provided by other more easily measurable variables. For example, this information can be used to provide an estimate of the controlled output. The esti- mated values of the output can then be used for overall control of the plant. Control schemes based on the feed- back of estimated outputs are often termed "inferential control schemes". An ideal situation arises when the plant states are completely observable from the secondary outputs. Under such circumstances, Kalman filtering techniques can be employed to estimate plant states using the secondary output measurements. Estimates of the controlled output can then be computed by using its re- lationship with the states. Control of the plant is achieved by feedback of either the state estimates or the output estimates to appropriate controllers. Published literature on the above methods is extensive. The papers by Morari and Stephanopoulos (1980a,b) and Morari and Fung (1982) are amongst the more recent contributions describing the application issues of these techniques. However, the use of Kalman filters is confined to situations where the plant is completely observable from the secondary outputs. For most plants, such a set of secondary outputs can be dif- ficult to determine or, in some cases, may not even exist. Brosilow and co-workers (Cwiklinski and Brosilow, 1977; Joseph and Brosilow, 1978) have suggested an estimator design technique using an input-output representation of the plant. The design is approached by obtaining a least-squares-based static estimator, which can be used to infer the controlled output from secondary measurements

0 1988 American Chemical Society