On the Development of Predictive Models with Applications to a

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On the Development of Predictive Models with Applications to a Metallurgical Process Matias Waller* and Henrik Saxe´ n Heat Engineering Laboratory, A° bo Akademi University, Biskopsgatan 8, FIN-20500 A° bo, Finland

A methodology for constructing useful predictive models, primarily intended for use on line by the process supervisors as a basis for decision making, is presented. The methodology mainly relies on a sound choice of model structure for black-box modelssa choice based on a clear definition of the modeling goal. Some simple approaches to the treatment of a combination of event-driven and continuous information in predictive models are also proposed. Furthermore, directional considerations in the tracking of time-variant parameters are illustrated to be of considerable practical importance for obtaining accurate models, and such considerations are possible to implement without prior knowledge of the underlying dynamics. The topics treated are finally applied to a prediction problem of industrial importance: The success of the case study clearly motivates the careful consideration of different modeling options. 1. Introduction The task of predicting important quality factors is a relevant and often crucial issue in the implementation of different control strategies. The predictive models can be used to implement automatic feedback control, but accurate models can also form the basis of a good supervisory system if automatic feedback control is not a plausible option. This paper presents a methodology for developing predictive black-box or a type of graybox models, primarily intended to be used on line for supervisory purposes. The focus on models intended to assist the process supervisors does not imply that all of the discussed topics are irrelevant for controller synthesis and design but rather that some other important, but often overlooked, issues fundamental for obtaining successful models within the modeling framework are emphasized. The approach of this paper is rather practical; the objective of the real-world problem is to predict important quality factors of the pig iron produced in blast furnaces, the dominating process for producing iron for primary steel-making.1 This is a challenging task because of the complex nature of the process, and difficulties in the construction of predictive models are discussed and tackled. The treatment is not restricted to the blast furnace process, however, because the proposed methodology addresses issues in model development of quite general importance: The different options and related difficulties encountered are typical of processes exhibiting largely unknown time-variant behavior. The successful application of the methodology on data from an industrial blast furnace further illustrates the benefits possible to achieve by carefully considering different modeling options. This paper is organized as follows: A discussion on the choice of model structure is given in section 2, which includes some illustrative examples and critical views on the approaches adopted in earlier studies. For comprehensiveness, a method for directional considerations in the tracking of time-variant parameters, although treated in detail in a separate paper,2 is briefly * Corresponding author. Tel.: +358 2 2153211. Fax: +358 2 2154792. E-mail: [email protected].

explained in section 3. Some difficulties encountered when fusing information of fundamentally different characteristics, i.e., continuous versus event-driven signals, are discussed in section 4. The treated topics are then applied to an industrial blast furnace in section 5. Finally, the last section presents some concluding remarks and gives suggestions for future work. 2. Choice of Model Structure In the following, the discussion regarding the choice of model structure is restricted to a family of linear black-box models. This limitation is motivated by the fact that this family of models, written as

A(q) y(t) )

B(q) C(q) u(t - L) + e(t) F(q) D(q)

(1)

represents all linear models used in practice3 and is well suited for illustrating the use of different regressors. Here, the choice of regressors will be used to define the model structure, while parametric time variance and nonlinear transformations of the inputs will be considered as structural details. In the equation, y(t) denotes the output at time t, u(t) the input, e(t) an (unmeasurable) noise signal, L the time lag of the input, and, finally, q the (forward) shift operator. The predictor associated with eq 1 can be written3,4

yˆ (t|θ) ) θTφ(t,θ)

(2)

where θ denotes the vector of parameters and φ(t,θ) the regressors. The different types of regressors possible to consider are5 as follows: 1. u(t - k), associated with the B polynomial. 2. y(t - k), associated with the A polynomial. 3. yˆ u(t - k|θ), simulated outputs from past u only, associated with the F polynomial. 4. (t - k|θ) ) y(t - k) - yˆ (t - k|θ), prediction errors, associated with the C polynomial. 5. u(t - k|θ) ) y(t - k) - yˆ (t - k|θ), simulation errors associated with the D polynomial.

10.1021/ie990504+ CCC: $19.00 © 2000 American Chemical Society Published on Web 04/03/2000

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These regressors also form a basis worth considering for more problem-specific, parametrically nonlinear combinations of the regressors, so the discussion, despite the restriction to eq 1, should be of general interest. Within the chosen model structure, the lag L of the input(s) and the orders of each respective polynomial in q-1 are determined to provide a justified statistical description with respect to some criterion, often some norm of the prediction error. While this task is generally quite straightforward to accomplish, the choice of model structure, on the other hand, is often not trivial, and a generous attitude is advised when trying different possibilities.6 However, an often overlooked issue is that the most accurate statistical description, given some simple criterion, not necessarily provides a useful model. To illustrate this point, consider the model

yˆ (t + l) ) y(t)

(3)

for all future times, l > 0, which, in essence, is the model suggested for daily IBM stock prices, i.e., “the best forecast for all future time is very nearly the current value of the stock”.7 Although the model no doubt provides an accurate statistical description, it raises a natural and important question: What can the model be used for? If the modeling goal is to provide a statistical description of the signal, the goal is achieved, but if the model is to be used for another purpose, such as a basis for deciding whether to invest in the stock or not, the model is obviously quite useless, and other approaches must be considered. For the example, the model was based solely on the observed time series, which quite obviously limits the possibilities regarding the choice of model structure, but the example just proves that the observed time series alone might not be sufficient for providing a “useful” (linear) model, and the investigation of other possibilities is justified. The main application studied in this paper, i.e., the prediction of an important quality and operational factor of the blast furnace process, the silicon content of the pig iron, is a relevant issue in blast furnace ironmaking and has been the subject of several investigations. In the purely empirical models proposed,8-11 using only controllable inputs and the semiempirical models,12-16 with some measured and calculated (not directly controllable) variables as inputs, previous observation(s) of the silicon content have been included as regressors. This use of autoregressive (AR) model structures is quite natural because of the significant autocorrelation exhibited by the investigated signal. However, the purpose of the modeling should be emphasized when selecting an appropriate set of regressors. Accurate predictions of sudden changes as well as estimates of the duration of the disturbances are the most important features of a predictive model primarily intended for use on line by the process supervisors as a basis for decision making. Because an autoregressive model often will tend to make the predictions inert, i.e., produce predictions close to the previous observations, it could be warranted to consider a model based solely on the inputs u, i.e., a finite impulse response (FIR) model capable of predicting sudden changes. It might thus be motivated to apply a different criterion for “model suitability” than some simple norm of the prediction error.11 It could, however, prove difficult to develop an objective quantification of model suitability. As an alternative, a clearly defined modeling goal can be chosen and kept in mind when selecting an appropriate structure for the

model. The model structure is thus determined by introducing a mental weight in the stage of selecting an appropriate structure. Within the chosen model structure, “traditional” statistical measures that monitor the prediction error can then be used to evaluate the performance of the models. This strategy is applied in the current paper, and the success of an FIR model with respect to the modeling goals is illustrated and compared to that of a “statistically sound” AR model in section 5. 3. Directional Tracking of Time-Variant Parameters A common approach to estimate time-variant parameters is to apply recursive identification, which enables an update of the parameters as new measurements become available.17 Most common methods used in practice,18 however, lack the possibility of any individual considerations regarding the update of each respective parameter, i.e., directionality. With a suitable choice of design variables, which determine the well-known trade-off between fast tracking abilities and noise sensitivity, the structure of the celebrated Kalman filter can successfully be used for tracking time-variant parameters exhibiting different time-variant characteristics. In the absence of prior knowledge of the underlying time-variant dynamics, there seems to be a void of easily implemented methods to determine these individual gains and the advantage of directional capabilities cannot readily be utilized. In order to deal with this shortage, consider the (recursive) Kalman filter17 for the case of one-dimensional output depending linearly on the estimated vector of parameters, a case for which it is sufficient to determine the matrix R in the algorithm:

θˆ (t) ) θˆ (t - 1) + K(t) (yˆ (t|θˆ (t - 1)) - y(t)) K(t) )

P(t - 1)φT(t) 1 + φ(t) P(t - 1)φT(t)

P(t) ) P(t - 1) + R(t) - K(t) φ(t) P(t - 1)

(4) (5) (6)

A detailed treatment of different methods for estimating the design variable R is beyond the scope of the current paper and is provided in a separate paper.2 In order to make the current presentation more comprehensive, it can be noted that the novel approach proposed to determine R is achieved by minimizing the variance of the output,

V(R) )

1

N

∑ (yˆ (k|θˆ (R)) - y(k))2

Nk)1

(7)

with respect to the diagonal matrix R with nonnegative elements (on a data record of length N). This minimization can preferably be performed off line, i.e., in a “once and for all” way in order to reflect a certain degree of time variance. The results of this minimization, however, cannot necessarily be trusted to a very high degree of accuracy. Therefore, it is proposed to merely adjust the diagonal elements from using rI, with r a small scalar and I the identity matrix, which represents the case of no prior knowledge of the time-variant characteristics, to the magnitude suggested by the results of the optimization. In section 5 it is illustrated that such directional considerations, though deceptively of only minor magnitude, are indeed of significant importance.

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Figure 1. Schematic view of the blast furnace process.

4. Combining Event-Driven and Continuous Information A common characteristic of many industrial processes is the combination of continuous and discrete event activities. Often, the process can adequately be divided into different stages and each stage analyzed separately, e.g., startup, a continuous stage, and finally shutdown. For instance, this strategy can be applied to a batch crystallizer, for which the stages consist of filling the crystallizer and initializing the process, making the process follow a desired trajectory during the “challenging” stage, and finally emptying the contents of the crystallizer.19 It is not, however, always as appropriate as in the previous example to divide and treat each stage separately. The operation of a blast furnace (see Figure 1), a large countercurrent chemical reactor and heat exchanger,1 can serve as an interesting example of a different type of continuous semibatch process: At the top of the furnace, the main energy resource, coke, is charged along with preprocessed ore and fluxes in alternating layers; i.e., the furnace is charged in discrete event fashion. Descending through the furnace, the ore is heated and reduced by the ascending gases until molten and removed (tapped) at the bottom of the furnace in the form of liquid (pig) iron and lighter slag, also in a discrete event manner: The tapping usually follows a typical cycle in which iron and slag are allowed to accumulate in the hearth of the furnace with the taphole plugged, followed by a (longer) period with the taphole open, when the hearth is nearly drained. Above the iron and slag layers, preheated, often oxygen-enriched, air (blast) is continuously injected through nozzles called tuyeres. The blast reacts with the coke, producing the hot and reactive gases needed to heat and reduce the ore. Other fuels, e.g., oil or pulverized coal, can also continuously be injected along with steam in connection with the blast. In addition, other continuous operations include the top gas outlet and the cooling of the lining of the furnace. A “divide and conquer” strategy is clearly not equally applicable to a process operated in this manner. The discrete event nature of the charging of burden and,

especially, the tapping of the furnace is fundamental for the continuous operation of the process and should therefore be considered in the modeling. With respect to eq 1 and the presented regressors, the combination of continuous and discrete event information will, out of necessity, influence the choice of model structure, because observations of the discrete event signals are available only at the events. In practice, this implies that the observations of the discrete event signals are obtained at irregular intervals (according to some distribution) and, in general, less frequently than the (uniformly) sampled continuous signals. When the output signals are discrete event and uniformly sampled signals are utilized as inputs, this means that the regressors y(t - k), (t - k|θ), and u(t - k|θ) are obtained in a manner substantially different compared to u(t - k) and yˆ u(t - k|θ). Although this notion could warrant the use of more complex modeling schemes, e.g., multiple models, simpler approaches are, according to the try simple things first principle, considered in the current work: Either focus on developing purely discrete event models with, possibly, observations of other discrete event signals as inputs and any other of the presented regressors or use only uniformly sampled signals u (and perhaps yˆ u) in the vector of regressors (i.e., an FIR model) and base the estimate of the parameters on the available prediction errors. Both of these strategies are applied and compared with respect to the modeling purpose in section 5. A combination of the different signals in a single model would necessitate an interpolation of a priori unknown intervals of time. It can be noted that the difficulty in combining these fundamentally different types of signals has often been avoided in earlier studies by, e.g., considering periods when the time between observations of the discrete event signals quite well can be approximated as constant.8,10,12,13,15 However, this is not an appealing alternative for models intended for use on line by the process supervisor(s), because a disturbance such as a delay in an event driving the process often creates a pressing need for good predictive models. Such ad hoc approaches to deal with the combination of event-driven and continuous information apparently overlook a fundamental feature of the investigated process, so it seems more motivated to apply a strategy which tries to deal with this feature, e.g., the strategies described in the previous paragraph. 5. Applied Predictive Modeling 5.1. The Blast Furnace in Brief. The objective of the operation of a blast furnace is to reach a low energy consumption and high production rate while maintaining good and consistent quality of the pig iron. The furnace is mainly controlled by the ore-to-coke ratio, in addition to blast-related quantities, e.g., blast temperature, the amount of steam, and/or fuel additives injected with the blast. After being charged at the top, the ore and coke reach the high-temperature zone close to the tuyere level (see Figure 1), where the quality of the product is largely determined, in some 4-8 h. This large time delay, and the quite slow response to any control actions, introduces a need to predict important quality and operational quantities, i.e., mainly the pig iron composition and its temperature. The prediction of these quantities, however, is a difficult task because of the complicated nature of the process, where heat and mass transfer and chemical reactions between several


metal, slag delay, etc., and chemical factors such as slag composition, pig iron carbon, sulfur and manganese content, was eliminated by examining the statistical significance of each respective signal. The explanatory significance and order of each signal and the most statistically justified of all possible variations of eq 1 were determined by minimizing Rissanen’s criterion24

U ) ln V +

Figure 2. Observations (+) of the silicon content and predicted values (O) using eq 8 for a period of 100 h.

phases occur under conditions difficult to determine because of complex multiphase flows and spatial distributions of composition, temperature, and pressure.1,20,21 On the other hand, many investigations have been carried out with the purpose of gaining further insight into this complex process.22,23 For the purpose of predicting the pig iron silicon content, the investigations have resulted in several interesting models providing some information of the internal state of the furnace. The information is often quantified by different indices, describing phenomena such as bed permeability, furnace energy reserve, combustion temperature, and production rate. These indices are typically calculated based on uniformly sampled signals, such as top gas measurements, blast parameters, heat losses, and (averaged) charged raw materials, while variables which refer to the pig iron only are obtained in connection to the taps. A main task is, therefore, to gather and utilize the available information in a useful and statistically justified way, which is explained in the next subsections. 5.2. Discrete Event Modeling. The notion that the output signal is available only in connection to the taps makes it appealing to consider a discrete event model for predicting the silicon content. In addition to a clearly measurable first-order autoregressive trend in the signal, however, the only other significant (causal) explanatory variable among other discrete event signals (for the data record described in the next subsection) turned out to be the (previous) value of the pig iron temperature. The predictive model thus obtained is given by

yˆ (t) ) ay(t - 1) + bϑ(t - 1)

(8)

where y is the output signal (the silicon content), ϑ is the pig iron temperature, t denotes the events, i.e., the tapping instants, and the parameters a and b are estimated using eqs 4-6. The poor predictive capabilities with respect to sudden changes, i.e., the inert behavior of a linear autoregressive model, are clearly seen in Figure 2, which illustrates the predictions (O) for a period of 100 h. As observations of the silicon content (+), the tap mean values were used and 15 min time labels were, for the sake of clarity, used in Figure 2, though obviously not necessary for the discrete event model. With respect to the use of continuous and eventdriven information, this strategy represents the case of constructing a discrete event model. The other approach, i.e., an FIR model based on uniformly sampled signals, is described in the next subsection. The use of other discrete event signals, e.g., physical quantities such as tapping speed, amount of slag and

n ln N N

(9)

where N is the length of the data record and n is the number of parameters in the model. Given a model structure, Rissanen’s description length has appeared quite adequate for determining a suitable model, but some ambiguity for penalizing model complexity, e.g., regarding the structure, still remains.25 In the present study, however, the models considered aresin a parametric sensesquite simple, and other criteria, e.g., Akaike’s criterion,26 yielded similar results. As argued in section 2, alternatives to the statistically appealing AR-type models need to be considered in order to meet the modeling needs. The modeling purpose and the vast amount of information available, both as directly measurable signals and as signals calculated based on insight into the process, motivate the effort needed to find useful FIR models. A major task in constructing suitable FIR models is naturally to find an appropriate set of explanatory inputs, treated in the next subsection. 5.3. Developing FIR Models. In the case of identifying the relationship between manipulatable variables and the output, the issue of determining inputs is limited to finding suitable time lags and orders of the (known) inputs. In the present case, however, the goal is to detect changes in the output, whether these are caused by control actions or disturbances, which allow for several hundred candidates as possible inputs. In order to obtain a manageable amount of variables, as much process knowledge as possible must be used in order to eliminate variables that are not descriptive of the mechanisms influencing the silicon transfer to the pig iron. In the present study, this left some 40 variables, which were subjected to further statistical elimination. For this elimination, different single-input, time-variant models for time lags up to 15 h and sixth order were studied with respect to minimizing U (eq 9). The most promising inputs thus detected were as follows: 1. u1, an index reflecting the energy reserve in the furnace. 2. u2, a heat index reflecting the heat of the combustion zone. 3. u3, the calculated production rate of pig iron. 4. u4, a bed permeability index calculated from the pressure drop. As noted in the brief description of the blast furnace process (section 5.1), these indices are calculated based on averaged charged and injected materials, cooling losses, top gas analysis, pressure drop, etc. Information regarding control actions is thus indirectly included in the (calculated) inputs given above. These inputs were found to be more suited for predictive purposes although the transformations make the relationship between the manipulatable variables and the output less translucent. It can also be noted that (linear) cross-correlations were taken into account in the statistical elimination

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Table 1. Optimal Combination of Inputs and Values for σ and U model

inputs

L

σ (%)

U

1 2 3 4

u1, u3 u2, u4 u1, u2, u3 y, ϑ

2, 3 1, 2 2, 2, 3 1, 1

0.1232 0.1190 0.1195 0.1238

-4.177 -4.247 -4.227 -4.167

Table 2. Directional R for the Models of Table 1 and Corresponding Values for σ and U model

diag(R)

σ (%)

U

1 2 3 4

(10-2, 10-5) (10-2, 10-3) (10-2, 10-2, 10-5, 10-4) (10-10, 10-4)

0.1178 0.1184 0.1141 0.1205

-4.274 -4.264 -4.335 -4.229

although not completely eliminated as explained in the next paragraph. As inputs, 15 min averages logged at blast furnace no. 1 of Rautaruukki Steel, Raane, Finland, were considered sufficiently frequent to obtain a satisfactory tracking of the signal to be predicted. In the study, a data set stretching over 125 days, which included 1373 taps, was considered. Although high redundancy was avoided in the choice of possible inputs, a certain degree of cross-correlation between the inputs was unavoidable, because the calculations of the inputs, in part, are based on the same measurements. Because of these correlations, different possibilities for combining the inputs were examined by further minimization of U to yield the optimal combinations. The most promising models with two and three inputs (the use of all four inputs listed above did not significantly improve model accuracy) are given in Table 1. In order to illustrate the model differences, each respective value for U and the standard deviation of the prediction errors, σ ) xV, are given in the table. For the sake of comparison, the previously presented discrete event model (eq 8) is included in the last row, and it can be noted that the corresponding standard deviation of the silicon content for the studied period was 0.1390%. In the table, the lags for the fourth model refer to the tapping instances and those for the other models to the (uniform) sampling rate of 15 min. Every model was of first order with respect to each input except for u3 of the third model, which was of second order. It can be observed that the arguments above apply to parametrically linear (timevariant) combinationssthe exploration of more general, parametrically nonlinear models remains. The values for U and σ given in Table 1 were obtained with no directional considerations in the tracking of the parameters according to eqs 4-6; i.e., R ) 10-2I was used for all models. As stated in section 3, significant improvement can be obtained when some directional considerations are implemented as suggested by the minimization of V(R) of eq 7. For half of the available data record, (N ) 74, ..., 723, where the beginning of the data record was ignored in order to eliminate the effect of initialization of the recursive algorithm on the results), this minimization suggested the use of R given in Table 2 for each respective model. For the sake of comparison, σ and U are also presented in the table. Clearly, even these minor directional considerations in the tracking of the parameters are of significant influence, and major issues of practical importancesthe choice of a suitable modelsare affected. It can be noted that the smallest decrease in U was obtained for model 2, which exhibits the smallest degree of directionality.

Figure 3. Estimated b1 (s) and b2 (- -) for model 1 with R ) 10-2I (upper) and with R of Table 2 (lower) for a period of 20 days.

Figure 4. Estimated b1 (s), b3 (- -) (upper) and b2 (s), b4 (- -) (lower) of eq 10 for a period of 30 days.

This model also changed from being the optimal model (in Table 1) to the poorest choice of FIR model given directional considerations (in Table 2), because of the significantly larger improvement obtained in the other models. To provide a qualitative explanation of the effects of implementing these directional considerations, the estimated parameters b1 and b2 of model 1 are illustrated in Figure 3 for both R ) 10-2I and R of Table 2. As the figure illustrates, the smaller degree of time variance allowed for estimating b2 (10-5 compared to 10-2) enforces a negative b2, rather than the occasional positive estimates that can be observed in the upper panel. A negative b2 is apparently more descriptive of the relation between u3 (the production rate of pig iron) and the silicon content, which explains the improved model accuracy. This observed negative correlation agrees well with the physical understanding of the process: A higher production rate will yield a shorter residence time for the iron droplets in the high-temperature region, resulting in less silicon being transferred to the pig iron. The estimated parameters of model 3,

yˆ (t) ) b1u1(t - 2) + b2u2(t - 3) + b3u3(t - 3) + b4u3(t - 4) (10) which is optimal (in Table 2) with regards to U as well as exhibiting favorable lags, are depicted in Figure 4 for a period of 30 days. The figure also clearly illustrates the difference in time variance in the estimates of b1 and b2 compared to that of b3 and b4 as can be expected from the use of the matrix R of Table 2. These estimates also agree well with operational experience: A highenergy reserve (u1) and hot combustion zone (u2) both, in general, lead to a higher silicon content reflected by


Figure 5. Observations (+) of the silicon content and predictions (s) using eq 10.

the (mainly) positive estimates of b1 and b2. Again, the effect of a higher production rate, qualitatively calculated by, e.g., the (primarily) negative sum of the estimates of b3 and b4, is seen to be a lower silicon content. The predictions obtained with the model of eq 10 are illustrated for two segments of data in Figure 5. As the time index in the figure reveals, the segments are of different length but illustrate the same number of taps, which can be explained by the two longer breaks (shutdown periods) in the continuous operation present in the upper panel. Despite the quite different characteristics of the two data sets, separated by more than 3 weeks, the model successfully predicts the evolution of the silicon content for both sets. The upper segment was, in part, used in the illustration of the discrete event model (Figure 2), and the difference between the models, especially at the large upsets, should be noted. The FIR model clearly succeeds in predicting sudden changes, as well as estimates of the duration of the disturbances, thus making the careful consideration of different model structures worthwhile. As Tables 1 and 2 reveal, the omittance of the statistically most significant regressor, i.e., y(t - 1) in eq 8, has not compromised the statistical significance of the proposed model. Rather, the choice of an FIR structure, which is in accordance with the modeling goals but not statistically intuitive, has yielded predictive properties superior to those of an AR structure. 6. Conclusions Some topics essential for the construction of predictive models useful for decision making have been treated. The choice of model structure has been demonstrated to be an issue of fundamental importance and often neglected in favor of choosing the most readily available statistically promising regressors. Some difficulties encountered in the construction of models that depend on information of both discrete event and continuous type were discussed, and two distinct possibilities for constructing models utilizing this information were suggested and illustrated. Directionality in the update of the time-variant parameters was further shown to significantly improve model accuracy and influence the choice of the optimal model. With the used method, such directional considerations were implemented without the need for prior knowledge of the time-variant characteristics and with no increase in the computational

complexity of the recursive algorithm. The successful application to an industrial blast furnace clearly illustrated the benefits of these considerations. To summarize, a methodology for constructing relevant black-box predictive models has been presented. The focus on a useful methodology has, however, left several interesting possibilities unexplored. The use of more general, parametrically nonlinear models, for instance, was not considered and might be a fruitful alternative considering the complexity of the process studied. Such models could be incorporated within the framework of using multiple models, thus combining relations that adequately can be described by linear (time-variant) models as well as relations appropriately described by less translucent, parametrically nonlinear models. Multiple models can also retain relevant, past information as an alternative to recursive identification of the parameters of the models, which, in essence, forgets old information. A multiple model approach would thus form an appealing alternative for the study of the blast furnace process, which, in addition to showing slowly varying conditions, is known to exhibit different operational states as well as complex nonlinear behavior. Because few physical guidelines to determine the state of the furnace are known, the use of multiple models could further serve as a tool for detecting such operational states. These challenges and interesting possibilities will form the future focus of the research. Acknowledgment The National Graduate School in Chemical Engineering, Finland, is gratefully acknowledged for its financial support. The authors also extend their gratitude to Rautaruukki Oyj for making industrial process data available. Literature Cited (1) Biswas, A. K. Principles of Blast Furnace Ironmaking; Cootha Publishing House: Brisbane, Australia, 1981. (2) Waller, M.; Saxeń, H. Estimating the Degree of TimeVariance in a Parametric Model. Automatica 2000, 36, 619. (3) Ljung, L. System IdentificationsTheory for the User; Prentice-Hall: Englewood Cliffs, NJ, 1987. (4) So¨derstro¨m, T.; Stoica, P. System Identification; PrenticeHall: London, 1989. (5) Sjo¨berg, J.; Zhang, Q.; Ljung, L.; Benveniste, A.; Delyon, B.; Glorennec, P.-Y.; Hjalmarsson, H.; Juditsky, A. Nonlinear Black-box Modeling in System Identification: a Unified Overview. Automatica 1995, 31, 1691. (6) Ljung, L. System Identification Toolbox User’s Guide; The Math Works Inc.: Natick, MA, 1995. (7) Box, G. E. P.; Jenkins, G. M.; Reinsel, G. C. Time Series Analysis: Forecasting and Control, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1994. (8) Phadke, M. S.; Wu, S. M. Identification of MultiinputMultioutput Transfer Function and Noise Model of a Blast Furnace from Closed-Loop Data. IEEE Trans. Autom. Control 1974, 19, 944. (9) Unbehauen, H.; Diekmann, K. Application of MIMO Identification to a Blast Furnace. Identif. Syst. Parameter Estim., Proc. IFAC Symp. 1982, 235. (10) Wu, M. W.; Chiou, J. C.; Chang, T. N. Pole-Assignment Self-Tuning Control for CSC no. 2 Blast Furnace Operation. Autom. Mining, Miner. Met. Process. Proc. IFAC Symp. 1986, 235. (11) Yan-Jiong, Z.; Liang-Tu, Y.; Zhou, M. Adopt Three Criterions to Choose MISO Prediction of Blast Furnace Operation. Autom. Mining, Miner. Met. Process. Proc. IFAC Symp. 1986, 241. (12) Chung, Y. H.; Chou, R. J.; Chang-Chiang, G. M.; Chen, G. L. An Autoregressive Model for the System Dynamics of Blast Furnace. I. China Inst. Chem. Eng. 1986, 17, 107.

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Received for review July 12, 1999 Revised manuscript received January 20, 2000 Accepted January 26, 2000 IE990504+

On the Development of Predictive Models with Applications to a

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