Model-Based Control of Strip Temperature for the ... - IEEE Xplore

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 2, MARCH 1998

Model-Based Control of Strip Temperature for the Heating Furnace in Continuous Annealing Naoharu Yoshitani, Member, IEEE, and Akihiko Hasegawa

Abstract—This paper presents the model of the heating furnace in continuous annealing processes for use in design of self-tuning control systems. A simplified mathematical model is derived from first principles. The model parameters are recursively estimated with an algorithm called recursive parameter estimation with a vector-type variable forgetting factor (REVVF). The REVVF algorithm was developed for such cases where some knowledge on parameter variability can be obtained beforehand. The control system of strip temperature presented here is hierarchical. The upper level is called “optimal preview control,” which performs preset control. It previews the approaching setup change, which is the change of strip size or reference temperature, and optimizes the line speed and the strip temperature trajectory. Next, the lower level is called “temperature tracking control,” which performs closed-loop control using the above trajectory as the control target. At this level, the generalized pole-placement self-tuning control was first employed; and later, the generalized predictive self-tuning control was introduced. These control methods were applied with some practical modifications and with the above mentioned REVVF. The control has been working successfully in several real plants. Index Terms—Adaptive control, continuous annealing, identification, modeling, pole assignment, predictive control, recursive estimation, temperature control.

I. INTRODUCTION

A

CONTINUOUS annealing process is a highly efficient heat treatment process after cold rolling in steel works. It produces steel strips of high tensile strength and high formability. A continuous galvanizing line also requires this process before zinc coating. The process consists of the heating, the soaking, the cooling and the overaging furnace, as shown in Fig. 1. Strip temperature is measured and controlled at the outlet of each furnace. Among these furnaces, the heating furnace (HF) influences large effects on production rate, strip quality, stability of operation, etc. On the other hand, the strip temperature at the outlet of the heating furnace shows some complicated characteristics because of slow dynamics, dead time, thermal interactions between the strip and hearth rolls which support the strip, and setup changes, that is, changes of strip thickness, strip width, or reference temperature. These make it difficult for a conventional control system with such as proportional, inteManuscript received January 1997; revised October 1997. N. Yoshitani is with the Department of Materials Science and Engineering, Teikyo University, Utsunomiya, 320 Japan. A. Hasegawa is with the Department of Computer Control Technology, Plant Engineering and Technology Center, Nippon Steel Corporation, 20-1 Shintomi, Futtsu, 293 Japan. Publisher Item Identifier S 1063-6536(98)02069-7.

Fig. 1. Outline of a continuous annealing process.

gral, and differential (PID) controllers to achieve satisfactory control performances. To overcome this problem, several control systems have been proposed in recent years [1], [2]. Most of these employ physical mathematical models of the plant with fixed parameters for on-line control. We have developed a control system using a simplified mathematical model of ARX (Auto-Regressive and eXogenous) form derived from a physical model [3]. This model is advantageous over a physical model in the easiness of applying control theory and parameter estimation algorithm such as the least squares method. In addition, it represents complicated characteristics of the plant as well as the physical model from which it was derived, by including a nonlinear gain and various disturbance terms. In order to keep good control performances under any changes of plant characteristics, we applied two types of selftuning control (STC), namely, the generalized pole-placement STC [4] and the generalized predictive STC [5]. For parameter estimation, we developed an algorithm called the recursive parameter estimation with a vector-type variable forgetting factor (REVVF) [11], a modified version of the least squares algorithm, and applied this to the above STC’s. This paper presents the modeling, control, and parameter estimation of this strip temperature control system. II. THE PLANT

AND ITS

CHARACTERISTICS

Fig. 1 shows a typical continuous annealing process called C.A.P.L. (continuous annealing and processing line). Here, the material for annealing is a cold-rolled strip coil, which is put on a pay-off reel on the entry side of the line. The head end of the coil is then pulled out and welded with the tail end of the preceding coil. Then the strip runs through the process with a certain line speed. On the delivery side, the strip is cut

1063–6536/98$10.00  1998 IEEE

YOSHITANI AND HASEGAWA: MODEL-BASED CONTROL OF STRIP TEMPERATURE

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(a)

(b)

(c)

(d)

Fig. 2. Strip heat pattern for continuous annealing.

into a product length by a shear machine and coiled again by a tension reel. The heat pattern of the strip, as shown in Fig. 2, is determined according to the composition and the product grade of the strip. The actual strip temperature must be within the defined ranges from the heat pattern to prevent quality degradation. The value of the heat pattern at the outlet of the heating furnace is the reference temperature for the control. In most cases, the strip in the heating furnace is heated indirectly with gas-fired radiant tubes. The heating furnace is 400 to 500 m in strip length and is split into several zones. The furnace temperature and fuel flow rate are measured at each zone, while the strip temperature is measured only at the outlet of the furnace with a radiation pyrometer. It takes a few minutes for a point on the strip to go through the furnace. Fig. 3 shows observed characteristics of the strip temperature at the outlet of the heating furnace (expressed as “outstrip temperature”, or simply “strip temperature” hereafter) in a real plant. Fig. 3(a) shows static characteristics, while Figs. 3(b)–(d) show dynamic ones. As seen from Fig. 3(b), the response of out-strip temperature against the change of fuel flow rate has a dead time and a large time constant. Next, Figs. 3(c) and (d) show the response against the change of strip thickness and the line speed, respectively. The first-order lag response in both figures is due to the heat transfer between the strip and hearth rolls. The ramp-like change of the temperature in Fig. 3(d) is caused by the ramp-wise change of the strip heating time due to the stepwise change of the line speed. In addition to these figures, it is observed that a change in furnace temperature causes an almost proportional change in out-strip temperature and that the change in strip width hardly causes any recognizable change in that. III. OUTLINE

OF THE

CONTROL

Fig. 4 schematically illustrates the control system. The system, installed in a process control computer, is hierarchically structured. It consists of “optimal preview control” at a higher level, and “strip temperature tracking control” and “parameter estimation” at a lower level. The higher level performs preset control. It calculates the speed reference after the approaching setup change, the

Fig. 3. Characteristics of strip temperature.

Fig. 4. Diagram of strip temperature control system.

optimal timing of speed change and the target trajectory of strip temperature . It uses predetermined production order sent from a business computer for production management. The production order includes the size, the grade, and the reference temperature of the strips. Next, the lower level performs feedback control and parameter estimation. The controller calculates the manipulated variable, which is the command value of the total fuel flow rate in the heating furnace (denoted by “fuel flow rate” hereafter), at a certain control interval (typically 1 min). is then distributed to each zone of the furnace to achieve a desired profile of the furnace temperature, which is not dealt with in this paper. The controller uses measured values of plant variables and employs as the control target. Here, includes and ; where and denote the outstrip temperature, the mean value of the furnace temperature over all zones of the furnace (expressed simply as “furnace temperature” hereafter), and the line speed, respectively. is not chosen as a manipulated variable of temperature feedback

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control, because frequent speed changes sometimes cause fluctuations in controls at other furnaces. Normally is changed, if necessary, only at setup changes. Estimated values of unknown parameters of the plant model are also calculated on-line at this level. The plant model is used at both levels.

Static model of strip temperature (15) where (16)

IV. PLANT MODEL The plant model was developed so that it represents all the characteristics described in Section II. The model consists of the dynamic model of strip temperature, the dynamic model of furnace temperature and the static model of strip temperature. The dynamic models describe dynamic relationships between the out-strip temperature or the furnace temperature, and the manipulated variable (fuel flow rate in the heating furnace), with additional disturbance terms which include the strip size and the line speed. On the other hand, the static model describes characteristics of out-strip temperature in steady state where the abovementioned disturbances, the strip size and the line speed, remain constant. A. Model Equations The model equations are written as follows: Dynamic model of strip temperature (1) where (2) (3) (4) (5) (6) (7) (8) Dynamic model of furnace temperature

(9) where

(17) (18) (19)

time [min] (one sampling period min); dead time [min]; total fuel flow rate in the heating furnace [ Nm /h] (“Nm ” (normal m ) denotes “m ” at 0 C); out-strip temperature of the heating furnace [100 C]; nonnegative integer (to be determined experimentally); out-strip temperature in steady state [100 C]; strip temperature at the inlet of the heating furnace (constant) [100 C]; furnace temperature of the heating furnace [100 C]; strip width [m] and thickness [mm]; line speed [ m/h]; average line speed during ( is the heating time of the strip at the outlet of the heating furnace); : disturbance terms; : average values of , respectively, in normal operations; : unknown model parameters (to be estimated on-line). As seen in the above equations, the dynamic models are chosen to have predictive forms derived from ARX models for the convenience of controller design. and from the static model are used in the dynamic model of strip temperature; where an increment of is included in and , and is used as a nonlinear gain for fuel flow rate and strip-mass flow rate. The future values of disturbance terms ( , ) can be predicted in advance because processing schedule is known beforehand. The accuracy of the dynamic models is improved by including the disturbance terms and the nonlinear gain.

(10) (11) (12) (13) (14)

B. Derivation of the Static Model The structure of the static model was determined through curve-fitting approximation of the rising pattern of calculated strip temperature [ C] in the heating furnace. In steady state, it can be assumed that the strip is heated mostly through radiation and that the strip temperature profile in the thickness


149

Now,

can be expressed by (22)

[kcal/deg] is the thermal capacity of the heating where furnace. Next, in normal operations where neither nor has abnormally large changes, and can be regarded to be linear to and to , respectively. Then can be approximated to be linear to because is observed to have linear relationship with as described in Section II. Besides, is proportional to the strip mass-flow rate. In addition, can be expressed as the weighted sum of the time series of the fuel flow rate for . Therefore, , , and are approximated by (23) (24) (25)

Fig. 5. Simulation of strip heating in the heating furnace.

direction is almost uniform. Therefore, the following heat transfer equation:

is calculated with

(20)

where are unknown parameters and is a nonnegative integer. In steady state, is equal to , which changes with the change of as mentioned in Section II; therefore (26)

where furnace temperature [ C]; Stefan–Boltsmann constant ( [kcal/m h deg ]); overall coefficient of radiative heat absorption ( ) (determined as 0.17 from actual data); strip specific heat [kcal/kg deg] (varying with the strip temperature); strip mass density [kg/m ]. Fig. 5 shows simulated results of (20). It represents the same characteristics as in Fig. 3(a), because the variables of the horizontal axis in both figures (i.e., and ) have the in the heating process. The curve in same influence on Fig. 5 was evaluated to be close to the exponential one, hence the static model was determined as (15)–(17).

Then, by using (26), (22) can be approximately expressed in discrete form as (27) The basic part of the dynamic model of strip temperature is derived by substituting (23)–(25) and (27) into (21) and by using (2), (3), (7), and (8)

(28) where

C. Derivation of the Dynamic Model of Strip Temperature 1) Basic Part of the Model for Steady State (When Strip Thickness and Strip Speed are Both Unchanged): This part, which is expressed by (1) without and , was derived from the dynamic heat balance equation in the heating furnace. The heat balance during one sampling interval ( [min]) is written as [kcal/min]

(21)

: [thermal inflow by gas combustion] [thermal outflow by exhaust gas]; : thermal increase in the heating furnace (that is, in the furnace atmosphere and on the inner surface); : thermal outflow by the strip; : thermal loss through the furnace body.

(29) are unknown parameters made and where up of and . 2) Complete Dynamic Model of Strip Temperature: The out-strip temperature has transient behaviors as shown in Fig. 3(c) and 3(d) after changes of strip thickness or the line speed. In order to describe these behaviors, the auxiliary part, , were added to the right-hand side of (28) to give the minimal realization of the complete dynamic model. Here, represents the first-order lag due to the thermal inertia of hearth rolls, while represents stepwise or ramp-wise changes, as shown in Fig. 6. Finally, the prediction form (1) is derived by eliminating from the following simultaneous

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Step 5) Determine the most appropriate value of (and ) by considering the values of Akaike’s information criteria (AIC) [6]. The values of and in (9) were determined in the same way. E. Comments on the Plant Model Thus, the plant model is derived from both physical theory and actual plant behaviors to establish the reliability of the model. The model structure is simplified in order to allow feasible control design and parameter estimation, without sacrificing the model accuracy in normal operations. The model accuracy is confirmed satisfactory with actual plant data under various operating conditions. Fig. 6. Disturbance terms

w1 ;

w2

.

V. OPTIMAL PREVIEW CONTROL equations:

The optimal preview control calculates , and mentioned in Section III by previewing the approaching setup change. An example of calculated results is illustrated in Fig. 8. (30)

Through this calculation, each into , and in the prediction form, the following equations:

and

is transformed in (1) satisfy (31)

The dynamic model of furnace temperature (9) is derived in the same way.

A. Speed Reference Under the assumption that the furnace temperature remains constant, is calculated by using the static model so as to bring the strip temperature equal to its reference temperature in steady state after the setup change. has a possibility of change only at setup changes. The above assumption, desirable for stable operation, approximately holds when fuel flow rate remains constant. B. Optimal Timing of Speed Change

D. Identification of Fixed Parameter Values The dynamic model of strip temperature (1) has two parameters whose values are fixed in the control: that is, the dead time and the model order . In order to identify these values, actual operation data as shown in Fig. 7 under various operating conditions were collected and investigated. First, the value of can be determined to be the time difference between the point where fuel flow rate was altered and the point where the change of temperature began to appear. In some plants, was found to be almost constant and the value was easily fixed. In the other plants, was found to vary according to operating conditions and the value was determined together with the value of . The value of and, for the latter plants, the value of were determined through the following steps. Step 1) Set an initial value of (and ). Step 2) Determine the other parameter values in (1) by applying the least squares method to the actual data that was considered to be the truest representation of actual operating conditions. Step 3) Calculate the sum of squared errors of (1) using parameter values obtained in the previous step. Step 4) Change the value of (and ) and repeat Step 2) and Step 3) until all the possible values of (and ) are investigated.

Let us assume that remains constant and that well before the setup change is equal to the reference temperature . Then a certain timing of speed change before the setup change generates a certain temperature trajectory. is the one which is limited within a permissible range as shown in Fig. 8 and which generates the trajectory that minimizes under the constraint defined below the cost function (32) and must be within the defined upper 1) Constraint: and the lower limits. and denote temperature deviations from just before and after the setup change, respectively. denotes a relative grade factor of the strip before the setup change . should be chosen to be close to one if the grade is higher before the setup change than after, and close to zero in the opposite case. is calculated through iteration and search method starting from an arbitrary initial value. C. Target Trajectory of Strip Temperature If the timing is not is the same as . If modification, denoted by to to obtain . First,

at the end of the range it is, an appropriate trajectory , should be calculated and added the value of just at the


151

Fig. 7. Actual operation data for identification.

setup change, denoted by , is calculated as the difference between and the temperature which minimizes with the constraint. Then is formed with a slope and a flat period before the setup change as in Fig. 8. and are appropriately chosen design parameters, where is introduced to compensate a delay in temperature tracking control with pole placement. After the setup change, rapidly approaches . The trajectory of actual strip temperature corresponding to is realized by changing the furnace temperature. VI. STRIP TEMPERATURE TRACKING CONTROL A. Necessary Conditions Temperature tracking control requires the following conditions. C1) To make use of predicted and desired values of future temperature (controlled variable). C2) To be robust against variations of the order, the dead time and other parameter values of the plant model. C3) To be capable of dealing with a nonminimum phase plant. C1) is to cope with the plant having dead time and a large time constant by taking control actions beforehand, especially before a setup change. C2) is aimed to maintain stable and nondeteriorated control performances under any changes of

Fig. 8. Optimal preview control.

plant characteristics. In this control, plant characteristics tend to have slow changes, so do model parameters, due to gas

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calorie change, equipment aging, etc. Therefore, adaptive control such as self-tuning control is more appropriate than controls with fixed parameter values such as control. This is because adaptive control uses recursively estimated values of model parameters, which are always more accurate than fixed ones when parameter values change slowly with time. C3 is required because the controlled plant is nonminimum phase when it is represented by (1). This is due to the length of time, typically several minutes, required for strip to travel from the inlet to the outlet of the heating furnace. Besides, due to indirect heating with radiant tubes, values of the manipulated variable in the past tend to provide greater contributions to strip temperature than immediate ones. This means that the coefficients in model equation (1) tend to become for some and , making the equation nonminimum phase. Considering all these conditions, we first employed the generalized pole-placement STC (self-tuning control) [4], with modification in offset removal; and later used the generalized predictive STC [5] incorporated with furnace temperature control to prevent excessive furnace temperature. B. Generalized Pole-Placement STC (GPP-STC) The generalized pole-placement STC (GPP-STC) was proposed by Allidina and Hughes [7]. It can keep closed-loop poles at preassigned locations under any changes of plant dynamics and can follow the changes of output references. These are nice properties for the temperature tracking control and the cost function of GPP-STC for the control [4], denoted by , is given by

In the temperature tracking control, mined as

and

are deter-

when which sets

to be 1. Then

(39)

is given by

(40) is a desired closed-loop characteristic polynowhere mial. was first chosen as unity to give a dead-beat and the fastest temperature response. Then, however, the fuel rate was found to have fairly large fluctuations. Therefore, was modified as to suppress the fluctuations. The appropriate range of is found experimentally to be [ ]. C. Generalized Predictive STC (GP-STC) The generalized predictive control (GPC) was proposed by Clarke et al. [9], [10]. Its self-tuning version (GP-STC) was also suggested. GPC minimizes a quadratic cost function consisting of predicted control errors and increments of control inputs, in the finite future. Like GPP-STC, GP-STC has nice properties of high tracking performance, high stability, and easiness in design parameter adjustment, suitable for strip temperature control. The cost function of GP-STC for strip temperature control [5], denoted by , is given by

(33) where (reference of

(34) (35)

(41)

(36) (37)

where

(38)

(reference of

(42)

where estimated value; time-shift operator; value of in steady state (introduced for offset removal); nonnegative integers. has the same form as the cost function of the generalized minimum variance STC (GMV-STC) [8]. The control laws for both STC’s are also the same. However in GPP-STC, the coefficients of the weighting polynomials and their orders are determined according to pole-placement condition; while and in GMV-STC are arbitrarily chosen design parameters.

and reference of ; prediction horizon for strip temperature: prediction horizon for furnace temperature: control horizon: weights for furnace temperature deviations and for fuel variations, respectively . The second term of (41) was added to keep furnace temperature below the upper limit . The reference is set a little below and the weight increases from zero to one as increases from to (see Fig. 9). That is, strip temperature control smoothly shifts to furnace temperature control to prevent excessive furnace temperature as furnace


153

where (51) (52) (53) and

Fig. 9. Weight

Wf versus. furnace temperature.

temperature approaches its upper limit. is set about 50 below . The control horizon is set at one to prevent large changes in fuel flow rate and to avoid on-line inverse matrix calculation. This causes no problem in control performances. Reference [9] also states that provides appropriate control performances for the process having no unstable or oscillatory poles. and are set approximately at the rise time of strip and furnace temperature respectively. , set about 0.6, is not critical as far as and are set as above. VII. RECURSIVE PARAMETER ESTIMATION Unknown parameters of both the dynamic and the static models are recursively estimated. The estimation for the dynamic model is performed at every sampling instance (typically every 1 min) with REVVF [11], [12]. REVVF assigns a different value of forgetting factor to each unknown parameter and thus makes it possible to adjust the adaptation rate for each estimate independently. REVVF is an extended algorithm from the one with a scalar variable forgetting factor proposed by Cordero and Mayne [13] and its deterministic convergence has been proved [11], [12]. In REVVF, parameter estimates of the dynamic model (1) are updated at time in the following way: (43) (44) (45) (46) (47)

(48)

(49) if trace otherwise (50)

vector of parameter estimates; signal vector: forgetting factor for the th parameter; parameter variability index for the th parameter (constant); positive large constant (e.g., ). The initial values such as , and design parameters such as , are determined beforehand. In actual calculation, the adaptive gain matrix should be updated using a factorization method such as U-D factorization [14] to keep the matrix positive definite and to improve numerical accuracy. REVVF is advantageous when there are differences in the rate of change among unknown time-varying parameters and the differences are known beforehand, at least roughly, through theoretical insight or experiment. If the th component of , denoted by , is known to have a larger rate of change than other components, then corresponding is set to be larger than other ’s for , which eventually let the estimate have a larger rate of change. Thus, the estimation accuracy is improved. In the estimation of the dynamic model (1), corresponding to is set far larger than other ’s for , because is considered to include unknown disturbances and therefore to have larger and more frequent changes than other parameters. The UD factorization method [14] is applied to the update of in (49)–(50). REVVF has been proven effective through numerical simulations [11]. In the estimation of the static model (15)–(17), the parameter estimates are updated once for each steady state. The steady state is detected when has decreased to a very small value. The recursive estimation is started after the transient state of heating-up of the heating furnace, and is temporarily stopped in the following cases. 1) The value of some plant variable(s) such as , , , is not reliable, or not in the effective range of the or plant model. This occurs, for example, in the following cases. a) The strip at the outlet of the heating furnace is not a product, but a strip which is repeatedly used for test operations or other purposes, because this kind of strip is likely to have rough surface to cause inaccurate temperature measurement. b) The junction of two strips is near to the outlet of the heating furnace at the sampling instance, because the strip thickness fluctuates at the junction to cause strip temperature fluctuations. 2) The rate of change of some plant variable(s) is abnormal.

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Fig. 11. Zeros of strip temperature model.

Fig. 10. Recursive estimation with actual operation data (points are plotted at every 5 min).

3) The magnitude of prediction error, in (43), is abnormally large. In real plants, to stop the estimation in these cases is essentially important to prevent deterioration of parameter estimates.

VIII. APPLICATION

TO A

REAL PLANT

The control and parameter estimation algorithm described so far were applied to real plants. Fig. 10 shows behaviors of parameter estimates for the dynamic model of strip temperature along with important plant variables plotted at every five observations (minutes). After about 100 observations, parameter estimates have converged to reasonable values, which illustrates the validity of the model structure and the estimation algorithm. After the initial estimation phase as shown in Fig. 10, each value of parameter estimates is limited within a physically reasonable range; thus, an abnormal value is prevented. Remark: In the initial phase, values of parameter estimates should not be limited, otherwise, estimation would go away from the least squares estimation and parameter convergence would be badly affected. Next, a complex plane in Fig. 11 shows locations of four estimated zeros of the transfer function from to in (1), i.e., roots of “ ,” at various times after the initial estimation phase in a real plant. As can be seen from the figure, some zeros are outside the unit circle, indicating that the model is nonminimum phase.

Fig. 12 shows an example of actual control of a real plant under GPP-STC. There are two setup changes in the figure, where both strip thickness and strip temperature reference are changed. The line speed pattern and the target trajectory of strip temperature are determined by the optimal preview control. The actual strip temperature follows its target with allowable differences and both underheating and unnecessary overheating are prevented. Since GPP-STC showed such a satisfactory performance, later-developed GP-STC have been installed in the plants which were not equipped with GPP-STC. Therefore, both STC’s have not been tested or implemented for use in the same plant. We have a general impression that GP-STC is a little more accurate in temperature tracking than GPP-STC, although we have not done comparative experiment or detailed analysis regarding these two STC’s. GP-STC does not need in Fig. 8 for the compensation of a delay in tracking control, while GPP-STC is computationally simpler than GP-STC. IX. CONCLUSION The strip temperature control at the heating furnace in continuous annealing processes has been developed using self-tuning control schemes. The control has the following important features. Plant model: Both dynamic and static models are developed and used in the control. The dynamic model of strip temperature includes the static model for forming disturbance terms and for gain scheduling, to increase model accuracy. Optimal preview control: In order to take control actions well before a setup change, the speed reference, the optimal timing of speed change and the optimal temperature trajectory (target trajectory) are calculated for the approaching setup change. Temperature tracking control: GPP-STC or GP-STC is employed for the control to cope with sluggish, nonminimum phase and slowly time-varying characteristics of the plant. In GP-STC, the cost function is modified to prevent overheating of the furnace. Parameter estimation: The model parameters are recursively estimated with REVVF, which uses the knowledge on variability of each parameter for better estimation.


Fig. 12.

155

Actual control.

The plant model is accurate enough to cope with most of operational changes, that is, changes of strip size, strip temperature or the line speed, using fixed parameter values. On the other hand, parameter estimation is to cope with slow changes of plant characteristics. It is our belief that, even in adaptive control, the model should be constructed to be as accurate as possible by including such as nonlinear gains or disturbance terms; and at the same time, to be of a suitable form for applying control theory. The control algorithm described so far has been incorporated into a software package for computer control of continuous annealing and processing line (CAPL) and continuous galvanizing line (CGL). It has been implemented in several real plants and is planned to be applied to some more ones, in Nippon Steel Corporation, Japan, and some other steelmanufacturing companies. In real plants, the control has been working quite successfully. Regarding the merits of the control in comparison with those of conventional PID control, the largest merit is production rate increase of 3 to 5%, as the control is accurate enough to relieve operators from frequent intervention of slowing down the line speed for fear of underheating of the strip. Other merits are energy conservation, quality improvement, et al. The economical benefit depends on each plant. The detailed figures are withheld here, but the benefit well justifies development and implementation of the control. As mentioned in Section III, the line speed is not chosen as a manipulated variable of temperature tracking control, because frequent speed change sometimes causes undesired effects on other parts of the process. However, as seen from Fig. 3(d), temperature response would be much faster if the line speed was used for the tracking control. Also, it is desirable to change the speed adequately for the optimization of entire process

operation. Therefore, it remains for future work to develop the control and optimization algorithm for changing the line speed to achieve the entire optimization including faster temperature tracking control.

ACKNOWLEDGMENT The authors would like to express their deepest gratitude to all the people who contributed to the development of the control in Nippon Steel Corporation, Japan, and to the reviewers of this paper for their helpful comments. The first author wishes to thank Prof. D. Q. Mayne and Mr. M. Rafiq for their suggestions and encouragements in the research on REVVF at Imperial College, U.K. REFERENCES [1] I. Ueda, M. Hosoda, and K. Taya, “Strip temperature control for a heating section in CAL,” in Proc. IECON’91, 1991, pp. 1946–1949. [2] K. Yahiro, K. Hirohata, T. Ooi, M. Haruna, K. Kuramoto, and K. Nakanishi, “Development of strip temperature control system for continuous annealing line,” in Proc. IECON’93, 1993, pp. 481–486. [3] N. Yoshitani, “Modeling and parameter estimation for strip temperature control in continuous annealing processes,” in Proc. IECON’93, 1993, pp. 469–474. [4] , “Optimal and adaptive control of strip temperature for a heating furnace in CAPL,” in Proc. 5th IFAC MMM Symp., Tokyo, Japan, 1986, pp. 380–385. [5] A. Hasegawa, “Development of a strip temperature control system with adaptive generalized predictive control,” in Proc. IEEE CCA’94, 1994. [6] H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in Proc. 2nd Ind. Symp. Inform. Theory, Akademiai Kiado, Budapest, Hungary, 1972, pp. 268–281. [7] A. Y. Allidina and F. M. Hughes, “Generalized self-tuning controller with pole-assignment,” Proc. Inst. Elect. Eng., pt. D, 1980, vol. 127, no. 1, pp. 13–18. [8] D. W. Clarke and P. J. Gawthrop, “Self-tuning control,” Proc. Inst. Elec. Eng., vol. 126, no. 6, pp. 633–640, 1979.

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[9] D. W. Clarke, C. Mohtadi, and P. S. Tuffs, “Genaralized predictive control—Part I, the basic algorithm,” Automatica, vol. 23, no. 2, pp. 137–148, 1987. , “Generalized predictive control—Part II, extensions and inter[10] pretations,” Automatica, vol. 23, no. 2, pp. 149–160, 1987. [11] N. Yoshitani, “A self-tuning regulator with multiple and variable forgetting factors,” EE-CON Rep. 82.13, Imperial College, U.K., 1982. , “A recursive parameter estimation with a vector-type variable [12] forgetting factor and its application to a real plant,” Trans. Soc. Instrument Contr. Eng., Japan, vol. 25, no. 5, 1989, pp. 71–77 (in Japanese). [13] A. O. Cordero and D. Q. Mayne, “Deterministic convergence of a selftuning regulator with variable forgetting factors,” Proc. Inst. Elec. Eng., pt. D, vol. 128, no. 1, pp. 19–23, 1981. [14] G. J. Bierman, “Measurement updating using the U-D factorization,” Automatica, vol. 12, no. 4-F, pp. 375–382, 1976.

Naoharu Yoshitani (M’95) received the B.Eng. and the M.Eng. degrees in electronic engineering from Kyoto University, Japan, in 1973 and 1975, respectively. From 1980 to 1982, he studied control theory and engineering at Imperial College, U.K., and received the M.Sc. degree. He received the D.Eng. degree on self-tuning control from Kyoto University, Japan, in 1990. He joined Nippon Steel Corporation, Japan, in 1975 and worked at Nagoya Steel Works and then at the Instrument and Control Laboratory until March 1996. Since April 1996, he has been an Associate Professor at Teikyo University, Japan. He is coauthor of Self-Tuning Control (Tokyo: SICE, 1996, in Japanese). His research interests include identification, adaptive control, and process control. He served as Associate Editor of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY from 1992 to 1995. He received a technical award from the Society of Instrument and Control Engineers (SICE), Japan, in 1988.

Akihiko Hasegawa received the B.Eng. degree in electric engineering and the M.Eng. degree in information engineering from Nagoya University, Japan, in 1980 and 1982, respectively. He joined Nippon Steel Corporation, Japan, in 1982, and worked at Nagoya Steel Works and then at the Plant Engineering and Technology Center. His research interests lie in intelligent control, adaptive control, and process control. Mr. Hasegawa is a member of the Society of Instrument and Control Engineers, Japan.