EP1624982B2 - Method for regulating the temperature of a metal strip, especially for rolling a metal hot strip in a finishing train - Google Patents

Abstract

The invention relates to a method for controlling and regulating the temperature of a metal strip in a finishing train of a hot rolling mill. A target function is formed by comparing a desired temperature gradient with an actual temperature gradient. The target function measures deviations from desired indications positioned in various places on the finishing train. The speed of the strip and the flow of the cooling agent are adjusted by predicting with the aid of a method of non-linear optimization with auxiliary conditions and are regulated and controlled online by solving a quadratic optimization problem with linear auxiliary conditions, preferably with the aid of an active set strategy.

Description

Method for controlling and / or regulating a metal strip in a finishing train

The present invention relates to a method for controlling and / or regulating the temperature of a metal strip in a finishing train.

From the WO 01/47648 A2 a method for controlling and / or regulating the temperature of a metal strip in a downstream of a finishing mill cooling section is known. In the context of this method, a desired temperature profile is compared with an actual temperature profile in order to determine control signals for actuators of the cooling section. Taking secondary conditions into consideration, at least one target function is formed for the actuators. Furthermore, a temperature profile for individual band points of the metal strip is determined.

From the DE 197 17 615 A1 Also, a method for controlling and / or regulating the temperature of a metal strip in a downstream of a finishing mill cooling section is known. In this method, a desired temperature is compared with an actual temperature to determine actuating signals for actuators of the cooling section. Taking secondary conditions into consideration, at least one target function is formed for the actuators.

From the US 4,274,273 A For example, there is known a method of controlling and / or controlling the temperature of a metal strip in a finishing train in which the metal strip is rolled from an initial thickness to a final thickness. To determine actuating signals, a desired temperature profile is compared with an actual temperature profile. It is determined a temperature profile for individual band points of the metal strip. There are determined control signals for the coolant flow, which is applied to the metal strip between the rolling stands. Furthermore, it is possible to determine actuating signals for the mass flow with which the metal strip passes through the finishing train.

The US 6,220,067 B1 describes a method that regulates the temperature of a metal strip at the exit side of a rolling line, ie the final rolling temperature. With such a method, phase transformations of the steel in the rolling train, which are particularly important in the case of two-phase rolling for the material properties of the rolled metal strip, can not be sufficiently influenced in a targeted manner. A comparable procedure, which is used to calculate a pass schedule, is described in the EP 1 014 239 A1 described.

The material properties and the structure of a rolled metal strip are determined by chemical composition and process parameters, in particular during the rolling process, such as e.g. determines the load distribution and the temperature control. Actuators for the rolling temperature, in particular the final rolling temperature are, depending on the type of plant and operating mode usually belt speed and interstand cooling.

It is an object of the invention to improve the control or regulation of the temperature of a metal strip, in particular in a finishing train, in such a way that disadvantages known from the prior art are avoided and in particular the control or regulation of the aforementioned actuators is improved.

The object according to the invention is achieved by a method for controlling and / or regulating the temperature of a metal strip in a finishing train, in which the metal strip is rolled from an input thickness to a final thickness, wherein a nominal temperature profile is compared with an actual temperature profile for the purpose of determining actuating signals is, wherein a temperature profile for individual band points of the metal strip is determined. Control signals for the mass flow are determined with which the metal strip passes through the finishing train. Alternatively or additionally, control signals for the coolant flow are determined, with which the metal strip is acted upon between the roll stands. Furthermore, at least one target function for actuators of the system in the finishing train is formed taking into account secondary conditions. The objective function is formed by solving an optimization problem, taking into account actuation limitations of the actuators when solving the optimization problem.

In determining the temperature profile for individual band points, the path and preferably additionally properties such as the temperature of individual band points are advantageously tracked. In this way, the accuracy of the control or regulation is significantly improved.

Advantageously, when the optimization problem is solved, technical boundary conditions, such as, in particular, control limits of the actuators, are taken into account in an extremely favorable manner, whereby, in particular, the greatest possible freedom for changing the actuators is ensured and the computing time required for the control or regulation is kept very low.

Advantageously, a target temperature is specified at the end of the finishing train. Alternatively or additionally, at least one desired temperature is predetermined in the finishing train. The control or regulation is thus substantially improved with regard to the material properties of the metal strip and with regard to its structural composition.

Advantageously, the actual temperature profile of the metal strip is determined with the aid of at least one model. Thus, an improved control or regulation of the temperature of the metal strip is made possible, even if the actual strip temperature can not be measured at locations relevant for the control, in particular in the finishing train.

Advantageously, the model is adapted online. In this way, an existing system drift can be taken into account and realistic results, in particular for the metal strips to be rolled next, can be determined.

Advantageously, to solve the objective function, an optimization problem with linear constraints online, i. especially in real time, solved. Adjustment limits are set up in particular in the form of equation or inequality constraints. The optimization solution advantageously delivers the values of the manipulated variables for a next controller cycle. Thus, a clear, uniform and plant configuration-independent built control is provided, which works reliably and quickly.

Advantageously, a quadratic optimization problem is solved. The optimization problem can be solved very quickly.

Advantageously, the optimization problem is solved by means of an active-set strategy. The optimization problem can be solved very effectively in real time.

Advantageously, an online capable stitching algorithm is precalculated by non-linear optimizations with constraints. The duration of the stitch plan calculation is thus kept extremely low. In particular, the stitch plan calculation provides optimal set-up values for the controller working online. Thus, the controller has sufficient degrees of freedom for belt temperature control.

The inventive method for controlling or regulating the temperature of a metal strip is particularly suitable for rolling strips with a thickness wedge, as used for example in semi-endless rolling at finished strip thicknesses below 1 mm. When rolling strips with thickness wedge, additional constraints on the actuators become active.

Further solutions of the above-described object are specified in claims 10 to 12. The advantages described for the method according to the invention apply accordingly.

FIG 1

den prinzipiellen Aufbau eines Walzwerks,

FIG 2

den schematischen Aufbau einer modell-prädiktiven Regelung für die Fertigstraße,

FIG 3

eine schematische Darstellung zur modellprädiktiven Regelung,

FIG 4

den Stell- bzw. Prädiktionshorizont für den Kühlmittelfluss, und

FIG 5

den Stell- bzw. Prädiktionshorizont für den Massenfluss.

Further advantages and details emerge from the following description of several embodiments of the invention in conjunction with the drawings. Here are an example:

FIG. 1

the basic structure of a rolling mill,

FIG. 2

the schematic structure of a model-predictive control for the finishing train,

FIG. 3

a schematic representation of the model predictive control,

FIG. 4

the setting or prediction horizon for the coolant flow, and

FIG. 5

the setting or prediction horizon for the mass flow.

FIG. 1 shows a plant for the production of metal strip 6, which includes a roughing 2, a finishing train 3 and a cooling section 4. Such systems are typical for the steel and metal industry. Behind the cooling section 4, a reel device 5 is arranged. From her is rolled down in the streets 2 and 3 preferably hot rolled and cooled in the cooling section 4 metal strip 6. The streets 2 and 3, a band source 1 is arranged upstream, which are heated, for example, as a furnace in the metal slabs, or, for example, as a continuous casting, in the metal strip 6 is generated is formed. The metal strip 6 is made of aluminum or steel, for example.

The system and in particular the roads 2, 3 and the cooling section 4 and the at least one reel device 5 are controlled by means of a control method which is carried out by a computing device 13. For this purpose, the computing device 13 is coupled with the individual components 1 to 5 of the plant for steel or aluminum production control technology. The computing device 13 is programmed with a computer program designed as a control program, based on which it carries out the inventive method for controlling or regulating the temperature of the metal strip 6.

According to FIG. 1 The metal strip or slab 6 leaves the strip source 1 and is then first rolled in the roughing train 2 to an input thickness for the finishing train 3. Within the finishing train, the belt 6 is then rolled by means of the rolling stands 3 'to its final thickness. The subsequent cooling section 4 cools the belt 6 to a predetermined reel temperature.

To ensure desired mechanical properties of the belt 6, a suitable temperature profile for the finishing train 3 and the cooling section 4 must be maintained. Since almost no widening of the rolled strip 6 takes place during the rolling process, strip length and, provided the mass flow remains constant, the strip speed also increases as a result of the rolling process.

FIG. 2 represents the finishing train 3 with its rolling stands 3 'closer and illustrates the inventive model-predictive control of the finishing train. 3

Within the finishing train 3, the contact times of the hot metal strip 6 with the relatively cold work rolls of the rolling stands 3 'and the inter-frame cooling devices 7 are the most important influencing factors on the temperature of the metal strip 6. The actuators of the control of the strip temperature in the finishing train are accordingly the mass flow 16 and the coolant flow 8. In FIG. 2 For ease of explanation of the embodiment, two band points P ₀ , P _{1 of} the metal strip 6 are exemplified.

The finishing train 3 is limited by its beginning x _A and its end x _E. The system dynamics in the finishing train 3 is characterized in terms of Temperatur.durch relatively large dead times 105. Thus, for example, the influence of a change in the coolant flow 8 to the temperature at the end x _{A of} the finishing train 3 can only be observed when the first belt point P ₀ , P _1, which was influenced by this change, leaves the last rolling stand 3 '. This is one reason why, according to the invention, the strip temperature control 17 is designed as a model-predictive control.

The computing device 13 for controlling the plant of the steel industry and in particular for controlling the finishing train 3 has a belt temperature model 12 and a belt temperature control 17. The belt temperature model 12 and the belt temperature control 17 preferably operate cyclically in control steps.

The strip temperature control 17 has a control device 14 which controls or regulates the coolant flow 8 of the intermediate-frame cooling devices 7 and the mass flow 16 of the metal strip 6, that is, in particular its speed v. The control device 14 is preceded by a linearized model 15, which is processed by means of a quadratic programming.

The module 12 for online determination of the strip temperature has an online monitor 9 for determining the current strip temperature, a module for online adaptation 10 and preferably a module for predicting 11 the temperature T ^j _{k = 0.1} selected band points P ₀ , P ₁ on.

The online monitor 9 uses a model for determining the current strip temperature and preferably the phase state of the metal strip 6 within the finishing train 3. The module 12 for online determination of the strip temperature therefore has a not shown in detail in the drawing belt temperature model. The band temperature model makes it possible, for example, to predict the final temperature of band points P ₀ , P ₁ , ie in particular the temperature of the band points PO, P1, at location x _E. Based on this, a linearized model 15 is created, which determines the strip temperature for an operating point of the finishing train 3 for a given change in the coolant flow 8 and / or given change in the mass flow 16.

By minimizing the quadratic deviation of the output of the linearized model 15, new correction values for coolant 8 and mass flow 16 are determined, taking into account setpoint values for strip intermediate temperatures, preferably within the finishing train, or given setpoint values for the final temperature of the strip 6 in the finishing train 3 become. The linearization of the belt temperature model results in a quadratic programming problem that can be solved sufficiently fast for on-line control of the belt temperature.

The purpose of the online monitor 9 is to change the current state, i. In particular, all the intermediate temperatures required for the control or regulation of the metal strip 6 of the finishing train 3 to determine. The data 102 present at the output of the online monitor 9 preferably also include real-time model corrections.

Tape data 101 actually measured in the finishing line and in particular temperatures may not always be present and as a rule only at a few specific locations, sometimes only at the locations x _A and x _E. The online adaptation 10 uses data 102 calculated by the online monitor 9, in particular temperatures determined by the online monitor 9, and preferably measured temperatures 101.

With the help of the online adaptation 10 correction factors are determined, which are used in particular for the correction of model errors in the online monitor 9. In this case, preferably actually measured temperatures 101 are compared with calculated temperatures 102. The online adaptation 10 is coupled both to the online monitor 9 and to the module 11 for the prediction of the temperature of selected band points.

Preferably, data originating from the output side of the online adaptation 10 is present at the input side of the module 11 for prediction of the strip temperature. The module 11 can further process data determined by the online monitor 9. The belt temperature calculated by the module 11 is forwarded to the belt temperature control 17. The belt temperature prediction module 11 also uses the belt temperature model of the belt temperature online module 12.

Input variables of the strip temperature control 17 or of the linearized model 15 are the actual temperature profile determined by the strip temperature model and a predetermined target temperature profile. The desired temperature profile is specified depending on the type of installation, the operating mode, the respective job and the desired properties of the metal strip 6.

The belt temperature control 17 uses input data 103 calculated by the belt temperature model 12. In this case, control specifications can be used particularly flexibly, since the online monitor 9 can determine any intermediate temperature of the belt 6 within the finishing train 3, even if there are no corresponding measured values.

FIG. 3 schematically illustrates problems relevant to model-predictive control, such as arise when metal is to be rolled in ferrite phase state. In addition to the temperature target specification T ^d ₂ at the end X _{E of} the finishing train 3 is preferably used further temperature setpoints T ^d ₀ , T ^d ₁ within the finishing mill 3. If, for example, the rolling operations of the two first rolling stands 3 'of the finishing train 3 in Austenite area, the other rolling operations, ie the rolling operations of the downstream rolling stands 3 ', but carried out in the ferrite range, you need at least three as in FIG. 3 shown target temperatures T ^d ₀ , T ^d ₁ , T ^d _2nd

The first setpoint temperature T ^d ₀ after the second rolling stand is to ensure that the temperature of the rolling operations in the first two rolling stands is above the transition temperature between the phase state areas. The second temperature setpoint T ^d ₁ is to ensure the phase transition before the third rolling stand of the finishing train 3. If possible, a final temperature T ^d ₂ at the end X _{E of} the finishing train 3 should be maintained.

The required predicted temperatures T ^j _{k = 0,1,2} are provided by the module 11 for prediction of the strip temperature by means of a model preferably for several band points P ₀ , P ₁ , P ₂ . The belt temperature control 17 can also respond to short-term temperature fluctuations, which are caused for example by the oven automation. However, this is preferably done by changing the coolant flow 8, and not by changing the belt speed v or the mass flow 16. Short-term temperature fluctuations, for example, local unevenness or folds of the metal strip 6 condition.

Long-term temperature fluctuations, which may be caused for example by a rolling train not shown in the drawing preceding the finishing train 3, are preferably compensated by acceleration a of the metal strip 6, that is to say by a change in the mass flow 16. The prediction horizon 106 is adjusted accordingly.

To do that in FIG. 3 It is preferable to solve this problem with the aid of the linearized model 15 solved as a minimization problem. Preferably, for this purpose, the control variables corresponding to the mass flow 16 and the coolant flow 8 are changed such that they correspond to the weighted quadratic error of the predicted temperatures T ^j _{k = 0,1,2} for the band points P ₀ , P ₁ , P ₂ with respect to the target Temperatures T ^d _{k = 0,1,2} minimize (see equation I ). Thus, at the individual valves 7, a coolant flow Q ₀ , Q ₁ or Q ₂ , collectively referred to as 8 causes, as far as possible from the technical limits of the inter-frame cooling devices 7, which are preferably designed as coolant or water valves 7 , lies away. Thus, the greatest possible freedom is achieved at the inter-frame cooling devices 7 to later, ie in subsequent control steps, to be able to respond to short-term temperature fluctuations.

The following flow limits of the inter-frame cooling devices 7 must be taken into account: The coolant flow Q ₀ , Q ₁ , Q _{2 of} a valve 7 can only be changed at a speed which corresponds to the dynamics of the respective valve 7 and must not outside technically conditioned minimum Q. ^max _i or maximum values Q ^max _i are. The mass flow 16 must also be within technical limits, which are determined in particular by a maximum or minimum speed of the metal strip when leaving the finishing train 3. With regard to the mass flow, a lower and an upper limit of the acceleration a of the metal strip 6 must also be taken into account.

kann dann ausgedrückt werden als $T_k^j+\mathrm{\Delta }{T}_k^j,$

wobei gilt: $$\mathrm{\Delta }{T}_k^j=\mathrm{\Delta }{T}_k^j\left(\mathrm{\Delta }{u}_{i_j}^j,\mathrm{\Delta }{u}_{i_{j+1}}^j,\dots \mathrm{\Delta }{u}_{j_{\mathit{kj}}}^j,\mathrm{\Delta }a,\mathrm{\Delta }s\right)\mathrm{.}$$

By means of the module 12, a prediction temperature T ^j _k for given coolant flow 8 and mass flow 16 and for an adaptation coefficient given for the corresponding control step are calculated with the aid of the belt temperature model. For further predictions, the adaptation coefficient is preferably frozen. In order to calculate the manipulated variables for the control for the next control steps, the current coolant flow 8 and the current mass flow 16 are set as the operating point. The new forecast temperature ${\tilde{T}}_k^j$

can then be expressed as $T_k^j+\mathrm{\Delta }{T}_k^j.$

where: $$\mathrm{\Delta }{T}_k^j=\mathrm{\Delta }{T}_k^j\left(\mathrm{\Delta }{u}_{i_j}^j.\mathrm{\Delta }{u}_{i_{j+1}}^j....\mathrm{\Delta }{u}_{j_{\mathit{kj}}}^j.\mathrm{\Delta }a.\mathrm{\Delta }s\right)\mathrm{,}$$

Finally, the following objective function is preferably expressed in the variables Δu ^j _i , Δa and Δs, in connection with FIGS Figures 5 and 6 is discussed in more detail, taking into account the previously mentioned Stellbegrengungen solved: $$\begin{array}{r}{\displaystyle \underset{j=0}{\overset{J-1}{\Sigma }}}\begin{array}{c}{\displaystyle \underset{k=0}{\overset{K-1}{\Sigma }}}\begin{array}{c}\frac{w_k^j}{2}{\left|T_k^j+\mathrm{\Delta }{T}_k^j-T_k^d\right|}^2\end{array}\end{array}+\frac{\delta }{2}{\displaystyle \underset{j=0}{\overset{J-1}{\Sigma }}}{\displaystyle \underset{i=i_j}{\overset{{}^{i}K-1.j}{\Sigma }}}{\left|Q_i^{\mathit{act}}+\mathrm{\Delta }{u}_i^j-\frac{Q_i^{\mathrm{Max}}+Q_i^{\min }}{2}\right|}^2\\ \frac{\alpha }{2}{\displaystyle \underset{j=0}{\overset{J-1}{\Sigma }}}{\displaystyle \underset{i=i_j}{\overset{{}^{i}K-1.j}{\Sigma }}}{\left|\frac{\mathrm{\Delta }{u}_i^j}{\mathrm{\Delta }t}\right|}^2+\frac{\mathit{\beta }}{2}{\left|\frac{\mathrm{\Delta }a}{\mathrm{\Delta }t}\right|}^2+\frac{\mathit{\gamma }}{2}{\left|\frac{\mathrm{\Delta }s}{\mathrm{\Delta }t}\right|}^2\end{array}$$

As FIG. 3 shows, the band temperature is predicted so far into the future until a band point P _{0 reaches} the last temperature setpoint T ^d ₂ . As a rule, this is at the end x _{E of} the finishing train 3, where preferably a not shown in detail in the drawing pyrometer measures the actual temperature of the metal strip 6. The model-predictive prediction always takes place for individual control steps .DELTA.t.

The FIGS. 4 and 5 illustrate the different setting horizon for the coolant flow (see FIG. 4 ) and the mass flow (see FIG. 5 ). In both figures, the abscissa represents a time axis.

The mass flow 16 is preferably influenced by the belt speed v, wherein the control horizon is preferably limited to a single control step. Subsequently, offset Δs and acceleration change Δa are preferably assumed to be constant (see FIG. 5 ). On the other hand, short-term temperature fluctuations are preferably influenced by the coolant flow Q _j . For this purpose, temperature prediction values are preferably used for band points P _j , which lie in front of the corresponding intermediate-frame cooling device 7 in the mass flow direction, so that the band points P _{j reach} the corresponding intermediate-frame cooling device only after the dead time 105 of the corresponding valve 7 plus the computing time has expired ,

(siehe Figur 4) bis zum Ende des Steuerungshorizonts vorgenommen wird, erfolgt die Aktualisierung des Kühlmittelflusses Q^act _{i j} nur unter Zuhilfenahme der ersten Korrektur $\mathrm{\Delta }{u}_{i_j}^j\mathrm{.}$

Um mögliche Oszillationen zu mindern, werden die aktualisierten Werte für $\mathrm{\Delta }{u}_{i_j}^j$

Δa und Δs ggf. mit einem Relaxationsfaktor 0 < χ ≤ 1 multipliziert.Although minimizing (II) taking into account all future coolant flow corrections $\mathrm{\Delta }{u}_i^j$

(please refer FIG. 4 ) is performed until the end of the control horizon, the updating of the coolant flow Q ^act _{ij is carried out} only with the aid of the first correction $\mathrm{\Delta }{u}_{i_j}^j\mathrm{,}$

To reduce possible oscillations, the updated values for $\mathrm{\Delta }{u}_{i_j}^j$

Δa and Δs may be multiplied by a relaxation factor 0 <χ ≤ 1.

Minimizing the equation (II) taking into account the corresponding control limits, especially those mentioned above, means solving a problem of non-linear programming, which is usually extremely computationally intensive and needs to be accelerated in order to be on-line. Control steps .DELTA.t can take place according to the invention, for example, every 200 milliseconds.

In order to achieve an acceleration, the procedure is preferably analogous to the Gauss-Newton method and the predicted temperature change is linearized around the operating point: $$\mathrm{\Delta }{T}_k^j\approx {\displaystyle \underset{i=i_j}{\overset{i_{\mathit{kj}}}{\Sigma }}}{S}_{\mathit{ki}}^j\mathrm{\Delta }{u}_i^j+{\tilde{S}}_k^j\mathrm{\Delta }a+{\tilde{S}}_k^j\mathrm{\Delta }s$$

${\tilde{S}}_k^j$

und ${\bar{S}}_k^j$

werden durch finite Differenzen wie folgt angenähert: $$S_{k,i_j}^j=\frac{T_k^j{|}_{Q_{i_j}^{\mathit{act}}+\mathrm{\Delta }}-T_k^j{|}_{Q_{i_j}^{\mathit{act}}}}{\mathrm{\Delta }}$$

$${\tilde{S}}_k^j=\frac{T_k^0{|}_{a^{\mathit{act}}+\mathrm{\Delta }}-T_k^0{|}_{a^{\mathit{act}}}}{\mathrm{\Delta }}$$

$${\bar{S}}_k^j=\frac{T_k^0{|}_{h_{\mathrm{exit}}{v}_{\mathit{exit}}^{\mathit{act}}+\mathrm{\Delta }}-T_k^0{|}_{{\mathrm{\nu }}_{\mathrm{exit}}{v}_{\mathit{exit}}^{\mathit{act}}}}{\mathrm{\Delta }}$$

The sensitivities $S_{\mathit{ki}}^j.$

${\tilde{S}}_k^j$

and ${\tilde{S}}_k^j$

are approximated by finite differences as follows: $$S_{k.i_j}^j=\frac{T_k^j{|}_{Q_{i_j}^{\mathit{act}}+\mathrm{\Delta }}-T_k^j{|}_{Q_{i_j}^{\mathit{act}}}}{\mathrm{\Delta }}$$

$${\tilde{S}}_k^j=\frac{T_k^0{|}_{a^{\mathit{act}}+\mathrm{\Delta }}-T_k^0{|}_{a^{\mathit{act}}}}{\mathrm{\Delta }}$$

$${\tilde{S}}_k^j=\frac{T_k^0{|}_{H_{\mathrm{exit}}{v}_{\mathit{exit}}^{\mathit{act}}+\mathrm{\Delta }}-T_k^0{|}_{{\mathrm{\nu }}_{\mathrm{exit}}{v}_{\mathit{exit}}^{\mathit{act}}}}{\mathrm{\Delta }}$$

${\tilde{S}}_k^j$

bzw. ${\bar{S}}_k^j$

zu ermitteln, muss das Bandtemperatur-Modell zusätzlich zur Vorhersage der Temperatur T^j _k nochmals gelöst werden. Gemäß der Gauß-Newton-Methode wird die Linearisierung (III) in den quadratischen Fehler der Zielfunktion (II) eingesetzt. Es ergibt sich folgende Näherung: $$\begin{array}{l}{\left|{T_k}^j+\mathrm{\Delta }{T_k}^j-{T_k}^d\right|}^2\approx \\ \approx {\left|T_k^j-T_k^d\right|}^2\\ +2\left(T_k^j-T_k^d\right){\displaystyle \sum _{i=i_j}^{i_{\mathit{kj}}}}{S}_{\mathit{ki}}^j\mathrm{\Delta }{u}_i^j\\ +2\left(T_k^j-T_k^d\right){\tilde{S}}_k^j\mathrm{\Delta }a\\ +2\left(T_k^j-T_k^d\right){\bar{S}}_k^j\mathrm{\Delta }s\\ +2{\tilde{S}}_k^j\mathrm{\Delta }a{\displaystyle \sum _{i=i_j}^{i_{\mathit{kj}}}}{S}_{\mathit{ki}}^j\mathrm{\Delta }{u}_i^j\\ +2{\bar{S}}_k^j\mathrm{\Delta }s{\displaystyle \sum _{i=i_j}^{i_{\mathit{kj}}}}{S}_{\mathit{ki}}^j\mathrm{\Delta }{u}_i^j\end{array}$$

$$\begin{array}{l}+2{\bar{S}}_k^j{\tilde{S}}_k^j\mathrm{\Delta }s\mathrm{\Delta }a\\ +{\displaystyle \sum _{i=i_j}^{i_{\mathit{kj}}}}{\displaystyle \sum _{l=i_j}^{i_{\mathit{kj}}}}{S}_{\mathit{ki}}^j{S}_{\mathit{kl}}^j\mathrm{\Delta }{u}_l^j\mathrm{\Delta }{u}_i^j+{\left|{\tilde{S}}_k^j\right|}^2{\left|\mathrm{\Delta }a\right|}^2+{\left|{\bar{S}}_k^j\right|}^2{\left|\mathrm{\Delta }s\right|}^2\mathrm{.}\end{array}$$

To the sensitivities $S_{\mathit{ki}}^j.$

${\tilde{S}}_k^j$

respectively. ${\tilde{S}}_k^j$

To determine the band temperature model in addition to the prediction of the temperature T ^j _{k must} be solved again. According to the Gauss-Newton method, the linearization (III) is used in the quadratic error of the objective function (II). The result is the following approximation: $$\begin{array}{l}{\left|{T_k}^j+\mathrm{\Delta }{T_k}^j-{T_k}^d\right|}^2\approx \\ \approx {\left|T_k^j-T_k^d\right|}^2\\ +2\left(T_k^j-T_k^d\right){\displaystyle \underset{i=i_j}{\overset{i_{\mathit{kj}}}{\Sigma }}}{S}_{\mathit{ki}}^j\mathrm{\Delta }{u}_i^j\\ +2\left(T_k^j-T_k^d\right){\tilde{S}}_k^j\mathrm{\Delta }a\\ +2\left(T_k^j-T_k^d\right){\tilde{S}}_k^j\mathrm{\Delta }s\\ +2{\tilde{S}}_k^j\mathrm{\Delta }a{\displaystyle \underset{i=i_j}{\overset{i_{\mathit{kj}}}{\Sigma }}}{S}_{\mathit{ki}}^j\mathrm{\Delta }{u}_i^j\\ +2{\tilde{S}}_k^j\mathrm{\Delta }s{\displaystyle \underset{i=i_j}{\overset{i_{\mathit{kj}}}{\Sigma }}}{S}_{\mathit{ki}}^j\mathrm{\Delta }{u}_i^j\end{array}$$

$$\begin{array}{l}+2{\tilde{S}}_k^j{\tilde{S}}_k^j\mathrm{\Delta }s\mathrm{\Delta }a\\ +{\displaystyle \underset{i=i_j}{\overset{i_{\mathit{kj}}}{\Sigma }}}{\displaystyle \underset{l=i_j}{\overset{i_{\mathit{kj}}}{\Sigma }}}{S}_{\mathit{ki}}^j{S}_{\mathit{kl}}^j\mathrm{\Delta }{u}_l^j\mathrm{\Delta }{u}_i^j+{\left|{\tilde{S}}_k^j\right|}^2{\left|\mathrm{\Delta }a\right|}^2+{\left|{\tilde{S}}_k^j\right|}^2{\left|\mathrm{\Delta }s\right|}^2\mathrm{,}\end{array}$$

$${\underline{b}}^{\mathit{lower}}\le \underline{\chi }\le {\underline{b}}^{\mathit{upper}}$$

Inserting the right-hand side of (VII) into (II), the quadratic programming problem arises in the following form: $$\min =f+{\underset{}{G}}^t\underset{}{\chi }+\frac{1}{2}\underset{}{{\chi }^t}\underset{}{\underset{}{H}}\underset{}{\chi }$$

$${\underset{}{b}}^{\mathit{lower}}\le \underset{}{\chi }\le {\underset{}{b}}^{\mathit{upper}}$$

Here, f is a scalar, H a symmetric, positive semidefinite NxN matrix, which is positive definite, if the positive parameters α, β, and γ are chosen to be sufficiently large. The remaining variables are n-dimensional column vectors. The inequality (IX) is to be understood component by component.

In order to solve the quadratic optimization problem, an active set strategy is preferably used.

According to the invention, in particular travel diagrams for the rolling speed v and / or for the water ramps or coolant ramps of the interstand cooling (7) are particularly advantageously calculated and maintained with particularly high accuracy.

In addition to the above and in particular initially discussed advantages of the invention, according to the invention in the control and / or regulation of the temperature of a metal strip 6 for the first time in a simple manner also allows a different weighting of the relevant control parameters in terms of prioritization.

According to the invention there is provided a flexible control method which is also applicable to other parts of the plant, such as e.g. in particular the roughing 2 or the cooling section 4, can be used. A more than one part of the system 1 to 5 cross-application of the invention is possible. Particularly advantageous is the use of the invention in two-phase rolling and driving a thickness wedge during the rolling of a semi-endless slab.

Claims (12)

1. Method for controlling and/or regulating the temperature of a metal strip (6) in a finishing train (3), in which the metal strip (6) is rolled from an input thickness to an end thickness,

   - a setpoint temperature profile being compared with an actual temperature profile in order to determine control signals,

   - and at least one target function for the system control elements in the finishing train (3) being formed by taking constraints into account,

   - a temperature profile being determined for individual strip points (P₀, P₁, P₂ or P_j) of the metal strip (6) ,

   - control signals being determined for the mass flow (16) with which the metal strip (6) passes through the finishing train (3), and/or control signals being determined for the coolant flow (8) which acts upon the metal strip (6) between the rolling stands (3'),

   - the target function being formed by solving an optimization problem, control limits of the system control elements being taken into account during the solving of the optimization problem.
2. Method according to Claim 1, characterized in that a setpoint temperature (T^d ₂) at the end of the finishing train (3) is specified.
3. Method according to one of the preceding claims, characterized in that at least one setpoint temperature (T^d ₀, T^d ₁) in the finishing train (3) is specified.
4. Method according to one of the preceding claims, characterized in that the actual temperature profile of the metal strip (6) is determined by utilizing at least one model (9 or 12).
5. Method according to Claim 4, characterized in that the model (9) is adapted online.
6. Method according to one of the preceding claims, characterized in that an optimization problem with linear constraints is solved online in order to solve the target function.
7. Method according to Claim 6, characterized in that a quadratic optimization problem is solved.
8. Method according to Claim 6 or according to Claim 7, characterized in that the optimization problem is solved with the aid of an active set strategy.
9. Method according to one of the preceding claims, characterized in that an online-capable pass schedule algorithm is precalculated by nonlinear optimization with constraints.
10. Computer program product comprising program code means suitable for carrying out the steps of a method according to one of the preceding claims, when the computer program product is run on a computing device.
11. Computing device (13) for carrying out the method according to one of Claims 1 to 9, the computing device (13) directly and/or indirectly influencing the temperature of the metal strip (6), characterized in that the computing device is programmed with a computer program product according to Claim 10.
12. Computing device according to Claim 11, characterized in that it comprises a module (12) for online determination of the strip temperature with the aid of a model and a module (17) for the strip temperature regulation.

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