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:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application claims priority to the German applications No. 10308222.0, filed Feb. 25, 2003 and No. 10321791.6, filed May 14, 2003, and to the International Application No. PCT/EP2004/001366, filed Feb. 13, 2004 which are incorporated by reference herein in their entirety. 
   FIELD OF INVENTION 
   The invention relates to a method for controlling and regulating the temperature of a metal strip, e.g. of steel or aluminum, in a finishing train for rolling a metal hot strip. 
   BACKGROUND OF INVENTION 
   U.S. Pat. No. 6,220,067 B1 describes a method which regulates the temperature of a metal strip at the output end of a mill train, i.e. the final rolling temperature. A method of this type cannot adequately selectively influence phase changes, which especially in dual-phase rolling are of significance for the material properties of the rolled metal strip, in the steel in the mill train. A comparable method, which serves for calculating a pass schedule, is described in EP 1 014 239 A1. 
   SUMMARY OF INVENTION 
   The material properties and the structure of a rolled metal strip are determined by chemical composition and process parameters, especially during the rolling process, such as e.g. load distribution and temperature management. Final control elements for the rolling temperature, in particular the final rolling temperature, are, depending on the type of plant and mode of operation, generally speed of the strip and inter-stand cooling. 
   An object of the invention is to improve the control or regulation of the temperature of a metal strip, especially in a finishing train, such that disadvantages known from the prior art are avoided and in particular that the control or regulation of the aforementioned final control elements is improved. 
   The object according to the invention is achieved in a method for controlling and/or regulating the temperature of a metal strip, especially in a finishing train, whereby, in order to determine adjustment signals, a desired temperature gradient is compared with an actual temperature gradient, whereby a temperature gradient for individual strip points on the metal strip is determined and whereby, taking into account auxiliary conditions, at least one target function is formed for final control elements of the plant in the finishing train. 
   In determining the temperature gradient for individual points on the strip, the path and preferably also properties such as the temperature of individual points on the strip are advantageously traced. In this way, the precision of the control and regulation is significantly improved. 
   Advantageously, the target function is solved by solving an optimization problem. Here, technical constraints such as in particular adjustment limitations of the final control elements are taken into account in an extremely favorable manner whereby, in particular, as much scope as possible is provided for changing the final control elements and whereby the computing time needed for controlling and regulating is kept very low. 
   Advantageously, a desired temperature at the end of the finishing train is predetermined. Alternatively, or in addition, at least one desired temperature in the finishing train is predetermined. Control and regulation are in this way substantially improved with regard to the material properties of the metal strip and with regard to its structural composition. 
   Advantageously, the actual temperature gradient of the metal strip is determined with the aid of at least one model. In this way, improved control or regulation of the temperature of the metal strip is enabled, even if the actual temperature of the strip cannot be measured at points, especially in the finishing train, relevant for control or regulation. 
   Advantageously, the model is adapted online. In this way, any plant drift that exists can be taken into account and realistic results, especially for the next metal strips to be rolled, can be determined. 
   Advantageously, adjustment signals are determined for the flow of the cooling agent. 
   Advantageously, actuating signals are determined for the flow of the material. 
   Advantageously, in order to solve the target function, an optimization problem with linear auxiliary conditions is solved online, i.e. in particular in real time. Adjustment limitations are established here, in particular in the form of equality or inequality auxiliary conditions. Solution of the optimization advantageously returns here the values of the adjustment variables for a next controller cycle. This provides regulation that is structured clearly, uniformly and independently of the plant configuration and that works reliably and fast. 
   Advantageously, a quadratic optimization problem is solved. The optimization problem can in this way be solved particularly fast. 
   Advantageously, the optimization problem is solved with the aid of an active set strategy. The optimization problem can in this way be solved particularly effectively in real time. 
   Advantageously, an online-capable pass schedule algorithm is calculated in advance by means of non-linear optimizations with auxiliary conditions. The length of time for calculating the pass schedule is in this way kept extremely small. The calculation of the pass schedule returns set-up values which are in particular optimally matched to the controller operating online. In this way the controller has sufficient scope to influence the temperature of the strip. 
   The inventive method for controlling and for regulating the temperature of a metal strip is in particular also suitable for rolling strips with a thickness wedge, as is used for example in semi-continuous rolling with finished strip thicknesses below 1 mm. When rolling strips with a thickness wedge, additional auxiliary conditions with regard to the final control elements become active. 
   Further embodiments are included in the remaining independent and dependent claims. The advantages described for the method according to the invention apply analogously. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further advantages and details will emerge from the description below of several exemplary embodiments of the invention and from the associated drawings in which by way of example: 
       FIG. 1  shows the basic structure of a rolling mill, 
       FIG. 2  shows the schematic arrangement of a model-predictive control for the finishing train, 
       FIG. 3  shows a schematic representation relating to the model-predictive control, 
       FIG. 4  shows the adjustment or prediction horizon for the flow of the cooling agent, and 
       FIG. 5  shows the adjustment or prediction horizon for the flow of the material. 
   

   DETAILED DESCRIPTION OF INVENTION 
     FIG. 1  shows a plant for the production of metal strip  6 , comprising a roughing train  2 , a finishing train  3  and a cooling stretch  4 . Plants of this type are typical for the steel and metal industry. A reeling device  5  is arranged downstream of the cooling stretch  4 . The metal strip  6  which is rolled preferably hot in the trains  2  and  3  and cooled in the cooling stretch  4  is reeled in by said reeling device. A strip source  1  is arranged upstream of the trains  2  and  3 , which strip source is fashioned for example as a furnace in which metal slabs are heated or for example as a continuous casting plant in which metal strip  6  is produced. The metal strip  6  consists for example of aluminum or steel. 
   The plant and in particular the trains  2 ,  3  and the cooling stretch  4  and the at least one reeling device  5  are controlled by means of a control method which is executed by a computing device  13 . To this end, the computing device  13  has control engineering links to the individual components  1  to  5  of the plant for steel or aluminum production. The computing device  13  is programmed with a control program fashioned as a computer program, on the basis of which it executes the method according to the invention for controlling and regulating the temperature of the metal strip  6 . 
   In accordance with  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 . Inside the finishing train, the strip  6  is then rolled by means of the rolling stands  3 ′ to its final thickness. The subsequent cooling stretch  4  cools the strip  6  to a predetermined reeling temperature. 
   In order to ensure desired mechanical properties in the strip  6 , a suitable temperature gradient has to be observed for the finishing train  3  and the cooling stretch  4 . Since virtually no widening of the rolled strip  6  occurs during the rolling process, the length of the strip and—provided the flow of material remains constant—the speed of the strip increase through the rolling process. 
     FIG. 2  presents in detail the finishing train  3  with its rolling stands  3 ′ and illustrates the model-predictive regulation of the finishing train  3  according to the invention. 
   Inside the finishing train  3 , the times of contact of the hot metal strip  6  with the relatively cold working rolls of the rolling stands  3 ′ and the inter-stand cooling devices  7  are the most important factors influencing the temperature of the metal strip  6 . The final control elements for controlling and regulating the temperature of the strip in the finishing train are accordingly the flow of the material  16  and the flow of the cooling agent  8 . In  FIG. 2 , two strip points P 0 , P 1  on the metal strip  6  are highlighted by way of example in order to simplify the explanation of the exemplary embodiment. 
   The finishing train  3  is delimited by its start x A  and its end x E . The plant dynamics in the finishing train  3  are characterized in terms of temperature by relatively long idle times  105 . Thus, for example, the influence of a change in the flow of the cooling agent  8  on the temperature at the end x A  of the finishing train  3  can be observed only when the first strip point P 0 , P 1  which was influenced by this change leaves the last rolling stand  3 ′. That is one reason why regulation of the strip temperature  17  according to the invention is fashioned as model-predictive regulation. 
   The computing device  13  for controlling the steel industry plant and in particular for controlling the finishing train  3  has a strip temperature model  12  and a strip temperature regulation  17 . The strip temperature model  12  and the strip temperature regulation  17  operate preferably cyclically in regulating steps. 
   The strip temperature regulation  17  has a regulating device  14  which controls and regulates the flow of the cooling agent  14  of the inter-stand cooling devices  7  and the flow of the material  16  of the metal strip  6 , i.e. in particular the speed v of said metal strip. Upstream of the regulating device  14  is a linearized model  15  which is processed with the aid of quadratic programming. 
   The module  12  for determining the strip temperature online has an online monitor  9  for ascertaining the current strip temperature, a module for online adaptation  10  and preferably a module for predicting  11  the temperature T j   k=0, 1  of selected points P 0 , P 1  on the strip. 
   The online monitor  9  uses a model for determining the current strip temperature and preferably the phase status of the metal strip  6  inside the finishing train  3 . The module  12  for determining the strip temperature online therefore has a strip temperature model, not shown in detail in the drawings. The strip temperature model makes it possible for example to predict the final temperature of strip points P 0 , P 1 , i.e. in particular the temperature of the strip points P 0 , P 1 , at the position x E . Taking this as a starting point, a linearized model  15  is set up which determines the strip temperature for a working point of the finishing train  3  for a given change in the flow of the cooling agent  8  and/or a given change in the flow of the material  16 . 
   By minimizing the quadratic deviation of the output of the linearized model  15 , new correction values are determined for flow of the cooling agent  8  and flow of the material  16 . Given desired values for interim strip temperatures preferably inside the finishing train or given desired values for the final temperature of the strip  6  in the finishing train  3  are taken into account in determining these correction values. Through linearization of the strip temperature model, a quadratic programming problem is produced which can be solved sufficiently fast to allow online control of the strip temperature. 
   The task of the online monitor  9  is to determine the current status, i.e. in particular all the interim temperatures needed for control and regulation, of the metal strip  6  in the finishing train  3 . The data  102  available at the output of the online monitor  9  preferably also contains real-time model corrections. 
   Strip data  101  actually measured in the finishing train, and in particular temperatures, will possibly not always be available and generally only at a few defined points, sometimes only at the points x A  and x E . Online adaptation  10  uses data  102  computed by the online monitor  9 , in particular temperatures determined by the online monitor, as well as preferably measured temperatures  101 . 
   With the aid of the online adaptation  10 , correction factors are determined which are used in particular for correcting model errors in the online monitor  9 . Here, temperatures actually measured  101  are preferably compared with calculated temperatures  102 . The online adaptation  10  is linked both to the online monitor  9  and to the module  11  for predicting the temperature of selected points on the strip. 
   Data originating from the output end of the online adaptation  10  is preferably available at the input end of the module  1  for predicting the strip temperature. The module  11  can process further data determined by the online monitor  9 . The strip temperature calculated by the module  11  is passed on to the strip temperature regulation  17 . The module  11  for predicting the strip temperature also uses the strip temperature model of the module  12  for determining the strip temperature online. 
   Input variables of the strip temperature regulation  17  and of the linearized model  15  are the actual temperature gradient determined by the strip temperature model and a predetermined desired temperature gradient. The desired temperature gradient is predetermined depending on the plant type, the operating mode, the respective job and the desired properties of the metal strip  6 . 
   The strip temperature regulation  17  uses input data  103  calculated by the strip temperature model  12 . Here, control specifications can be used particularly flexibly since the online monitor  9  can determine any interim temperature of the strip  6  inside the finishing train  3 , even if no appropriate measured values are available. 
     FIG. 3  illustrates schematically problems relevant to model-predictive regulation, such as arise, for example, when metal in the ferrite-phase status range is to be rolled. Besides the desired temperature indication T d   2  at the end x E  of the finishing train  3 , further desired temperature values T d   0 , T d   1  inside the finishing train  3  are preferably used. If, for example, the rolling operations of the first two rolling stands  3 ′ of the finishing train  3  are to occur in the austenite range, but the remaining rolling operations, i.e. the rolling operations of the downstream rolling stands  3 ′, in the ferrite range, at least three desired temperatures T d   0 , T d   1 , T d   2 , as shown in  FIG. 3 , are needed. 
   The first desired 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 lies above the transition temperature between the phase status ranges. The second desired temperature value 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 also to be met. 
   The predicted temperatures needed T J   k=0, 1, 2  are provided by the module  11  for predicting the strip temperature with the aid of a model preferably for multiple points P 0 , P 1 , P 2 , on the strip. The strip temperature regulation  17  can also respond to short-term temperature fluctuations that are caused, for example, by the furnace automatic control. However, this preferably takes place as a result of a change in the flow of the cooling agent  8  and not by a change in the strip speed v or in the flow of the material  16 . Short-term temperature fluctuations may, for example, cause local unscheduled irregularities or folds in the metal strip  6 . 
   Long-term temperature fluctuations, which may be caused, for example, by a rolling operation preceding the finishing train  3  and not shown in detail in the drawings, are preferably compensated for by acceleration a of the metal strip  6 , i.e. by a change in the flow of the material  16 . The prediction horizon  106  is adapted accordingly. 
   In order to solve the problem shown in  FIG. 3 , it is preferably solved as a minimization problem with the aid of the linearized model  15 . To this end, the control variables corresponding to the flow of the material  16  and the flow of the cooling agent  8  are preferably changed such that they minimize the weighted quadratic error of the predicted temperatures T j   k=0, 1, 2  for the strip points P 0 , P 1 , P 2  with reference to the desired temperatures T d   k=0, 1, 2  (see equation I). Thus, at the individual valves  7 , a coolant flow Q 0 , Q 1  and Q 2 , jointly referred to as  8 , is effected which lies as far as possible from the technical limits of the inter-stand cooling devices  7 , which are preferably fashioned as coolant valves or water valves  7 . In this way, the maximum possible tolerance is achieved at the inter-stand cooling devices  7  so as later, i.e. in subsequent regulating steps, to be able to respond to short-term temperature fluctuations. 
   The following adjustment limitations of the inter-stand cooling devices  7  must be taken into consideration: the coolant flow Q 0 , Q 1 , Q 2  of a valve  7  can be changed only with a speed which matches the dynamics of the respective valve  7  and must not lie outside technically determined minimum Q max   i  and maximum Q min   i  values. The flow of the material  16  must also lie within technical threshold values which are determined in particular by a maximum and a minimum speed of the metal strip upon leaving the finishing train  3 . As far as the flow of the material is concerned, a lower and an upper limit on the acceleration a of the metal strip  6  must also be observed. 
   A predicted temperature T j   k  for a given flow of the cooling agent  8  and flow of the material  16  and for a given adaptation coefficient for the regulating step concerned is calculated by the module  12  with the aid of the strip temperature model. The adaptation coefficient is preferably frozen for further predictions. In order to calculate the adjustment variables for control for the next control steps, the current flow of the cooling agent  8  and the current flow of the material  16  are set as a working point. The new predicted temperature {tilde over (T)} k   j  can then be expressed as T k   j +ΔT k   j , the following applying: 
   
     
       
         
           
             
               
                 
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   Finally, the target function reproduced below in the variables Δu j   i , Δa and Δs, more details of which will be given in connection with  FIGS. 5 and 6 , is preferably solved, taking into account the adjustment limitations specified previously: 
   
     
       
         
           
             
               
                 
                   
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                 ( 
                 II 
                 ) 
               
             
           
         
       
     
   
   As  FIG. 3  shows, the strip temperature is predicted into the future until such time as a point on the strip P 0  reaches the last desired temperature value T d   2 . As a rule, this lies at the end x E  of the finishing train  3 , where a pyrometer, not shown in detail in the drawings, preferably measures the actual temperature of the metal strip  6 . The model-predictive prediction is carried out constantly for individual regulating steps Δt. 
     FIGS. 4 and 5  illustrate the different adjustment horizon for the flow of the cooling agent (see  FIG. 4 ) and for the flow of the material (see  FIG. 5 ). In both Figures, the abscissa represents a time axis. 
   The flow of the material  16  is preferably influenced by the strip speed v, the adjustment horizon preferably being restricted to a single regulating step. Offset Δs and change in acceleration Δa are then preferably assumed to be constant (see  FIG. 5 ). Short-term temperature fluctuations, by contrast, are preferably influenced by the flow of the cooling agent Q j . For this, temperature prediction values are preferably used for strip points P j  which, viewed in the direction of flow of the material, lie upstream of the corresponding inter-stand cooling device  7 , so that the strip points P j  do not reach the corresponding inter-stand cooling device until the idle time  105  of the corresponding valve  7  plus the computing time have expired. 
   Although the minimization (II) is carried out, taking into consideration all future coolant flow corrections Δu i   j  (see  FIG. 4 ) until the end of the setting horizon, the coolant flow Q act   ij  is updated only with the aid of the first correction Δu i     j     j . In order to reduce possible oscillations, the updated values for Δu i     j     j Δa and Δs are where applicable multiplied with a relaxation factor 0&lt;χ≦1. 
   Minimizing the equation (II) taking into account the corresponding adjustment limitations, especially those mentioned previously, means solving a non-linear programming problem which is as a rule extremely computation-intensive and which, in order to be online-capable, has to be accelerated. Regulating steps Δt can, according to the invention, be carried out, for example, every 200 milliseconds. 
   In order to achieve an acceleration, the procedure followed is preferably analogous to the Gauss-Newton method and linearizes the predicted temperature change about the working point: 
   
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
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                     k 
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                         ki 
                       
                     
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                     s 
                   
                 
               
             
             
               
                 ( 
                 III 
                 ) 
               
             
           
         
       
     
   
   The sensitivities S ki   j , {tilde over (S)} k   j  and  S   k   j  are approximated by finite differences as follows: 
   
     
       
         
           
             
               
                 
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                 ( 
                 IV 
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                       k 
                       0 
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             
                               
                                 h 
                                 exit 
                               
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                           exit 
                         
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                   Δ 
                 
               
             
             
               
                 ( 
                 VI 
                 ) 
               
             
           
         
       
     
   
   In order to determine the sensitivities S ki   j , {tilde over (S)} k   j  and  S   k   j , the strip temperature model, in addition to the prediction of the temperature T j   k , has to be solved once again. According to the Gauss-Newton method, the linearization (III) is inserted in the quadratic error of the target function (II). The following approximation is produced: 
   
     
       
         
           
             
               
                 
                   
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                         T 
                         k 
                         j 
                       
                       + 
                       
                         Δ 
                         ⁢ 
                         
                             
                         
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                       - 
                       
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                         k 
                         d 
                       
                     
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                           d 
                         
                       
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                     2 
                   
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                     ⁢ 
                     
                       ( 
                       
                         
                           T 
                           k 
                           j 
                         
                         - 
                         
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                           k 
                           d 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           
                             i 
                             j 
                           
                         
                         
                           i 
                           kj 
                         
                       
                       ⁢ 
                       
                         
                           S 
                           ki 
                           j 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           u 
                           i 
                           j 
                         
                       
                     
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           T 
                           k 
                           j 
                         
                         - 
                         
                           T 
                           k 
                           d 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         S 
                         ~ 
                       
                       k 
                       j 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           T 
                           k 
                           j 
                         
                         - 
                         
                           T 
                           k 
                           d 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         S 
                         _ 
                       
                       k 
                       j 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     s 
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       
                         S 
                         ~ 
                       
                       k 
                       j 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           
                             i 
                             j 
                           
                         
                         
                           i 
                           kj 
                         
                       
                       ⁢ 
                       
                         
                           S 
                           ki 
                           j 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           u 
                           i 
                           j 
                         
                       
                     
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       
                         S 
                         _ 
                       
                       k 
                       j 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     s 
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           
                             i 
                             j 
                           
                         
                         
                           i 
                           kj 
                         
                       
                       ⁢ 
                       
                         
                           S 
                           ki 
                           j 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           u 
                           i 
                           j 
                         
                       
                     
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       
                         S 
                         _ 
                       
                       k 
                       j 
                     
                     ⁢ 
                     
                       
                         S 
                         ~ 
                       
                       k 
                       j 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     s 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                   
                   + 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         
                           i 
                           j 
                         
                       
                       
                         i 
                         kj 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           
                             i 
                             j 
                           
                         
                         
                           i 
                           kj 
                         
                       
                       ⁢ 
                       
                         
                           S 
                           ki 
                           j 
                         
                         ⁢ 
                         
                           S 
                           ki 
                           j 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           u 
                           i 
                           j 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           u 
                           i 
                           j 
                         
                       
                     
                   
                   + 
                   
                     
                       
                          
                         
                           
                             S 
                             ~ 
                           
                           k 
                           j 
                         
                          
                       
                       2 
                     
                     ⁢ 
                     
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           a 
                         
                          
                       
                       2 
                     
                   
                   + 
                   
                     
                       
                          
                         
                           
                             S 
                             _ 
                           
                           k 
                           j 
                         
                          
                       
                       2 
                     
                     ⁢ 
                     
                       
                          
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           s 
                         
                          
                       
                       2. 
                     
                   
                 
               
             
             
               
                 ( 
                 VII 
                 ) 
               
             
           
         
       
     
   
   If the right-hand side of (VII) is now inserted in (II), then the quadratic programming problem presents itself in the following form: 
   
     
       
         
           
             
               
                 min 
                 = 
                 
                   f 
                   + 
                   
                     
                       
                         g 
                         _ 
                       
                       t 
                     
                     ⁢ 
                     
                       χ 
                       _ 
                     
                   
                   + 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       
                         
                           χ 
                           t 
                         
                         ⁢ 
                         
                           H 
                           _ 
                         
                         ⁢ 
                         χ 
                       
                       _ 
                     
                   
                 
               
             
             
               
                 ( 
                 VIII 
                 ) 
               
             
           
           
             
               
                 
                   
                     b 
                     _ 
                   
                   lower 
                 
                 ≤ 
                 
                   χ 
                   _ 
                 
                 ≤ 
                 
                   
                     b 
                     _ 
                   
                   upper 
                 
               
             
             
               
                 ( 
                 IX 
                 ) 
               
             
           
         
       
     
   
   Here f is a scalar, H a symmetrical, positive semi-definite N×N matrix which is positively definite when the positive parameters α, β and γ are chosen sufficiently large. The remaining variables are n-dimensional column vectors. The inequality (IX) is to be understood in component terms. 
   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 inter-stand cooling (7) are particularly advantageously calculated and matched with especially high precision. 
   In addition to the advantages of the invention hereinabove and especially described in the introduction, the invention enables for the first time in the control and/or regulation of the temperature of a metal strip  6  in a simple manner a different weighting, in the sense of a prioritization, of the indications relevant for said control. 
   According to the invention, a flexible controlling and regulating method is provided which can also be used for other plant parts such as e.g. in particular the roughing train  2  or else the cooling stretch  4 . A use of the invention covering more than one part of the plant  1  to  5  is possible. Use of the invention is particularly advantageous in dual-phase rolling and in the travel of a thickness wedge during the rolling of a semi-continuous slab.