Patent Publication Number: US-10307806-B2

Title: Rolling control method for metal strip, rolling control apparatus, and manufacturing method for rolled metal strip

Description:
TECHNICAL FIELD 
     The present invention relates to a rolling control method for controlling the profile of a metal strip after rolling, a rolling control apparatus that performs the rolling control method, and a manufacturing method for a rolled metal strip. 
     BACKGROUND ART 
     Various methods have been proposed as technology for predicting the profile of a metal strip, such as a sheet or a plate, after rolling. 
     For example, Japanese Patent Application Laid-Open (JP-A) No. 2008-112288 describes technology that improves the prediction precision for an extrapolation region for which actual data does not exist, and also corrects errors in a rolling model. Specifically, a database of actual results, in which manufacturing conditions of previously manufactured products are stored associated with manufacture outcome information, is employed to compute a degree of similarity between respective samples in the database of actual results and request points (prediction target points), and to generate a prediction formula for the vicinity of the request points using weighted regression weighted by the degree of similarity. The prediction precision for the extrapolation region is improved by the prediction formula. 
     JP-A No. 2005-153011 describes technology that predicts the profile of a metal strip by splitting elongation strain (stress) that is distributed in a strip width direction of a metal strip during rolling into elongation strain that is geometrically transformed into a wave profile during buckling, and elongation strain still present in the metal strip after buckling. 
     Moreover, JP-A No. 2012-218010 describes technology that predicts the profile of a metal strip by measuring characteristic amounts of the profile of the metal strip at exit from a rolling mill, and also finding elongation strain present in the metal strip during measurement, then superimposing the elongation strain on the profile characteristic amounts, and measuring this as true profile characteristic amounts applied by the rolling mill. Note that positions in a strip passing direction of the strip and a width direction of the strip, and height direction displacement, are measured on exit from the rolling mill as geometric values. Moreover, profile, steepness, and elongation strain difference are found as the profile characteristic amounts. 
     SUMMARY OF INVENTION 
     Technical Problem 
     However, in the method described in JP-A No. 2008-112288, no consideration is given to non-linear phenomena such as buckling of the metal strip, and such non-linear phenomena cannot be reflected in the prediction formula. Moreover, modelling errors arise when no consideration is given to non-linear phenomena, and so the profile of the metal strip after rolling cannot be accurately predicted. 
     In the inventions described in JP-A Nos. 2005-153011 and 2012-218010, consideration is given to buckling of the metal strip when predicting the profile of the metal strip, thereby improving the prediction precision in comparison to cases in which buckling is not taken into consideration. However, careful investigation by the inventors has revealed that there is still room for improvement in improving the prediction precision, as explained below. 
     In consideration of this point, an object of the present invention is to predict the profile of a metal strip after rolling with good precision, and to give excellent control of the profile of the metal strip. 
     Solution to Problem 
     In order to achieve the above object, the inventors investigated methods for predicting the profile of a metal strip after rolling, and controlling the profile of a metal strip based on the predicted profile of the metal strip. The inventors reached the following findings. 
     As described in JP-A No. 2005-153011, technology is known in which rolling direction elongation strain distributed in a strip width direction of a metal strip is split into elongation strain that is geometrically transformed into a wave profile by buckling, and elongation strain still present in the metal strip after buckling. Moreover, the invention described in JP-A No. 2012-218010 expands on the invention described in JP-A No. 2005-153011, and determines a true elongation strain distribution by finding the elongation strain distribution that is not transformed into a wave profile and is still present in the metal strip after buckling, and superimposing this on the elongation strain distribution that is transformed into a wave profile of the metal strip measured on exit from the rolling mill. The profile of the metal strip is then controlled using feedback control. 
     The present invention expands further on the inventions described in JP-A Nos. 2005-153011 and 2012-218010. The inventors discovered that there is correlation between rolling load difference distribution and elongation strain difference distribution in the strip width direction of a metal strip that undergoes changes due to buckling. By quantitatively establishing this correlation, the inventors found that it is possible to find a true elongation strain difference distribution of the metal strip. Namely, out of the elongation strain difference distributed in the strip width direction of the metal strip, when the elongation strain difference that is transformed into a wave profile so as to cause out-of-plane deformation is transformed into a wave profile by actual buckling of the metal strip, the load distribution corresponding to the elongation strain difference is further transformed into an elongation strain difference present in the metal strip. Namely, it was found that the true elongation strain difference of the metal strip is greater than hitherto imagined. Predicting the true elongation strain difference of the metal strip in this manner enables the profile of the metal strip to be controlled with greater precision. The gist of the present invention is as follows. 
     A first aspect of the present invention provides a rolling control method including: finding a critical buckling strain difference distribution, which is a distribution in a strip width direction of differences in a critical strain at which a metal strip will buckle, based on a strip thickness of the metal strip, a strip width of the metal strip, tension acting on the metal strip at exit from a rolling mill, and a provisional elongation strain difference distribution which is a distribution of differences in the strip width direction of elongation strain along a rolling direction of the metal strip during rolling under specific rolling conditions, and which is found under conditions in which out-of-plane deformation of a metal strip is restrained; in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution, finding a true elongation strain difference distribution by adding the difference between the provisional elongation strain difference distribution and the critical buckling strain difference distribution to the provisional elongation strain difference distribution; and rolling the metal strip without changing the specific rolling conditions in cases in which the provisional elongation strain difference distribution does not exceed the critical buckling strain difference distribution, and rolling the metal strip under rolling conditions set based on the true elongation strain difference distribution in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution. 
     A second aspect of the present invention provides the rolling control method of the first aspect, further including finding the provisional elongation strain difference distribution. 
     A third aspect of the present invention provides the rolling control method of either the first aspect or the second aspect, wherein, when finding the true elongation strain difference distribution, a converted tension is found by converting a difference between the provisional elongation strain difference distribution and the critical buckling strain difference distribution into tension acting on the metal strip at exit from the rolling mill, and the true elongation strain difference distribution is found by adding an elongation strain difference distribution corresponding to the converted tension to the provisional elongation strain difference distribution. 
     A fourth aspect of the present invention provides the rolling control method of the third aspect, wherein, when finding the true elongation strain difference distribution, a second order differential with respect to the strip width direction of a rolling load difference distribution in the strip width direction of the metal strip corresponding to the converted tension is found as an elongation strain difference distribution corresponding to the converted tension. 
     A fifth aspect of the present invention provides a rolling control method including: under conditions in which out-of-plane deformation of a metal strip is restrained, finding a provisional rolling load difference distribution, which is a distribution of differences in rolling load in a strip width direction of the metal strip during rolling under specific rolling conditions, and finding a provisional elongation strain difference distribution, which is a distribution of differences in the strip width direction in elongation strain along a rolling direction of the metal strip during rolling; finding a critical buckling strain difference distribution, which is a distribution in the strip width direction of differences in a critical strain at which the metal strip will buckle, based on the provisional elongation strain difference distribution, a strip thickness of the metal strip, a strip width of the metal strip, and tension acting on the metal strip at exit from a rolling mill; in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution, finding a critical buckling load difference distribution, which is a rolling load difference distribution corresponding to the critical buckling strain difference distribution, from a correlation between the provisional rolling load difference distribution and the provisional elongation strain difference distribution, finding a difference between the provisional rolling load difference distribution and the critical buckling load difference distribution, and finding a true elongation strain difference distribution by adding a strain difference distribution corresponding to the difference to the provisional elongation strain difference distribution under the assumption that there is no crown ratio change in the metal strip between exit from and entry to the rolling mill; and rolling the metal strip without changing the specific rolling conditions in cases in which the provisional elongation strain difference distribution does not exceed the critical buckling strain difference distribution, and rolling the metal strip under rolling conditions that are set based on the true elongation strain difference distribution in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution. 
     A sixth aspect of the present invention provides a rolling control method including: under conditions in which out-of-plane deformation of a metal strip is restrained, finding a provisional rolling load difference distribution, which is a distribution of differences in rolling load in a strip width direction of the metal strip during rolling under specific rolling conditions, and finding a provisional elongation strain difference distribution, which is a distribution of differences in the strip width direction in elongation strain along a rolling direction of the metal strip during rolling; finding a critical buckling strain difference distribution, which is a distribution in the strip width direction of differences in a critical strain at which the metal strip will buckle, based on the provisional elongation strain difference distribution, a strip thickness of the metal strip, a strip width of the metal strip, and tension acting on the metal strip at exit from a rolling mill; in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution, finding an out-of-plane deformation load difference distribution corresponding to an out-of-plane deformation strain difference distribution, which is a difference between the provisional elongation strain difference distribution and the critical buckling strain difference distribution, from a correlation between the provisional rolling load difference distribution and the provisional elongation strain difference distribution, deriving a new rolling load difference distribution by superimposing the out-of-plane deformation load difference distribution on the provisional rolling load difference distribution, finding a new elongation strain difference distribution based on the new rolling load difference distribution under the assumption that there is a change in a crown ratio of the metal strip, and further finding a new critical buckling strain difference distribution based on the new elongation strain difference distribution, the strip thickness and the strip width of the metal strip, and tension acting on the metal strip at exit from the rolling mill; finding a difference between the new elongation strain difference distribution and the new critical buckling strain difference distribution, and finding a true elongation strain difference distribution by adding this difference to the new elongation strain difference distribution; and rolling the metal strip without changing the specific rolling conditions in cases in which the provisional elongation strain difference distribution does not exceed the critical buckling strain difference distribution, and rolling the metal strip under rolling conditions that are set based on the true elongation strain difference distribution in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution. 
     A seventh aspect of the present invention provides the rolling control method of the sixth aspect, wherein finding the out-of-plane deformation load difference distribution is performed plural times by taking the new elongation strain difference distribution as the provisional elongation strain difference distribution, and taking the new critical buckling strain difference distribution as the critical buckling strain difference distribution found. 
     An eighth aspect of the present invention provides the rolling control method of the first aspect to the seventh aspect, wherein the metal strip undergoes out-of-plane deformation at entry to the rolling mill. 
     A ninth aspect of the present invention provides the rolling control method of any one of the first aspect to the eighth aspect, further including: employing a profile meter installed at exit from the rolling mill to measure the profile of the metal strip after rolling; and correcting the provisional elongation strain difference distribution based on a difference between an actual elongation strain difference distribution that has been transformed into out-of-plane deformation found from a measured profile of the metal strip, and an elongation strain difference distribution predicted to be transformed into out-of-plane deformation. 
     A tenth aspect of the present invention provides a rolling controller including: a computation section that finds a critical buckling strain difference distribution, which is a distribution in a strip width direction of differences in a critical strain at which a metal strip will buckle, based on a strip thickness of the metal strip, a strip width of the metal strip, tension acting on the metal strip at exit from a rolling mill, and a provisional elongation strain difference distribution which is a distribution of differences in the strip width direction of elongation strain along a rolling direction of the metal strip during rolling under specific rolling conditions, and which is found under conditions in which out-of-plane deformation of a metal strip is restrained, and the computation section, in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution, finding a true elongation strain difference distribution by adding the difference between the provisional elongation strain difference distribution and the critical buckling strain difference distribution to the provisional elongation strain difference distribution; and a control section that rolls the metal strip without changing the specific rolling conditions in cases in which the provisional elongation strain difference distribution does not exceed the critical buckling strain difference distribution, and that rolls the metal strip under rolling conditions that are set based on the true elongation strain difference distribution in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution. 
     An eleventh aspect of the present invention provides a manufacturing method for a rolled metal strip, the manufacturing method including: finding a critical buckling strain difference distribution which is a distribution in a strip width direction of differences in a critical strain at which a metal strip will buckle, based on a strip thickness of the metal strip, a strip width of the metal strip, tension acting on the metal strip at exit from a rolling mill, and a provisional elongation strain difference distribution, which is a distribution of differences in the strip width direction of elongation strain along a rolling direction of the metal strip during rolling under specific rolling conditions that is found under conditions in which out-of-plane deformation of a metal strip is restrained; in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution, finding a true elongation strain difference distribution by adding the difference between the provisional elongation strain difference distribution and the critical buckling strain difference distribution to the provisional elongation strain difference distribution; and rolling the metal strip without changing the rolling conditions in cases in which the provisional elongation strain difference distribution does not exceed the critical buckling strain difference distribution, and rolling the metal strip under rolling conditions set based on the true elongation strain difference distribution in cases in which the provisional elongation strain difference distribution exceeds the critical buckling strain difference distribution. 
     Advantageous Effects of Invention 
     According to the present invention, out of the elongation strain difference distribution in the strip width direction of the metal strip (namely, the elongation strain difference distribution of the first step), the out-of-plane deformation strain difference distribution that is transformed into a wave profile and causes out-of-plane deformation (namely, the difference between the elongation strain difference distribution of the first step and the critical buckling strain difference distribution of the second step) is added to the elongation strain difference distribution. This thereby enables precise and accurate prediction of the true elongation strain difference distribution of the metal strip. Accordingly, setting the rolling conditions based on the true elongation strain difference distribution enables excellent control of the profile of the metal strip after rolling. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a drawing illustrating an elongation strain difference distribution Δε(x) and a rolling load difference distribution ΔP(x) of a steel strip in a case in which the steel strip is rolled under conditions in which out-of-plane deformation of the steel strip is restrained. 
         FIG. 2  is a drawing illustrating a critical buckling strain difference distribution Δε cr (x) and an out-of-plane deformation strain difference distribution Δε sp (x) configuring an elongation strain difference distribution Δε(x), and a critical buckling load difference distribution ΔP cr (x) and an out-of-plane deformation load difference distribution ΔP sp (x) configuring a rolling load difference distribution ΔP(x), in a case in which a steel strip is rolled under conditions in which out-of-plane deformation of the steel strip is restrained. 
         FIG. 3  is a drawing illustrating a state after an out-of-plane deformation strain difference distribution Δε sp (x) and an out-of-plane deformation load difference distribution ΔP sp (x) have disappeared in a case in which out-of-plane deformation of a steel strip is permitted. 
         FIG. 4  is a drawing illustrating a situation in which metal flows into a reduced load region within a roll-bite and an elongation strain difference distribution in a steel strip increases. 
         FIG. 5A  is an explanatory diagram schematically illustrating a relationship between elongation strain difference and rolling load in a steel strip in plan view, and illustrates an elongation strain difference distribution Δε(x). 
         FIG. 5B  is an explanatory diagram schematically illustrating a relationship between elongation strain difference and rolling load in a steel strip in plan view, and illustrates a critical buckling strain difference distribution Δε cr (x) and an out-of-plane deformation strain difference distribution Δε sp (x). 
         FIG. 5C  is an explanatory diagram schematically illustrating a relationship between elongation strain difference and rolling load in a steel strip in plan view, and illustrates a true elongation strain difference distribution Δε′(x). 
         FIG. 6  is a flowchart illustrating a steel strip rolling control method of a first exemplary embodiment. 
         FIG. 7  is a diagram illustrating a situation in which an elongation strain difference distribution Δε(x) does not exceed a critical buckling strain difference distribution Δε cr (x). 
         FIG. 8  is a diagram illustrating a situation in which an elongation strain difference distribution Δε(x) exceeds a critical buckling strain difference distribution Δε cr (x). 
         FIG. 9  is a diagram illustrating the concept of a true elongation strain difference distribution Δε′(x). 
         FIG. 10  is a graph to explain advantageous effects of the first exemplary embodiment. 
         FIG. 11  is a graph to explain advantageous effects of the first exemplary embodiment. 
         FIG. 12  is a flowchart illustrating a steel strip rolling control method of a second exemplary embodiment. 
         FIG. 13  is a diagram illustrating a correlation between a rolling load difference distribution ΔP(x) and an elongation strain difference distribution Δε(x). 
         FIG. 14  is a flowchart illustrating a steel strip rolling control method of a third exemplary embodiment. 
         FIG. 15  is a diagram illustrating a new rolling load difference distribution ΔP 2 (x). 
         FIG. 16  is a graph to explain advantageous effects of the third exemplary embodiment. 
         FIG. 17  is a diagram schematically illustrating a rolling line provided with a rolling mill, a rolling controller, and a profile meter. 
         FIG. 18  is a flowchart illustrating a flow of processing executed by a rolling controller according to an exemplary embodiment of the present invention. 
         FIG. 19A  is a model diagram for a deflection function. 
         FIG. 19B  is a model diagram for a deflection function. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Explanation follows regarding exemplary embodiments of the present invention, with reference to the drawings. In the present specification and the drawings, configuration elements having substantially the same function as each other are allocated the same reference numerals, and duplicate explanation is omitted. Note that in the present exemplary embodiment, explanation is given regarding a case in which a steel strip is employed as a metal strip. The following explanation deals with strain and load distribution in a roll-bite of the steel strip. 
     Principles of Steel Strip Elongation Strain Occurrence 
     First, explanation follows regarding principles of the occurrence of elongation strain in a rolling direction (referred to below as “elongation strain”) when a rolled steel strip buckles (when out-of-plane deformation occurs in the steel strip), with reference to  FIG. 1  to  FIG. 4 , and  FIG. 5A  to  FIG. 5C .  FIG. 5A  to  FIG. 5C  correspond to  FIG. 1  to  FIG. 4 , and are explanatory diagrams schematically illustrating relationships between elongation strain difference and rolling load difference in a steel strip in plan view. Note that in the following explanation, explanation is given regarding a center wave occurring in the steel strip. The center wave refers to out-of-plane deformation in a wave profile that occurs at a strip width direction central portion of the steel strip, and is also referred to as center stretching. Here, the explanation deals with respective parameters acting on the steel strip on a conceptual level only. Details relating to methods for computing the respective parameters, for example, will follow later in an exemplary embodiment of a steel strip rolling control method. 
     As illustrated in  FIG. 1 , a steel strip H is rolled using a rolling mill  10  including a pair of rollers. The Y direction in  FIG. 1  indicates the rolling direction of the steel strip H, and the steel strip H is conveyed and rolled in the Y direction from a negative direction side toward a positive direction side. The X direction in  FIG. 1  indicates the strip width direction of the steel strip H.  FIG. 1  illustrates half of the steel strip H in the strip width direction, namely from a center H c  to an edge H e  in the strip width direction of the steel strip H. 
       FIG. 1  illustrates an elongation strain difference distribution Δε(x) in the strip width direction of the steel strip H in a roll-bite, and a rolling load difference distribution ΔP(x) acting in a vertical direction of the steel strip H (Z direction) across the strip width direction, in a case in which the steel strip H is rolled under a condition in which out-of-plane deformation of the steel strip H is restrained (namely, a condition in which out-of-plane deformation of the steel strip H is not permitted). The elongation strain difference distribution Δε(x) is a distribution of the elongation strain difference at a strip width direction position x relative to elongation strain at the strip width direction center H c  of the steel strip H. Similarly, the rolling load difference distribution ΔP(x) is a distribution of the rolling load difference at a strip width direction position x relative to rolling load at the strip width direction center H c  of the steel strip H. Moreover, the elongation strain difference distribution Δε(x) and the rolling load difference distribution ΔP(x) have a 1:1 correspondence in the strip width direction. In  FIG. 1 , since out-of-plane deformation of the steel strip H is restrained, compressive stress is generated in the rolling direction immediately after the roll-bite on exit (see the large arrows in  FIG. 1 ). A relationship between the elongation strain difference distribution Δε(x) and the rolling load difference distribution ΔP(x) illustrated in  FIG. 1  is schematically illustrated in  FIG. 5A . 
     As illustrated in  FIG. 2 , the elongation strain difference distribution Δε(x) is split into an elongation strain difference distribution Δε cr (x) that is still present in the steel strip H after buckling (referred to below as the critical buckling strain difference distribution Δε cr (x)), and an elongation strain difference distribution Δε sp (x) that is transformed into wave shaped out-of-plane deformation after buckling (referred to below as the out-of-plane deformation strain difference distribution Δε sp (x)). Of these, the critical buckling strain difference distribution Δε cr (x) is a strain difference distribution of the limit at which the steel strip H would buckle were the strain difference to increase any further. In other words, the critical buckling strain difference distribution Δε cr (x) is a distribution in the strip width direction of differences in the critical strain at which the steel strip H will buckle. Similarly, the rolling load difference distribution ΔP(x) is split into a rolling load difference distribution ΔP cr (x) (referred to below as the critical buckling load difference distribution ΔP cr (x)) that has a 1:1 correspondence in the strip width direction with the critical buckling strain difference distribution Δε cr (x), and a rolling load difference distribution ΔP sp (x) (referred to below as the out-of-plane deformation load difference distribution ΔP sp (x)) that has a 1:1 correspondence in the strip width direction with the out-of-plane deformation strain difference distribution Δε sp (x). Note that the critical buckling strain difference distribution Δε cr (x), the out-of-plane deformation strain difference distribution Δε sp (x), the critical buckling load difference distribution ΔP cr (x), and the out-of-plane deformation load difference distribution ΔP sp (x) illustrated in  FIG. 2  are schematically illustrated in  FIG. 5B . 
     Then, when out-of-plane deformation of the steel strip H is permitted, as illustrated in  FIG. 3 , the out-of-plane deformation strain difference distribution Δε sp (x) is transformed into out-of-plane deformation and disappears. Moreover, the compressive stress illustrated by the large arrows in  FIG. 1  decreases, and apparent tension acting in the rolling direction of the steel strip H increases (see the large arrow in  FIG. 3 ). When this occurs, rolling load matching this tension, namely the out-of-plane deformation load difference distribution ΔP sp (x) corresponding to the out-of-plane deformation strain difference distribution Δε sp (x), disappears. When the out-of-plane deformation load difference distribution ΔP sp (x) disappears, as illustrated in  FIG. 4 , metal flows in the strip width direction toward a reduced load region, namely from the edge H e  toward the center H c  of the steel strip H (see the large arrow in  FIG. 4 ). As a result, due to the principle of constant volume, the elongation strain at the center H c  of the steel strip H increases according to the amount of metal that flows in along the strip width direction. Namely, an increase in elongation strain difference occurs corresponding to the disappearance of the out-of-plane deformation load difference distribution ΔP sp (x) (see the thinner arrow in  FIG. 4 ). Accordingly, as illustrated in  FIG. 5C , a true elongation strain difference distribution Δε′(x) of the steel strip H can be obtained by adding an elongation strain difference distribution Δε n (x) that has increased corresponding to the disappearance of the out-of-plane deformation load difference distribution ΔP sp (x) (this is referred to below as the buckling exacerbation strain difference distribution Δε n (x)) to the elongation strain difference distribution Δε(x) when out-of-plane deformation of the steel strip H is restrained, illustrated in  FIG. 1 . The buckling exacerbation strain difference distribution Δε n (x) is an elongation strain difference distribution arising as a result of buckling of the steel strip H, and is an unobserved strain difference distribution in cases in which out-of-plane deformation of the steel strip H is restrained since buckling does not occur. Note that the out-of-plane deformation strain difference distribution Δε sp (x) and the buckling exacerbation strain difference distribution Δε n (x) are both elongation strain difference distributions corresponding to the out-of-plane deformation load difference distribution ΔP sp (x), and are equivalent distributions to each other. However, they are referred to by different terms for the sake of convenience. 
     As described above, as a result of careful investigation by the inventor into rolling load difference distribution and elongation strain difference distribution in the strip width direction of the steel strip H that undergoes changes as a result of buckling, it has been found that when out-of-plane deformation of the steel strip H is restrained, there is correlation between the rolling load difference distribution ΔP(x) and the elongation strain difference distribution Δε(x) illustrated in  FIG. 5A , and there is also correlation between the rolling load difference distributions ΔP cr (x), ΔP sp (x) and the elongation strain difference distribution Δε cr (x), Δε sp (x) illustrated in  FIG. 5B . Based on this, it has been found that when out-of-plane deformation of the steel strip H is permitted, there are correlations between the rolling load difference distribution ΔP cr (x) and the elongation strain difference distributions Δε cr (x), Δε sp (x), Δε n (x) illustrated in  FIG. 5C , and these correlations have been quantitatively established. Moreover, it has also been found that the true elongation strain difference distribution Δε′(x) illustrated in  FIG. 5C  increases more than the elongation strain difference distribution Δε(x) obtained under conditions in which out-of-plane deformation is restrained, as illustrated in  FIG. 5A  and  FIG. 5B , by an amount corresponding to the buckling exacerbation strain difference distribution Δε n (x), leading to the derivation of Equation 1 below. Note that the elongation strain difference distributions described in JP-A Nos. 2005-153011 and 2012-218010 are the same as the elongation strain difference distribution Δε(x) illustrated in  FIG. 5B . The true elongation strain difference distribution Δε′(x) derived using the method represented by Equation (1) in the present invention is closer to the actual elongation strain difference distribution than the elongation strain difference distributions derived using the known methods.
 
Δε′( x )=Δε( x )+Δε n ( x )  (1)
 
     First Exemplary Embodiment 
     Next, explanation follows regarding a first exemplary embodiment of a method for controlling the profile of the steel strip H after rolling, based on the findings described above.  FIG. 6  is a flowchart illustrating a rolling control method for the steel strip H in the first exemplary embodiment. 
     First, under conditions in which out-of-plane deformation of the steel strip H is restrained, a provisional elongation strain difference distribution Δε(x) in the strip width direction of the steel strip H during rolling under specific rolling conditions is found (step S 10  in  FIG. 6 ). The provisional elongation strain difference distribution Δε(x) may be computed using a known method, such as a Finite Element Method (FEM), a slab method, physical modeling, or a regression formula from experimentation or computation. Step S 10  is known technology. 
     The modeling used to predict the rolled profile at step S 10  is already in use. Strip crown prediction formulas that are necessary during real operations are respectively found for individual rolling mills using statistical methods, based on computed results using numerical analysis methods. For example, as described in Document 1 below, a method exists that employs a strip crown prediction formula for exit from a general rolling mill to derive a strip crown by separating factors dependent on only elastic deformation conditions of the rolling mill from factors dependent on plastic deformation conditions of the rolled material. 
     Document 1: Shigeru Ogawa, Hiromi Matsumoto, Shuichi Hamauzu, Toshio Kikuma: Plasticity and Technology (Journal of the Japan Society for Technology of Plasticity), Vol. 25, No. 286 (November 1984), 1034-1041. 
     Employing this method enables the strip crown at entry to and the strip crown at exit from the rolling mill to be found. Moreover, it is possible to find an elongation strain difference Δε by multiplying a shape change coefficient ξ found through separate experimentation by a crown ratio change (Ch/h−CH/H). Namely, the elongation strain difference Δε can be expressed using Equation (2) below.
 
Δε=ξ·( Ch/h−CH/H )  (2)
 
     wherein CH is the crown on entry to the rolling mill, H is the strip thickness at entry to the rolling mill, Ch is the crown at exit from the rolling mill, and h is the strip thickness at exit from the rolling mill. At step S 10 , the provisional elongation strain difference distribution Δε(x) can be found based on Equation (2). 
     Next, the critical buckling strain difference distribution Δε cr (x) in the strip width direction of the steel strip H is found based on the provisional elongation strain difference distribution Δε(x) found at step S 10 , the strip thickness and strip width of the steel strip H, and the tension acting on the steel strip H at exit from the rolling mill (step S 11  in  FIG. 6 ). Specifically, the critical buckling strain difference distribution Δε cr (x), which is the strip width direction critical elongation strain difference distribution at which the steel strip H will buckle, is computed by FEM or flat strip buckling analysis employing the provisional elongation strain difference distribution Δε(x), the strip thickness and strip width of the steel strip H, and the tension acting on the steel strip H. 
     Note that flat strip buckling analysis is, for example, performed employing buckling modeling formulated using a known triangular residual stress distribution (critical buckling strain difference distribution) described in the Journal of the Japan Society for Technology of Plasticity: Plasticity and Technology, Vol. 28, No. 312 (January 1987), pp 58 -66 (referred to below as Document 2) or alternatively, by following the method described in JP-A No. 2005-153011 using a distribution arrived at by discretization in a chosen manner. In particular, the method described in JP-A No. 2005-153011 is formulated so as to enable analysis even using a stress distribution resulting from residual stress distributed in a chosen manner in the width direction, and so as to enable buckling analysis even for residual stress discretized at each position in the strip width direction. 
     Moreover, buckling modeling employing, for example, the method described in the collected papers from the 63rd Japanese Joint Conference for the Technology of Plasticity (November 2012: Akaishi, Yasuzawa, and Ogawa) (referred to below as Document 3) enables critical buckling strain (stress) to be computed by inputting strip thickness, strip width, and tension, and a residual strain (or residual stress) having a distribution in the strip width direction and being uniform in the rolling direction. 
     JP-A No. 2005-153011 and Document 3 discuss methods for finding buckling strain and buckling modes using buckling analysis, and using the results of thereof to make flatness predictions for out-of-plane deformation after buckling, and to estimate residual strain after out-of-plane deformation. Explanation follows regarding the methods described in JP-A No. 2005-153011 and Document 3. 
     The methods make the following assumptions. 
     (a) That a metal strip is a thin flat strip and that residual plastic strain in the strip width direction is uniformly distributed in the rolling direction and in the thickness direction. 
     (b) When considering unit tension, even if residual stress generated as a result of plastic strain is distributed, integrating in the strip width direction matches a unit tension. 
     (c) That plastic strain should consider rolling direction strain, and other components may be ignored. 
     These methods employ an energy method in order to solve a buckling problem for a flat strip with plastic strain in line with the above assumptions. The energy method employed in buckling analysis is determined by a Trefftz determination standard. Moreover, the contents of Document 2 are utilized for the necessary relationships and basic logic regarding stress, strain, displacement, strain energy, potential energy, and the like. Additional considerations in order to predict the buckled shape using these methods in cases in which non-uniform plastic strain is generated in the strip width direction are given below. Note that in the coordinate system employed, the x axis is the rolling direction, the y axis is the strip width direction, and the z axis is the strip thickness direction. 
     (A) The strip width direction y axis is divided into elements, and residual strain for evaluating the buckled shape is allocated in a chosen manner to each element i as plastic strain ε x *(i). 
     (B) In order to consider non-uniformity in the plastic strain in the strip width direction, a deflection function employs a beam element having two nodal points such as part A in  FIG. 19A  and  FIG. 19B , and a deflection amount in the strip width direction is expressed by the three-dimensional function of Equation (3).
 
 w ( y )= a   1   +a   2   y+a   3   y   2   +a   4   y   3   (3)
 
     Moreover, since displacement in the rolling direction generally has a periodic sine waveform, a sine wave function is used as a multiplier to give Equation (4).
 
 w ( x,y )= w ( y )·sin(π x/L )  (4)
 
wherein L is a half-cycle pitch (half the wavelength) of the sine wave.
 
     The analysis using these methods includes discretizing the plastic strain and displacement functions into respective elements as described above, performing a variant operation of δ(δ 2 π) on the second variant δ 2 π of the total potential energy based on the governing equation in Document 2, and finding an answer that satisfies F=0 for the following Equation (5), namely finding buckling stress and a buckling mode as an answer for a particular problem. 
     
       
         
           
             
               
                 
                   
                     
                       
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     wherein the suffix 1 is a small increment in displacement after buckling, ε x * is plastic strain, ε m * is an average value of ε x * in the strip width direction, H is the strip thickness, σ f  is the unit tension stress, E is the Young&#39;s modulus, ν is the Poisson&#39;s ratio, and D=EH 3 /12 (1−ν 2 ). As a result, this enables the critical buckling strain difference distribution Δε cr (x) to be found. 
     Next, determination is made as to whether or not the steel strip H will buckle (step S 12  in  FIG. 6 ). Specifically, determination is made as to whether or not the provisional elongation strain difference distribution Δε(x) found at step S 10  and the critical buckling strain difference distribution Δε cr (x) found at step S 11  satisfy the following Equation (6).
 
Δε( x )&gt;Δε cr ( x )  (6)
 
     As illustrated in  FIG. 7 , if Equation (6) is not satisfied at step S 12 , and determination is made that the provisional elongation strain difference distribution Δε(x) found at step S 10  does not exceed the critical buckling strain difference distribution Δε cr (x) found at step S 11 , then it is presumed that the steel strip H will not buckle and will be flat. In such cases, the profile of the steel strip H is controlled by rolling the steel strip H with the rolling conditions left as they are, unchanged (step S 13  in  FIG. 6 ). Note that  FIG. 7  is a diagram illustrating an elongation strain difference distribution in the strip width direction, similarly to  FIG. 1  to  FIG. 4 , and  FIG. 5A  to  FIG. 5C , taking the elongation strain at the strip width direction center H c  of the steel strip H as 0. Accordingly, when illustrated as in  FIG. 7 , the elongation strain at the edges H e  of the steel strip are negative values. Similar also applies in  FIG. 8 . 
     However, as illustrated in  FIG. 8 , if Equation (6) is satisfied at step S 12 , and determination is made that the provisional elongation strain difference distribution Δε(x) found at step S 10  exceeds the critical buckling strain difference distribution Δε cr (x) found at step S 11 , it is presumed that the steel strip H will buckle. In such cases, the difference between the provisional elongation strain difference distribution Δε(x) found at step S 10  and the critical buckling strain difference distribution Δε cr (x) found at step S 11  is found. This difference is the buckling exacerbation strain difference distribution Δε n (x) illustrated in  FIG. 5C  (Δε n (x)=Δε(x)−Δε cr (x)). Then, as illustrated in  FIG. 9 , Equation (1) is used to find the true elongation strain difference distribution Δε′(x) by adding the buckling exacerbation strain difference distribution Δε n (x) to the provisional elongation strain difference distribution Δε(x) found at step S 10  (step S 14  in  FIG. 6 ). 
     Next, the profile of the steel strip H is controlled by setting rolling conditions based on the true elongation strain difference distribution Δε′(x) found at step S 14 , and rolling the steel strip H (step S 15  in  FIG. 6 ). Specifically, the rolling conditions are set such that, for example, the true elongation strain difference distribution Δε′(x) becomes equal to or lower than the critical buckling strain difference distribution Δε cr (x). Accordingly, the steel strip H does not buckle, and is flat after rolling. The rolling conditions include, for example, rolling load, and roller bend moment that controls deflection of the rollers. Note that the rolling conditions can be set in a chosen manner, and the true elongation strain difference distribution Δε′(x) may be determined using the present algorithm to control the profile of the steel strip H after rolling as necessary. 
     According to the first exemplary embodiment, the true elongation strain difference distribution Δε′(x) of the steel strip H is found by adding the buckling exacerbation strain difference distribution Δε n (x) found at step S 14  to the provisional elongation strain difference distribution Δε(x) found at step S 10 . By finding the elongation strain difference distribution in this manner, the prediction precision of the elongation strain difference distribution can be increased in comparison to hitherto. Accordingly, setting the rolling conditions based on the true elongation strain difference distribution Δε′(x) enables excellent control of the profile of the steel strip H after rolling. 
       FIG. 10  and  FIG. 11  are graphs explaining advantageous effects of the first exemplary embodiment. The horizontal axes in  FIG. 10  and  FIG. 11  indicate the distance from the center of the steel strip, and the vertical axes indicate elongation strain difference in the rolling direction of the steel strip. Note that the elongation strain differences in  FIG. 10  and  FIG. 11  are values relative to the center of the steel strip (taking this as zero). The up-down asymmetrical model in  FIG. 10  and  FIG. 11  is an FEM model for rolling under conditions in which out-of-plane deformation of the steel strip H is permitted, and elongation strain differences found using this rolling model are actual elongation strain differences. By contrast, the up-down symmetrical model in  FIG. 10  is an FEM model for rolling under conditions in which out-of-plane deformation of the steel strip H is restrained. The new model in  FIG. 11  is a rolling model of the first exemplary embodiment, and is a model reflecting the true elongation strain difference distribution Δε′(x) described above. Simulations of rolling steel strip were performed using each model. 
     As illustrated in  FIG. 10 , the elongation strain difference distribution found using a known up-down symmetrical model differs from the elongation strain difference distribution found using the up-down asymmetrical model. By contrast, as illustrated in  FIG. 11 , the elongation strain difference distribution found using the new model of the first exemplary embodiment is almost the same as the elongation strain difference distribution found using the up-down asymmetrical model. It can therefore be seen that the first exemplary embodiment enables the elongation strain difference distribution of the steel strip to be predicted more precisely and accurately than hitherto. 
     Further investigations by the inventors revealed that when the profile of the steel strip was controlled using the method described in the first exemplary embodiment, yield due to profile was improved by 1% in comparison to hitherto. 
     Note that in the first exemplary embodiment, the true elongation strain difference distribution Δε′(x) may be found based on tension fluctuations caused by buckling at exit from the rolling mill. Specifically, at step S 14  the found buckling exacerbation strain difference distribution Δε n (x) is converted into tension acting on the steel strip H. A change ΔP n (x) in the rolling load difference distribution in the strip width direction arising due to tension fluctuations at exit from the rolling mill is found, and then, as in Equation (7) below, a second order differential is taken of ΔP n ′(x) with respect to the strip width direction x to find the elongation strain difference distribution Δε n ′(x). Then, as in Equation (8) below, the elongation strain difference distribution Δε n ′(x) found with Equation (7) is added to the provisional elongation strain difference distribution Δε(x) found at step S 10  to find the true elongation strain difference distribution Δε′(x).
 
Δε n ′( x )= d   2   ΔP   n ( x )/ dx   2   (7)
 
Δε′( x )=Δε( x )+Δε n ′( x )  (8)
 
     In this manner, converted tensions from converting the buckling exacerbation strain difference distribution Δε n (x) into tension are initially found, and then the elongation strain difference distribution Δε n ′(x) corresponding to the converted tensions is found, such that the found elongation strain difference distribution Δε n ′(x) closer approximates to reality. Moreover, when finding the elongation strain difference distribution Δε n ′(x), a second order differential is taken of the change ΔPn(x) in the rolling load difference distribution, thereby getting even closer to reality. This thereby enables the true elongation strain difference distribution Δε′(x) of the steel strip H to be predicted even more precisely. 
     Note that in the present exemplary embodiment, the provisional elongation strain difference distribution Δε(x) is found at step S 10 . However, step S 10  may be omitted in cases in which the provisional elongation strain difference distribution Δε(x) is already known, or in cases in which a previously found value may be employed. In such cases, the known provisional elongation strain difference distribution Δε(x) is employed at S 11  to find the critical buckling strain difference distribution Δε cr (x). 
     Second Exemplary Embodiment 
     Next, explanation follows regarding a second exemplary embodiment of a method for controlling the profile of the steel strip H after rolling.  FIG. 12  is a flowchart illustrating a rolling control method of the steel strip H in the second exemplary embodiment. 
     First, under conditions in which out-of-plane deformation of the steel strip H is restrained, a provisional rolling load difference distribution ΔP(x) in the strip width direction, and a provisional elongation strain difference distribution Δε(x) in the strip width direction of the steel strip H during rolling under specific rolling conditions, are found (step S 20  in  FIG. 12 ). Similarly to at step S 10 , the provisional rolling load difference distribution ΔP(x) and the provisional elongation strain difference distribution Δε(x) may be computed using a known method, such as an FEM, a slab method, physical modeling, or a regression formula from experimentation or computation. 
     Next, the critical buckling strain difference distribution Δε cr (x) in the strip width direction of the steel strip H is found based on the provisional elongation strain difference distribution Δε(x) found at step S 20 , the strip thickness and the strip width of the steel strip H, and the tension acting on the steel strip H at the exit from the rolling mill (step S 21  in  FIG. 12 ). Step S 21  is performed using a similar method to step S 11  above. 
     Next, determination is made as to whether or not the steel strip H will buckle (step S 22  in  FIG. 12 ). Step S 22  is performed using a similar method to step S 12  above. 
     At step S 22 , in cases in which determination is made that the provisional elongation strain difference distribution Δε(x) found at step S 20  does not exceed the critical buckling strain difference distribution Δε cr (x) found at step S 21 , then it is presumed that the steel strip H will not buckle. In such cases, the profile of the steel strip H is controlled by leaving the rolling conditions as they are, without any changes, and rolling the steel strip H (step S 23  in  FIG. 12 ). 
     However, in cases in which, at step S 22 , determination is made that the provisional elongation strain difference distribution Δε(x) found at step S 20  exceeds the critical buckling strain difference distribution Δε cr (x) found at step S 21 , it is presumed that the steel strip H will buckle. In such cases, the correlation between the provisional rolling load difference distribution ΔP(x) and the provisional elongation strain difference distribution Δε(x) found at step S 20  is found, as illustrated in  FIG. 13 . Based on this correlation, the critical buckling load difference distribution ΔP cr (x) that corresponds to the critical buckling strain difference distribution Δε cr (x) found at step S 21  is found. Then, the out-of-plane deformation load difference distribution ΔP sp (x), which is the difference between the provisional rolling load difference distribution ΔP(x) found at step S 20  and the critical buckling load difference distribution ΔP cr (x) found at step S 24 , is found (ΔP sp (x)=ΔP(x)−ΔP cr (x)). Moreover, making the assumption that there is no crown ratio change in the metal strip between exit from and entry to the rolling mill, a known method such as an FEM, a slab method, physical modeling, or a regression formula from experimentation or computation is employed to find the out-of-plane deformation strain difference distribution Δε sp (x) from the out-of-plane deformation load difference distribution ΔP sp (x). Note that the correlation between the provisional rolling load difference distribution ΔP(x) and the provisional elongation strain difference distribution Δε(x) found at step S 20  may be employed when finding the out-of-plane deformation strain difference distribution Δε sp (x) from the out-of-plane deformation load difference distribution ΔP sp (x). Then, the true elongation strain difference distribution Δε(x) is found by adding the out-of-plane deformation strain difference distribution Δε sp (x) to the provisional elongation strain difference distribution Δε(x) found at step S 20 , as in Equation (9) below (step S 24  in  FIG. 12 ).
 
Δε′( x )=Δε( x )+Δε sp ( x )  (9)
 
     Next, the profile of the steel strip H is controlled by setting rolling conditions based on the true elongation strain difference distribution Δε′(x) found at step S 24 , and rolling the steel strip H (step S 25  in  FIG. 12 ). Step S 25  is performed using a similar method to step S 15  above. 
     The second exemplary embodiment is a modified example of the first exemplary embodiment described above. The method for computing the increase in the elongation strain difference distribution from the provisional elongation strain difference distribution Δε(x) differs between the first exemplary embodiment and the second exemplary embodiment. At step S 14  of the first exemplary embodiment, the increase in the strain difference is found from the difference between the provisional elongation strain difference distribution Δε(x) and the critical buckling strain difference distribution Δε cr (x). However, at step S 24  of the second exemplary embodiment, the increase in the strain difference is found from the difference between the provisional rolling load difference distribution ΔP(x) and the critical buckling load difference distribution ΔP cr (x). Accordingly, the second exemplary embodiment can enjoy similar advantageous effects to the first exemplary embodiment. Namely, the true elongation strain difference distribution Δε′(x) of the steel strip H can be predicted more precisely and more accurately than hitherto. Moreover, setting the rolling conditions based on the true elongation strain difference distribution Δε′(x) enables excellent control of the profile of the steel strip H after rolling. 
     Third Exemplary Embodiment 
     Explanation follows regarding a third exemplary embodiment of a method for controlling the profile of the steel strip H after rolling.  FIG. 14  is a flowchart illustrating a rolling control method of the steel strip H in the third exemplary embodiment. 
     In the third exemplary embodiment, steps S 30  to S 33  in the flowchart illustrated in  FIG. 14  are similar to the respective steps S 20  to S 23  of the second exemplary embodiment. Note that steps S 30  to S 34  are performed repeatedly, as described below, and so, for ease of explanation, the number of times of repetition is appended as a suffix of each parameter. For example, when step S 30  is performed for the first time, a rolling load difference distribution ΔP 1 (x) and an elongation strain difference distribution Δε 1 (x) are found, and when step S 31  is performed for the first time, a critical buckling strain difference distribution Δε cr1 (x) is found. 
     Step S 34  is processing performed in cases in which, at step S 32 , determination is made that the provisional elongation strain difference distribution Δε 1 (x) found at step S 30  exceeds the critical buckling strain difference distribution Δε cr1 (x) found at step S 31 , and that the steel strip H will buckle. In such cases, the correlation is found between the provisional rolling load difference distribution ΔP 1 (x) and the provisional elongation strain difference distribution Δε 1 (x) found at step S 30 , as illustrated in  FIG. 13 . Moreover, an out-of-plane deformation strain difference distribution Δε sp1 (x) is found, which is the difference between the provisional elongation strain difference distribution Δε 1 (x) found at step S 30  and the critical buckling strain difference distribution Δε cr1 (x) found at step S 31 , (Δε sp1 (x)=Δε 1 (x)−Δε cr1 (x)). Based on the above correlation, an out-of-plane deformation load difference distribution ΔP sp1 (x) corresponding to the out-of-plane deformation strain difference distribution Δε sp1 (x) is found. Then, as illustrated in  FIG. 15 , the out-of-plane deformation load difference distribution ΔP sp1 (x) is superimposed on the provisional rolling load difference distribution ΔP 1 (x) found at step S 30  to compute a new rolling load difference distribution ΔP 2 (x) (step S 34  in  FIG. 14 ). Namely, the new rolling load difference distribution ΔP 2 (x) can be expressed by Equation (10) below.
 
Δ P   2 ( x )=Δ P   1 ( x )+Δ P   sp1 ( x )  (10)
 
     Note that when buckling has occurred, the out-of-plane deformation load difference distribution ΔP sp1 (x) disappears, and so in practice, in order to find ΔP 2 (x), processing is performed to subtract ΔP sp1 (x) from ΔP 1 (x). 
     In the third exemplary embodiment, it is assumed that there is a change in the crown ratio of the metal strip between exit from and entry to the rolling mill. Namely, when there is a fluctuation in rolling load acting on the steel strip H, it is assumed that the deflection of the rollers of the rolling mill  10  fluctuates due to the fluctuation in the rolling load, and the elongation strain of the steel strip H also fluctuates. Moreover, an average rolling load is added to the new rolling load difference distribution ΔP 2 (x) found at step S 34  to find a new rolling load difference distribution, and processing returns to step S 30  and a new elongation strain difference distribution Δε 2 (x) is computed based on the new rolling load difference distribution. Then, at step S 31 , a new critical buckling strain difference distribution Δε cr2 (x) is found based on the new elongation strain difference distribution Δε 2 (x), the strip thickness and strip width of the steel strip H, and the tension acting on the steel strip H at exit from the rolling mill. Then, after going through step S 32 , a new rolling load difference distribution ΔP 3 (x) is again computed at step S 34 . Note that the correlation between the rolling load difference distribution ΔP 1 (x) and the elongation strain difference distribution Δε 1 (x) employed on the first occasion at step S 34  may be found as the correlation between the rolling load difference distribution and the elongation strain difference distribution, and this correlation may be employed repeatedly from the second occasion onward. 
     Steps S 30  to S 34  are performed M times (M being a positive integer) so as to finally compute an elongation strain difference distribution Δε M (x) and a new critical buckling strain difference distribution Δε crM (x). A buckling exacerbation strain difference distribution Δε nM (x), which is the difference between the elongation strain difference distribution Δε M (x) and the new critical buckling strain difference distribution Δε crM (x), is then found (Δε nM (x)=Δε M (x)−Δε crM (x)). Then, the true elongation strain difference distribution Δε′(x) is found by adding the buckling exacerbation strain difference distribution Δε nM (x) to the elongation strain difference distribution Δε M (x), as in Equation (11) below (step S 35  in  FIG. 14 ).
 
Δε′( x )=Δε M ( x )+Δε nM ( x )  (11)
 
     Next, the profile of the steel strip H is controlled by setting rolling conditions based on the true elongation strain difference distribution Δε′(x) found at step S 35 , and rolling the steel strip H (step S 36  in  FIG. 14 ). Step S 36  is performed using a similar method to step S 25  above. 
     In the third exemplary embodiment, steps S 30  to S 34  are performed repeatedly, under the assumption that there is a change in the crown ratio of the metal strip between exit from and entry to the rolling mill. This thereby enables the precision of the buckling exacerbation strain difference distribution Δε nM (x) to be improved, and enables the true elongation strain difference distribution Δε′(x) of the steel strip H be predicted with even greater precision. 
       FIG. 16  is a graph to explain advantageous effects of the third exemplary embodiment. In  FIG. 16 , the horizontal axis indicates the number of repetitions M of steps S 30  to S 34 , and the vertical axis indicates the accuracy ratio when predicting the profile of the steel strip. The “accuracy ratio” here refers to a ratio of the steepness of the steel strip obtained by simulation against the steepness of a steel strip actually manufactured (computed steepness/actual steepness). Note that “steepness” is an index indicating the extent of center stretching, edge stretching, and the like, and is a value expressing the ratio of a wave height against the pitch of the wave as a percentage. It can be seen from  FIG. 16  that the accuracy ratio of profile prediction improves as the number of repetitions M increases. 
     Note that the number of repetitions M can be set as desired, and, for example, a predetermined number of repetitions may be set, or alternatively, processing may be repeated until the buckling exacerbation strain difference distribution Δε nM (x) converges. 
     Other Exemplary Embodiments 
     The first exemplary embodiment, the second exemplary embodiment, and the third exemplary embodiment described above are each implemented using the rolling line  1  illustrated in  FIG. 17 . The rolling line  1  includes the rolling mill  10  described above, and a rolling controller  20  that controls the rolling mill  10 . The rolling controller  20  includes a computation section  21  and a control section  22 . The computation section  21  performs computation for the steps S 10  to S 14  of the first exemplary embodiment, the steps S 20  to S 24  of the second exemplary embodiment, and the steps S 30  to S 35  of the third exemplary embodiment. The control section  22  sets rolling conditions based on the computation results of the computation section  21 , namely based on the true elongation strain difference distribution Δε′(x). These rolling conditions are output to the rolling mill  10 , and the rolling mill  10  is controlled so as to control the profile of the steel strip H after rolling. 
       FIG. 18  is a flowchart illustrating an example of a flow of processing executed by the rolling controller  20 . 
     At step S 101 , the computation section  21  receives input of provisional rolling conditions set for the rolling controller  20 . 
     At step S 102 , the computation section  21  finds the provisional elongation strain difference distribution Δε(x) in the strip width direction of the steel strip H during rolling based on the received input of rolling conditions. 
     At step S 103 , the computation section  21  finds the critical buckling strain difference distribution Δε cr (x) in the strip width direction of the steel strip H based on the provisional elongation strain difference distribution Δε(x) found at step S 102 , the strip thickness and strip width of the steel strip H, and the tension acting on the steel strip H at exit from the rolling mill. 
     At step S 104 , the computation section  21  performs buckling determination. Specifically, the computation section  21  determines whether or not the provisional elongation strain difference distribution Δε(x) found at step S 102  and the critical buckling strain difference distribution Δε cr (x) found at step S 103  satisfy Equation (6). In cases in which the computation section  21  determines that Equation (6) has been satisfied (in cases in which it is presumed that buckling will occur), processing transitions to step S 106 , and in cases in which the computation section  21  determines that Equation (6) has not been satisfied (in cases in which it is presumed that buckling will not occur), processing transitions to step S 105 . 
     At step S 105 , the computation section  21  notifies the control section  22  that there is no need to change the input provisional rolling conditions that were received at step S 101 . 
     At step S 106 , the computation section  21  finds the difference between the provisional elongation strain difference distribution Δε(x) found at step S 102  and the critical buckling strain difference distribution Δε cr (x) found at step S 103  as the buckling exacerbation strain difference distribution Δε n (x) (Δε n (x)=Δε(x)−Δε cr (x)). The computation section  21  then uses Equation (1) to find the true elongation strain difference distribution Δε′(x) by adding the buckling exacerbation strain difference distribution Δε n (x) to the provisional elongation strain difference distribution Δε(x). The computation section  21  then supplies the true elongation strain difference distribution Δε′(x), derived as described above, to the control section. 
     At step S 107 , the control section  22  derives new rolling conditions based on the true elongation strain difference distribution Δε′(x). For example, the control section  22  derives new rolling conditions such that the true elongation strain difference distribution Δε′(x) becomes equal to or lower than the critical buckling strain difference distribution Δε cr (x). Note that the new rolling conditions may be derived by the computation section  21 . 
     At step S 108 , in cases in which the control section  22  has received notification from the computation section  21  that there is no need to change the rolling conditions, the control section  22  outputs the original rolling conditions to the rolling mill  10  and controls the rolling mill  10 , thereby controlling the profile of the steel strip H after rolling. However, in cases in which the control section  22  has derived new rolling conditions at step S 107 , the control section  22  outputs the new rolling conditions to the rolling mill  10  and controls the rolling mill  10 , thereby controlling the profile of the steel strip H after rolling. 
     At step S 109 , the control section  22  determines whether or not to end rolling. The control section  22  returns processing to step S 101  in cases in which the control section  22  has determined not to end rolling, and ends the present routine in cases in which the control section  22  has determined to end rolling. 
     Note that in the flow of processing of the rolling controller  20  illustrated in  FIG. 18 , explanation has been given regarding an example corresponding to the rolling control method according to  FIG. 6  (the first exemplary embodiment). However, the rolling controller  20  may be configured to execute processing corresponding to the rolling control method according to  FIG. 12  (the second exemplary embodiment) or  FIG. 14  (the third exemplary embodiment). 
     A profile meter  30  may be installed at the exit from the rolling mill  10  in the rolling line  1 . The profile meter  30  measures the profile of the steel strip H after rolling. The profile of the steel strip H is measured by positions in the rolling direction and positions in the strip width direction of the steel strip H, and the height displacement at these positions. The measurement results of the profile meter  30  are output to the rolling controller  20 . In the computation section  21  of the rolling controller  20 , the out-of-plane deformation strain difference distribution Δε sp (x) is corrected based on the measurement results of the profile meter  30 , accompanying which the true elongation strain difference distribution Δε′(x) is also corrected. Correction of the true elongation strain difference distribution Δε′(x) is performed using the method described in JP-A No. 2012-218010. Namely, first, an actual out-of-plane deformation strain difference distribution Δε sp (x) is found based on the measurement results of the profile meter  30 . The actual out-of-plane deformation strain difference distribution Δε sp (x) and an out-of-plane deformation strain difference distribution Δε sp (x) predicted using an exemplary embodiment described above are compared against each other, and a difference (error) E therebetween is taken as the model error. Based on the error E, learning is performed and the provisional elongation strain difference distribution Δε(x) (rolling load difference distribution ΔP(x)) found at step S 10 , S 20 , or S 30  is corrected. Specifically, the error E is added to the provisional elongation strain difference distribution Δε(x) (rolling load difference distribution ΔP(x)) found at step S 10 , S 20 , or S 30 , and then the respective subsequent processing is performed in order to find the true elongation strain difference distribution Δε′(x). Then, the control section  22  corrects the rolling conditions based on the corrected result of the true elongation strain difference distribution Δε′(x) by the computation section  21  such that the profile of the steel strip H will achieve a target profile. In this manner, the rolling conditions are feedback controlled based on the measurement results of the profile meter  30 . The inventors found from their investigations that performing such feedback control improves yield due to profile by a further 0.5%. 
     The present invention may also be applied in cases in which the steel strip H undergoes out-of-plane deformation on entry to the rolling mill  10 . The inventors found from their investigations that in cases in which the steel strip H undergoes such out-of-plane deformation on entry to the rolling mill, the elongation strain difference distribution of the steel strip H after rolling increases in comparison to cases in which the steel strip H does not undergo out-of-plane deformation on entry to the rolling mill. In other words, the prediction precision of the profile of the steel strip becomes even poorer when using known methods. By contrast, in the present invention, since the elongation strain difference distribution corresponding to the amount of out-of-plane deformation at entry to the rolling mill can be included in the out-of-plane deformation strain difference distribution Δε sp (x), there is no effect on the prediction of the true elongation strain difference distribution Δε′(x) of the steel strip H. This thereby enables the profile of the steel strip H to be appropriately controlled even when the steel strip H undergoes out-of-plane deformation at entry to the rolling mill. 
     Note that in the exemplary embodiments described above, the present invention has been explained using an example in which a center wave is generated in the steel strip. However, the present invention may also be applied in cases in which edge waves or quarter waves are generated. 
     Explanation has been given regarding preferable exemplary embodiments of the present invention with reference to the attached drawings. However, the present invention is not limited to these examples. It would be clear to a person skilled in the art that various modifications or adjustments may be made within the scope of the concepts recited in the scope of claims, and a person skilled in the art would understand that these would obviously fall within the technical scope of the present invention. 
     INDUSTRIAL APPLICABILITY 
     The present invention is useful in cases in which the profile of a metal strip, for example a sheet or a plate, after rolling is predicted, and the profile of the metal strip is controlled based on the prediction results. 
     The disclosure of Japanese Patent Application No. 2014-187290, filed on Sep. 16, 2014, is incorporated in its entirety by reference herein. All cited documents, patent applications, and technical standards mentioned in the present specification are incorporated by reference in the present specification to the same extent as if each individual cited document, patent application, or technical standard was specifically and individually indicated to be incorporated by reference.