SHEET WIDTH CONTROL METHOD IN HOT ROLLING

Abstract: This invention provides a sheet width control method, in hot rolling, that calculates a sheet width distribution of a rolling material in a longitudinal direction and its temperature on the entry side of a vertical rolling mill set as the object for controlling the sheet width, further calculates a distribution of sheet width change amounts of the rolling material along the longitudinal direction occurring on the downstream side to the vertical rolling mill along a rolling direction, and calculates and controls roll gap of the vertical rolling mill in the longitudinal direction of the rolling mill on the basis of a sheet width distribution of the rolling material determined from the distribution of the sum of the calculated sheet width change amounts along the longitudinal direction and on the basis of a final target sheet width.

Description: DESCRIPTION SHEET WIDTH CONTROL METHOD IN HOT ROLLING Technical Field This invention relates to a sheet width control method in hot rolling and a learning method for a sheet width change estimation system. Background Art A sheet width in hot rolling changes depending on various factors in a roughing rolling mill train, a finishing rolling mill train, between finishing rolling mill train, and on a run-out table from the finishing rolling mill train to a coiler. A known sheet width control method measures, or calculates by estimation, the sheet width change amounts described above, controls gap of a vertical rolling mill disposed on the entry side of a roughing horizontal rolling mill or a finish rolling mill and thus controls the sheet width. For example, the method described in Japanese Unexamined Patent Publication (Kokai) No. 60-203315 measures the sheet width by a sheet width meter arranged at the entry side of a coiler or on the delivery side of a finishing rolling mill, controls gap of a roughing vertical rolling mill on the basis of the measurement value, and controls the sheet width. The method described in Japanese Patent No. 2,968,637 divides the sheet width change in finishing rolling into a change resulting from rolling deformation and a change resulting from creep deformation, and calculates the sheet width by use of a sheet thickness of a rolling material, a reduction, a tension between stands, deformation resistance of the rolling material, a rolling material temperature, a strip running time between stands and a sheet crown ratio change amount as estimation parameters of the sheet width change of rolling material. Further, the method described in Japanese Unexamined Patent Publication (Kokai) No. 62-68617 estimates and learns a width change of a successive rolling material by use of a width change amount due to necking on a run-out table and actual measurement values of a position and a length, and expands the width of a corresponding portion of a bar to be rough rolled to an amount corresponding to the width change by using a vertical rolling mill. According to these conventional methods that conduct feedback control on the basis of the sheet width meter, however, a control lag occurs in accordance with the distance between a detection end and a control end. Therefore, high precision sheet width control is almost impossible. Therefore, Japanese Unexamined Patent Publication (Kokai) No. 54-149357, for example, discloses a sheet width control method that calculates a sheet width change in a rolling mill train on the basis of a rolling condition and a rolling material temperature, controls roll gap of a vertical rolling mill so that a product width becomes constant on the delivery side of a finishing rolling mill train, and controls the sheet widt . However, even when the product width is rendered constant on the delivery side of the finishing rolling mill train, the sheet width changes, too, on the subsequent run-out table and in a cooling process from a coiler to a room temperature. To keep the sheet width of the hot-rolled products constant, therefore, it becomes necessary not only to estimate and control all the sheet width changes occurring on the downstream side of the vertical rolling mill set as the control objet but also to calculate the sheet width change in the rolling mill train. However, this prior art does not concretely mention such methods and does not provide a practical solution. According to these conventional methods, a thermometer and a sheet width meter arranged on the entry side of a vertical rolling mill measure the temperature of the rolling material and its sheet width on the entry side of the vertical rolling mill set as the control object for controlling the sheet width. Therefore, it is essentially necessary to arrange the thermometer and the sheet width meter on the entry side of the vertical rolling mill set as the control object. Further, Japanese Unexamined Patent Publication (Kokai) No. 11-285718 discloses a method that estimates and calculates all the sheet width changes occurring in a final roll stand of roughing rolling mills on the downstream side to a vertical rolling mill set as the control object and on a run-out table at each position of a front end portion, a tail end portion and a middle portion in a longitudinal direction, and sets and controls in advance a distribution of a gap pattern of the vertical rolling mill in the longitudinal direction. Among the conventional methods described above, the method that executes the feedback control on the basis of the sheet width control cannot conduct the high precision sheet width control of the front end and the tail end portions and at a skid mark portion because the detection end is spaced apart from the control end a control lag occurs. The method that executes the feed-forward control on the basis of the estimated calculation values must estimate, very accurately, all the sheet width changes occurring in roughing finishing rolling mill train on the delivery side to the vertical rolling mill for controlling the sheet width, and on the run-out table, and must set in advance the gap of the vertical rolling mill on the basis of the estimated values. In this case, the sheet width estimation formula used for the sheet width control is the one that is determined from the examination results of specific steel kinds or typical steel kinds by, for example, experiments in a laboratory, experiments in an actual setup or numerical analysis. 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CD Φ 3 i CO 0 ιQ 3 Hi CD μ- rt 3 tr o CD 3 * l-i H 3 P 3 rt H 3 l-i rt μ- t- 1 ^ » 0 rt 3 ft JX 0 3 Hi -- ■ --• CD rt Φ rt TJ CD CD l-i rt φ CD Ω ex 0 Hi Ω ex t- 1 3 i ti CD μ- tr 93 rt cx μ- 0 μ- tyH- Ω 0 Φ 0 3 --" 0 Ω 3 CD t- 1 s3 H 0 Φ μ- CD Ω Φ μ 1 CD φ μ- μ- β 3 CD Hi 3 ro Ω ct Hi ti μ- ex μ- Φ rt 0 -- ■ μ- μ- 0 3 3 rt CD Ω CD rt 3 H Ω CD sQ Φ CD μ- μ- 3 H ex -- ■ μ- β Ω 3 3 rt o -Q Φ " O 3 ex H rt < 0 rt --" O rt rt -- ■ O rt PJ rt μ- 0 t μ- ti O 3 β CO rt ti Hi H Ω 3 μ- μ- tr Φ -- ■ μ- 3 ty3 3 ty3 tr 3 0 3 rt CD rt tr ct 0 3 μ- β rt CD H σ 3 3 ct -Q ro CD s3 ti ro 0 μ> sQ CO CD CD tr 1 rt μ- CD -- ■ H. μ- Φ β ct μ- Q CD ct μ- O en en 3 rt μ- " rt N 3 μ- φ " tr -- ■ CD O ex Ω rt μ- 3 l-i rt CD Φ cx 0 0 tyCD 3 3 μ- 0 μ- O CX 3 μ- Φ rt -- ■ Φ rt μ- o sQ iQ tr tyti rt -- ■ rt ro rt μ- 3 Hi 3 3 3 μ- --" μ- 0 3 Φ Φ ro μ- tr μ- tyCD rt H" -Q -Q Φ 0 rt ιΩ CD 3 μ- 0 0 3 rt Φ CD 3 ro 3 tr rt Hi tr 3 sQ 3 Hi 3 rt CX φ tr φ -Q Φ CO CO cπ o cπ o cπ cπ rt Ω CD sx CD Ω CD Hi Ω 0 3 3 Ω Ω rt O μ> CX Ω CD rt < CD rt }-• s3 rt rt Ω ct o hi ct t rt Φ 0 μ- μ- 3 CD l-i 0 Hi μ- CD CD 0 tytyHi 0 μ- O Ω tr φ tr o μ- tr Φ o tyΩ 0 tr tyΦ 3 3 (X H O O 0 3 rt 3 ro ro 3 co 3 Ω φ i O φ 3 ex Φ 3 3 ro Ω Φ ro 3 TJ rt φ Φ β Ω 3 3 rt rt μ> Φ Ω t ti (+ -Q ct TJ 0 Ct 3 sQ t TJ Ct β 1 TJ Φ H Ω 3 β tQ i tyti β ti < tyμ- ι-i ti hi CO μ- s CD μ- tr -- • Φ H < hi μ- CX Φ ti 0 0 ct rt rt 0 ro CD μ- 0 φ rt ro rt μ- μ- cx H tyΩ tyrt 0 hi 0 Φ ti 3 μ- 0 i CD μ> Hi μ- t ty3 CD CD i tr β cr CD μ- 3 ro CD t ro β cx 3 CD hi μ- sQ CΩ 3 CD rt 0 0 rt tr ro < cx μ j rt M ct φ ti cx β Φ 3 Φ tyΦ CX μ- Q rt μ> rt 3 ct sQ ct β μ- rt 3 Hi μ- Φ μ- Φ μ- μ- μ- 0 μ- rt CD -Q rt rt ro rt μ- CD μ- β μ- μ- s 3 hi μ- β ι-i 3 ty3 rt 3 t-i cr CD 3 3 Ω H -- ■ 3 μ- tr t 3 ct ct ι-5 3 Ω CD μ- rt hi Φ -Q ro o rt -Q ct φ ι-3 rt Φ μ- 1 i-3 iQ CD 0 CD O rt rt Φ 3 0 CD H β Φ sQ CD μ- rt σ β φ Ω tr ti 3 μ- rt 0 H μ" μ- 3 ty0 μ- 0 μ- h- 1 μ- cx μ> 3 Φ β (X CX rt < Ω Φ ct SD TJ rt Ω s3 3 ct rt 3 ro CD cx h- 1 3 cx cr μ- ex t i rt μ- ex μ- 3* Φ β tr 3 Φ tr (D Φ i-3 * t ti μ- -Q ex O rt tr ct μ- cQ rt ex β 3 μ- tyhi φ μ- μ- 3 μ- CΩ Φ H H ti φ CD t Φ -- ■ Φ Φ ro 0 3 μ- Hi CD tr φ tr 3 μ- tr μ- rt CD co ro 0 CD CD o CD t rt H 0 < CD 3 ct μ< iQ 3 ti rt Φ φ • iQ ct i μ- 1 rt μ» Ω h-" 3 μ 1 rt i CD μ- μ- CX Φ rt CD i tr rt CO CD Φ t Φ rt β cx Φ 0 H CD tr ti μ- tr Ω 3 " μ- H β tyO rt φ Φ tr μ- 3 ct Ω tyTJ Hi 3 μ- Ω 3 μ- tr μ- CD o (X μ- σ φ CD -Q μ- CO CO H ro 3 * 3 Φ 3 CD Φ rt ro CD μ- μ- μ- co rt μ- cr φ 3 ti h- 1 Hi μ- cr β φ 3 rt φ Φ Φ Φ rt TJ Φ iQ rt H μ- H μ- 3 rt μ- o H β φ -Q o o i β rt ct μ- 1-3 ti rt μ- ti Φ rt Φ μ- O hi 0 en *x CD H 0 Hi Φ rt rt 3 rt Φ rt μ- ti 3 μ- CX 3 H CD H 3 H CD 3 O H rt rt μ- 3 Ω μ- 3 μ» vQ tyΩ μ- 0 s3 O 3 cr μ- μ- s sQ O 3 CD S3 μ- μ- " !- ■ • • tr CD σ rt ct 0 s3 μ- μ- ro rt o 3 μ- μ> Φ CD β H CD μ- CO rt μ- !D CD -- ■ μ- φ β CO tyμ- 3 μ- 3 rt μ- 3 CO cx CD rt rt φ rt ex 3 " < β ex CD 3 μ- 3 3 Ω ft μ- t o ro 0 ex i-3 tr CO o rt μ- Ω Φ μ- Ω i rt μ- μ- Φ ι-i rt CX 3 φ < 0 H μ- 3 o rt Φ tr 3 o 0 tr 3 tr H O ct μ- tr 3 ti Φ tr CO Φ 0 sQ CD Φ 3 CD Φ 0 3 H Hi tr CD TJ φ Hi Hi -Q μ- 3 μ- cr 1 cQ en φ 3 rt en 3 rt en Ω 3 Φ o en φ CD H 1 φ to 0 K CD O β CD Φ ex P rt μ- iQ ty3 β rt ti rt CD 0 rt 0 rt CO 0 rt 0 rt rt 3 3 0 " 0 3 rt -- • CD 3 μ- t- 1 co ro CD H μ- 0 rt μ- 0 CO -- ■ tr 3 CD 3 tr tr μ- H Hi μ- 0 φ μ- ex cn 0 CD rt rt rt μ- 0 ty0 Hi β μ- Ω Φ CD -Q s3 3 Φ Φ t+ μ j CD CO 0 3 rt μ- rt 3 CO CD tr rt Φ 3 3 ro 3 H 3 rt CD 0 μ- μ- φ tr μ- rt 0 3 -Q μ" H i iQ φ φ ii -Q " -» 3 ct φ LQ h- 1 H ty3 rt ex CD ι-i H φ 3 O tyCD en φ μ- rt O 3 μ- Φ 0 O tyex Ω 0 ro rt β rt CD 0 O en cQ 3 ro 3 0 rt CO Ω cr rt ty3 rt TJ CD rt ct rt cr 3 ro 3 β tr rt tr β H -- ■ cr φ sQ Φ Hi tr 0 rt β " ro ti Φ h- 1 tytr tr <_l. rt CD -- ■ h- 1 cr Φ tyμ- l-i μ> H" CD t TJ CD CD φ rt 3 μ- rt Φ rt CD H. ro φ 0 φ rt H 0 rt CD μ- CD ro 3 Ω Φ μ- μ- CD CD rt tr CD rt tr O μ- O tr 3 CD CD cx Ω 3 * o Φ ct 3 CD o CD tr cx 3 3 μ- CD CO tr φ β tr l-i Φ rt 3 O i-i cr Φ CD rt -- ■ en rt rt Φ rt hi Φ iQ μ- cr β μ> CD * «• -Q <-3 cn en CO φ φ H Φ O tr 3 O CJ. < β O tr tr μ- tr μ- (X eo l_I. T3 3 rt φ μ* 0 Φ μ- φ β Φ H 3 φ μ- 3 Hi Φ μ- Φ JD 3 φ CD ex iQ (D 3 3 0 ct 0 cx H " σ 3 0 Ω TJ H Φ sQ φ H O 3 3 hi CD o Ω rt μ- Φ 3 CD CD Hi ty3 0 s O μ- l_l. β Hi μ- rt CD CO rt fx tr ti rt sQ Φ 3 rt Hi rt H i-s ex rt rt ro 3 μ- 3 φ TJ Φ 3 rt Φ CX rt 0 ti cr CD cx Φ Φ Φ CD φ Φ rt ct v-5 !X CD iQ Ω en CD rt iQ Hi H μ- tys μ- rt Ω ^ 3 ^ μ- ti rt Hi CD Ω 3 Ω hi H, tr O tr μ- ct 3 μ- rt rt Ω tr O Φ CX en ro μ- 3 O CD o rt μ- 3 * o 3 rt o CD μ- μ- Φ cr φ rt 3 * CX 3 cx ι-i tr φ ex H CD μ- rt ex < hi 3 CD rt Ω 3 tyCD φ l-i μ- β \- > SD CD 1—1. β Q μ- Hi Φ μ- 3 ti H ct φ O rt μ- Φ CD <Q ro -- ■ CD 0 3 Ω .- ■ φ β CX Ω i 0 CD i ti l-i Φ H- tr 3 H ti α i μ- 3 rt β Ω TJ μ- tr 3 φ ti 3 0 O Φ CO Ω cr rt (- ■ 0 Φ μ- Ω ct CD CD ex 1 CD μ- (+ CD 3 SD CD Ω μ 1 H Ω μ- rt β μ- μ- h- 1 CD β tytr Φ o CD H 3 rt CD 3 rt rt en * rt cx μ- rt O 3 O -- ■ μ ro Φ o Hi rt o Hi H -Q Φ μ- μ- μ- H- μ- Φ 0 μ- 3 iQ μ- Hi CD Φ 3 o μ- 3 cr 0 Φ Φ H 0 ex 3 3 0 3 0 3 rt r+ -Q Hi rt 3 -Q 0 ti CD H- 3 Φ iQ -Q 3 0 3 sQ rt Φ tyifl rt 3 CD " -!• Hi tr ro tr H Φ transverse and longitudinal directions so calculated to thereby calculate the sheet width distribution of the rolling material along the longitudinal direction on the entry side of the vertical rolling mill set as the object for controlling the sheet width. The fourth invention comprises the steps of: measuring the sheet width distribution of the rolling material along the longitudinal direction and the temperature distribution of the rolling material in both longitudinal and transverse directions when the rolling material exists on the upstream side of a vertical rolling mill set as the object for controlling the sheet width along the rolling direction; calculating the temperature distributions of the rolling material in both longitudinal and transverse directions in each rolling mill and between the rolling mills on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction from the temperature distributions of the rolling material in both longitudinal and transverse directions so measured; and calculating the distribution of the sheet width change amount of the rolling material along the longitudinal direction occurring in each rolling pass on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction on the basis of the temperature distributions of the rolling material in both longitudinal and transverse directions so calculated and the sheet width distribution of the rolling material along the longitudinal direction so measured to thereby calculate the sheet width distribution of the rolling material along the longitudinal direction on the entry side of the vertical rolling mill set as the object for controlling the sheet width. The sheet width distribution of the rolling material along the longitudinal direction on the entry side of a vertical rolling mill set as the object for controlling the sheet width can be calculated highly accurately by measuring the temperature distribution along the longitudinal direction as the temperature information of the rolling material in the second invention, by measuring the temperature distribution in the transverse direction as the temperature information of the rolling material in the third invention, and by measuring the temperature distributions in both longitudinal and transverse directions as the temperature information of the rolling material in the fourth invention. The fifth invention comprises the steps of: measuring a temperature distribution of a rolling material in at least one of longitudinal and transverse directions and a sheet width distribution of the rolling material along a longitudinal direction when the rolling material exists on the upstream side of a vertical rolling mill set as the object for controlling the sheet width along a rolling direction; calculating the temperature distribution of the rolling material in at least one of the longitudinal and transverse directions in each rolling mill and between the rolling mills on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along a rolling direction from the temperature distributions of the rolling material in at least one of the longitudinal and transverse directions so measured; calculating a distribution of a sheet width change amount of the rolling material along the longitudinal direction occurring in each rolling pass on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction on the basis of the temperature distribution of the rolling material in at least one of the longitudinal and transverse directions so calculated and the sheet width distribution of the rolling material along the longitudinal direction so measured to thereby calculate the sheet width distribution of the rolling material CO C to tv> cπ o cπ o cπ ti ct ct ct cx Ω Hi CD rt 3 ti en O ex μ- O Hi 0 tr μ- Φ tyμ- O μ- H- 1 tr CD μ- O O Hi μ- 3 hh hi H Φ cn 3 ro CO 3 3 o Φ rt cx ti O H rt TJ rt CX (D 3 Φ rt Ω ft φ CD rt 3 μ- μ> ti Φ μ> ti β -Q CO ti tr μ- CD tr Ω t ty¬ 3 0 μ- hi 0 μ- h3 Ω tyμ- 3 h- 1 φ rt ro rt Q 3 σ CD 3 tr tyct rt rt ro CD (D -Q Ω μ- tr -Q rt sQ β ro μ- CD tr Φ h- 1 β H 0 Φ rt φ 3 μ- rt β μ- rt 3 hi Φ rt 0 3 O 3 Φ μ- ct μ- H ct μ- Hi sQ LQ 0 3 μ- CD μ> en CD 3 Ω β 0 Φ β 0 μ- φ " S 3 -Q rt Ct TJ O μ- ex 3 ex 3 hh Ω rt 0 μ- Φ μ- en Φ μ- μ- μ- μ- <+ O 3 ex ct rt CX 3 0 O hi H" CD 3 O 3 3 0 tr 3 CO -Q rt tr tr CD ->• iQ 3 CD φ a CD H CD H ct tr μ- tr Φ φ φ Ω Φ rt hi ex -- ■ O r- 1 μ- ti φ <+ rt Ω 3 CD β t ti ct 3 0 φ β Ω cr CD CD --> O i-S rt cr CD tr 3 CD tr < t CX tr CD 0 CD μ< ct Ω Hi Φ O Φ 3 φ CD 3 Φ Φ μ- CD CO 3 CD Ω φ β rt CX rt cx 3 CD 3 3 μ- -Q β hS ct ex ct s3 i μ- i rt O μ- CD i-3 co μ- rt μ- tr μ- ty¬ Φ ct O O ct 0 μ- cx h-" φ rt tf CD 5D rt φ CD ro Φ hi 3 ti μ 4 0 CD rt 0 β Φ rt (- ■ φ r+ 3 CD *»« CD H 1 3 CO tr CD Hi cx μ- cx μ- 1 H Ω 3 μ- 3 μ- μ- 3 μ- O 3 CD O μ- O rt CO 3 CD en 3 3 Ct 0 H O Ct 3 σ -Q CD 3 cr O tr < v-3 3 < sQ Φ O Hi Φ β 3 * CD <_I. 0 3 sQ β h- 1 φ φ ex Φ CD Ω 3 Φ φ H 3 ex μ- rt μ- hi 3 H 3 CO CD rt rt rt Ω 0 a rt μ- 3 hi w CD Ω co CD β t tyμ- CD CX cx rt rt β 0 sQ 0 φ ct CD Φ rt H. rt ro 0 μ- μ- μ- ct tr 3 H φ H Φ Φ CD 3 0 CD ti Hi tyφ μ- Hi H hi Ω ex H. CD μ- H Hi rt φ O v-3 ro 3 0 μ- μ- μ- μ- β μ- μ- 3 O CD H Ω hi CD CO CD Hi 3 3 ti CD μ- 1 i CD Ct H 0 rt μ- rt TJ (- ■ tf μ- Q Φ μ- CD Φ -- ■ tr ct h- 1 3 * tr μ- Ω O φ rt CD Ω rt Ω Φ tr μ- Ω Φ β 0 0 0 3 φ P tr r 3 ct μ- φ ct μ- Φ 3 CD rt 3 3 H -Q rt 3 φ μ- μ- 3 en μ- 3 rt sQ H i-i μ- rt μ- ex TJ μ> 0 O φ i-i Ω O O O l-i (+ ct S3 hi O μ 1 3 CD rt 3 P 3 O 3 β H 3 Hi O tyβ μ- rt O μ- en en rt tr CD rt TJ CD μ* -- ■ ro ex cx ti h- 1 3 φ φ rt CD μ- o rt -- ■ μ- ct P 1 rt 0 μ- M CD --• H Φ rt 3 Hi tf μ- < 3 tr 3 μ- 3 3 φ rt CD φ CD sQ H Φ -Q Φ 3 φ P CD 3 O CD Φ CD rt CD μ- fx rt -Q H h- 1 < -Q 3 c+ φ en 3 rt CO β T5 SD 3 tr ti ct μ- Φ tr CD rt TJ tr rt ti P CD Φ O rt μ- cx co H 3 ft Φ Ω Φ Φ Φ ^* 3 rt H 3 * Ω μ- rt CO CD tr tr 0 ti 0 CX Φ en I- 1 Φ CD ti H Φ rt Φ 3 CD 3 CD hi β μ- -- ■ φ μ- Φ Φ rt Φ 3 CD μ- 3 3 03 Ω σ ti tr β ex SD cQ tr rt β μ- (D 0 t O H 0 Φ μ- ct CO Hi Φ Hi H Φ 0 μ- μ- rt 3 0 co 3 CO CO to t cπ o cπ o cπ π μ- μ- S3 H S3 μ- Ω w rt (X rt CD CO CD ct 3 3 μ- hh Ω hh ct < cr - 3 Ω h rt rt o CX 3 3 μ- o O μ- 3 O μ- tr μ- tytyφ i φ φ P φ 3 μ- 0 hi tr φ Φ 0 μ- CD μ- tr tr tr o < < rx μ- 3 cx < 3 sx Φ CD ro ro ct Φ P 3 rt CD < 3 μ- o φ ti rt 3 (X 4 3 φ Φ S Φ φ Ct Q rt φ rt φ ct Φ CO TJ Φ CO ro μ- h- 1 3 rt S3 u3 rt Ω μ- Φ 3 3 3 tr μ- μ- tf 3 i ti - 1 rt CD 3 rt φ ι-i β 3 CO φ CD μ- φ μ- tr β CO Ω Hi Ω CO ct rt H • 3 rt rt O O O μ- O Hi CD Φ hi μ- ti rt H tr ti rt tyΩ φ rt " tr o μ- ft rt μ- μ- 3 s β ex μ- H 3 σ 3 tr CD 0 CD CD Φ μ- 3 tf ro CD 3 β Ω CD μ- 3 hi 0 0 cx μ- 0 Q β sQ μ- φ rt CD 3 rt h- 1 CO 0 TJ CD Φ Φ μ- 1 ex tr rt TJ μ* CD Hi Φ 3 3 ct 3 μ- CD 3 μ- rt μ- rt μ- tyβ Φ β 3 ct 0 3 rt rt μ- P Φ 0 Φ M o CD tr μ- 3 rt 3 tr rt μ- rt rt CD ro hi i CD rt tyμ- cx Hi ti 3 * 3 3 μ- hi H 3 Ω rt φ ti Ω sQ ro β O β tr μ- Φ 0 Φ tr ct ro 3 μ- s3 o Φ v-3 3 3 ti O tf μ- CD ex 3 CX . 0 ex hh 0 Φ 0 ct μ- 3 μ- " φ 0 ct 0 Ω CD 3 ti 3 σ 3 ft < μ- μ- (X rt cr 3 3 • 3 CD & ti i • 3 0 μ- rt 0 Φ CD sx β tyφ 3 O 3 μ- tr LJ. 0 rt μ- cQ CO ct Φ rt μ- 0 JX P Φ cx h- 1 3 cx ti β t φ μ- rt ti ro ι-i CD Hi CD CD Φ 3 tyCD tr tr rt rt tr 3 μ- 3 ct Φ 0 sQ tf rt i μ- Φ rt H rt μ- Ω ro rt rt (0 φ tr tyCD Q ti o TJ ■ μ- cn ti H tr 0 Φ 0 ex CO μ- rt H 3 rt rt ti tyφ O ro rt 3 μ- φ β hi ty3 rt o 0 CO ex CD Ω 3 β tyΩ CD tr μ- < 3" μ- ro ct Hi cx CD V- 1 3 3 Ω 3 φ ro h- 1 Hi cr CD ct Ω ro CD 3 φ μ- cr Φ h Φ O tx μ- N 3 0 μ- cQ rt rt Ω hi 3 CD tyCD μ- 0 Φ φ ex ι-i β 3 0 3 β M s Hi CD 0 CX 3 μ- tn μ- φ ex μ- CD ro Ω rt O Hi rt ti φ rt rt hi Φ sQ rt o μ- ct Ω 3 sQ μ- 1 3 0 CD h N 3 rt Ω ty3 rt H rt 0 Ω μ- μ- 3 μ- μ- 3 (X tr Ω Φ 0 μ- 3 0 φ o 0 0 LQ < Φ CO 0 ro rt tys3 O ti rt o O Ω rt rt o -Q rt 0 Hi ct CD Hi h- 1 ti 3 hh Φ 3 μ- hi o tr ro μ- μ> CD 1 μ- 3 3 0 H β 3 μ- 3 * μ- hi H tyΦ μ- 0 KJ Φ Φ rt i CO x cx O 3 Φ ex 3 μ- O 3 ► <: rt 3 ex H Hi ro rt CD Ω rt Hi tyt μ- ct μ- cr Φ rt μ- CO 3 3 0 ex rt μ- o β (X < μ- 0 μ- Ω tyct hh μ- ro μ- O tr 3 l_l. rt ι-5 en tr 3 sQ Hi Φ ti CD 3 hh ex μ- φ 3 3 3 hi o β ro tf ct i 3 Ω 3 -Q φ ty O t « -Q φ CD CO O μ- CD μ- co 3 sQ μ- o CD 3 i Φ tyo μ- CO CD μ- Ω Φ h- 1 μ- ti 3 3 ft Ω --• t- 1 rt 3 rt ct t CD h- 1 ti 1-5 ro 3 en tr μ* cr 3 ft rt 3 3 CO CD CX tr hi t- 4 Φ ty(D i-s μ- ct tytr h- 1 rt rt μ- o ex tr ro φ < 0 Φ μ- CD h3 μ- φ rt Φ μ- μ- P ro μ- 0 o ro μ- μ- tr tr 3 μ- Hi rt μ- Φ ti rt φ Hi 3 3 rt tr H Φ rt cr 3 0 3 cr 3 3 3 φ Φ ιΩ en μ- 3 * 3 rt 0 S3 3 CD 0 t sQ μ- φ cx hi tyti Φ Q Hi !X i cx β " -« CD a sQ iQ H- rt Hi Φ cQ Φ ct 3 ti 0 H μ- μ- ro 0 0 μ- <+ 3 0 O α μ- 3 hi rt s3 Φ μ- •<: *< 3 3 φ CO H CD h- 1 ct CD rt ti μ- rt •"-. μ- 1-5 ex σ o 3 vQ μ- tr Ω ti μ- μ- 3 0 Ω CD Hi φ Φ μ" rt H P tr H μ- φ 0 tr μ 1 0 μ- l_l. 3 cr 0 0 cx 3 3 0 0 CO rt Φ 0 rt Ω Φ H- σ Φ <s P μ- Ω 3 Φ 0 φ H φ 3 Φ 3 β μ- μ- ct cQ ti •» * 3 3 μ- Φ hi ti rt μ- 3 3 o φ 3 3 rt 3 hi h- 1 Φ Ω CD CD P rt 3 h- 1 tr 0 φ rt cx ti H Φ CD μ- 3 TJ -Q < CO H CD vQ μ- 0 CO Φ μ- Ω ft rt Ω ft μ- < Φ μ- Ή 3 H rt hi Φ μ- 0 • * CO 0 Φ Φ tr rt < 0 Hi μ- rt 3 rt hi 3 * φ 0 φ h{ 3 μ- tr 0 0 CD ti 3 CD H 3 Φ μ- ro 3 3 X 0 0 Q μ- Hi Φ hi 3 3 iQ μ- μ- φ Hi H O rt rt CD rt CD CD 3 Φ Ω H CD rt rt Hi 0 0 CD ti μ- rt rt 3 -- ■ 3 Hi 0 tr tr rt rt φ rt CD CD rt CD 3" tr rt 3 3 i •<« 3 0 SD 0 μ- O 3 iQ CD ct μ- CD Hi φ Φ 0 -- ■ β Φ P H- 1 φ Φ 3 Φ rt tyμ- -- ■ H Hi 0 μ- CD cn Φ ty3 rt M 3 Φ H ti CO -3 ti CX μ- tf ro Ω CD -- 1 3 h- 1 Φ 3 < ro sQ tr O ct CO 0 P φ μ- β μ- hi CX μ- i 3 Φ μ- μ- 0 μ- μ- P rt N rt μ- Φ Φ 3 tyμ- cr rt CD CD ti ex O μ- P rt o < Ω 3 3 cx 3 tr Ω Ω o 3 Hi t LQ ro l-J. tr rt φ rt l-i 3 * Φ Hi 0 rt rt Φ ιΩ o Φ CD 0 rt 3 Cϋ ct μ- tr < rt φ Φ CO tr " φ Φ μ* 3 μ- o 0 ct H 3 3 0 hi Φ CD CO tr i φ Φ rt CO tr Ω 0 μ- Ω μ- μ- rt hi -- ■ tyo o TJ s CO -- ■ CD CD H tr tr rt Φ 3 h- 1 rt 3 rt 3 3 μ- CD μ- ft ro Hi tyμ- μ- Hi rt O ct CO ct φ φ 3 Φ 0 tr -Q μ- cQ 0 ft 3 tr CD Ct ro 3 3 ti ty¬ Hi tr μ- φ Hi rt 3 ro o CD 3 i-3 φ μ- rt CO tyφ iQ o ro Φ Ω rt O hi O -Q 3 3 rt o 3 tr ro rt ct 3 CD φ CD i *< Hi μ- CO 3 LQ Φ CD 0 rt μ< φ 3 ex finish rolling mill train and thus controls the sheet width. According to the seventh invention, as the sheet width control by means of the tension control between the rolling mills of the finishing rolling mill train supplements the sheet width control by a vertical rolling mill set as the object for controlling the sheet width, the sheet width can be controlled more precisely. In the method according to any one of the first invention through the seventh invention, the eighth invention arranges the control of the upstream side of the vertical rolling mill set as the control object to the first roll stand of the horizontal finishing rolling mill train along the rolling direction. According to the eighth invention, as the vertical rolling mill set as the control object is arranged on the upstream side to the first horizontal roll stand of the finishing rolling mill train along the rolling direction, width reduction by the vertical rolling mill can be carried out at a stand where the sheet thickness of the rolling material is greater, and buckling deformation of the rolling material resulting from the width reduction can be prevented. Therefore, the sheet control can be conducted more efficiently. In the method according to any one of the first invention to eighth invention, the ninth invention calculates dividedly the sheet width change amounts of the rolling material in each rolling pass for a front end portion, a middle portion and a to tail end portion. In the method according to any one of the first invention to eighth invention, the tenth invention calculates dividedly the sheet width change amounts of the rolling material in each rolling pass on the entry side, inside and the delivery side of the roll bite of rolling mill. The sheet width change amounts of the rolling material in each rolling pass can be estimated more co co t to h- 1 Cπ o cπ o cπ o cπ Hi hi rt Hi rt t μ- μ- CD cr Φ Ω 3 ct Ω H- Hi ct s3 rt CO Ω Ω 0 TJ μ- t rt σ TJ CX TJ μ- 0 tr 0 tr tf 3 3 3 Φ CD 0 φ tr tr φ 0 Φ tyS3 tyo 0 3 H. 3 tytr Φ o μ- H a -- ■ Φ hi φ φ < < CX Hi ct H rt μ- C CO H H ro Φ ro 3 hi Φ o ro ro Ω ii < Φ μ- 3 Φ φ O μ- ti tr hi 3 β 3 3 hi φ TJ ti < Φ CD ct μ- Ω CO μ- (D β Ω CO 3 3 CD ti 3 Φ O ct Q h- 1 β Φ < ct 0 Φ μ- 3 ex Φ β μ- sx μ- H tr 3 Ω O t t rt H φ CD Ω ex φ H Φ rt Hi μ- Φ H 3 Ω o rt H Φ 3 CO O Φ cn 3 μ- iQ rt CD H Φ μ- μ- 3 < ct ct Φ 3 μ- CD o 3 rt 3 s Φ rt ti Φ μ- 3 ft φ 3 sx φ 3 β ti TJ o 0 CD rt μ- μ- S3 3 i 3 hi tr μ- 3 μ- CD 0 μ- H ct sQ 3 CD β Φ en 3 3 rt H 1 tr 0 0 3 * ct rt Φ LQ CD rt rt IX rt o cr 3 t < ^ rt ^ ^ tr μ- h- 1 cn Ω tyβ Φ 3 3 φ tr tr CD hi Ω tyμ- tr rt 3 o SD tr Φ tyct φ i Φ rt O TJ rt ro φ ti Φ β HI Φ o ro 3 Φ tr Φ ct t φ ii CO ro tr ct cr 0 3 CX μ- Hi hi O TJ Hi ct Φ μ- H H < μ* rt tr CO O "< μ- φ tr Φ 3 h- 1 CD Φ 0 " "* • 0 3 CD tr O Φ μ- 3 3 rt O CX hi Ω Φ 3 ro Φ Φ -. tr 3 ct ro Ω φ CD μ- 3 < ct Φ Hi CD H hi 3 < Φ μ- 3 μ- Φ 0 3 Φ CO 3 H φ ct φ CD Φ Φ 3 CD t μ- CD cn 3 -- ■ μ- tr rt rt en 3 3 Φ rt 3 <! Ω hi rt rt rt Φ 3 0 φ Φ ct μ- 3 μ- cn β tr 3 3 β rt φ sx μ- tr Φ φ β ct 3 tr sQ TJ μ- ft H μ- 3 * μ- 3 Hi ct 3 tr O rt ex tr CD o L CX hi rh Φ CD Φ h{ 0 hi Hi tyct 0 -- ■ ex μ- φ 0 0 3 rt Φ ct 0 Φ Hi tr sx φ φ $x φ tr ti H en rt cx rt CD 0 ro μ- cx H SD Φ 3 Ω 3 ex CD CD X SD tr ex h- 1 Φ 3 0 <χ Φ 3 3 φ rt hi hi 0 ti CD ex sQ rt rt TJ μ- 0 rt μ- Φ ct ft CD μ- i μ- CD φ O tyCD CD Ω 3 CD 0 rt μ- TJ CD μ- 0 hi Ω ex μ- CD Hi ty3 tr Ω 1 < en O 3 3 Hi ro 3 H Ω O Ω 3 μ- •<! rt o H Ω 0 Hi Φ 0 rt 3 Ω ro < TJ s3 Φ Ω H O CD tr Hi sQ cn rt CO Φ 0 ti TJ Ω Ω tr 3 0 Ω 3 CO ti tr < Ω t Φ 0 μ- 0 • 3 H Φ tr tr $D TJ Hi ti i ti 0 Ω Ω Φ 0 ct CO H φ 0 tf H 3 ii cx 3 hi β Φ ct ct φ Ω tr 0 ex ti φ 0 hi ti (X SD Ω μ- ti Hi tr φ φ Ω 3 hi Φ O ct rt ct μ- cx rt φ rt tr tr φ μ- Ω CD ti μ- Φ Ω < cx Φ Φ CD o ex cx o Φ cx Ω O ft cx μ- μ- tr 3 μ- tr Φ Φ rt 3 O CO 3 <! Ω rt μ- μ- Φ Hi Ω tr φ Φ μ- ti rt 3 μ- μ- H o O rt 3 Φ o s3 < ti φ CD μ- t μ- <x 3 TJ 0 β Φ Hi CD 3 3 H CD μ- rt 0 3 O 3 3 Ω 3' cQ Hi μ- CO Ω ?3 φ (X rt sx μ- 0 φ sQ hi h- 1 Φ Hi iQ β 0 CD 0 H 3 LQ cr tr CX ex tr 0 μ- 3 μ- ct μ- Φ 3 3 CD SX 3 CD rt μ- CD h- 1 h- 1 3 O "* μ- Ω CD CD μ- rt Φ rt rt φ ti cx rt 3 i O sx LQ ct Φ SD Ct Ω ct CD Hi t ct CD 3 3 3 O tr tr φ ti rt μ- iQ CD 3 -- ■ Ω CD O Hi rt Φ s3 μ- CD 0 μ- β Ω ct O cr Φ cx LQ μ- ro ct Φ tr O 3 ■ ! rt O 0 μ- (X μ- φ tr 0 3 3 0 3 tr μ- Ω Φ Φ CD < Ω Ω 3 rt CD 0 tyΦ CO rt ti o 3 Φ ct H LQ Ω Φ φ φ CD rt β rt 3 3 Φ cn 0 r Ω O Hi Hi Ω ro H tr tr 3 3 Hi rt rt Φ Φ ct Hi rt 3 φ μ- tr CD rt ■ : ti ty3 μ- μ- O o CD Hi Φ Φ CD o tr rt 3 rt 3 μ- Hi tf φ ^< • 3 CD Φ 3 μ- *<: ro rt sx O 3 rt CD hi rt CD μ- Φ rt CD hi (D ct tf CD 0 μ- O h- 1 CO t 0 0 o φ hi ct 3 rt 3 * 3 3 tyΩ tr Ω rt rt μ- 3 Ω 3 S tr Φ rt 3 Ω ex φ 0 μ- Φ rt β 3 3 CO rt 0 tr i Φ •<! CD ro β φ μ- Φ 0 ex CD tr SX μ- Φ CD μ- < 3 en SD 3 Φ μ- h- 1 Ω O rt φ Φ -3 3 3 CD ex μ- CO ti φ Hi φ Φ Φ μ- rt Ω cx ≤3 Ω 0 Hi 0 μ- H. CD rt 3 μ- rt • CD CO 3 rt rt 3 3 μ- o 3 0 3 μ- CD O Φ μ- 0 tr Φ 0 3 0 0 ft rt cx tr tr LQ tytr < Φ SD Hi ct hi ct 0 ct ct Hi h- 1 Hi $χ 3 CD Hi 3 β Φ 3 h- 1 Φ 3 rt CO φ Φ ro φ Φ -- ■ tr Hi tr Φ 0 Ω 0 rt 3 Hi Φ ti • M ex μ- SD tr μ- tyφ Ω 3 rt Ω CD β Φ ex 3 H β rt Hi tr rt LQ μ- rt rt 0 H- <x 3 3 ro ct Φ Ω 0 ct cr 0 3 en μ- ct μ- SX μ 1 tr tr φ Ω 3" Φ H 3 Hi ct CX Ω < Φ CD o 3 μ- rt SD 3 ex μ- 3 tr ti < rt SD φ ct σ φ μ- O Φ sQ O tr o Φ rt s3 rt ti rt 0 0 CD ct 3 <! . Φ 0 μ- TJ tr rt tyΦ φ φ CX 3 rt ti t 3 3 μ- μ- ti ti 3 φ Φ CD iQ φ sx O Φ Φ rt ro rt r CD 3 rt tr 3 Ω tr rt rt 3 (X 3 Φ 0 -< cr ex 3 3 Hi " Φ i CO Φ s CD ft ct Ω tf Φ CD CD tr Φ hi μ- μ- rt SD Ω Φ ct φ ct μ- sx rt Hi 3 φ cn μ- O rt CD O 0 ex tr rt rt ct 0 CD μ- μ- H cr μ> μ- ti rt φ μ- 3 CD 3 rt Φ <! 3 h- 1 3 rt μ- μ- 3 tr H 3 rt 0 CO μ- ; 0 O tr 3 CO CD 3 TJ Φ hi CD cQ * tr a 0 0 Φ φ 0 0 tr 3 ct ct 3 3 rt cx hi 3 μ- -- ■ Φ tr 3 3 ct μ> CD Hi φ Φ o rt rt 0 μ- — . μ- rt CD β rt CD tr μ" hi 3 tyHi 0 o co tr (- ■ φ tr 3 O Φ SD CD Φ ro 3 hi φ ro s cx 3 3 3 CD ro sx CX SX invention to twelveth invention, the fifteenth invention provides a method of a sheet width control method comprises the steps of; learning the correction coefficient and/or the correction term of the sheet width change estimation formula used in the sheet width control on the basis of the actual measured value of the temperature and/or the actual measured value of the sheet crown between the finishing rolling mills and/or on the delivery side of the finishing rolling mill. Here, the definitions of the terms are as follows. When the kind of rolling mills is to be discriminated, the term "horizontal rolling mill" or "vertical rolling mill" is used. When the term "rolling mill" is merely employed, it means only the horizontal rolling mill or only the vertical rolling mill, or both of the horizontal and vertical rolling mills. When the term "finishing rolling mill train" is employed, it means a plurality of only horizontal roll stand of finishing rolling mills, or a plurality of horizontal and a vertical roll stand of finishing rolling mill train. The vertical rolling mill set as the object for controlling the sheet width may be arranged at an arbitrary position of the rough rolling mill train or the finish rolling mill train. For example, the vertical mill set as the object for controlling the sheet width may be an arbitrary vertical rolling mill inside the rough rolling mill train or an edger between stands inside the finish rolling mill train. Further, a width reduction machine such as a sizing press may be set as the object for controlling the sheet width. When the term "temperature" is merely employed, it means a mean temperature or a temperature at an arbitrary point. Brief description of the drawings FIG. 1 shows a flow of operation of claim 1 of the present invention; FIG. 2(a) and Fig. 2(b) show a flow of operation of claim 5 of the present invention; FIG. 3 shows an outline of a hot rolling line used in Examples 1 to 8 of the present invention; FIG. 4 shows an outline of a hot rolling line used in Example 9 of the present invention; FIG. 5 shows an outline of a hot rolling line used in Example 10 of the present invention; FIG. 6 is a structural view of an apparatus for explaining the present invention; FIG. 7 is a graph for explaining a method of dividing a sheet width and a temperature measurement value in a longitudinal direction; FIG. 8(a) - FIG. 8(c) are graphs showing the difference between the measured value of the sheet width change amount and the determined one from a rolling condition, and contents of component elements of a material. FIG. 9 shows an outline of a hot rolling line used in Example 11 of the present invention. The most preferred embodiment The inventors of the present invention have conducted numerous theoretical and experimental studies on the behavior of a sheet width change in a hot rolling process, and have acquired the following knowledge. The sheet width change in the hot rolling process occurs in each rolling pass and between rolling mills, on a run-out table from a final roll stand of finishing rolling mill train to a coiler, and in a zone from the coiler to room temperature. When the sheet width change is estimated by dividing the process into each of these process steps, the sheet width change behavior in the hot rolling process can be estimated very accurately. This sheet width change is greatly affected by the conditions on the rolling mill side such as a draft schedule, a crown schedule, a diameter of each roll, a roll peripheral speed of each rolling mill, tension between the rolling mills, rigidity of each rolling mill, etc, by the conditions on the side of a cooling apparatus on the run-out table such as a quantity of cooling water and a cooling pattern, by the conditions on the coiler side such as a coiling speed of the coiler, and by the conditions on the side of a rolling material such as a kind of a steel (components), a sheet width, a sheet thickness, a sheet crown, a temperature, a distance of a corresponding portion from a front end or tail end of the rolling material, and so forth. Incidentally, not only the mean temperature but also the temperature distribution along a longitudinal direction and along a transverse direction exert great influences as the temperature conditions of the rolling material. Even when these conditions are equal, the sheet width change behavior is different between the front end portion of the rolling material, its middle portion and its tail end portion. Further, as to rolling deformation, not only the sheet width change inside a roll bite but also deformation which occurs at the entry side of roll bite (pre-deformation) and deformation which occurs at the delivery side of roll bite (post-deformation) on the roll bite are great. In addition, the roll gap of the vertical rolling mill and the tension between the rolling mills can be the control end of the sheet width control but within the sheet control range where the rolling operation of the vertical rolling mill is greater than the tension between the rolling mills, buckling deformation of the rolling material occurs with width reduction as the sheet thickness of the rolling material becomes smaller. Furthermore, the following result is obtained after careful examination. When the sheet width change amount calculated and measured in the actual rolling mill are compared for each material component, and influences of the material components on the sheet width change are examined. The sheet width change amount in the actual rolling mill is determined from the difference of the sheet width measured by a sheet width meter arranged on the entry side and the delivery side of the finishing rolling mill train. The sheet width change amounts at each stand and between the stands are calculated the sheet width change amounts is compared with the actual measured value of the sheet width change amount described above. Incidentally, the deformation resistance value of the rolling material is inversely calculated from the rolling load that is actually measured. FIGS. 8(a) to 8(c) show the relation between the difference the actual measured value and the calculated value of the sheet width change amount and the content of the component elements of the material. It can be understood from these graphs that the difference between the actual measured value and the calculated value of the sheet width change amount increases or decreases depending on the amounts of various component elements. The deformation resistance value in the model formulas uses the inverted calculation value from the actual measurement load. It is therefore believed that the difference depending on the component elements mainly results from the difference of creep deformation characteristics. In other words, the sheet width change between the stands is the change that results from the creep deformation. To accomplish the estimation with higher accuracy, therefore, it is necessary not only to consider the influences of the material components as the carbon amount and the change resistance value in the estimation of the sheet width but also to conduct calculation while taking the influences of the material components on the creep deformation characteristics into account. However, it is not realistically possible to conduct the creep tensile tests for all the kinds of steels used for actual operations and to .prepare the estimation formula for each steel kind because an enormous expense and much time would be necessary. On the basis of the observation described above, the present inventors have accomplished a sheet width control method having higher accuracy than conventional methods, and have invented an invention for improving sheet width accuracy and yield. Hereinafter, a sheet width control method of the present invention will be explained with reference to the flowchart shown in FIG. 1. To begin with, the distribution of the sheet width of the rolling material along the longitudinal direction and the temperature of the rolling material are measured at the point of time when the rolling material exists on the upstream side relative to the vertical rolling mill set as the object for controlling the sheet width along the rolling direction. Next, the temperatures of the rolling material at each rolling mill and between the rolling mills on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction are calculated from the temperature information of the rolling material thus measured. At this time, as described in the second invention, the temperature distribution of the rolling material along the longitudinal direction is measured, and the temperature distribution of the rolling material along the longitudinal direction in each rolling mill and between the rolling mills upstream of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction is preferably calculated. Also, the temperature distribution of the rolling material along the transverse direction is preferably calculated, and the temperature distribution of the rolling material along the transverse direction in each rolling mill and between the rolling mills on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction is preferably calculated as described in the third invention. Further, as described in the fourth invention, the temperature distributions of the rolling co co to to cπ o Cπ o cπ cπ ft hi < rt TJ Ω φ 0 rt < SD rt tr TJ CD rt μ- sx 0 β μ- 0 ϋ hi CD 0 μ- CD Ω 0 3 t- 1 SD 3 tr μ- 0 Φ Φ 0 0 CO 3 tr Φ μ-" tr Φ 0 cr tr 3 μ- cr TJ 3 Hi 0 O ϋ σ 3 ii 0 Hi μ- O μ- hi CD φ sx ii 3 μ- ti rt φ Φ H O φ rt i 0 μ- hh li CD h- h- 1 ii < φ 3 μ* 3 CD Φ ct rt rt TJ 3 ti μ- ct 3 s3 rt < Ω O φ φ rt φ rt CD φ Φ r ct μ" sQ rt Φ rt tr μ- μ- φ c+ Φ 3 0 cn μ- <-3 CO Φ μ- Φ x- ii Ω Ω ti SD tr μ- 3 Ω 3 TJ hi tr μ- ii TJ H φ 3 Ω ti Ω CD Hi t Ω tr Φ 0 • ti 3 ct rt Φ Ω Φ H 3 CO cQ ct rt ii 0 Φ P rt μ- ii μ- 3 SD iQ CD SD Cfl rt rt Φ SD rt Φ 3 3 Φ CD μ- SD tr ^ cQ ft φ μ- φ 3 β σ Φ TJ rt O Φ μ- rt Φ tr φ CD rt 0 hh 3 hi Φ CD cx hh O Hi < Hi 1 Φ O 3 β sx O 3" rt φ rt rt O --3 CO μ- 3 O ti O ex 3 O 3 φ μ- Φ μ- rt Φ ti 3 μ- hi hi -- ■ 3 Φ H tr Hi tf O ti CD 0 SX μ- (X o ti «-» H 3 ϋ cr 3 μ- H μ- CD sQ μ> 0 Φ 0 3 s3 O S3 Φ Φ μ- 3 μ- h- 1 μ- ϋ 3 CD LQ ct Φ CD o CD 3 ct μ" 3 CD ! Φ μ- O μ- rt 3 3 ti Ω ex μ 1 μ- co Φ O Ω rt tr μ- rt 3 tr β ft Ω sQ t SD CO sx 3 sx ii 3 * Hi o Φ 0 Φ μ- 3 rt Ω Hi ct 0 tr ct Ω s3 CO h- 1 tr i tr cn μ- CD β h- 1 rt rt μ- -Q rt O Φ ft O Hi 3 3 LQ ii rt tr 3 φ • : tr CD Φ P • 0 Φ Φ φ 3 3 SD n β μ- tr 3 μ- tr tr H Ω rt O iQ μ- μ- ct φ rt φ Φ 3 0 t rt Q CO (D φ 3 LQ ct h-" H Φ 3 rt H Hi 3 cr O tf hi < Ω 3 SX Hi 3 tr Ω ϋ cr CD μ- β a μ- O SD tr h- 1 O 3 m β 3 φ β 0 φ CD CO hi φ CD 0 CD 3 φ CD μ < 0 t cn 3 sx ty 3 H 1 CD rt φ Ω rt μ- ct ct • TJ ti μ> tr 0 rt rt rt CD 3 --" cn μ- rt * Hi μ- μ- μ- CD LQ h- 1 tf μ- β tr φ μ- hh CO h- 1 rt Ω φ μ" tr hi tr CD 0 h-" 3 O O 3 3 μ- φ O ti μ- 1 μ- Φ μ- 1 H 0 μ- ft μ- μ- β φ μ" Φ P Φ β 3 CD μ- rt 1 φ μ> σ rt 3 CD > SD cQ 3 3 Φ 3 0 CD 3 μ- 3 3 hi 3 Ω H ct μ- 3 hi i-3 μ j 3 tr CD Φ • tr h- 1 Φ μ- LQ rt rt LQ < SD CO μ- φ LQ P P 3 hi CD ϋ Φ μ- CO sQ Φ 0 CD CD Φ < Ω CD μ 1 Ω Φ Φ 3 CD CD H rt s3 i- 0 < 0 ex rt 0 3 β en t CD β Φ sx CD 3 S SD μ- CX rt H cx 0 tr 3 rt φ μ- H φ β SX 0 ϋ rt tr rt 1 rt μ- 3 CO CD μ- -- ■ 3 tf rt P Hi μ- tr ti cx sx 3 hi h- 1 SX tr μ- cr rt Φ Φ Φ β CD H 0 rt <x Ω LQ 0 Φ μ- σ 3 CO φ 0 . ct μ- μ- CO μ- CD μ- φ ti C_i. tr O 3 φ rt CD φ β SD Φ rt β 3 Ω φ 0 t sQ μ- tr r- 1 3 φ 3 3 3 Φ φ Φ 3 3 TJ Φ CO Ω 3 h{ ti tr μ* 3 CO CD rt 3 tr sx CO h^ LQ LQ cx SD Ω Ω Ω CD Φ φ Φ 0 SX rt rt Φ μ- P SD rt tf h- 1 s3 i φ ti φ tr μ- > CD cx h-" CD rt rt β rt CD ti Hi • rt μ- CD Ω rt rt t Φ φ O Φ 3 CD μ" CO 3 μ- 3 rt h- 1 μ- TJ 0 CO CD tr o O CD tr Φ Φ Φ Φ ii φ rt CO 0 Φ iQ 0 φ μ- H SD tr Ω 0 Hi CD rt Hi β rt ct ro 3 Hi cx CD CX ii rt 0 3 ty tr μ> hh rt cx 3 ct φ rt Φ 3 β 3 O rt tyi β tr H cx μ- 3 μ- cr h- 1 Φ φ μ- 3 φ iQ " Ω Φ IX t- 1 * ϋ H ro t Φ ii Φ 3 0 O rt Φ 3 LQ σ SD CD 3 rt φ 3 rt S3 μ- CO CD rt i rt SD Φ ty3 Φ Ω σ 3 tycx Φ ^ -- ■ CO μ- μ- tf t- 1 rt LQ tr μ- μ 1 Ω ct CO μ- μ- Φ rt Ct Ω CD TJ ro Φ en μ- L_l. ro « * Φ μ- 3 Φ 0 Φ ex ii tr 0 0 3 ii Φ h3 O 3 0 3 CD tr ex φ rt CD CD rt P to rt LQ 3 S3 3 rt μ- φ rt 3 3 h- 1 TJ P (X ty3 μ- CO rt rt Φ Φ Ω tyϋ rt Ω 3 tr 3 tr ti LQ μ- μ- Hi tr S cr tr CD Φ 3 ro ct en 3 tyφ 3 rt ro O 3" tr o Φ sx 0 3 0 H- (X μ- tr Φ ϋ φ rt μ- ii CD O ϋ μ- ct ro < Φ rt rt Φ β Hi CD μ- t-> rt rt hi μ- μ- sx O tr μ- 3 P < 3 cn 0 sx φ CD SD CD Hi φ hi 3 3 μ- " h- 1 β tr CO CO Ω 0 Φ 3 ct Φ tr H Φ Ω rt μ* Ω S3 μ- 1 0 3 μ- ex o rt Φ rt rt 0 H μ- ex ct rt tr μ- cr tr β ti rt Φ H β tr μ- h-" ti rt 3 μ- CD rt co tr 3 3 μ- Ω H TJ 3 μ- l_l. β φ 0 ti CD tr Φ μ- 0 0 s Φ sx ^ H LQ CO H tr Φ LQ CO LQ 3 tr CD tr ti cn 3 φ TJ CD rt Φ φ Φ ct 3 Hi cn μ- TJ rt Ω *< rt μ- 0 0 CO φ P CD μ- 0 φ φ rt iQ Ω CO Ω tr iQ φ ex CD 0 tr O 3 hi 3 Hi ex tr rt rt rt 3 3 3 hi hh rt tr rt rt tf SX cr -3 < i rt ct μ- CD 3 CO CD μ- sQ ω φ tr μ- i μ- φ Φ SX ii μ- CD μ- ct φ tr 3 SD rt rt μ- rt σ rt sx φ 3 Φ CD cx φ μ- N i SD μ- Hi Φ ti ii CO (X tr ti rt Φ rt 3 ii ex Φ 3 tr Φ rt TJ CO μ- 3 0 CD cn Φ hi 0 CD 0 Φ μ- rt φ ct 0 CD SD sx h- 1 0 φ i rt μ- ro CO Φ H CO ti CD 3 tr μ- φ ii 3 H- 1 Ω en 3 * μ- 3 Ω 3 Φ μ- μ- H Ω ϋ 0 rt Φ 3 ct rt t LQ Ω rt cn Ω rt (X tr CD ct CD 0 CD O μ> CO ti SD h- 1 tr 0 Ω O tr SD •<! tr tr rt CO μ- μ- O (D tr CD ty• tr cn μ- Hi 3 CD μ- ft Φ 3 rt β φ μ" Φ rt μ- μ- 3 O Hi 3 Φ μ» ro rt Φ rt 3 P 3 β μ- μ- 3 tr O (X sQ 3 (X φ tr σ iQ rt 0 3 φ Φ ϋ 3 rt 0 rt 3 φ CO SD rt φ Φ tr Hi cx ex Φ sQ tr 3 CO rt Φ Φ least one of the set value of the roll gap of the vertical rolling mill, its rolling load and the rolling load of the horizontal roll stand of roughing rolling mill train. Further, as described in the sixth invention, at least one of a sheet width meter and a thermometer is arranged on the entry side of the vertical rolling mill set as the control object, and the estimation value of at least one of the distribution of the sheet width of the rolling material along the longitudinal direction and the distribution of the temperature of the rolling material along the longitudinal direction is preferably corrected by the respective measurement value. Further, the temperature information of the rolling material in the vertical rolling mill set as the object for controlling the sheet width in each rolling mill and between the rolling mills on the downstream side of the vertical rolling mill set as the object for controlling the sheet width along in the rolling direction, in a zone between the delivery side of the finish rolling mill train and the coiler and in a zone from the coiler to the cooling finish point is calculated on the basis of the temperature information of the rolling material on the entry side of the vertical rolling mill set as the object for controlling the sheet width calculated as described above. The sheet width change amounts of the rolling material occurring in the vertical rolling mill, in each rolling mill as the control object and between the rolling mills on the downstream side of the vertical rolling mill set as the object for controlling the sheet width along in the rolling direction, in the zone from the delivery side of the finishing rolling mill train to the coiler and in the zone from the coiler to the cooling finish point are calculated for the entire length on the basis of the temperature information of the rolling material and its sheet width information thus calculated. Further, when the sheet width change amounts of the same portion at these points are added, the sheet width change amount on the downstream side of the vertical rolling mill set as the object for controlling the sheet width along in the rolling direction is calculated, and the estimated value of the final sheet width of the rolling material is computed. At this time, as described in the fifth invention, the temperature distribution of the rolling material in at least longitudinal and transverse directions is preferably calculated in the vertical rolling mill set as the object for controlling the sheet width, in each rolling mill and between the rolling mills on the downstream side of the vertical rolling mill set as the object for controlling the sheet width, in the zone from the delivery side of the finishing rolling mill train to the coiler and in the zone from the coiler to the cooling finish point. As described in the ninth invention, the estimation formula of the sheet width change amount is preferably formulated dividedly as to the front end portion, the middle portion and the tail end portion. As described in the tenth invention, the estimation formula of the sheet width change amount is preferably formulated dividedly at to regions on the entry side of the roll bite, inside the roll bite and on the delivery side of the roll bite. The estimation formula of the sheet width change amount is expressed as functions of the conditions on the rolling mill side such as the reduction, the roll diameter, the peripheral speed of the roll, the tension between the rolling mills and rigidity of each rolling mill, the conditions on the side of the cooling machine on the runout table such as the quantity of cooling water and the cooling pattern, the conditions on the coiler side such as the coiling speed, and the conditions of the rolling material such as the kind of a steel (components), the sheet width, the sheet thickness, the sheet crown, the temperature and the distance of the given portion from the front end or the tail end. For example, it is co co to cπ o cπ o cπ cπ Ω ct SD Hi 3 3 < CD ct CO μ- Ω H ti ct Ω H CD rt Ω rt H CD Ω O rt ct Hi H μ- h3 -: s3 TJ 0 SD 3 β μ- μ- CD CD tr rt 3 CD 3 0 tr 0 0 β tr 0 tr 0 3 0 tr tr tr μ- 0 3 tr h3 μ- 0 3 π CX 3 1 H "< φ CD < 3 φ 3 h- 1 Hi φ μ- Φ SD H CJ. Φ Φ 3 h- 1 Φ tr cx tn CO φ Ω H r-> β 3 μ- rt rt hh 3 μ> μ" ti φ P " i- 1 0 φ rt cn μ- ct rt φ CO SX rt σ tr μ- li hi μ- μ- li Ω hh μ- Φ Ω ti φ μ- LO hj 0 tr μ- CX t > tr μ- rt 0 CD μ- tr φ Φ μ- 3 O O 3 Ω -S 0 μ- μ- 3 Φ CD ft 0 CO 3 ι-3 ∞ *<! ti σ Φ tr CO Φ 0 ti hh rt Φ 0 cn co cQ h- 1 LQ μ- tr h- cx 3 sQ hi TJ rt ct iQ tr -- * Ω H ti φ 3 SD 0 φ Hi CD φ φ Φ CD 0 hh μ- CD Φ • O tr Φ SD ex CO CD μ- ft Hi μ- rt rt Ω s 3 μ- μ- 3 3 3 μ- 3 3 rt 3 0 3 H 3 O P CD rt μ- Φ tr 3 tr CO ft tr Ω μ- 3 co μ- rt 3 rt μ- tf SX H LQ CD LQ SD φ 3 3 3 ct μ- 3 CO φ μ- Φ rh rt tr Φ O > - μ j sQ μ- -- ■ ft LQ cn μ- 1 CD μ- SD rt φ rt CD ex LQ O 0 Hi Ω φ 3 CD tr 3 tr φ 3 «« CD ► < tr S3 tr 3 3 Ω TJ μ- rt Φ ct μ- Φ 3 H ti rt 3 Hi Φ φ μ- Ω TJ 3 β ^ Φ 3 μ- φ LQ 0 0 hi μ- rt TJ β β μ- o cx μ- Ω Ω tr tr rt rt μ- TJ tr SD rt φ μ- rt 3 O 3 CO μ- 3 rt ti CD cn Φ cr Ω hi 3 hi φ ? o CD μ- tr ti TJ ex ^ Hi <+ tr ct cn tf TJ rt Hi tr P Φ 3 Φ rt 3 Φ i sx t SD Φ CO 3 hi 3 CD Φ CD Φ μ- φ Φ O H < φ ct Φ Ω O 3" Ω sx o φ tr ex CD Φ μ- sQ tr μ- CO φ CO 3 3 ti rt s3 μ- hi O rt CD φ μ- rt β < Φ Φ S i Φ β CD CO N Φ φ CO 3 3 Ω Φ CD μ- 3 * μ- 3 -- ■ rt h- 1 tr h- 1 rt rt O 0 μ- 3 P CD μ- 3 tr Ω ti CO 0 ex tr O Φ H SD h- 1 CD φ sx Ω SD μ- Φ β tr 3 Hi Ω rt ii TJ 3 rt CO O ct < 3 0 • Φ CD Hi 3 μ- 3 h- 1 t hi rt 0 μ- φ s3 β φ μ- ti 0 CO 0 φ i μ- CD rt Hi φ cn rt σ CD rt * rt tr Φ rt 3 3 < μ- D < rt tn o Φ Hi rt Φ rt μ- 0 μ- 3 CD rt rt Φ Φ SD SD SD Φ iQ Φ μ- ex P tr 0 β β CO tr rt * N 3 3 cx y-> rt H sx tr ex cx 0 hi ti 0 cn li μ- ii CD rt cx h- 1 Φ Hi 3 cn Φ CD Φ Φ < 0 tr 3 3 φ Φ Hi iQ sQ hh φ •»» CD ft rt tr Φ β 3 3 β < Φ SD 3 Ω rt ct ϋ Φ Ω μ- CD tr CD φ Φ sx tr μ- s t Φ H tf CD Φ rt Ω Φ SD TJ h- 1 t 0 tr tr 0 μ- sx Ω Hi ^ cr rt t rt ct rt H φ Ω hi 0 Φ i h- 1 ti cn tr «- ■ h- 1 β SD 3 μ- φ H H (X rt ϋ μ- 0 tr tf CO tr φ SD φ Hi li 0 0 CD μ- rt μ- φ β rt Φ h- 1 ct CO μ» 0 Φ tr μ- 3 rt <s Φ cn < Φ 0 φ sx CO h- 1 SD 3 Hi 3 r N μ- 3 3 < Φ tr CD ti CO μ- 3 tr tr Φ tr CD β tr t rt μ- φ 3 < CD φ • Φ H 0 Ω tr 3 t-> rt Ω Φ Φ ti φ h- 1 li rt ii Ω φ ii φ tr 3 ct CD » ex CD Φ ct 3 o O SD φ LQ μ- CD 0 0 φ β O tr O Φ Φ O hi φ Φ tr 3 O rt ti φ rt O Hi CO φ 3 3 CO ct CD ct Φ h-^ SD μ- 1 sx r+ h- 1 SX Φ CX h- 1 μ- μ- rt hi tr i hh φ rt 3 LQ H rt μ- rt Φ h- 1 μ" rt ^ rt hi 3 0 μ- μ- μ- rt β μ- < ti 3 CD 3 O s3 0 μ- s3 μ- 3 * 0 μ- Hi ω μ- 3 0 CD μ- LQ tr CD 3 rt Ω O 3 CO 3 LQ μ- Hi 3 rt LQ CD μ- 3 SD h- 1 CΩ μ- rt 3 h3 3 H 3 μ- Φ rt Ω 0 ti μ 1 O H rt μ- Φ cx sQ tr SD 3 <x LQ 3 - 1 3 Φ LQ CD Hi < 3 CD rt 0 CD rt tr 0 ^ « TJ rt rt Φ TJ CX rt μ- Ω CD Φ 3 tr 0 (X Ω φ SD Ω 3 μ- Ω s3 CD sx tr 3 φ O 3 tr tr 3 tr 3 rt 3 0 -- ■ f- 1 -- ■ H Φ o 3 hi CX 0 tf 3 3 μ- μ- Hi rt Ω φ (D Φ O S μ- tr iQ 3 φ Φ 3 Hi 3 rt o 3 CD • ex O CD c+ CO Ω tr 0 Ω ft CO Hi tr rt Φ TJ CO H β μ- TJ μ- CD s3 0 CD 3 Hi 3" φ rt 0 Φ 3 cn Φ rt Φ O 3 SD tr 3 H- co μ" 3 0 0 tr 3 Hi LQ rt rt < tr 3 rt 3 tr H μ- rt 3 rt CD ti φ CD rt • SD φ 3 3 φ \ 0 Φ --, tr φ Φ tr φ φ rt CO ti 3 Φ μ- 3 tf rt CD SD rt φ φ rt φ μ > CD CD φ ^ φ CD rt Hi Φ φ SD 3 μ- 3 ti tr 0 0 Φ CD CD Φ t tr φ hi Φ IX rt μ- sx 3 rt tr tr ft Φ Ω CD iQ ft tr 0 φ rt rt h- 1 rt tr Φ rt LQ ti rt O 0 CD ft μ- CD φ Φ tf SX Ω tr μ- tr tr o Φ μ- < φ φ μ- β σ 3 Hi CD 3 rt TJ CD Φ h- 1 tr O Φ ti ct o σ S3 μ- O φ SD rt P μ- μ- rt ■ <; Ω tr Φ ii Ω Φ SD Hi 3 ti μ- μ- tr Hi φ μ- cn 3 H Hi CO 4 rt (X Φ TJ rt 0 μ- 0 μ- O tr CD CO 3 μ- 3 N Φ s3 cx rt 0 < • tr rt 50 CD tr ti 3 hi Ω Ω h- 1 Φ CO sQ 3 Ω CD 0 rt μ- rt CD rt < μ- H rt CD tr 0 0 cQ φ ii ct sx 0 h- 1 ti rt Φ CD tr Ω φ 3 S3 cx tr Ω tr Ω 3 tr t- 1 t Hi Φ Φ 0 Φ 3 3 μ- 0 rt H SD Ω 3 rt Φ rt Φ tr h- 1 CD Φ Φ β SJ tr 0 -- ■ CO Ω ti Φ rt 3 O O 3 β rt CD Φ tr μ- 0 β -- ■ ii φ tr φ Hi . μ- ^J tr ct en H sQ H Hi cQ li μ- 3 φ Hi Φ " -» φ SD 3 O φ μ- rt CO 0 μ- Φ CD 0 < μ- 3 rt TJ i φ 0 0 3 Ω 3 Φ 0 co tr P 0 rt 3 LQ >< • cx Hi φ 3 hh coefficient α and a correction term β as the functions of the contents of the component elements of the rolling material as expressed by the following equations (1) and ( 2 ) , and the sheet width change amount is corrected by using either one, or both, of the correction coefficient and the correction term so that high accuracy estimation of the sheet width change amount can be accomplished for all kinds of steels: Ct — 0.(1-,^, 2 , « • , L--N • . • (-1) β = β(C lf C 2 , . . , C N ) . . . (2) Here, C x to C N are the contents of 1 to N component elements contained in the rolling material, and represent the contents of the component elements C, Si, Mn, etc, shown in FIGS. 8(a) to 8(c). Either one, or both, of the correction coefficient ct and the correction term β are added to, or multiplied by, the sheet width change estimation formulas (1) and (2) as shown in the following equations (3) to (5) to correct the calculated sheet width change amount: ΔW F ' = ct F Δw F . . . (3) ΔW F ' = ΔW F + β F . . . (4) ΔW F ' = α F ΔW F + β F . . . (5) Here, ct F is the correction coefficient expressed as the function of the steel kind component element of the sheet width estimation formula in finish rolling, β F is the correction term described above, ΔW F " is the calculated sheet width change amount corrected by either one, or both, of ct F and β F . As described in the twelveth invention, a sheet width change estimation having higher accuracy becomes possible by separating the sheet width change resulting from the plastic deformation from the sheet width change resulting from the creep deformation, each having a different mechanism, and calculating the correction coefficient and the correction term of the sheet width change estimation formula. In this case, when the correction coefficient of the influence term resulting from the plastic deformation in the sheet width change estimation formula in finish rolling is FP , the correction term is β FP , the correction coefficient of the influence term resulting from the creep deformation is cc FC and the correction term is β FC , the formula corrected by using them is expressed by the following equation (6): ΔW F ' = (ct FP ΔW FP + β FP ) + (ctF c Δw FC + β FC ) . . . (6) Here, ct FP , β FP , α FC and β FC are the functions of the contents of the component elements as defined in the equations (1) and (2). This correction may also be conducted by using only one of the correction coefficient and the correction term as expressed by the equations (3) and (4). In connection with the sheet width change behavior resulting from the plastic deformation, the correction coefficient and the correction term are defined for each sheet width change estimation formula for the entry side of roll bite and inside of roll bite, and the correction is conducted respectively. In this way, estimation of the sheet width change with higher accuracy can be expected. In the sheet width change estimation formula in rough rolling, too, the correction is conducted as represented by the following equation (7) by using the correction coefficient and the correction term expressed by the functions of the component elements in the same way as in the finish rolling described above. The sheet width change estimation formula in roughing rolling mill may be conducted in accordance with the estimation formula described, for example, in The Proceeding of the 1979 Japanese Spring Conference for the Technology of Plasticity, pp. 489-496. In this case, the calculation is naturally conducted while the rolling conditions such as the reduction ratio, the sheet thickness, the sheet width, the deformation resistance, etc, are taken into consideration. ΔW R ' = ct R Δw R + β R . . . (7) Here, ct R is the correction coefficient expressed as the function of the component elements of the steel kind in the sheet width change estimation formula in rough rolling, β R is the correction term and ΔW R is the sheet width change amount calculated from the sheet width change estimation formula in rough rolling. ΔW R ' is the calculated sheet width change amount corrected by ct R and β R . Incidentally, in the sheet width change estimation formula in rough rolling described above, it is normal to calculate the sheet width change for the front end portion, the middle portion and the tail end portion of the rolling material in accordance with the different sheet width change estimation formulas, respectively, and sheet width change estimation with higher accuracy can be expected when the correction coefficient and the correction term are defined and corrected again for each estimation formula. The sheet width change occurring on the run-out table between the finishing rolling mill and the coiler is believed to result from the creep deformation that occurs at 600 to 900 °C in the phase transformation region. Therefore, the estimation formula based on the creep formula in this temperature range is used for the sheet width change estimation formula. In this case, the correction formula is similarly given by the following equation (8) where ct ROT is the correction coefficient of the sheet width change estimation formula on the run-out table and β R0T is its correction term: ΔW R0T ' = α ROT Δw ROT + β R0I . . . (8) Here, Δw R0T is the sheet width change amount calculated from the sheet width change estimation formula on the run-out table and ΔW R0T ' is the calculated sheet width change amount corrected by ot ROT and β R0T . As a result of studies by creep experiments, the present inventors have confirmed that the creep deformation behavior is drastically different before and after the phase transformation depending on the steel type, and the sheet width change estimation formula has to use creep formulas before and after the phase transformation. The correction coefficient and the correction term, too, must be different before and after the phase transformation. In this way, higher precision estimation of the sheet width change can be expected. Therefore, the thirteenth invention separates the correction coefficient and the correction term before the phase transformation of the rolling material from those after the phase transformation, and discloses a method of calculating the correction coefficient and the correction term of the sheet width change estimation formula. In this case, the correction coefficient and the correction term are given by the following equation (9): CROT — .RO T 1 ( C-L , (_ 2 1 . . . , N ) ( J-ROT — - L τ ) C'ROT — "ROT2 V ^i ' ^-2 ' • • • i ) V ■ '-ROT ■ " ■ !_. ) PROT — P Rθ τι ( ^-i f C 2 r . . . , C N ) ( T R0T ≤ l τ ) p R0T = P ROT2 ( C 1 , C 2 , • • • , C N ) ( T R0I > T τ ) . . . ( 9 ) Here, T R0T is the temperature of the rolling material on the run-out table, T τ is the phase transformation temperature of the rolling material, ct R0T1 and β ROT1 are the correction coefficient and the correction term before the phase transformation, and α-,^ and β R0T2 are the correction coefficient and the correction term after the phase transformation. Furthermore, it is possible to estimate a highly accurate sheet width change by learning the correction co co to to cπ o cπ o cπ cπ Hi I- 1 tr rt Ω Ω φ rt μ> tr Ω μ- er Ω l- 1 Ω CD -- ■ CD ct tr Ω CD ii ii μ- t 3 Ω Ω μ- μ> 0 tr tr 0 -α tr o 0 tr CD 0 co μ- 0 φ 0 o ti tr o ti ti 0 0 Φ 0 0 3 tr Φ tr 0 3 CX) ti φ SD Φ β φ O ti CD CO 3 «• ex 3 ii --J O φ to ii SD ii 0 rt β Φ SD SD Φ μ- μ- 3 Hi SD μ- 3 β sQ rt LQ Ω hi 3 φ CD μ- μ- H LQ rt CD 3 Hi co CD N hi LQ Hi rt < SD N LQ 3 μ- rt tr μ- 0 Φ 3 rt CD 3 3 N μ- P tr μ- tr sx β LQ Hi tr 3 0 O φ μ- o S μ- φ 3 O φ μ- ct tr rt 3 Ω tr 1 CD Φ iQ 0 3 CD μ- 3 Φ φ li Φ μ- μ- ex 3 β Ω 3 (1 o ti SX 3 3 β φ 3 β CO rt Φ 3 s3 rt Φ 3 sQ 3 v-3 h- 1 Φ Ω 3 ft sQ φ μ- H " 3 rt t CD Q sx φ ex μ- μ- CD CO Φ μ- CX P rt rt iQ hh μ- (X Φ μ- sQ σ CD tr CO Φ rt μ- rt CD 3 μ- CO rt μ- 0 ϋ tr rt IX CD SD 3 tr 3 0 < CO φ Φ μ- ft 3 tr 1 -. — . Ω tr h- 1 O 3 tr φ 3 Φ 3 3 o Φ rt CO 0 CO P Φ ii μ- β Φ < rt 3 hi rt 3 μ- rt Φ SD φ β φ ii CD ii μ- s3 H tf 3 tr Ω 0 ii li SD μ- rt O s3 H LQ 3 h- 1 ti 3 CO φ μ* φ rt 3 3 rt Φ H tr O ct • : h- 1 3 μ- 1 φ 0 SD SD cr CX O rt 0 t μ> ex Φ LQ * 3 tr ft Φ 0 μ- σ μ> Φ β CD φ ii rt 3 SD ti Φ sx t ex • H d t Φ Φ tr ct - 1 3 C . μ- μ- Φ CD Φ ct 3 μ- 3 h- 1 O μ- ex CD Ω o Hi μ 1 CX ft μ- 3 o μ- 3 0 H to rt Φ μ" φ Φ 3 3 3 μ- μ- CX 3 O μ- CD 0 β tr H μ- H Hi o Φ 0 S3 Ω iQ Ω rt ex σ O sQ rt CO 3 rt CD II 3 hi CD Φ CD Φ ex O φ t σ • P ii cr φ μ- CO H 1 ft tr Φ Φ 3 rt tr rt μ- tr μ- 3 ct rt Ω rt 3 Ω tr ► rt H L_l. ti rt O 3 β ct tr 3 φ SD 3 hh φ o tr 3 CD CD rt ct tr rt Φ tr CT. li --> Φ ct rt SD CTi Hi μ- μ- O s3 CO φ μ- 3 cQ O Hi Φ LQ rt 3 0 rt μ- rt μ- β φ • SD t Ω ti tr 3 0 μ- 3 Hi Φ ^ Hi cx ti Ω μ- cx tr 0 (D tr 0 CO CD 3 σ rt • : CX 0 ii 1 3 < Φ CO Ω H μ- 3 3 0 rt ,-, CD 3 0 rt Φ 3 rt Φ 3 tr Φ Ω LQ -« 3 CD 3 LQ Φ rt 3 rt 0 3 O μ- β hi tr Ω μ- 3 O tr φ 0 Φ hh CD Φ Ω rt 3 tr Φ ti t SD Hi 1 ti μ- -Q H Hi Φ Ω rt Φ φ 0 ii SX CD 0 μ- rt μ> CD 0 i P rt Φ ct 3 ti hi SD Φ CD + β --" SD SD μ- tr ti rt ct hh H H ex Φ 3 CD μ- tr Φ SD rt Ω μ- 3 rt cx co Φ t tr Φ O rt Φ ti ~J ii rt μ- Hi O Hi Φ tr Ω μ- tr tr rt μ- ct Φ > H μ> o tr O Ω ii CD s3 ct Ω 3 tr Ω β li 3 μ- 3 μ- P ct 3 0 Φ hi 3 μ- Λ -a φ o β Φ LQ β g |μ- * rt μ- tr rt φ 0 0 μ> O 3 0 " 3 • 3 Hi CD μ- li O β -_ (X t 3 h- 1 (1 rt cx φ μ- rt ti 3 Hi o Hi 1 CD μ- μ- φ 0 SD μ- H μ- rt 3 - • • c+ ti σ CD H " tr CO μ- rt 0 tr 3 rt CX3 0 P - 1 CD 3 sx 3 3 N μ- 3 tr rt P CD O 0 rt μ- co tr o 3 Φ O H rt hh β μ- IX tr μ- cx 0 3 Φ rt μ- H O β 3 Φ _ sx tr 3 0 3" CD μ- rt 3 < < Φ tn O ct 3 μ- μ> Φ o + Φ Hi LQ φ CX μ. μ- Φ 3 Φ Ω Φ Φ h- 1 φ 3 3 LQ φ Φ rt ti tr 3 Φ rt rt o o tr hi 3 tr < Φ μ- φ 0 ti rt SX μ- rt ti hi - O ti tr CD tr CO O 3 > Ω rt μ- CO <! II μ- 0 SD CD ct Φ rt Φ μ- <! CD 5D rt rt rt to ϋ ft 3 φ h- 1 μ- H - — . o tr 3 tr SD co Hi CD rt Hi i ti 3 Φ tf σ tr μ- μ- 0 tr 3 μ- 0 μ> hi Φ sQ CD t— - μ- β ϋ s3 H *< iQ li φ -• φ Ω Ω Ω μ- Φ 0 Ω i LQ μ- N hh o o hi TJ β O t ti μ- μ- μ- μ* rt rt ti Φ P CD O 3 Hi 0 0 3 O — • φ Ω H Φ Φ (+ 3 tr Φ cr ex Ω CO t rt μ- tr •»« CD h- 1 3 sQ μ- CO μ- H H 3 rt *. --• ' Ω 0 O CO O Φ SX β rt μ- μ- μ- 1 tr Ω Φ hi tr CD 3 tr rt H H O ct rt tr . rt 3 μ- TJ rt tr φ ex Φ P ii O φ hi H μ- 3 iQ φ tr φ tr (D Φ rt . " »» rt h- 1 3 0 a O Ω σ μ- 3 Φ CD 3 h- 1 Φ 0 0 sx μ- φ Φ hi CO h-" φ h- 1 tr . H μ- H 4 μ- 0 •<; O 3 rt 3 CO 0 Ω rt h- 1 h- 1 Φ 3 rt ft μ- CD Φ rt 0 3 rt 3 μ- 3 Φ O CX tr ti 3 μ- 0 ti μ- CO μ> CD 3 H hi tr -^ tr iQ tr rt + rt H rt rt P Hi Φ 0 Φ 3 μ- μ- r-> S tr o 3 LQ μ- O φ Ω - 1 Φ Φ tr Φ tr CD φ 3 CD Φ h- 1 rt iQ μ- 3 CD CD P μ> μ- φ ! cx 3 φ o o rt 3 φ 3 CD H φ h- 1 ii SX rt rt Φ Φ sQ rt CO CO sx Φ 3 CD μ" rt ϋ — Hi CD μ- Hi t ^ O tr CO μ- ti 3 li rt P tr rt rt CD 0 μ- H H 0 ϋ μ- s3 + 0 CO 3 μ> rt φ tr S 3 μ- tr 3 3 0 ex CD tr ti hh CO S3 φ μ- LQ -- ■ 3 μ- 3 O tr LQ t 3 ^ Φ μ- sQ μ> μ> •-« μ- ex rt Φ 3 s3 φ rt μ- Ω Φ CD ex 3 iQ ~J φ o Φ Ω Φ ex o o μ> CO P μ- ex rt O rt rt -- ■ rt *. φ rt 0 ct rt 3 o ^j (D μ> Ω ^ Ω rt 3 rt μ- s3 H tr rt ti ¬ tr s3 μ- 3" μ- o ii 0 ct rt hi SX tr 0 μ- 0 CD . tr ct Φ μ- μ 1 ^ *- μ- hi β tr CD 3 3 hh μ- φ tr μ> φ r- 1 CD o σ CD μ- 0 iQ 3 H ii CD t 0 Φ 0 h-" 3 Hi ct Hi ex 3 learned. More concretely, the constants of the correction coefficient and the correction term of each sheet width change estimation formula are calculated so that the values of both sides of the equation (10) become equal. The constants calculated afresh in this way are then multiplied by a certain gain and are added to the constants before learning to conduct learning. In this instance, distribution to each estimation formula may be uniform or may have a certain weight. Needless to say, learning can be done with higher accuracy if the actual measured values of the sheet width meter immediately close to the delivery side of the finishing rolling mill and between the rolling mills can be used conjointly. Next, it is possible to estimate a highly accurate sheet width change by learning the correction coefficient and the correction term of the sheet width change estimation formula from the actual measured temperature between the rolling mills and/or on the delivery side of the finishing rolling mill train, and/or the actual measured value of the sheet crown as described in the fifteenth invention. Considering the case where the correction coefficient and the correction term are supplemented by learning by use of the thermometer 121 and the sheet crown meter 122 disposed on the entry side of the coiler 107 on the delivery side of the finishing rolling mill train, as shown in FIG. 6, the temperatures in the final horizontal roll stand of the roughing rolling mill train 103, in the final horizontal roll stand of the finishing rolling mill train 118 and between the final horizontal roll stand of the finishing rolling mill and the coiler 107 and the distribution of the sheet crown along the longitudinal direction are calculated again on the basis of the actual measured values of the temperatures by the thermometer 121 on the entry side of the coiler 107 and the sheet crown distribution along the longitudinal direction, and are compared with the recalculated sheet width change amounts as represented by the following equation (11) to conduct learning of the correction factor and the correction item of the sheet width change estimation formula: M c (i) - M R (i> = ΔW R " !i> + Δw F ' ,(i) + ΔW ROT ,,(i> ... (11) Here, ΔW R " U) , Δw F " (i) and ΔW R0T " (i) represent the sum of the calculated sheet width change amounts of the ith division point along the longitudinal direction in the horizontal roll stand of roughing rolling mill train, in the horizontal roll stand of finishing rolling mill train and between the horizontal stand of the finishing rolling mill and the coiler, that are calculated again on the basis of the actual measured value of the temperature on the delivery side of the finishing rolling mill and the sheet crown actual measured value. As explained above, the present invention measures the distribution of the sheet width of the rolling material along the longitudinal direction and its temperature at the point where the rolling material exists on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction, calculates the temperatures of the rolling material in each rolling mill and between the rolling mills on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction from the measured temperatures of the rolling material, and calculates the distribution of the sheet width change amount of the rolling material along the longitudinal direction that occurs in each rolling pass on the upstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction on the basis of the calculated temperature of the rolling material and the distribution of the measured sheet width of the rolling material along the longitudinal direction to thereby calculate the distribution of the sheet width of the rolling material along the longitudinal direction on the entry side of the vertical rolling mill set as the object for controlling the sheet width. Further, the present invention calculates the temperatures of the rolling material in each rolling mill and between the rolling mills on the downstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction, in the zone from the final stand of the finishing rolling mill to the coiler and in the zone from the coiler to the cooling finish point, calculates the distribution of the sheet width change amounts of the rolling material along the longitudinal direction occurring in each rolling pass and between the rolling mills on the downstream side of the vertical rolling mill set as the object for controlling the sheet width along the rolling direction, in the zone from the final stand of the finishing rolling mill to the coiler and in the zone from the coiler and the cooling finish point, calculates the roll gap of the vertical rolling mill set as the object for controlling the sheet width along the longitudinal direction of the rolling material on the basis of the distribution of the sum of the calculated sheet width change amounts of the rolling material along the longitudinal direction, the distribution of the measured sheet width of the rolling material along the longitudinal direction and the final target sheet width of the rolling material, and conducts the control so as to attain this roll gap. Therefore, the present invention provides the effect that sheet width accuracy and the yield can be drastically improved in comparison with the conventional methods. In addition, the present invention uses the sheet width change estimation formula corrected by the correction coefficient and the correction term that take the influences of the component elements into consideration, and learns the correction coefficient and the correction term from the actual measurement values of the sheet width. In consequence, the present invention can estimate the sheet width change occurring in the roughing rolling mills, in the finishing rolling mills and on the run-out table for various steel kinds. Since the present invention sets and controls the roll gap pattern of the vertical rolling mill on the basis of these sheet width change estimation values, the present invention can accomplish the sheet width control with higher accuracy than the conventional methods, and can accomplish the improvement of the yield. [Examples] <Example 1> The explanation will be given on the application of the present invention to a hot rolling line including a roughing rolling mill train 2 having r stages, in total, of vertical and horizontal roll stand of roughing rolling mills, a finishing rolling mill train 5 having f stands, in total, of vertical rolling mills 3 and horizontal roll stand 4, a cooling machine 6 on a run-out table, a coiler 7 , a sheet width meter 8 and a thermometer 9 arranged on the entry side of the roughing rolling mill train 2, a roughing rolling mill train controller 10, a vertical roll stand of finishing rolling mill controller 11, a horizontal roll stand of finishing rolling mill train controller 12, a cooling controller 13, a coiler controller 14 and an arithmetic unit 15. This explanation will be given hereinafter n the case where the vertical stand of vertical rolling mill 3 of finishing roll train 5 is used as the object for controlling the sheet width. The rough rolling mill train controller 10 has a roll gap control function and a roll peripheral speed control function for each vertical roll stand of roughing rolling mill train, a crown/shape control function, a reduction control function and a roll peripheral speed control function for each horizontal roll stand of roughing rolling mill train. The vertical roll stand of finishing rolling mill controller 11 has a roll gap control function and a roll peripheral speed control function for the vertical roll stand of finishing rolling mill train. The finish horizontal rolling mill train controller 12 has a tension control function between the rolling stands, a crown/shape control function, a reduction control function, a roll peripheral speed control function and a looper control function of the finishing rolling mill train. The cooling controller 13 has a water quantity pattern control function of the cooling machine on the run-out table. The coiler controller 14 has a coiler peripheral speed control function are inputted to the arithmetic unit 15 and the sheet width measurement value by the sheet width meter 8 , the temperature measurement value by the thermometer 9 and the information on the rolling material 1 such as the steel kind that is transferred from an another arithmetic unit (not shown) are outputted. The control information for the roughing rolling mill train controller 10, the vertical roll stand of finishing rolling mill controller 11, the finish horizontal rolling mill train controller 12, the cooling controller 13 and the coiler controller 14 are outputted. Next, the sheet width control method of the present invention will be explained with reference to the flow shown in FIG. 1. In the setup calculation, the arithmetic unit 15 decides the draft schedule in the roughing rolling mill train 2, the draft schedule, the crown schedule, the roll peripheral roll speed of each rolling mill, the tension between the rolling mills in the finishing rolling mill train 5, the cooling condition of the cooling machine 6 on the run-out table and the coiling speed of the coiler 7 from the target sheet thickness of the rolling material 1 on the delivery side of the finishing rolling mill train, the target sheet crown, the target coiling temperature of the coiler, and so forth. The arithmetic unit 15 divides the rolling material into n L elements in the longitudinal direction. The sheet width measurement value W 0 (i) (i = 1 to N L ) corresponding to each element measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2 is transferred to, and stored in, the arithmetic unit 15. The temperature T 0 of the rolling material 1 measured by the thermometer 9 arranged on the entry side of the roughing rolling mill train 2 is transferred to the arithmetic unit 15, and is stored as the temperature of each element described above. On the basis of the sheet width W 0 (i) and the temperature T 0 , the rough rolling condition decided by the setup calculation, the arithmetic unit 15 calculates the temperatures of the rolling material 1 in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and the temperature on the entry side of the vertical roll stand 3 as the control object, in the following description, the temperature of the rolling material 1 in the kth roll stand (k = 1 to r) of the roughing rolling mill train 2 is ^ lk) , the temperature of the rolling material 1 between the (k-1 )th roll stand and the kth roll stand is T RI (k) and the temperature of the rolling material 1 on the entry side of the vertical roll stand 3 as the control object is T ent . The arithmetic unit 15 calculates the sheet width change amounts in each rolling mill and between the rolling mills of the roughing rolling mill train 2 on the basis of the calculated temperature information of the rolling material 1 and the sheet width distribution of the rolling material 1 along the longitudinal direction measured by the sheet width meter 8 arranged on the entry side to the roughing rolling mill train 2. Hereinafter, the sheet width change amount of the ith element of the rolling material 1 in the kth stage rolling mill of the rough rolling mill train is ΔW RR (k> (i) , the sheet width change amount of the ith element of the rolling material 1 between the (k-l)th roll stand and the kth roll stand is Δw RI ( ) (i) and the sheet width change amount of the ith element of the rolling material 1 between the rth stand of the roughing rolling mill train 2 and the vertical rolling mill 3 as the control object is Δw ent (i) calculated by following formula (12). The arithmetic unit 15 adds the sum ΔW R (i) of the sheet width change amounts in each rolling mill and between the rolling mills of the roughing rolling mill train 2 calculated by the calculation described above and the measurement value of the sheet width: W 0 (i) measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2, and calculates the sheet width W ent (i) of the rolling material 1 along the longitudinal direction on the entry side of the finish vertical rolling mill 3 as the control object. W ent (±) = w 0 (i) + W R (i) . . . (12) AW R, ( S- ) " 1 Δθ(i) + w k) i) + ΔWL nt (i) k-l JcZϊ Further, the arithmetic unit 15 calculates the temperatures of the rolling material at each point in each rolling mill and between the rolling mills of the finishing rolling mill train 5 inclusive of the vertical rolling mill 3, in the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7 and in the zone from the coiler 7 to the cooling finish point on the basis of the sheet width W ent (i) and the temperature T ent of the rolling material 1 on the entry side to the finish vertical rolling mill 3 as the control object vertical. Hereinafter, the temperature of the rolling material 1 in the kth (k = 1 to f) stand of the finishing rolling mill train 5 is T RP t > , the temperature of the rolling material 1 between the (k-l^th roll stand and the kth roll stand is T RI (k) and the mean temperature in the kth (k = 1 to N) zone formed by dividing the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7 into N zones in the longitudinal direction is T R0T (k> . The arithmetic unit 15 calculates the sheet change amounts of the rolling material at each point in each rolling mill and between the rolling mills of the finishing rolling mill train 5 inclusive of the finish vertical rolling mill 3, in the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7, and in the zone from the coiler 7 to the cooling finish point on the basis of the temperature information of the rolling material 1 and its sheet width distribution along the longitudinal direction. Hereinafter, the sheet width change amount of the ith element of the rolling material 1 in the kth (k = 1 to f) finish rolling mill group 5 is ΔW FR ^ 1 ( i ) , the sheet width change amount of the ith element of the rolling material 1 between the (k-l)th roll stand and the kth roll stand is Δw FI < ) (i), the sheet width change amount of the ith element of the rolling material 1 in the kth (k = 1 to N) zone when the zone from the delivery side of the finish rolling mill train 5 to the coiler 7 is divided into N zones in the longitudinal direction is Δw R0T (k) (i), and the sheet width change amount of the ith element of the rolling material 1 from the coiler 7 to the cooling finish point is ΔW c2 (i). The sum Δw F (i) of the sheet width change amounts in each rolling mill and between the rolling mills of the finishing rolling mill train 5 calculated as above, the sum ΔW cl (i) of the sheet width change amounts in the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7, the sum Δw c2 (i) of the sheet width change amounts in the zone from the coiler 7 to the cooling finish point and the sheet width ΔW ent (i) on the entry side of the vertical rolling mill 3 are added for the ith element of the rolling material 1, and the estimated value W oal (i) of the final sheet width of the rolling material is calculated by following formula (13). + Δw cl (i) + Δw c2 (i) Δ^.(i) = Δ^ OT w (i) --1 The estimation formulas of the sheet width change ratio ΔW-^'fi) of the ith element of the rolling material 1 in the kth stand of the roughing rolling mill train 2, the sheet width change amount ΔW RI ( ) (i) of the ith element of the rolling material between the (k-l)th roll stand and the kth roll stand, the sheet width change amount Δw ent (i) of the rolling material between the rth roll stand of the roughing rolling mill train 2 and the vertical rolling mill 3 as the control object, the sheet width change amount Δw-^'^i) of the ith element of the rolling material 1 in the kth (k = 1 to f ) roll stand in the finishing rolling mill train 5, the sheet width change amount Δw FI (k) (i) of the ith element of the rolling material 1 between the (k-l)th roll stand and the kth roll stand, the sheet width change amount Δw R0T < ) (i) of the ith element of the rolling material 1 in the kth (k = 1 to N) zone when the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7 is divided into N zones and the sheet width change amount Δw c2 (i) of the ith element of the rolling material 1 from the coiler 7 to the cooling finish point are expressed as the functions of the conditions on the rolling mill side such as the reduction in each rolling mill, the roll diameter of the rolling mills, the roll peripheral speed, the tension between the rolling mills, the rigidity of each rolling mill, etc, the conditions on the cooling machine side on the run-out table such as the cooling water quantity and the cooling pattern, the conditions on the coiler side such as the coiling speed of the coiler, and the conditions of the rolling material such as the steel kind (components), the sheet width, the sheet thickness, the sheet crown, the temperature and the distance of the corresponding portion from the front end or the tail end. For example, the estimation formulas of various sheet width change amounts described in "Theory and Practice of Flat Rolling" edited by Rolling Theory Committee of the Japan Steel and Iron Institute of Japan in 1984, page 73, Table 3.1. The estimated value W cal (i) of the final sheet width of the rolling material 1 thus determined is compared with the final target sheet width W aim . If W cal (i) is greater than W aim , the roll gap of the vertical rolling mill 3 for the corresponding portion is decreased and if the former is smaller than the latter, the roll gap of the vertical rolling mill 3 for the corresponding portion is increased. In this way, the roll gap of the finish vertical rolling mill 3 is controlled so that the estimated value W oal (i) of the final sheet width of the rolling material 1 coincides with the final target sheet width W aim . Sheet width control accuracy of the system described above is compared with that of the conventional method. One hundred rolling materials are used for each method, and the sheet width of the coil is measured in the successive process. As a result, the standard deviation of the difference between the actual measured value of the sheet width defined by the entire length of the rolling material along the longitudinal direction and the target value is 2.5 mm in the conventional method, whereas it is 1.5 mm in the sheet width control method of the present invention. It has thus been confirmed that the method of the present invention can improve sheet width accuracy, and the effect of the novel sheet width control method of the present invention has been verified. The explanation given above represents the case where the vertical rolling mill set as the control object arranged on the first horizontal roll stand of the finishing rolling mill train is not particularly limited to the vertical roll stand of finishing rolling mill train in the present invention, but may be the vertical rolling mill in the roughing rolling mill train or the edger in the finishing rolling mill train, for example. The explanation given above represents also the case where the width reduction device is the vertical rolling mill, by way of example, but is not particularly limited thereto, and may be a sizing press, for example. Further, in the explanation given above, the sheet width meter 8 and the thermometer 9 are arranged on the entry side of the roughing rolling mill train 2 by way of example, but the present invention does not particularly limit the arrangement of the sheet width meter and the thermometer on the entry side of the roughing rolling mill train. In other words, they may well be arranged on the upstream side to the vertical rolling mill set as the object for controlling the sheet width along the rolling direction. [Example 2] In the application of the present invention to the hot rolling line shown in FIG. 3, in the same way as in Example 1, the thermometer 9 arranged on the entry side of the roughing rolling mill train 2 measures the temperature distribution of the rolling material 1 along the longitudinal direction, and transfers the result to the arithmetic unit 15. The arithmetic unit 15 stores a temperature T 0 (i) corresponding to each element of the rolling material 1 divided into n L elements along the longitudinal direction. The arithmetic unit 15 further calculates the temperature distribution of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical roll stand 3 set as the object for controlling the sheet width. Next, the arithmetic unit 15 calculates the distribution of the sheet width change amounts of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical rolling mill 3 as the vertical rolling mill set as the object for controlling the sheet width on the basis of the temperature information of the rolling material 1 calculated as described above and the sheet width distribution of the rolling material 1 along the longitudinal direction measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2. As a result, estimation accuracy of the sheet width change amounts of the rolling material in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical rolling mill 3 as the vertical rolling mill set as the object for controlling the sheet width can be improved. It has thus been confirmed that the standard deviation of the difference between the actual measured value of the sheet width defined by the entire length of the rolling material along the longitudinal direction and the target value is 1.3 mm, and the sheet width control method of this example can further improve sheet width estimation accuracy. [Example 3] In the application of the present invention to the hot rolling line shown in FIG. 3 in the same way as Example 1. In this Example, the thermometer 9 arranged on the entry side of the roughing rolling mill train 2 measures the temperature distribution of the rolling material 1 along the transverse direction, and transfers the result to the arithmetic unit 15. The arithmetic unit 15 stores a temperature T 0 (j) corresponding to each element of the rolling material 1 divided into n c elements along the transverse direction. The arithmetic unit 15 further calculates the temperature distribution of the rolling material 1 along the transverse direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the finish vertical rolling mill 3 set as the object for controlling the sheet width. Next, the arithmetic unit 15 calculates the distribution of the sheet width change amounts of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical rolling mill 3 set as the object for controlling the sheet width on the basis of the temperature information of the rolling material 1 calculated as described above and the sheet width distribution of the rolling material 1 along the longitudinal direction measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2. As a result, estimation accuracy of the sheet width change amounts of the rolling material in each rolling mill and between the rolling mills 1 of the roughing rolling mill train 2 can be improved. It has thus been confirmed that the standard deviation of the difference between the actual measured value of the sheet width defined by the entire length of the rolling material along the longitudinal direction and the target value is 1.3 mm, and the sheet width control method of this example can further improve the sheet width estimation accuracy. [Example 4] In the application of the present invention to the hot rolling line shown in FIG. 3, in the same way as in Example 1, the thermometer 9 arranged on the entry side of the roughing rolling mill train 2 measures the temperature distribution of the rolling material 1 along both longitudinal and transverse directions, and transfers the result to the arithmetic unit 15. The arithmetic unit 15 stores a temperature T 0 (i,j) corresponding to each element of the rolling material 1 divided into n L along the longitudinal direction of the rolling material 1 and n c elements along the transverse direction. The arithmetic unit 15 further calculates the temperature distributions of the rolling material 1 along both longitudinal and transverse directions in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical rolling mill 3 set as the object for controlling the sheet width. Next, the arithmetic unit 15 calculates the distribution of the sheet width change amounts of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the finish vertical rolling mill 3 set as the object for controlling the sheet width on the basis of the temperature information of the rolling material 1 calculated as described above and the sheet width distribution of the rolling material 1 along the longitudinal direction measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2. As a result, estimation accuracy of the sheet width change amounts of the rolling material 1 in each rolling mill and between the rolling mills 1 of the roughing rolling mill train 2 can be improved. It has thus been confirmed that the standard deviation of the difference between the actual measured value of the sheet width defined by the entire length of the rolling material 1 along the longitudinal direction and the target value is 1.2 mm, and the sheet width control method of this example can further improve sheet width estimation accuracy. [Example 5] In the application of the present invention to the hot rolling line shown in FIG. 3, in the same way as in Example 1, the thermometer 9 arranged on the entry side of the roughing rolling mill train 2 measures the temperature distribution of the rolling material 1 along both longitudinal and transverse directions in accordance with the flow shown in FIGS. 2(a) and (b), and transfers the result to the arithmetic unit 15. The arithmetic unit 15 stores a temperature T 0 (i,j) corresponding to each element of the rolling material 1 divided into n L along the longitudinal direction of the rolling material 1 and n c elements along the transverse direction. The arithmetic unit 15 further calculates the temperature distributions of the rolling material 1 along both longitudinal and transverse directions in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical rolling mill 3 set as the object for controlling the sheet width. Next, the arithmetic unit 15 calculates the distribution of the sheet width change amounts of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical rolling mill 3 set as the object for controlling the sheet width on the basis of the temperature information of the rolling material 1 calculated as described above and the sheet width distribution of the rolling material 1 along the longitudinal direction measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2. Next, the arithmetic unit 15 calculates the temperature distribution of the rolling material along both longitudinal and transverse directions at each point in each rolling mill and between the rolling mills of the finishing rolling mill train 5 inclusive of the vertical rolling mill 3, in the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7 and in the zone from the coiler 7 to the cooling finish point. Further, the arithmetic unit 15 calculates the distribution of the sheet width change amounts of the rolling material along the longitudinal direction at each point in each rolling mill and between the rolling mills of the finishing rolling mill train 5, in the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7 and in the zone from the coiler 7 to the cooling finish point on the basis of the calculated temperature information of the rolling material 1 and the sheet width distribution of the rolling material 1 along the longitudinal direction. As a result, estimation accuracy of the sheet width change amounts of the rolling material 1 in each rolling mill and between the rolling mills of the roughing rolling mill train 2, in the zone from the delivery side of the finishing rolling mill train 5 to the coiler 7 and in the zone from the coiler 7 to the cooling finish point can be improved. It has thus been confirmed that the standard deviation of the difference between the actual measured value of the sheet width defined by the entire length of the rolling material 1 in the longitudinal direction and the target value is 1.0 mm, and the sheet width control method of this example can further improve sheet width estimation. [Example 6] This example provides a method that is suitable when the control of the roll gap of the vertical rolling mill alone cannot sufficiently accomplish the final target sheet width W aim . In the application of the present invention to the hot rolling line shown in FIG. 3, in the same way as in Example 1, the tension control between the rolling mills of the horizontal roll stand of the finishing rolling mill train 4 supplements the sheet width control by the roll gap control of the vertical rolling mill 3. Incidentally, the tension change between the rolling mills of the final horizontal stand of the finishing rolling mill train 4 invites the change of the rolling load of each horizontal stand of the finishing rolling mill train 4, the change of the sheet thickness co to t cπ o cπ o Cπ cπ estimation accuracy of the sheet width change amounts of the rolling material 1 such that the standard deviation of the difference between the actual measured value of the sheet width defined by the entire length of the rolling material 1 along the longitudinal direction and the target value is 1.2 mm. [Example 9] This example provides a method that will be suitable when the vertical rolling mill set as the object for controlling the sheet width is the vertical rolling mill 17 of the roughing rolling mill train 2. The explanation will be given on the application of the present invention to the hot rolling line including a roughing rolling mill train 2 having r stands, in total, of vertical roll stands inclusive of the final vertical roll stand 17 and horizontal roll stands, a finishing rolling mill train 4 having f stands, in total, of horizontal roll stands, a cooling machine 6 on a run-out table, a coiler 7, a sheet width meter 8 and a thermometer 9 arranged on the entry side of the roughing rolling mill train 2, a roughing rolling mill train controller 10, a finish horizontal rolling mill controller 12, a cooling controller 13, a coiler controller 14 and an arithmetic unit 15, shown in FIG. 4. Here, the explanation will be given on the case where the vertical rolling mill 17 is set as the object for controlling the sheet width. In this example, the thermometer 9 arranged on the entry side of the roughing rolling mill train 2 measures the temperature distribution of the rolling material 1 along the longitudinal direction. The temperature distribution so measured is transferred to the arithmetic unit 15. The arithmetic unit 15 stores a temperature T 0 (i) corresponding to each element of the rolling material 1 that is divided into n L elements along the longitudinal direction. The arithmetic unit 15 calculates the temperature distribution of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2 on the upstream side to the vertical rolling mill 17 in the rolling direction, and on the entry side of the final vertical roll stand of rough rolling mill 17 as the vertical rolling mill set as the object for controlling the sheet width. The arithmetic unit 15 further calculates the distribution of the sheet width change amounts of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2 on the upstream side to the vertical rolling mill 17 in the rolling direction, and on the entry side of the vertical rolling mill 17 set as the object for controlling the sheet width on the basis of the calculated temperature information of the rolling material 1 and the sheet width distribution of the rolling material 1 along the longitudinal direction measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2. Next, the arithmetic unit 15 calculates the temperature distribution of the rolling material 1 along both longitudinal and transverse directions at each point in each rolling mill and between the rolling mills of each of the roughing rolling mill train 2 on the downstream side to the final vertical roll stand of roughing rolling mill 17 in the rolling direction and in each rolling mill of the finishing rolling mill train 4, in the zone from the delivery side of the finishing rolling mill train 4 to the coiler 7, and in the zone from the coiler 7 to the cooling finish point. In this example, the thermometer 9 arranged on the entry side of the roughing rolling mill train 2 measures the temperature distribution of the rolling material 1 along the longitudinal direction. The temperature distribution so measured is transferred to the arithmetic unit 15. The arithmetic unit 15 stores a temperature T 0 (i) corresponding to each element of the rolling material 1 that is divided into n L elements along the longitudinal direction. The arithmetic unit 15 further calculates the temperature distribution of the rolling material 1 along the longitudinal direction on the entry side of the vertical roll stand 17 set as the object for controlling the sheet width. Furthermore, the arithmetic unit 15 calculates the distribution of the sheet width change amounts of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of the roughing rolling mill train 2, and on the entry side of the vertical rolling mill 16 set as the object for controlling the sheet width on the basis of the calculated temperature information of the rolling material 1 and the sheet width distribution of the rolling material 1 along the longitudinal direction measured by the sheet width meter 8 arranged on the entry side of the roughing rolling mill train 2. Next, the arithmetic unit 15 calculates the temperature distributions of the rolling material 1 along both longitudinal and transverse directions at each point in each rolling mill and between these rolling mills of each of the roughing rolling mill on the downstream side to the vertical roll stand 17 of roughing rolling mill train and the finishing rolling mill train 4, in the zone from the delivery side of the finishing rolling mill train 4 to the coiler 7 and in the zone from the coiler 7 to the cooling finish point. Furthermore, the arithmetic unit 15 calculates the distribution of the sheet width change amounts of the rolling material 1 along the longitudinal direction in each rolling mill and between the rolling mills of each of the roughing rolling mill train 2 on the downstream side to the vertical roll stand of roughing rolling mill 17 inclusive of the vertical roll stand in the rolling direction and the finishing horizontal rolling mill train 4, in the zone from the delivery side of the finishing horizontal rolling mill train 4 and the coiler 7 and in the zone from the coiler 7 to the cooling co co to to cπ o cπ o cπ cπ rt to l- 1 ii hi CO P li P σ. cn O Hi < ti Ω o o S3 < Ω Ω tr CD CO rt 3 s SX Ω ex 0 Hi tr tr O 0 O •-» li 0 3 rt Hi μ- Φ 0 0 cr 3 tr ts φ o 0 P ty tr P P μ- μ- P μ- Hi μ- φ 0 ^ li β sx 0 P 3 ii h- 1 3 I—I. φ X ii 3 3 to Φ Φ ti & α Hι t-> CD 3 s3 P P LQ 3 3 Hi μ- rt H CD φ t 3 P μ- rt hh Φ Φ ιΩ Φ rt Hi Ω rt rt μ- Ω 3 P μ- μ- 3 tf P (X μ- co μ- μ- μ- Ω tf 3 hh hi μ- rt rt rt Φ ti tr φ β hi tr 10 P 3 3 hi sQ μ- P 3 tr Ω 3 3 rt Φ ct t-3 TJ μ- o H tr rt μ- H μ 1 μ- φ tf CD μ- Ω sQ LQ 0 Φ 3 rt 0 μ- μ- P Q Φ tr tr tr μ- 1 φ 3 β s3 s3 P Φ CO P σ φ 3 O β CX sQ tr ti Hi co 3 ii φ hh Φ φ μ- φ ex φ CO μ- μ- < μ" φ 3 rt β li TJ μ- 3 3 sQ φ β tr LQ 3 Φ 0 3 CD . 3 ex ex sx P Hi Ω P φ rt 0 0 s3 n- μ- μ- tr 0 ti ii 3 Hi H μ- sx P ti rt tn h- 1 φ *<« tr rt t h- 1 -- ■ μ- Φ μ- H- 1 μ- tr H φ μ- 3 0 3 1 μ- < ii 0 TJ ti tr φ rt φ tr tr β 3 ti • 0 μ" 3 φ Q ii 3 0 o 3 φ 0 μ- TJ Ω • ! Φ X tr P φ Φ P Φ σ Φ 3 μ- rt ii . LQ rt μ> 3 β μ- ti μ> 3 h- 1 0 Φ P 0 3 3 Ω Ω 1 α φ CO 3 φ Ω ct rt tr μ- φ rt CD rt μ- rt μ- 3 CO ct 3 a. o o μ- 0 t β P LQ o cπ o H ϋ H Φ 3 rt tr μ- μ- 3 li rt Ω rt μ- TJ Ω 3 3 tn 3 cr S3 r-> 3 rt • 3 P P 0 iQ Φ ct μ- Ω 3 sQ P tr P ii cx s3 t- 1 0 rt O rt rt ιΩ ^ φ rt 0 3 tr rh μ- μ- Φ hi P 3 P LQ μ- φ rt 0 φ μ- φ Hi tr 3 li li t φ *-« 3 P rt φ H 3 3 μ" 3 3 tr LQ 3 3 μ- sx Φ hh o o . rt rt 3 ιΩ rt tr tn O μ- ct μ- vo 3 μ- tr o O ct TJ rt μ- h- 1 h- 1 cπ tf tf 3 Φ Φ < Φ Ω Ω 3 ti μ 1 φ hi ii μ- to o 3 μ- hh tr ii tr Φ H Φ Φ t tr rt ti φ H r- 1 O o LQ ^ μ< P •»» 0 0 μ- h- 1 ct 3 0 Φ Hi 3 3 3 3 tr φ 3 * μ- cr H Φ Φ 3 3 ti h- 1 CD tr 0 iQ rt o < Hi φ Φ Φ h- 1 Φ φ ti Φ P P ct •-. ti t ct 3 CD rt ϋ P - 1 - 1 P t H 3" Hi μ- TJ φ sx rt rt 0 3 Φ CO μ- H ti μ- μ- ti P μ- μ- rt P < 0 rt Φ sx ii Ω tr tr P 3 rt P P h- 1 μ- Ω rt μ» O 0 μ > ex P 3 Ω 3 3 ii 3 μ- Ct tr ct Φ Φ rt rt 0 O Ω cQ μ- Ω CO 0 - 1 tD P tr tt-- μ < Φ μ- ιΩ O ιΩ LQ P 3 - tr φ < tr CD CO tr CX ex Ω μ- H rt 3 Φ 3 Φ μ- μ- ιΩ μ- φ Φ Φ Φ 0 P O rt Φ β rt LQ P 0 P Φ φ Ω 0 (X 3 3 3 tr 3 CD hi P 3 Hi rt 0 H β P tr μ- 3 hh hi Φ 3 ti hi 0 Hi to φ μ- μ- 0 li LQ TJ tr rt hi rt Hi μ- ex SX Φ rt CX 0 X CX 3 -» 0 ti 1 H 1 ui ii ii φ μ- O 3 rt CD rt μ- μ- Φ β rt μ* TJ μ» μ* ft ct 3 1 H μ- CO H Φ φ Ω -- ■ Φ μ- tr tr rt 3 3 3 3 ra SX μ- tf -- ■ P to - 1 ti tr P CD tr N ft μ- CO ct P rt 3 Φ φ tr μ- <Ω P sQ Φ rt μ- rt Φ μ- P 3 -• -» 0 Φ rt *. rt P 0 P 3 φ h- 1 μ- tr <! φ φ CD M ct P P 3 CD 3 3 cn tr ti ) < 3 3 φ 3 s3 3 0 Φ 3 rt rt tr to 3 P rt LQ P P P P < tf φ P P μ- rt ex rt μ- ti ιΩ cx 3 0 TJ μ> 0 cx β SX 1 cn Φ rt H φ Φ Φ μ- P 3 P D μ- ex 0 rt < s3 hi • μ- o ti P tr 3 3 μ- μ- Ω Hi ti ti Φ Φ cn 3 3 LQ H 3 μ- rt H 3 rt μ- Φ μ- Φ ^J rt li Hi φ ti (X Φ TJ μ- 0 rt O μ- ct t ti tr ^ ex Ω 3 tr > P tr o CO tr Φ (X SX μ- φ Φ 3 tr O 3 H-> μ- ct φ H μ- μ > < . μ- t P 3 rt φ Φ Ω rt ii rt ii r- 1 3 h- 1 μ- O Ω s3 H φ P P 0 3 β φ 3 φ rt CD tr 3 § rt tr < CX φ P s3 Φ μ- D *. P μ- ^ rt CO μ 1 3 ιΩ hi tr tr rt Ω μ- Φ P Φ Ω s3 rt CO μ- rt 3 tr 1 (X Ω tr rt rt μ- rt μ- s3 P φ P P o 0 h- 1 <! rh μ- β μ- μ- LQ P rt cn S 0 o P μ- 0 3 μ- 3 P μ- CD φ Ω μ- Ω 3 3 ϋ β μ- μ- sx ii cn 1 Ω tr h5 tr μ- μ- 0 h{ 3 3 rt LQ o μ- H- 1 H 1 rt Ω 3 Ω <! o Φ P 0 t Φ φ Ω O <! 0 sx sx 1 μ- SX LQ P 3 H " tr β < o Φ P h- 1 rt 3 tr rt tr β 0 ti Φ t- > 3 Φ rt μ- N CD h- 1 P μ- Φ S3 H Φ ti 3 3 O μ- μ- Φ 3 3 μ- li μ" Φ tr 3 o • ^ 3 s3 tn tr Φ μ- P 3 ex rt SX μ- Hi o ct 3 P μ- ft N rt μ- rt 0 LQ 3 μ- ii μ- cn φ 3 sx Ω rt μ- μ- 3 3 tr Hi cn LQ rt ϋ 0 μ- 3 Φ Hi 3 rt μ- H 0 0 M φ 3 rt < μ- 3 0 ft sQ rt β 0 μ- 0 3 Ω LQ ii Φ 3 P 3 h-" Hi β H 1 rt φ cn tr o sQ 3 tr tr o CD hi rt < μ> h- 1 ct P rt rt P h- 1 CD LQ P β Ω 3 P Φ φ Hi 3 tr Φ cπ P 3 CO tr Φ Ω rt ^ tr σ P tn p. P . rt ,_, P φ 3 Φ μ- μ- Φ ii tr ti 0 Φ cn β rt 3 0 to rt rt ti μ" μ- o rt P hi P tr tr μ- 0 00 3 P rt Φ tr σ H rt φ Φ 0 3 φ h- 1 h-" tr (- ■ Φ ft tr φ 3 φ Φ φ rt object for controlling the sheet width. Incidentally, the roughing rolling mill train controller 10 has a control function of roll gap control function and a roll peripheral speed control function for each vertical roll stand of roughing rolling mill train, a crown/shape control function a reduction control function and a roll peripheral speed control function of each horizontal rolling mill train. The finish vertical rolling mill controller 11 has a control function of roll gap and a roll peripheral speed control function of the vertical rolling mill 3. The finish horizontal rolling mill train controller 12 has a tension control function between the rolling mills a crown/shape control function, a reduction control function, a roll peripheral speed control function and a looper control function of each horizontal roll stand of finishing rolling mill. The cooling controller 13 has a water quantity pattern control function of the cooling machine on the run-out table. The coiler controller 14 has a coiler peripheral speed control function. The sheet width measurement values by the sheet width meters 8 and 8 ' , the measured temperature value by the thermometer 9 and the information on the rolling material 1 such as the steel kind that is transferred from an another arithmetic (not shown) unit are outputted to the arithmetic unit 15. In this example, rolling is conducted by the same method as that of Example 1. The sheet width meter 8 ' arranged on the entry side of the vertical rolling mill 3 measures the distribution of the sheet width of the rolling material 1 along the longitudinal direction at the point at which the rolling material 1 reaches the entry side of the vertical rolling mill 3 with the progress of rolling, and corrects the sheet width W ent (i) of the rolling material 1 on the entry side of the vertical rolling mill 3. Here, the temperature of the rolling material 1 and its sheet width change amount are measured at each point in each rolling mill and between CO co to to h- 1 cπ o cπ o cπ o cπ h- 1 O -J P 5d CD Φ Hi CO Φ tr i 0 Φ 3 3 3 Ω Φ fj H 3 Q β • 3 Φ LO ti • P h- 1 μ> μ> --> ex Φ 3 o rt Φ to rt 3 * φ Ω 0 3 rt li 0 H Φ ti 0 Hi vertical rolling mill 102. Reference numeral 112 denotes the controller of final horizontal roll stand of roughing rolling mill 104. Reference numeral 113 denotes a finishing rolling mill train controller. Reference numeral 114 denotes a cooling controller. Reference numeral 115 denotes a coiler controller. Reference numeral 116 denotes an arithmetic unit. Incidentally, the vertical rolling mill controller 111 has a control function of roll gap of vertical roll stand 102 set as the control object, and the roughing horizontal rolling mill controller 112 has a crown/shape control function, a reduction control function and a roll peripheral speed control function of the final horizontal rolling mill 104. The finish horizontal rolling mill controller 113 has a tension control function between the rolling mills, a crown/shape control function, a reduction control function, a roll peripheral speed control function and a looper control function of the finishing rolling mill train 117. The coiler controller 115 has a coiler peripheral speed control function. In setup calculation, the arithmetic unit 116 decides the roll peripheral speed condition, the draft schedule, the crown schedule, the roll peripheral speed of each rolling mill and tension between the rolling mills in the first to seventh rolling mills of the finish horizontal rolling mill train 117, the cooling condition of the cooling machine on the run-out table and the coiler peripheral speed from the target sheet thickness of the rolling material 101 on the delivery side of the finishing rolling mill train 117, the target sheet crown, the target coiling temperature of the coiler, etc. The instructions of these conditions is given to the vertical rolling controller 111 the rough horizontal rolling mill controller 112, the finish horizontal rolling mill controller 113, the cooling controller 114 and the coiler controller 115. In the sheet width meter and the thermometer 109 arranged on the entry side of the vertical rolling mill 102 set as the object for controlling the sheet width, the sheet width of the rolling material and its temperature are measured in the entire length of the rolling material along the longitudinal direction and are transferred to the arithmetic unit 116. In the arithmetic unit 116, the temperature measurement value and the sheet width measurement value along the transverse direction are divided into x - 1 , n 2 - 1 and n 3 - 1 along the longitudinal direction for the front end portion of the rolling material 101, its middle portion and its tail end portion, and the sheet width and the temperature at each division point (i = 1 to n x + n 2 +n 3 ) along the longitudinal direction are stored. However, the values n x , n 2 and n 3 vary depending on the sheet width of the rolling material 101 and on the rolling conditions of the rough/finish rolling mills, and are set for each condition. On the basis of this temperature, the temperature of the rolling material 101 at the same point at each division point in each rolling mill of the vertical rolling mill 102 set as the object for controlling the sheet width, the rough rolling mill 104 and the finish rolling mill train 117, the temperature between these rolling mills, and the temperature in the zone from the finish seventh final horizontal rolling mill to the coiler 107 are calculated in the same way as the setup calculation. The correction coefficient and the correction term of the sheet width change estimation formula in the final horizontal roll stand of roughing rolling mill 114, the finishing rolling mill train 117 and in the zone from the finishing rolling mill train 117 to the coiler 107 are calculated from the contents of the component elements of the rolling material 101. In the arithmetic unit 116, the sheet width change amount of the rolling material 101 occurring in the horizontal rolling mill train 104, in each rolling mill and between these rolling mills of the finishing rolling mill train 117 and in the zone from the finishing rolling mill and the coiler 107 are calculated at each division point are calculated on the basis of the temperature distribution of the rolling material 101 along the longitudinal direction estimated at each point by the sheet width change amount estimation obtained by the correction coefficient and the correction term. In other words, on the bases of the distribution of the sum of the calculated sheet width change amounts along the longitudinal direction and the final target sheet width of the rolling material 101, the target sheet width of the rolling material 101 in the vertical rolling mill 102 set as the object for controlling the sheet width is calculated at each division point along the longitudinal direction, and the distribution of the gap control of vertical rolling mill 102 along the longitudinal direction, that takes the width change amount due to the dog bone shape and the deformation amount of the vertical rolling mill set as the object for controlling the sheet width into consideration, is calculated. The pattern of this gap set value is transferred to the vertical rolling mill controller 111. The vertical rolling mill controller 111 sets the gap pattern of the vertical roll of the vertical rolling mill 102. Here, the process steps described so far will be called the "first step" of the novel sheet width control method of the present invention. Further, the correction coefficient and the correction term of the sheet width estimation formula are learned by the formula (11) on the basis of the actual measured values of the sheet width, the temperature and the sheet crown along the longitudinal direction measured by the sheet width meter 120, the thermometer 121 and the sheet crown meter 122 arranged on the entry side of the coiler 107, and the sheet width control for reflecting the learning result on the next rolling material is conducted. This process step will be called the "second Co co to to cπ o cπ o cπ cπ P s ti H CO TJ s μ- 0 rt 3 - 1 rt 3 CO (X μ- 0 Φ Ω 3 ct 0 ii h3 rt P P rt CO CD CD 3 μ- O 0 tr li μ- 3 Hi tr tr Φ . tr P tr μ- 3 Hi CD o μ- tr 3 0 tr tr h- 1 σ tr φ tr rt O sx 3 φ Φ sx < P φ rt -" Φ rt Φ hh <5 ct φ ex φ φ φ 3 o φ Ω φ Φ β rt ιΩ φ Ω rt φ ft rt 3 tr Φ Φ hh φ rt μ- Hi rt rt 0 <. 0 Φ TJ 3 tr μ- μ- rt μ- tr 3 tr 0 3 a H rt Φ 3 tr 3 Hi tr Ω tr μ- 3 3 CO Φ TJ 3 ct ; rt *•« 3 rt CD 3 H rt φ tn rt sx § 0 μ- hi > rt Φ P μ- 0 Φ 3 φ β rt • ii (X LQ β S3 μ- Ω tr 3 μ- tr tr 3 P 3 φ CD μ- r+ Ω Ω 3 LQ P 3 Φ s3 cn 0 0 Ω sx μ- o o φ CX o ti φ Φ o S3 < μ- 3 o ti μ- μ- o < φ to cr P CD CO μ- Hi Hi P 3 μ- ex 3 3 β 3 o φ Hi tr Φ α Ω P 3 o o Φ 3 Φ 3 cx β φ r-> h3 Φ tiex ct h- 1 μ- 3 rt rt TJ CO < rt CO Φ 3 P rt Φ . < 3 3 rt 3 tiμ- ti H μ> tr 3 ro c+ 0 rt rt Ω μ" P tr CO li li rt tr Φ φ rt 3 t H tr ii φ rt ti rt ll ii Φ Φ rt TJ tr tr tr β H H tr 0 φ ti P μ 1 S Ω tr μ- 0 tr Φ h- 1 Hi 0 μ- ^ ro 3 o rt cn cn Φ Φ ex φ CO μ- CD μ- o Φ rt 0 3 Φ CD 0 P 0 Ω Φ Hi tr ii μ- Ω tr P to sx μ- φ Φ P CO ex 3 tr 3 ιΩ φ rt β CO hi 3 3 tn rt 3 Φ 0 3 0 0 Φ ti ti rt φ μ- CO rt 3 3 H σ tr rt ex TJ φ P Hi 3 tr 3 ex μ- P μ- μ- rt t h- 1 < Hi 3 Φ 0 O Φ rt ti rt Φ rt Φ φ tr H ft μ- φ rt φ β to μ- ex 0 tr CO Φ ct rt < CO Φ ii s3 rt P Φ φ CO φ tD tr 3 φ " φ rt Φ 3 ti Φ rt μ- 3 rt H φ μ- P Ω μ- μ- tr μ- < 3 rt P rt CD Ct to Φ Φ 3 rt P tr Ω Ω Φ Φ 3 rt tr o s3 H μ- P to rt tr sx 0 3 P Ω Φ Φ Φ tr rt ^ Φ 0 0 0 3 to 3 Φ ιΩ μ- ro -- ■ μ- 3 μ- β rt ex < μ- < s3 Ω TJ 3 TJ φ H rt tr s3 3 3 Hi ro β P μ 1 0 cn ιΩ ex rt o rt tr Φ μ" Φ μ- β ct φ o tr tr φ μ- P Ω SX rt P h- 1 rt 3 3 3 3 rt tr μ- tr 3 μ- μ- 3 P ii ex ti μ- --> ct 3 *< Φ ex 3 o β ti ct cn ct Φ W P o Φ tr φ 3 CO Φ 0 Ω 3 rt σ μ- rt P CO μ- LΩ CO rt SX ii Ω 0 tr β ti μ- ct σ < rt φ P rt 0 3 o μ- μ- Hi tr Ω 3 0 s3 μ- ct P rt tr H ct h- 1 Φ H 0 μ- 3 φ ■ ! ro tr Ω ct ct ti O 3 3 tr o μ- • β < Hi μ- rt tr Ω P rt Φ Φ CD Hi P cx ti 0 o φ μ- tr o rt O 3 μ- φ Ω CO Φ sx β Φ rt 3 Ω tr Ω sx 3 Ω IX μ- 1 CD μ- β 3 s3 ii tr l_l. rt Hi H ct rt sx 0 Ω Φ 3 t rt CX β SX o Φ rt Φ o rt P CO cn rt μ- μ- β φ tr 0 P * 3 P sx ct tr tr μ- Φ P P 3 μ- S3 rt μ- tr 3 * Hi β h- 1 μ- tr μ- H (X P rt Ω φ P h- 1 ii μ- rt 3 • μ- Φ 3 3 -- ■ H rt H 0 μ- tr " *< Φ 0 CD to 3 φ cn 0 rt μ 1 μ- ct o 3 ti o Ω P rt SX ϋ Φ 3 ct 0 φ hi Φ sQ φ tr 0 φ ti Φ CO 0 tr 3 3 0 μ- 3 O CD tr ii ft tD ex t Ω P 3 hh 3 0 Hi H H φ H 0 3 ti Φ SX H β ti0 tr tf Φ o ct 0 P Ω H 0 ft h- 1 Hi μ- P rt μ- <! rt sx Φ P Φ -- ■ ro β μ- - 1 ro Φ P μ- μ» tr s3 3 Ω 0 0 0 ti H Φ 3 ft 3 μ- co Φ ti μ- CO < 3 ct H ct tD o Φ Ω 3 φ μ- TJ Ω 3 3 Hi "< μ- Ω ιΩ Φ 3 tr -- ■ 0 ii -- ■ β μ- Φ 3 tn rt tr to P β rt ιΩ Ω 3 ft TJ rt TJ P β " φ Φ ti P rt μ- β P tr P β CD rt H ti ti P 0 CD ιΩ Hi ti tr H CO CD to Ω 3 Φ ct tr co 0 tD TJ μ- ro S3 to Ω σ *< 3 * Φ P 0 rt 3 μ- rt 0 0 0 0 Φ tf 3 rt sQ 3 μ- 0 Hi μ- TJ tn Φ μ- rt rt to tn CX Ω H tr to rt sx 3 tr ϋ < sx < rt (X Φ Φ μ- rt Φ 0 (X Ω 3 rt cx Φ β rt rt Ω ■ <! Φ tr ii Φ P P μ- Φ tr Φ rt 0 tr 3 3 0 rt sQ μ- φ rt Φ P P Φ 0 s3 3 Φ 0 ft rt o sx o sx β P rt tr 3 rt 0 3 tr Φ < s3 tr μ* t- 1 3 3 3 μ- Φ 1 Φ o φ σ μ- Hi CO 3 0 0 o Hi TJ ro ct ex P μ- rt cl- rt rt rt O rt H Hi hi Ω rt 3 IX S3 ex μ- Hi < Hi P tr μ 1 cx P ?r o μ- ti 3" tr tr 3 μ- μ- P P sQ rt P tr μ- co P rt ti to φ β rt μ- TJ P φ 0 Φ 0 LQ 33 3 P P t- 1 μ- tr 3 Φ μ- α S3 rt h- 1 rt tr φ tr t P tr 0 3 φ H CD h- 1 rt CX μ- μ- sQ Ω 3 P φ SX Φ to titr --> tf β 3 * Φ sx ro Ω tr ti3 sx ϋ Ω tr Ω • rt ex <! β μ- 3 er Φ • Φ Φ Φ Φ o φ ro 3 ιΩ P •<: ii 3 φ O β rt ft Φ P μ- 3 CO TJ rt o ti LO TJ S3 rt ϋ ex Φ μ- rt μ- Φ 3 IX tr tr H P LQ tr ii tr Ω . Ω φ ti o hi μ- H tD • rt rt CD μ- Ω tr rt Hi < μ- φ rt 0 rt φ Φ Φ 0 0 P 3 0 Hi φ rt s3 φ tr Φ tr 0 o Φ tr μ- Φ 3 Ω μ- 3 Φ P φ O 3 3 CO CD tr μ- Ω φ ti Φ -- 1 3 < cx o ii ti P tr to Ω sQ CO ct Φ Φ Hi ct h- 1 rt φ rt φ --3 -• to φ IX cn rt P tr P tr 3 Hi μ- 3 H μ- μ- μ- tr 3 rt rt μ- ct tr h- 1 o H rt μ- 3 φ h- 1 P P μ- rt Hi hi 0 rt 3 3 Φ rt tr tr 0 Φ t o 0 0 sQ φ LΩ Φ 3 LQ P 3 3 o o P Hi P 3 φ ct tr Ω φ μ- rt • 3 3 P rt SX sx direction occurring on the downstream side to of vertical rolling mill in a rolling direction, and calculates and controls roll gap of the vertical rolling mill in the longitudinal direction of the rolling material on the basis of the sheet width distribution of the rolling material along the longitudinal direction that is determined from the distribution of the sum of the calculated sheet width change amounts along the longitudinal direction and on the basis of a final target sheet width.

Claims: CO oo to to cπ o cπ o cπ S N ii s3 hi CD CO CD IX cx 3 o 3 ex 0 rt f- 1 s3 hj S3 hi ii ii μ- 0 0 μ- o P P Φ μ- μ- Φ hh P μ- σ tr 0 μ- 0 μ- 0 0 O cx 3 sx μ- μ- rt ii cn P t H ι_ι. Φ 3 ex SX rt Φ M rt (X ex φ rt cn cn Φ Φ Φ ιΩ rt - rt h- 1 tr tr μ- Ω H β P H Ω Ω β μ- tr μ- tr μ- μ- μ- Hi CD 3 hi ti CD rt μ- ti μ- μ- rt rh TJ rt 3 3 3 3 Ω hi rt μ- o o μ- cr Φ α P μ- CO β Ω Q μ- Q sQ tr 0 P 3 μ" rt 0 β IX 0 hh rt IX tr 3 P 3 3 3 tr 3 rt ti 3 0 ii μ- P 3 3 3 3 3 ex rt μ- μ- μ- Φ μ- rt o cn ti Φ 3 3 P P μ- μ- P ιΩ CO tr 3 3 o 0 O h- 1 O 0 P P Ω ct ft Φ P o Φ μ- 1 ιΩ O 3 a 3 Ω 3 φ φ ii φ μ- hh tr rt μ- Ω 0 hi 0 CO ii P sx H O 3 3 L_l. rt 0 tr 3 P rt a CD ex P μ- μ> cn μ- 3 hh 0 φ μ- P Φ tr H Φ h- 1 tr rt μ- μ- 3 P φ O P 0 Ω Ω μ- rt tiΩ Ω Φ ti Ω Φ ti ex ti 0 Ω μ- r 3 Ω β P 0 3 (- ■ ro P ct to Φ 3 β 0 ro φ β P 3 P 3 μ- μ- μ- P CD ii μ- Φ P σ P cr Ω 3 h- 1 CO ιΩ P ft μ- μ" rt Ω μ- CD 3 CO μ- Ω Hi 3 μ- • rt P P 0 c+ ft Ω 0 CO tr 3 Ω co β ro tr ιΩ 0 P β 0 ct α Φ Ct to μ- Hi μ- β SX φ β hi rt 3 hi li Ω H Φ μ- 3 0 0 H 3 μ- rt φ 0 P hi CX tr P •<- ti P μ- IX to iQ rt 3 Hi P Φ li tr β P P Hi t rt o μ- φ ft μ- c+ Ω 0 h- 1 P tr rt P φ Φ TJ Ω rt μ- 0 H tr 3 μ- 0 cn Ω P 0 rt Φ 0 CD μ- CO Ω CD tr μ- to 3 φ o φ 3 3 μ- β 3 Hi tr Ω P 3 β ct O ct 3 P sQ P μ- Ω tr Φ ιΩ rt cx μ- μ- CX φ < Ω μ- sQ ii μ- tr i hi ιΩ μ- 3 rt «_ι. (X P H φ 3 P 3 rt φ β cx Φ o (_l. Φ 0 sx ct Ω LΩ μ- Φ Q Ω Hi 0 rt ft tr CD ti ti P CX 3 Φ P h-" rt tr 0 0 Ω s3 tr β 0 Φ rt tr Φ tr rt ti hi Ω 3 M tr ii φ 0 3 3 rt 3 li Hi 3 tr φ φ μ- μ- o CX Hi rt μ- φ 0 h- 1 μ- "* CD ii ct μ- P rt Φ rt φ Ω 3 μ- ti CD 3 h- 1 sx μ- Hi ft 0 tr 3 ct rt tr CO Φ ct P ιΩ H co 0 Hi μ- ιΩ ct * μ- 3 μ- O -i φ LQ tr φ φ tr 3 μ- ci- 3 O sx Φ μ- to ιΩ 3 ti φ h- 1 ii Φ ti O Φ TJ 3 μ- 3 ti li Φ 3 3 3 ct rt P μ- rt μ- tn 3 Φ Φ μ- H 3 U3 μ- ct μ- TJ lΩ H Hi o P Ω 3 3 rt tr < P tr ct ti (X 0 tr tr Ω 0 Φ μ- μ- O sQ 3 * Φ Φ Φ μ- P rt H φ 3 β φ O Hi μ- H 3 σ 3 P N 3 CD ro hj Φ rt s3 rt tr P P rt 3 P P β μ- 0 rt μ- 3 CD ft μ. ct β μ- β μ- Ω rt μ- rt ci- rt P rt rt c+ to Ω 3 ti ex μ- rt tr μ- 3 cx ti μ- 3 tr φ 0 Φ ti tr 3 β Φ μ- tr 0 φ 0 Φ M Φ Φ Ω s3 μ- rt Φ 3 LΩ i 3 3 O Φ sx H hi 0 μ- 3 Φ P rt μ- 3 tr ti μ- TJ Φ μ- 3 TJ μ^ Hi H" 0 TJ rt r-> tr ex P 0 rt 3 0 P 0 φ < tr CO P 0 φ li μ- Hi P Φ Φ rt ex Hi 3" μ- M Hi ti μ- Φ Φ μ 1 0 μ- ti 0 3 3 li -3 ti tr μ- Φ M μ* P 3 H rt 0 Hi 3 3 sQ rt ex P μ- 0 H ex CD CO M μ- μ- P rt ιΩ rt s3 Hi μ- rt P tr rt 0 μ- rt P ti 3 3 β μ- Φ 3 rt ^* 3 P rt Φ tr β rt 3 ti H μ- 0 to ιΩ CO ti rt Ω Φ CD tr IX tr Φ ti tr μ- Φ μ- H φ rt tr Φ tr P 3 P rt φ Hi φ < rt Φ * * • 3 μ- Ω σ rt TJ tr φ φ μ- 1 μ- tr μ- μ- φ s3 CO rt ct β hi μ- P φ φ O CO sx φ to 3 3 CD ti Φ β μ- rt 0 3 P tD rt Hi CO P tr P tr rt Φ 0 3 rx o μ- -- ■ sQ to tD tr μ- Φ P Φ μ- 3 Hi μ- μ- 3 0 μ< O Φ cx Φ φ Ω 3 3 μ- rt 0 P Φ rt ct P P n 3 tr 3 μ- rt μ- 1 O LΩ φ (X CO co t to I— 1 cπ o cπ o cπ o cπ 3 ii ex 0 rt μ- 0 H P Φ 3 P P CD P hi CD t- 1 s3 0 P 3 cx CD Ω Ct Hi ti ¬ < cr φ 0 μ- σ tr 3 Hi 0 cn X ro μ" 3 P Ω 0 tr 0 μ- 3 μ- 0 μ- P 0 0 hi er Φ Φ 0 P M ϋ ro h- 1 μ- ti ¬ 0 (X μ- Ω H Φ 3 sx 0 3 co μ- 0 0 ro H rt 3 CO φ Φ ro w μ 4 ct α- ro 3 a 0 ro LQ rt t 3 LQ rt cx H tn 3 rt s3 LQ β μ- Ω Ω β P P μ- tr rt ii LΩ rt hi rt μ- 3 * tr cQ μ- hi μ- P cn μ- φ μ- ii 3 ct rt TJ Ω μ- 3 ro to tr hi sx to l£) rt ro CO rt μ- ii 3 μ- CO tr Ω φ ct Φ LQ μ- to tr sx ιΩ P rt ro 0 μ- P s3 β Ω rt ro β σ 0 LΩ sx P ro P 3 β SX o hh t 0 0 3 tr 3 TJ μ- tr σ tr rt IX β μ- Φ 3 3 O ti i H cx tr 3 cx ro rt LΩ . sx μ- P P ro μ- ct h- 1 Hi Ω IX ct ct μ- P hi Φ O o μ- Φ μ- rt 3 3 tD P 3 μ- μ- μ- 0 ii tr 3 P ct Hi P H φ rt CD h- 1 3 3 rt > tr P ιΩ μ- h- 1 CO P 0 3 3 μ- s3 0 Φ P 3 φ H Ω 3 ro Ω tr P 0 TJ LQ 0 CD Cfl 0 h- 1 3 LΩ μ- μ- μ- r-> ex hi O O μ- μ- Ω rt ro μ- 3 Φ to 0 3 rt CO ro 3 sx ii μ- 3 3 tD 3 3 ct sx ιΩ ti 3 Ω tr P o ιΩ tr ex 0 3 tr li P ti ¬ μ- o P ci- μ- LQ cΩ μ- Hi β μ- P P H ro μ- 3 Hi μ- ro μ- Hi P h- 1 er 3 μ> μ- rt ti sx Ω o 0 TJ rt rt ct ti ¬ 3 P ro Ω CD H 0 rt Ω ti tiTJ P ιΩ μ> li tr O ro 3 3 P 3 ti CO tr β β ro Φ μ- rt o P Φ β rt β o P ro to ro O 3 li μ- μ- Φ P φ μ- P * rt Φ ex hi ii P 3 3 μ- Ω 3 tr ex σ Ω P ti μ- sx 0 3 3 3 Ω rt ti ¬ Ω Ω ti H μ- Φ μ- CD s3 cx cx rt d- Φ μ- Ω rt μ- μ- 3 μ" μ- LΩ rt 0 rt μ- 0 ro β O ro 3 3 P β μ* μ- β μ- CD 3 ro β μ- sx P rt μ- rt μ- 3 ro 3 ti h-" 3 CD 0 P cx H sx Ω li 0 sx P Ω h- 1 o h- 1 3 tr 3 0 LQ 3 LQ rt P μ- P ci- 3 3 μ- μ- ti ¬ rt 0 3 0 μ- ct P 3 H 0 CO Φ μ- 3 TJ tr 3 P ct ti Φ co P 3 Ω er μ- t- 1 H CO ft 0 CD 3 ci- rt CO rt CD Ct Φ sx μ- 0 CD ti ¬ (X Ci- LQ O 3 h- 1 CO rt cx Hi μ- CD 0 ef P ϋ Φ o tf ii 3 * 3 μ- ro μ- ti o 3 Ω LQ μ- o to H μ- o 3 o μ" ct ro 3 O rt tn Ω Φ P CD < tr P cΩ μ> (X ti H μ- 3 rt TJ o 3 P μ- li ti vΩ μ- Ω tr CX Ω ct ro ro - 1 μ- ro ro σ cQ tr hi 3 Ω LQ Ω μ- cr ro Ω 3 P ro N P 0 β H β Cfl i rt 0 rt 3 Ω β φ μ- ft 0 P β Ω Ω H P iQ H 0 0 μ- cn 3 ii 0 ti tr rt 33 3 tr ιΩ rt tr rt rt rt tD H 3 3 h- 1 rt rt 0 0 h- 1 Ω cr 3 Hi 3 hi 3 ro ro μ- φ ιΩ ro 0 ro μ- μ- 3f CO μ- 0 ci- P Ω i μ- μ- 3 μ" Ω 3 β P ro cΩ ti ¬ rt μ- LQ ro Ω Φ rt 3 0 0 Φ tr 3 h- 1 ti ct β 0 0 o c+ β P " CO Hi er tr 3 μ- (X rt P 3 c+ rt tr < 3 3 Φ LQ o Φ -- ■ 3 3 H ti ¬ P μ- Hi μ- ex φ φ LQ rt μ- 3 * ro φ ro D φ 3 μ* ti P h- 1 0 ιΩ P ro ti ¬ co hi 3 μ- β CD s3 tn ro 3 hi P tr 0 0 ct rt CD μ- ti ¬ μ- 0 0 μ" P ti ¬ H ro o μ- ti o ex μ- cx ci- μ- hi P TJ tD rt μ- ^ Hi 3 tr ct to P ro 3 Hi Hi TJ ro μ- ex 0 3 co Φ tr 0 3 μ- ti ex 0 μ- h- 1 ro tr μ- sx iQ φ tr 0 h- 1 sx LQ μ- P Hi tr Ω ι_ι. s3 3 μ- t h- 1 ex o ti Φ Ω CD CD μ- μ- O ti ¬ CO 3 0 ••«• P to rt Φ 3 φ P tr tr 3 P Φ P ti P P rt sx CO CX CD P 3 er P LΩ Hi 3 tiP ii μ- Ω tn P β μ- ii LΩ rt ct h- 1 0 μ- μ- β rt ti ¬ CD P 3 P φ μ- P er μ- 0 o rt rt Ω rt P 3 o μ- β " (X tr ro μ- 3 SX ti ¬ sx rt P ro cx -- ■ 3 H tr ex μ- LQ >-> ct ti s3 hi μ- TJ 3 rt ex ro to tr o ti ¬ μ- Hi Φ μ- o 0 -- ■ β ro μ- 0 μ- CD ϋ 3 sx to 0 P hi β hi ro <S 3 er ti ¬ Ω μ- 0 P H ti 3 3 3 μ- sx ex μ- 3 tr 0 P μ- tr μ- 3 o ro U3 ro ro 0 3 μ- hi 3 o φ sQ μ- 3 μ- ex rt μ" LQ ro H - CO 0 0 P Hi P h- 1 CD H 3 μ- LQ 3 Ω 0 cQ 3 μ- tf μ- ro H ci- hh c+ rt μ- 0 tr rt rt CD TJ Ω tn ct Hi ct P cn 3 3 rt μ- ti «• rt 3 Hi μ- ro μ- tr tr ro Φ 3 ct o μ- μ- μ- tr 3 rt P ι P 3 μ- μ- H P P P 3 ro Ω CD ro H ti μ- tr 3 ex 3 0 CO ro tD μ- ϋ ti ¬ s3 ιΩ ti σ O μ- ti ¬ LQ rt P ro P r- 1 φ ci- Φ LQ 3 P φ (X μ- o 3 ro μ- φ β μ> 3 o er h- 1 rt c+ rt H" ti μ- ti rt μ- tr 3 μ- H sx 3 Ω rt H rt 3 φ 3 s3 β 0 N 0 o TJ CO cx 0 tn ti β cΩ μ- rt P rt μ- μ- CD P LQ P μ- ii s H ft 0 h- 1 Hi P 0 μ> P ro ct P tr ti ¬ μ- 0 3 P ti CO ticx 0 μ- ro ti ¬ H 3 tn tD 0 Ω μ- ft H ro 0 3 cQ μ- ιΩ ti ¬ tr ro rt ex CD er P Φ μ- titn μ- 3 ct 0 3 * CD ti 3 ex ro er φ ti tr rt ro μ- 3 er 3 rt μ- 3 Φ ro μ- o ct ro ro μ- μ- tr o 3 LΩ Φ P ιΩ tr o rt P Hi rt P 3 Hi 3 φ 3 μ- 1 LΩ SX oo t to I— 1 cπ o cπ o Cπ cπ a 0 ct s3 3 ti cx 0 rt μ- o SX 0 ft P P P CO P Ω tD 3 Ω ti ¬ CD P ti ¬ 0 ci- s3 μ- tr tr o μ- CD 0 μ- cr tr 3 Hi μ- tr tr 3 3 P Ω 0 μ- P P er er μ> er σ tf 0 μ- hi φ 3 ex P ti ro ii C . Φ SX 0 (X μ- Ω 3 ti ¬ h- 1 φ φ 0 φ Φ 3 sx Φ φ LQ ti ¬ CO I- 1 ro ro ro CQ ro ro 3 (X 0 t ro ro Ω φ 3 φ LΩ ti ¬ Ω Ω β μ- er β μ- 0 Ω β P P Ω Ω β CO ιΩ ti ¬ ti ii hi β H rt LQ rt Ω β μ- er rt t+ TJ rt hi 3 ct rt J Ω μ- rt t TJ P er hi IX co 0 rt μ- 0 Φ rt TJ ct μ- CO β Ω ro LΩ μ. CO tr ex μ- cn μ- P ro 0 μ- • 0 P P 3 s3 ti ¬ 3 to β Ω 0 hh ti ¬ tr sx o Hi rt o Hi ci- cx 3 ti ¬ cΩ μ- er TJ Hi ci- sx er 3 0 ll μ- P -.. 3 3 0 H ti i 3 o ti ci- ti ¬ ιΩ μ- ti ¬ ro μ- IX φ φ 0 ti μ- P hi ro 3 3 P ti ro 0 o -.. li φ ti ¬ ti ro μ- 3 er P rt rt ti ti Φ 3 3 0 P P LQ P ti ¬ Hi P P er P 3 3 rt > cQ ro tiβ er -- ■ P P P cΩ 3 Ω 3 1 Φ 3 ro H Ω 3 Ω 3 Φ 3 TJ LΩ o o er cx 0 rt Ω 3 -- ■ Φ 0 (X ti o o μ- μ- o ti CD ro CO ti ¬ < 3 ro μ- sx 3 β 0 ct 3 CO ex P μ- 3 3 CO 3 3 3 cn 3 < hi 3 Ω er er ro cQ 3 μ- LQ ti 3 to (X P tr rt μ- μ- 3 P ci- μ- LΩ Ω rt μ- 0 ro P P μ» ro ro ti to P CD μ- Φ rt μ- μ- 3 φ li sx H o Ω rt ti cx Ω ti cx 3 H t-t ti ¬ 3 P ro rt rt tr " ci- rt ti ex H o Ω 0 ro ro β P tr o ro 3 3 P O ro φ CD β ro Φ μ- rt CD μ- er Φ ti β sx o Φ Φ β P tr (- ■ Ω a P Φ μ- P ct φ hi ii P 3 tr Ω φ Φ sx μ- ex μ- h- 1 Ω 3 h-" P rt ct rt Ω μ» rt ti ¬ Ω ct Φ ro μ- CD 3 ro P rt μ- σ μ- CΩ rt ct t+ Ω to μ- 0 μ- to β 0 ti ¬ μ- 0 h- 1 ro β μ- O ii P β \-> μ- ro -- ■ h- 1 H β 3 ct μ- o μ- CΩ β μ- 3 o μ> 3 ro 3 ti μ> 3 μ- sx ti ». IX rt o s3 φ rt P ii 3 0 h-" to cΩ ti ¬ 3 0 P cΩ 3 LΩ rt P μ- P LΩ ti ¬ 3 H μ- μ- ti ¬ ti 3 μ- Ω μ- -- ■ μ- cΩ ti ¬ 3 0 P er Hi rt TJ " 3 P rt er er ro to P 3 Ω er s3 0 cΩ ex rt o σ er Hi rt O rt ro 0 μ- ti ¬ ro ti ¬ ro ex μ- ti ¬ φ φ Ω rt LΩ 0 μ- μ- ti ¬ μ- 3 β ti ¬ Φ 0 μ- Hi tr Ω CO 3 er ii er 3 er 3 rt ti o 3 Ω ex rt er o μ- rt er Ω CΩ 3 ro < Ω P LQ ro P ro <! cr P LΩ Φ < μ- μ- 3 ti ¬ TJ 0 ti ¬ μ- β 3 o ii μ- Φ < Ω P cΩ rt ro β μ- rt ro ro φ to 0 tr LΩ er ii 3 er 3 cx cx Hi Φ 0 Φ β μ- tr CD ϋ ti ex rt rt β CO H rt 0 rt CO ii P 3 β Φ μ- rt * cQ μ- μ- CD Ω 3 CD ϋ ti ti ¬ Φ tr rt ti tr ϋ ii tr rt s 3 er tr rt μ- rt ft O H 3 CO 0 CD rt tr rt i er ro μ- μ- ti Φ P ro ro μ- ro ιΩ ro Φ μ- cx tr μ- er O μ- 0 3 P ti ¬ P μ- 0 Φ μ- μ- ϋ ro rt ro Ω 3 0 3 ro Ω ro Φ Ω < o Φ er 3 μ- ts 3 μ- o hh Φ Ω 3 o φ ct P LQ sx CD cx rt P 3 rt tit P hi 3 φ ιΩ " μ- φ sx 3 rt P ιΩ cx 3 μ- < μ- h- 1 tr ro O CD -- ■ φ 3 μ- tr P tD μ 1 μ- TJ 3 μ- μ- CO ro CD s3 CD ro 3 s3 P 0 0 rt ti ¬ CD μ- β CD H CO P s3 μ- μ- tD Φ μ- hi 3 3 rt ti ci- μ- ti P TJ μ- H μ- Hi 3 er rt tD li rt β 0 O μ- μ- i 3 3 ct H sx o LQ ti CO ti sx o μ- rt ro ex O μ- lΩ s3 φ 3 * ro Φ μ- li h- 1 <x 0 cΩ ti P rt φ μ- ro μ- ti ¬ sx ti H t 3 to μ- μ- 0 rt Ω o Φ h- 1 Ω ti ¬ φ μ- rt tr CD 3 σ cr er P P tr cΩ CD P rt CD ex rt 3 ex μ- P li er μ 1 P 3 cr β μ- o P β ex β μ- ti 3 rt μ- tr μ- β rt ti ¬ P μ- 3 h- 1 o μ- Ω P β hi P 3 tr C-h rt μ- rt P 3 0 CD β P 3 3 CD ex IX 3 * ro μ- to 0 0 rt LQ Ω 0 3 er ti ¬ rt Φ h- 1 LΩ φ μ- ti μ- LΩ < ti -- ■ cΩ P CD μ- TJ 3 3 hh 0 β μ> 3 cΩ ro μ- 0 hi H 0 ro 0 0 CD ro O ti ¬ rt H 3 CD ti ¬ 3 μ- ti ii o CX 3 3 o μ- 3 Ω 3 3 3 μ- hi 3 3 ro o P μ- er er o CD ti ¬ P P 3 rt 3 0 μ- 3 μ- LΩ μ- P rt ιΩ μ- 3 CO sx cΩ μ- ti 3 h- 1 -- ■ CO o 0 ro 3 P er rt rt LQ er μ- h- 1 P tD 0 μ- o lΩ ro μ- μ- μ- rt Hi ci- μ- Φ φ Φ Φ μ 1 μ» o ci- ti ¬ H μ- hh 0 Hi rt to ft P μ- ex ϋ • • 0 ti ¬ cx ti ti ex 3 H" μ- H ti er 3 P 3 tr 3 sx rt tr -- ■ ti ¬ 3 μ- μ- ti σ er φ μ- P σ 3 P μ- ro CD iQ rt CO CD CO μ- μ- H φ cn er cΩ li σ 0 φ H tr P P ti ¬ P to cΩ μ" ti ¬ tr ro 0 tr CO P ro H ti μ- φ ro φ ro 0 ■ <! h- 1 3 ro CD φ o er β ti rt TJ 3 ro o μ- i rt ro tr ϋ rt X 3 3 Ω rt -> Ω φ μ 1 <χ ti μ- ct TJ 3 ro rt o P LQ α 0 O Ω β 0 μ- ro P rt μ- μ- rt 3 H P μ- D P cΩ μ- P CD CD h- 1 P rt rt P to ti ¬ ti ¬ μ- 0 3 ci- μ- h- 1 ti ¬ P P CD CO 0 H CD CD rt tr -- ■ D 0 μ- μ- H CD rt ro ro 0 3 cΩ Hi ti 3 0 er o to CΩ rt er 3 μ- tr ro μ- 3 0 0 μ- co H ti 3 0 •<: cΩ 3 φ Hi er ro CD 3 ti ¬ o φ ro 3 rt 3 3 3 ti ¬ μ- 0 ti cQ ti ¬ 0 ro ro ιΩ er 3 rt sQ tr iQ er 0 P Hi er 3 rt ro ro φ 3 Φ oo co to to cπ o cπ o c-π sx ci- o ex o 0 μ- ti Hi μ- Hi 3 CΩ P ii cQ ci- 3 CD φ O μ- ti O P Ω P rt μ- < μ- rt μ- β tr φ sx μ- ex ex β hi 0 μ- rt o H 3 H 3 μ- Φ o 0 P 0 0 3 SX 3 μ- μ- μ- sx 0 ti 3 rt 3 μ- hh Φ LΩ tr cΩ H Ω φ Φ CΩ rt 3 3 Ω P μ- P tr P rt μ- o c-h P rh μ- IX 3 Φ CD ro o to H μ- ti 3 H μ- CD μ- o to P P CO o O h-" o h- 1 til μ- Ω P P 3 3 P d- φ cQ 0 tr 0 P Ω 3 Φ 3 cn 3 β cΩ cQ β P rt H rt P cr φ σ Φ Φ rt o 3 o (X ti Φ rt TJ ti ¬ μ- cx 3 * Φ er rt P ii 0 P P ti ¬ 3 O rt ts rt P SX 3 β P tr μ" cΩ ii 3 Φ 0 rt μ- Φ CΩ ti 3 tr rt < Φ cΩ Φ β CX ro tr CX H- ti • - rt tΩ μ- CΩ CΩ tr tr 3 rt ro Ω φ Φ P hi P Φ h- 1 μ- P rt XX 3 Ω P β cx β s3 3 rt μ< μ- sx μ- P rx 0 rt rt 3 Φ tr CΩ CO co t to cπ o cπ o cπ cn Ω CD 3 Ω P rt CD CX o ct rt 3 Hi Ω t 3 0 s i 3 ct CD ti ¬ H Ω ct < P C+ 0 μ- P P tr 0 P μ- tr tr 0 μ- ti P ti 0 0 μ- 0 hh μ- 0 P tr P ts 0 0 tr φ er 3 sx ti ¬ 0 ro 3 μ- H ro 3 sx P rt 0 3 3 (X ti ¬ ro μ- P 3 ro ti 0 ro ti- ro ro Ω 3 ιΩ α φ ro cQ ti ¬ 3 Φ 3 ti ¬ rt cQ tΩ ti ¬ ro ii 3 H" TJ t 3 hi ti β cQ D μ- Ω Ω β μ- er CD ti ts tr μ- P er μ- ti 3 CΩ μ- hi tΩ μ- cQ cn O ct μ- tr rt H rt C+ TJ rt <! μ- ti ¬ 0 Φ P rt μ- 3 μ- 0 hi < 3 μ- cπ tr Ω tr 0 P P ti ¬ ro β 0 μ- D β Ω ro P er 3 β cx P cΩ P 3 0 ro LQ tΩ • ro P ti ¬ ro h- 1 ti ¬ er ro cx o th ti ¬ IX tr hi Φ μ- < ex SX r- > ro ti μ- Φ h- 1 er ro μ- ct ro ro rt μ- M 3 o ts μ- P CΩ μ- Φ μ- ii o 3 ti ¬ CΩ 3 3 rt Φ rt 3 tr P 3 μ- H ro 3 3 ro P ti ¬ 3 hi σ 3 0 3 μ- ro ro μ- ro P cΩ hi cΩ ro rt s3 P 3 0 P P cΩ ro LΩ ct Φ P t- 1 ιΩ H X ti 3 ti ¬ > s3 0 H s3 O tr o μ- h- 1 ιΩ 3 Ω 3 μ" ro sx o 3 μ- rt μ- cΩ ex ro ti ¬ μ- o μ- ti ¬ < 3 ro 3 cx o μ- 3 TJ rt Ω s μ- P co P μ- H er tΩ 3 (X er ro cQ LΩ ti ¬ P 3 ti3 CO IX P ii cQ ro tr P Φ P 3 CD ti- 3 3 ti μ- φ er ct μ- LQ rt φ ii CΩ μ- er 3 P er ci- μ- μ- 3 Φ hi ro h- 1 Φ 3 cQ li φ cn sx P ro P ro tr 3 μ- er rt ct tr rt ex rt Φ ti cx H o Ω Ω P P 3 ex Ω o rt ti ¬ Ω 3 tΩ ro . LΩ rt CD μ- tf φ β IX Φ O ro ro β P t+ rt rt CO hi 3 P μ* 0 P ro rt ro ti ¬ rt β SX tr Ω Φ ro cx μ- ci ¬ H tr Ω 3 - μ- β tr o CΩ ti ¬ P r- > t- 1 P 3 ti μ- P P ro 3 CX μ- ro P rt μ- CΩ ts μ- P ct rt rh Ω o t φ P ts ti ¬ Ω μ- CO CΩ μ- o CD TJ s3 μ- μ- CD ro -- ■ 3 ti ¬ P P CD μ- o μ- CΩ β 3 Φ ro φ μ- P ro β 3 ti ¬ er P 3 o β tΩ μ- 3 ti ¬ rt O s3 P ts 3 μ- 3 0 CΩ P c-t μ- 3 i cΩ ti ¬ er ro μ< CΩ 3 hi IX P ts H 3 μ- μ- CO to cQ ti3 0 P CD cx 3 CΩ μ- P er ro ro LΩ μ- o tiμ- 33 0 ιΩ sx σ < P er hh rt to ft μ- s3 cQ H < P rt ex φ rt P P 3 Hi er tΩ σ μ- μ- el ¬ cx β ro 0 ti ¬ ro 0 μ- 0 CΩ μ- o φ (- ■ μ- μ- β 3 P cΩ • • Φ (X β sx μ 4 rt s' μ- rt ii 0 Hi er Ω CΩ 3 o ti ¬ ex 3 -- ■ hi 3 ti 0 TJ s3 o SX ft Ω ct μ- ct t μ- β ti μ- CO 3 ro < Ω P cQ 3 3 ts rt μ- -> CΩ P LΩ Φ tr CO μ- 3 P o H μ- tr 3 ex cx ro 0 ro LΩ tiro β μ- Φ Φ μ- tr r-> μ- φ Ω t_I. ti ¬ sx cΩ P μ 1 3 P φ o *•«• LQ μ- μ- Ω 3 er CO li ti cx P P tr H 3 0 ti ¬ rt Φ ts tiro ti ¬ ti ¬ CD Ω 3 3 CΩ rt ex P Φ er rt ti CD 0 β P ιΩ sx 3 er μ- Ω ro er P CO P ro ts rt 3 P ti ¬ μ- 0 μ- rt ro μ- μ- hi sx β Hi rt H CO μ- LΩ ro 0 t+ P er Ω 3 0 tiμ- 0 μ- H ts o Hi ti tiro Ω 3 o μ- ti μ- 0 φ 3 ti 3 3 3 H ro rt TJ h- 1 er 0 Hi " μ- 3 ro ro rt P cΩ CD φ ti ¬ o 3 rt μ- φ P ti ¬ Hi ro 0 ro ro Φ 3 sx tr CO Ω Φ 3 μ- ti ¬ sx er 3 LΩ t- 1 Ω rt ro 0 CO ti3 rt o hi 3 cn μ- β CD P rt P TJ s3 μ- μ- ts * * • φ CO P ft 3 ti μ- ro cΩ 3 P ro o 0 P w hj rt 0 μ- μ- CD ro μ- ϋ 3 3 μ- P CΩ CΩ μ- μ> TJ ex ti μ- 33 ro rt ti ¬ σ 3 μ- ro Φ μ- cx o rt ti cx 0 ιΩ tr o 0 ro ro Ω ro t+ μ- β er -_ι. IX rt Ω 0 3 3 P ti ¬ ro β o Hi li rt o 3 P H 0 -3 β cx o hi 0 Φ ti ¬ rt 3 ro ti CD 0 rt er P 3 rt 3 0 tr 3 to tΩ P 3 t er SX ti ¬ Hi ro IX Ω er H P μ- P 0 3 β μ- Ω P μ- LQ to φ rh rt rt 0 ro μ- er rt ro 0 to 0 0 CO to Φ H P 3 er ti ¬ o μ- P " ti ¬ μ- β H 3 3 - sx μ- h- 1 3 Hi β 0 ro h- 1 lΩ ro 3 t μ- μ- o er 3 0 H 0 P P cx 0 μ- 3 Hi ro μ- ti ¬ H μ- 0 O ϋ ti β ex 3 tr φ 3 ro t- 1 CΩ μ> μ- 3 tΩ o 3 μ- er O CO ro 3 Ω Hi cx 3 3 o μ- o cx LQ φ ro < P CD cΩ ti ¬ tr i ti ¬ 3 ro 3 P sx ιΩ P μ- LΩ μ- t- 1 P Hi μ- ϋ φ β P ex μ- ro μ- sx ti ¬ μ- ts 0 ts cQ μ- rt CD }-• 3 0 Ω TJ Ω 0 μ- 3 ti cx μ- ts t+ μ- ti ¬ Ω " ! 0 rt cx rt 3 Ω tf titi ¬ μ- μ- P P t- 1 μ- rt CO er Hi tΩ cΩ rt H μ- β tr 0 3 cr tr o P β Φ ts er 3 P H ti ¬ rt μ- hi ro σ CX β ts 3 CO P C_l. φ hi ti ¬ H μ- ro CΩ LΩ CO μ- Φ Hι ts ti ti ¬ H ti ¬ Ω 0 Ω β μ- rt 0 ct μ- rt ro o ct ro P tr φ o 3 * P 3 Ω 0 Φ 0 er μ- er P rt rt 3 μ- h- 1 ti sx ro Ω φ μ* tr H ti ¬ β ti rt TJ 3 ro 3 LΩ rt ii P ro tr Φ μ- μ- P o μ" 0 ro H ft 3 1 ro μ- ro c+ o P LΩ ro cx μ- 3 β μ- 0 o h- 1 3 μ- μ- 1 μ- rt μ- ti P sx μ- P to rt 0 μ- rt CD 3 3 3 3 μ- rt P Hi hi 3 ro H 0 CD CD rt 3 CD 3 μ- er cΩ P 0 cΩ μ- 0 0 "< cQ tr P 3 μ- tr μ- Q 0 Φ tr 0 3 Hi 3 hi 3 3 ti0 ro ex 3 Φ Hi cQ ti ¬ SX 0 LΩ er 3 Φ rt P er Hi ro P ro CO oo to to cπ o cπ o cπ cπ tiH 0 s3 P ts CΩ 33 0 P 3 IX ex ci- 3 titiHi rt <i tr h- 1 S3 P μ- Ω CD 3 0 er 0 Hi 0 μ- Ω 0 er 0 μ- 3 H μ- μ- μ- ti P er er hi tr φ Φ 0 μ- 3 3 0 μ- μ- 0 Hi φ 3 3 rx Ω μ- Φ 3 cx 0 ts CΩ P rt ro ro o Φ ii rt 3 ex (X 3 3 cΩ CD iQ rt 0 ro cΩ tict 3 h-> ro ct 3 ro 3 ct 33 tΩ ti ¬ P ct ro ιΩ CO φ μ- P μ- er hi rt μ- er er cΩ Ω hi CΩ ti tr N CO μ- Φ μ- er μ- hi μ- P 3 ct μ- rt sx σ. cΩ rt ro CΩ rt μ- < μ- P o CO tr Ω Φ rt 3 t J o rt P rt μ- ft β α β sx μ- P -3 β Ω rh ro μ- σ ro P CD 3 P Φ P 3 β Ω 0 h- 1 o 3 β sx hi cx (X μ- 3 TJ μ- sx er tr er rt o β ti h- 1 μ- ro μ- Φ sx er P 3 • μ- hi μ- CD cΩ • (X μ- P P ro 3 rt CΩ CO sx ct CO μ- P ro μ- rt μ- li 3 0 3 ti ¬ ti ¬ 3 3 CD P μ- ro P Hi ti P 3 3 N 3 tr σ 3 0 CΩ P μ> P ts rt > er P LΩ μ- tΩ CD 0 0 H Hi s3 0 μ- P cQ 0 Hi LΩ φ Φ P μ- μ- 1 μ- O H 1 ro tΩ o o 3 α O Hi o μ- μ- t- 1 (X Φ 3 li rt --> μ 1 sx μ- tr CΩ 0 3 tiμ- 3 3 3 cx h- 1 ro o ti ¬ < s3 μ- Φ P 3 cx β P er Hi SX P o LΩ er Ω o H ιΩ tiμ- ti ¬ μ- ti ex P 3 er φ Φ P 3 3 cΩ μ- rt 3 φ μ- 3 Hi μ- φ P hh Φ er O CD er 3 0 μ- 3 Hi ro t$ Φ 3 ιΩ rt ex H μ- * ! φ Ω CD ti 0 rt Ω h- 1 Ω P ro P er cΩ h- 1 ti 0 Ω ϋ P rt 3 SX Ω 0 3 Φ o rt 0 P Φ β tiβ 0 P Ω tΩ t+ rt μ- P φ β P o CD μ- 3 P tiP Ω 3 O 3 μ- Ω 3 er ex tr β P μ- tiex Hi h- 1 3 μ- Ω 3 3 Hi tr Ω CO tiP tits rt rt 3 s3 (X sx rt rt ro μ- Ω μ- μ- o t- 1 ro μ- o μ- 3 rt ti- Ω μ- ro P P ts rh Ω er P Φ μ- 0 CD μ- β μ- CΩ 3 Φ β P ex 3 CD 3 Ω 3 3 ιΩ μ- ra β CO 3 ro μ- P φ β φ 3 ti 0 Hi ex Ω H 0 ex P Ω > ti ¬ CΩ P TJ o P cQ 0 t- 1 P μ- ct 3 ti t- 1 CΩ μ- 3 0 rt rt 0 3 0 μ- t- 1 rt P ro ti CD ro μ- 3 3 0 P μ- Ω H CD μ- P < < P CΩ hh er μ- H hh tΩ rt cx 0 CΩ rt hi μ- ti ¬ CO μ- Hi rt ex tr s3 0 ti < P ct Φ φ " P P 3 μ> CD tisx th μ- * * • O P ro ϋ er φ μ- 0 μ- μ- μ- 0 Φ μ- ii hi 3 μ- Ω Ω LΩ μ- 0 CO ll μ- o 3 μ< O rt ϋ 0 φ ti- Ω to 3 Ω Hi SX t- 1 ti 3 rt CΩ P sx sx 0 3 P μ- ti li cQ μ- Ω 3 β CO Ω P cΩ o μ- ti ¬ μ- μ" CD P LΩ μ- Φ M P 3 Ω ιΩ Ω μ- tr ro 3 P ro ti rt μ- 1 H P- β μ- μ- 3 er 3 μ- φ H Ω 0 ti ¬ ii μ- ti ¬ 0 P sx β Ω Ω ii cQ ro 0 μ- 0 CQ 0 li cx P μ> P cΩ 3 0 ti ¬ P cx 3 er 0 3 ts 3 3 r+ rt O 0 Ω 0 3 h- 1 3 t{ ro μ j P LQ ex 3 er μ 1 μ- LQ φ μ- 1 CΩ 0 tiP Ω H μ- μ- 3 3 β Hi cx tiLΩ tiμ- ti SX H 3 μ- LΩ Φ ti μ" t- 1 ts e+ β o 0 0 ci- P μ- er μ- er ti3 0 μ- ti 0 μ- 3 ti ti Φ P rt μ- μ» 0 Φ M 3 3 ti tiP tiΩ ro 3 3 φ er cΩ CD rt o 3 μ- Φ P rt 0 Ω rt φ 3 3 μ» li P M o cΩ ro tier rt μ- cΩ Φ μ > ti0 LΩ t- 1 Ω rt φ ct 3 cΩ rt Φ μ- tiμ- o o h- 1 P H ro ro ti Ω t- 1 0 μ- μ- ll μ- rt 3 μ- μ> TJ 0 tiCΩ P ro 3 Hi Hi TJ μ- sx μ- 0 t- 1 ex tr CX 3 3 μ- P μ- rt O CO μ- H TJ μ- 0 φ φ 3 er 0 H cx LΩ μ- P H σ o μ- L-i. 0 cQ tr 3 tr φ 0 Φ φ 3 3 P H P cπ o ^ « tiW 3 0 t- 1 P 0 β --• rt H φ S φ β Ω cΩ ro rt 0 3 P ii cΩ CΩ CD P ti ^ cx P P 3 er P LQ Hi 3 3 rt μ- o φ Ω 3 P 3 rt 0 3 CΩ CΩ P ct rt ro CΩ P 3 P ro μ- μ- sx LΩ μ- 3 Ω rt CO Ω P μ- o 3 ii P rt rh 3 P β ti a3 μ- 3 cx tisx ti< 3 μ- 0 cΩ CO rt fitr ti ¬ o μ- O CO tiμ- β μ- ii o ii μ- tr 3 e-h ex ro to er Φ tic+ 3 P μ- Hi ts ro 3 μ- t- 1 er 3 o H H φ 3 φ P ro o P ti β ti φ ti tier β Hi μ- o 0 Φ ii ti 3 H 1 tiΦ 3 Φ Φ H er μ- 3 o rt er ro (X O μ- sx 3 ii P 0 μ- o cΩ μ- er φ Φ 3 ex ro o P Hi P t- 1 to μ- ro μ- Hi 3 ^ 3 H P Hi rt 3 Φ sx P cx CΩ Φ 0 μ- P μ- ci- rt μ- μ" 0 er Ω CD 3 μ- Ω Ω t- 1 Hi o LΩ o Ω o μ- φ P Hi Ω H 3 rt 3 Hi μ- φ P er P CΩ CO 0 μ- 0 O μ- ti ¬ μ- 0 s3 er Hi CΩ rt to tio ti P P P 3 φ 0 ro h- 1 P er μ- 3 3 μ- 3 P er 3 P sx σ 3 ti ¬ β ti ¬ ts 3 ti0 μ- μ 1 H rt cΩ ct 3 ro μ- t-- ci- sx cΩ H ro μ- μ- O H ti ¬ ts P ϋ er μ- LQ er h- 1 3 o er hi cΩ rt P TJ ro ti ¬ ti Φ 0 CD Ω ti φ fio er μ- to Φ Φ tr ro rt 3 ro 3 s3 0 μ- 3 o H er 0 TJ 3 CO tr o ro Ω ts H φ o- cx β tiμ- CΩ P ιΩ P μ- rt s3 SX H μ- Φ rt P cΩ tr μ- Ω t+ Φ t- 1 β tirt er CΩ 3 P ti tΩ rt sx β μ- o 3 P o CO ro TJ ct P μ- rt er 0 μ- ro er LQ μ- LQ ti ¬ er ro ti ¬ μ- <χ H rt 3 N μ- CΩ ti ¬ ro 0 ro μ- Hl 3 3 μ- Φ 3 0 ro ex CD er ro H er 3 μ- ti(X 0 3 ct er rt μ- ts 0 0 cQ 0 3 Φ rt ro ro μ- LQ 3 er μ- 0 3 ιΩ tr P ro 3 3 ti 3 ti ¬ d- P P 3 3 μ- φ Φ 3 rt t- 1 cΩ 3 SX oo to to cπ o Cπ o Cπ cπ P rt sx H P ro P hi fiP rt CO 0 ti μ- ϋ s3 P TJ ti s3 P ci- Hi ti< P hh Ω s3 P 0 3 tr μ- 0 μ- 0 ts 3 er er 3 O 3 0 μ- Ω 0 0 μ- Ω ti μ- er ro Ω μ- O μ- Ω cr CX Φ CO Hi tito (- ■ P sx ro CD ro h- 1 CD H sx Ω ti Ω P 3 ro ϋ Ω 3 3 sx Ω l_l. tiβ er tr H μ- ro μ- ti0 rt H ti0 μ- P r 0 μ- titi0 ro D o ts μ- 3 ro ro μ- 3 Cf t 0 μ- sx μ- er ti μ- μ- er hi 3 CO μ- ti CΩ ts er hi Ω Φ σ μ- 3 Ω H ro 3 ro 0 Hi μ> 3 ro 3 (X h- 1 0 3 IX LD er Ω ex CX) tr o sx l rt rt c-j. tr LΩ rt ct LΩ ct 3 s3 --> Ω μ- o 3 cΩ Ω μ- . μ- tr ro P μ- . o μ- . φ β μ- 0 P s3 cΩ μ- Ω . P er 3 • ^ er 3 3 0 ro 3 hi H 1 Hi 3 h Ω rt 3 o 3 s3 3 3 ro μ- sx H 3 3 TJ P LΩ TJ P LΩ ti rh cΩ 0 μ- ιΩ 0 rt μ- P 3 ro μ- μ- cx ro ct ci- P μ- sx P 3 P P 3 tiμ- ii 3 CD hi 0 ti ¬ sx 3 β tf μ- > CD LΩ rt CΩ c rt > er N s3 o rt - LΩ P rt h > 3 ro o tit- 1 μ- cx 3 tiCΩ ro o 3 CO ro O ro o μ- o μ- μ- 0 Ω 0 ϋ Hi o er 3 to μ- Ω CΩ CO • er CΩ μ- CΩ 3 t- 1 m 3 P sx CD 0 ti 0 μ- ti rt P 3 er tr ro μ- P P tr ex μ- P P er ti rt tiμ- P er LΩ P tr 3 Hi P Ω ro O P μ- P P h- 1 ro CΩ 3 3 ro sx tD 3 3 ro o P er 3 3 ro c+ ti 3 ro ci ¬ Ω t- 1 O tr CO ex 3 ro sx o • ! φ i- 1 o < ro cΩ < ro 3 ro o "< ro ts o cΩ 3 o rt CΩ N cΩ rt rt ro Ω β rh Φ Ω β rt μ- μ- rt μ- 3 rt 0 3 P P titiμ- P O H sx ro 0 t- 1 P 3 o P 3 O μ- hi CD 3 o CD O tiTJ 3 ro er 3 μ- 3 o μ- s3 μ- rt 3 s3 TJ t- 1 rt ti S3 3 o μ- 3 s3 μ- μ- ti 33 μ> ts SX 3 * * P sx ro -- ■ H P .-> μ- < Ω CD μ- o Ω CD μ- LQ h- 1 P CD μ- o 3 ro μ- μ- 0 0 rt rt H CD 3 O sx ro β 0 sx ti β 0 sx H ts ex ci ¬ 3 LΩ sx 3 Hi P CΩ 0 μ- Ω Hi μ- Ω 0 " -» tihi Hi o tirt t- 1 Hi 0 rt sx ts 0 tits 0 ticΩ Hi 0 O ti 3 ct β er ^ P Hi er μ- P hh er μ- CD P CΩ Hi er P σ 3 hh er μ- tiΩ o 3 μ- O μ- 3 S3 ct CΩ 0 rt CΩ ii rt 3 ro μ- ro P ti¬ 3 er 0 Hi P 3 0 rt tr Ω CD ro P Ω Ω 3 Φ P Ω Ω ro P cΩ rt Ω Ω 3 ct rh Ω Ω er cΩ ro ti Hi ro 3 3 CΩ ro 0 μ- cx μ- h- 1 0 CX μ- O Ω 3 ro 0 . 33 ro μ> 0 D li Ω Ω 0 li CΩ μ- hi 3 cx sx P 3 P sx P 3 rt SX P P 3 ro ti P 3 rt < ro 0 o H P 0 o ro ct ro sx μ- ti3 SX μ- ci- μ- CΩ μ- tiro μ- μ- rt CΩ tr ro Ω 3 H 3 μ- μ- Ω Hi μ- ti μ- ti 3 ts cx μ- ts 3 ti 0 0 0 3 ts 3 P 3 hi er Φ H ft TJ hi β cn SX O Ω 3 0 o <! 0 CO 0 < 0 CΩ 0 3 Hi 3 tito 0 CΩ 0 ro rt μ- 0 ro β CΩ Hi μ- h- 1 P μ- μ- 1 h- 1 . er t- 1 CD ro CD μ- 0 3 Ω P Ω Hi o ■i P rt sx t- 1 h- 1 sx " Hi tiro t- 1 P μ- I- 1 rt tr Ω 3 ro t+ P μ- Hi ti μ- er 3 tiro μ- 3 rt Φ μ- 3 μ- er 3 μ- CΩ 3 Φ P 3 μ- Ω H 3 μ- sx ro ro er ex 3 rt φ P sx 3 ct ro 3 CD 0 rt ro (X ft CD s3 φ μ- tio O Ω P m 3 t ro --> ιΩ O rt μ- (- ■ Q O tiμ- σ O tiΩ o tiμ- ct rt 3 hi β H P cQ ti ex tr ^ tr t- 1 ^ er co β c_ι. er ti 0 er IX ti ro ro H μ- 0 μ- o H 3 CD O 3 oo O tr TJ ro 0 o 3 σ. o rt s3 0 3 Hi Φ P ra ex μ- CΩ cx 0 0 P »« sx Φ Hi P -. (X μ- CD Ω «-« t- 1 ci- *• sx er μ- t- 1 ^ ro P Ω tift 3 H tit- 1 3 ti3 o ti3 tict H" ti . ex 3 Ω tiro P ti μ- ts P ro s3 μ- CX ti ro S3 μ- cΩ ts μ- μ- o S3 μ- rt μ- P ro rt ro sx 3 o ro 3 μ- Ω t li er 3 ti r 3 ro th er 3 3 H er 3 tr 3 3 3 O CX ex P ιΩ tr Ω cr er μ- ro TJ P μ- ro H P o ro cΩ μ- ro cΩ sx liti 0 μ" Ω β o μ- ro P ti er 0 P ti er o 3 li ii tr ro ii er μ- en tr 3 0 μ- er 3 t+ ti t+ ro 0 ti Hi t- 1 ro 0 h- 1 ro 0 3 cx ro o o 3 P ro » Hi 3 P μ- sx ro ro μ- tirt hi μ- tih- 1 CO Ω μ- tiμ- μ- tiμ- o X ticΩ ti ti0 μ- 3 μ- 3 μ- 0 μ- 3 μ- μ- 0 3 cr 3 Ω μ> hh TJ β er Hi 0 ro 3 3 o ci- 3 ts 0 3 3 ts 3 sx 3 ts μ- 1 ^ ts P t- 1 H CΩ ro μ- 3 t- 1 ti LQ Hi ti rt o 3 titi0 LΩ ro citi0 CD rt 0 H cn ro ro 3 μ- μ- 0 • ! ro tr • ro er μ- ts er er -- ■ Ω to P n σ μ- μ- P Hi rt CD P ro t- 1 ro P ro 3 rt 0 ro μ> O ro t- 1 β ro μ- tn O P to 3 0 P CΩ Ω μ- 3 Ω μ- μ- o μ" μ- Hi μ- H rt cx ro Hi CΩ er cQ tiμ- μ- er CO 3 sx tr Ω 3 3 CO 3 P sx μ- μ- P er P sx ex er LΩ er cΩ tiμ- cQ O tr cΩ tiCD 3 3 t- 1 ro 3 ro ro ro er 3 P ro ro CD P LQ μ- o •<! « * ro ro ro cΩ μ- ro sx tn 0 t- 1 3 rt rt sx rt Hi μ> cΩ t to cπ o cπ cπ sx 3 P Ω Hi 0 ti 3 3 Ω Hi 0 3 σ φ Ω h-> Ω ti Hi μ- μ> φ φ Ω tr P 3 0 μ- ro er P 3 P ro CO 0 h- 1 tr Φ 0 rt > P rt P Ω ro h-" P P Ω ro rt Hi rt ti P CO ti ro -> μ- ω β 3 rt H w 3 t ro o μ- tj 0 3 β 3 3 < β P cΩ 0 O μ- CD β LQ 0 O H hi 3 ro li LΩ β s3 Φ ts h-" Φ hi hh - i 3 li ro hi Hi ι-> μ- ro P Ω -> Φ rt Hi tr μ> H φ cπ LΩ P ro -I-- P rt rt -> CO μ- P 0 ro to * ! ex 3 Φ P Ω • 3 sx ro P Ω • μ- p- μ- μ- to • li 3 ti ti • Φ CΩ 3 3 (X CΩ 3 3 0 o *» Φ cΩ P ro CΩ <: P rt ex P μ- < rt sx P 3 3 CD hi Ω μ- μ- P tΩ μ- \ μ- 0 P μ- μ- > S3 > β hh Φ 0 3 > sx r-> β 3 O 3 ii 3 0 3 P H μ- er 1 li ii Φ β ii P ti to to β P H CΩ CΩ Hi 0 rt Φ CΩ rt 0 SX ti CD CO Φ Φ ft tr ct O ro rt tr c+ ii ro ti tr μ- 3 μ- ro P tr O SX μ- to --> Φ H 3 μ- CΩ t- 1 CD Φ 3 3 Φ φ 3 < Ω μ- ro Hi σ 0 P --> φ P O O P h- 1 CD ti β μ- Φ cΩ TJ μ- rt sx ro Φ < 3 μ- rt μ- ci- Hi 3 μ- ct hh 3 rt sx μ- rt ch rt P α rt 3 tf cx rt tiP o Hi P Φ 3 Ω tr S3 μ- hh O S3 . Φ Ct hh O S3 er ii CD s3 ti CD sx LQ o s3 φ Φ β O Ω μ- tr 0 Ω μ- φ P P μ- 0 rt H ti μ- Φ Φ hi O h-> sx (X ro li 0 h- 1 cx H Ω μ- 3 μ- • : CD ti ex Hi 3 3 ti J-- ti ¬ Φ 3 hi oo tiTJ Φ 0 cx rt Ω P ro tiμ- P β ti * * er μ* CΩ β ti er er ti tr Ω Ω μ- Ω er 3 rt 3 φ μ- er ro P hi Ω ii P sx rt μ- tr ex P Ω 3 Ω < ro P Ω 33 Ω CΩ μ- Φ 0 Ω Φ φ μ- Ω ra Φ \ ct tr O φ ro rt er o φ Ω ti 0 Φ Hi Ω CO o 0 tr 0 0 μ- φ 3 hi ct 0 μ- CD 3 μ- rt ti 3 TJ 0 β tr 3 3 μ- hi hi 3 O ti ci ¬ ^ 3 0 H ci ¬ Ci- sx μ- Φ ti ¬ hi h- 1 ro ti ¬ 3 0 3 Φ ts 33 3 ro ts ti Φ 3 Ω ts cx 3 P ro hh ts cΩ H P ti ¬ μ- 0 to μ- tiμ- o P ex cΩ rt 0 Φ P ti ¬ rt P 0 er μ- 3 t- 1 μ- cx er μ- 3 3 μ- Hi rt ro Ω -- ■ ti μ- CΩ Φ ct sx rt ro ti ¬ CΩ ! CO 0 o μ- sx s3 rt 0 3 tr ro 3 ro er ro 3 th P 3 3 ti 0 μ- 0 3 ιΩ φ tr 3 φ Φ tr 3 ro CD 0 Ω μ- φ 3 3 th cx ti ro φ P P ei- O σ P P ti ¬ H P cx Hi rt P 0 rt tiμ- 3 ct CΩ 0 ti er Hi ro CΩ O ti er 3 h- 1 P tr rt P hi tr P er 3 μ- μ- Hi 3 O rh μ- Hi 3 o P Ω to Ω O μ- 3 3 o cΩ H Ω CΩ μ- sx Ci- s ra μ- ex rt β tr rt sx o sx P Ω SX sx H ii to 3 tf ro CΩ 3 μ- φ 0 3 tr 3 tΩ 0 0 P cΩ P ro ro O P iQ P o P φ ii P • P CO P CD P μ- s3 Hi μ- Ω 3 Hi μ- Ω 3 ti ¬ ct Ω tr 3 P Ω " P 3 sx to Ω Hi ex CΩ Ω ro P Ω CΩ ro LQ μ- Ω 3 P P 0 μ- ti ¬ ti ¬ P o o ex s3 3 0 tr ro ro o sx P tn μ- ii 3 er er CΩ μ- ti Hi μ- IX ti φ rt ti ci- \ Ω rt tf sx sx μ- ro ro er ex cx Hi ex SX φ ro Ω SX ti 0 rt Φ CD μ- CD ro μ- CΩ 0 ti ¬ CΩ μ- rt 33 CO 0 μ- P hi β 3 CD Ω 3 er H P ro Ω 3 P ti er P 3 μ- rt ti 3 μ- P TJ rt 0 cΩ μ- O Ω rt O LQ μ- μ- cΩ s3 sx μ- ti cQ 3 0 H Φ H 3 H rt H sx P Ω sx μ- rt 3 ro « 3 H 33 ti rt LQ r-> β 33 ti rt 3 * rt sx 3 * P Ω rt P μ- φ 0 μ- P μ- ro o hi < P 0 rt t+ rt o rh rt sx Ω 3 cx Ω o P 3 tr Ω μ- μ- tr β ci- rt P cQ ti ¬ rt P cΩ Ω tr 0 o Ω Φ ti tf μ- 3 er μ- 3 β Φ H P 3 3 H Φ 0 >< 0 μ- ro P 3 P 3 3 3 μ- LΩ μ- cΩ 3 ro 3

Inventors: HIGO TSUYOSHI, ISHII ATSUSHI, KISHI KAZUHIKO, OGAWA SHIGERU, TAKAMACHI YASUYUKI, YAMADA KENJI, HIGO, TSUYOSHI, ISHII, ATSUSHI, KISHI, KAZUHIKO, OGAWA, SHIGERU, TAKAMACHI, YASUYUKI, YAMADA, KENJI
Applicants: HIGO TSUYOSHI, ISHII ATSUSHI, KISHI KAZUHIKO, NIPPON STEEL CORP, OGAWA SHIGERU, TAKAMACHI YASUYUKI, YAMADA KENJI, HIGO, TSUYOSHI, ISHII, ATSUSHI, KISHI, KAZUHIKO, NIPPON STEEL CORPORATION, OGAWA, SHIGERU, TAKAMACHI, YASUYUKI, YAMADA, KENJI
Application year: 2001
Main classification: B21B
IPC codes: B21B1, B21B37