- Email: [email protected]
Roll Eccentricity Compensation Based on Anti-Aliasing Wavelet Analysis Method
Available onlineat www.sciencedirect.com
..-"
-.;- ScienceDirect JOURNAL OF IRO!':'l AND STEEL RESEARCH, INTERNXfIONAL. 2009, 16(2): 35-39
Roll Eccentricity Compensation Based on Anti-Aliasing Wavelet Analysis Method CHEN Zhi-ming ,
LUO Fei,
XU Yu-ge ,
YU Wei
(College of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, Guangdong , China) Abstract: Roll eccentricity is an important factor causing thickness variations during hot strip rolling and might define the limit of strip thickness control accuracy. An improved multi-resolution wavelet transform algorithm was proposed to compensate for the roll eccentricity. The wavelet transform method had good localization characteristics in both the time and frequency domains for signal analysis; however, the wavelet method had a frequency-aliasing problem owing to the less than ideal cut-off frequency characteristics of wavelets. This made its component reconstruction of an inaccurate signal. To eliminate inherent frequency aliases in the wavelet transform, fast Fourier transform (FIT) and inverse fast Fourier transform (IFFT) were combined with the Mallat algqrithm, This synthesis was described in detail. Then, the roll eccentricity component was extracted from rolling force signal. An automatic gauge control (AGC) system added with a multi-resolution wavelet analyzer was designed. Experimental results showed that the anti-aliasing method could greatly restrain the inverse effect of eccentricity and the thickness control accuracy was improved from ± 40 I'm to ± 15 um, Key words: hot strip rolling; roll eccentricity compensation; wavelet analysis; frequency alias
During the hot strip rolling process, the strip passes through the roll gap and is cast into a thin sheet, as depicted in the upper part of Fig. 1. The shape of the roll, then, has a great influence on the processed thickness and shape of the strip. Because the demand for strip quality is becoming more critical, especially the thickness, much attention must be paid to roll eccentricity compensation. Roll eccentricity, which defines the strip thickness control accuracy, is usually caused by inexact roll grinding, nonuniformed thermal roll expansion, or by deviations between the axis of the roll barrel and the axis of the roll neck[I.2] , as shown in the bottom part of Fig. 1. Roll eccentricity is not a stationary signal. Its frequency and amplitude might vary with the change of rolling speed and with the thermal expansion of the rolls. Typically, roll eccentricity affects the measured strip thickness of a six-stand hot rolling mill ranging from 25 f-lm to 50 f-lm.
Ef: /~ Axis of roll neck
Fig. 1
Eccentricity
or roll
Foundation Item: Item Sponsored by National Natural Science Foundation of China (60774032); Provincial Natural Science Foundation of Guangdong Province of China (06025724); Key Project of Guangzhou Scientific Program of China (2007Z2-D0121); Special Research Fund of Ministry of Education of China for College Doctoral Subjects (20070561006) E-mail: [email protected]; Revised Date: June 16,2008 Biograpby:CHEN Zhi-ming(l981-), Male, Doctor Candidate,
• 36 •
Journal of Iron and Steel Research, International
Extensive research has been done on roll eccentricity compensation. Fast Fourier transform (FFT) is a traditional method in this research area. Shiozaki and Takahashi proposed the Fourier analyzer of roll eccentricity (FARE) method using the relationship between roll eccentricity and rolling force variatiorr". The FFT algorithm can separate desirable and noise components from a background signal. But the FFT has local contradictions between the time and frequency domains, and is generally only suitable for a stationary signal. Because the frequency and amplitude of roll eccentricity always vary with the change of rolling speed and with the abrasion and thermal expansion of the rolls, this restricts the FFT's ability to separate the eccentricity component from its background signal-the rolling force signal. The wavelet transform analysis method, which is developed from the Fourier transform, has good localization characteristics in both the time and frequency domains, and is usually applied to analyze the signal component in specified signal frequency band and time window. The use of wavelet transform for roll eccentricity compensation is an innovation with great potentiaP·4,5]. An adaptive threshold method of wavelet transform was proposed in Ref. [5 J, in which the decomposition level and the threshold values were determined adaptively , and a form of soft-threshold was used to reconstruct the eccentricity signal. Huang accomplished the eccentricity compensation based on wavelet packet de-noising theorl 4 ] . But it should be pointed out that frequency aliasing and redundant images are present during the wavelet decomposition and reconstruction process because of the less than ideal cut-off frequency of the used wavelets. This means that the decomposition and construction do not guarantee that the reconstructed roll eccentricity component reflects the real situation, and this leads to unsatisfied compensation. In this study, an improved multi-resolution wavelet method that combines the Mallat algorithm with the FFT and inverse fast Fourier transform (lFFT) is proposed. The algorithm eliminates frequency aliasing and yields accurate eccentricity compensation and based on it, a controller is designed.
1 1. 1
Vol. 16
(W"J)(a,b)=
lal- 1/ 2 [
:
r~bl dr
J(ryq;[
(1)
where, (W
Ao[J(t) J = f
l
Ct)
="fH (2t- k)A j-
A j [J(t) J
J
1
[J(t) J
Dj[J(t) =I(;(2t-k)A j- 1 [J(t) k
(2)
J
where, t is the discrete time sequence, t = 1 , 2, ... , N; f Ct) is thepriginal signal; j is the decomposition level, j = 1 , 2, ... , J, and ] = 10g2 N; Hand G are the quadrature mirror filters for decomposition in time domain, i. e. , H is the low-pass filter, and G is the high-pass filter; A j is the wavelet coefficient of the approximation component, i. e., the low-frequency component of J(t) at level is and D, is the wavelet coefficient of the detail component, i. e. , the high-frequency component of J ( t ) at level j. This procedure is depicted in Fig. 2, where J. is the sampling frequency. In this manner, an integrated signal is decomposed into a sequence of subsignals with different frequency bands. The reconstruction algorithm can be described as follows:
J
A j [J(t) =2}'h (t-2k)Aj + 1 [J(t) k
J+
2Ig (t- 2k)D j + 1 [J(t) J
(3)
k
Reconstruction is the reverse of decomposition. 1. 2
Frequency aliasing and anti-aliasing
In the decomposition of A j - 1 , for its approximation A j and detail D j , there are two key steps: con-
Anti-Aliasing Wavelet Method Mallat algorithm
The definition of the wavelet transform of a signal J(r) is as follows:
Fig.2
Decomposition of I(t) and its frequency distribution
Issue 2
• 37 •
Roll Eccentricity Compensation Based on Anti-Aliasing Wavelet Analysis Method
volution of A j - I with Hand G, respectively, with the results being A' j and D' j ; down-sampling of the convolution results, i. e. , one sample is kept out of two from A'j and tr.; with the results being A j and
u.. Similarly, there are two key steps during the reconstruction: up-sampling of A j and D j , i. e. , putting one zero between each sample of A j and D, ; and convolution of up-sampled A j and D, with hand g, respectively. For the Mallat algorithm, the down-sampling procedure will halve the sampling frequency, whereas the up-sampling procedure doubles it at each transform level. In this way, frequency alias and redundant images are created. If wavelet transform is used only for compression and reconstruction of original signal, the reconstructed signal will be exactly the same as the original, since the up- and down-samplings are opposite procedures. However, when the Mallat algorithm is used to reconstruct a sub-band signal, redundant frequencies and images are caused by the convolution and down-sampling and up-sampling procedures because of the less than ideal cutoff fq~quency characteristics of the wavelet filters[7.8J. It means that each sub-band includes redundant frequency components that should be in other sub-bands after the decomposition, and redundant images are created and remain in the reconstructed signal. This is detrimental while trying to extract a specific signal component from its complex integrated background signal. One solution to this problem is to eliminate the redundant frequency components and images by using the FFT and IFFT in the wavelet decomposition and reconstruction procedures'" .8J . The procedure of eliminating the aliases when calculating the approximation coefficient A j is as follows: (1) To apply FFT to the convolution result of A j - I with H; (2) To set the amplitude to zero if the frequency is out of the frequency band of A j (~!,/2j+1 here) ; (3) To apply IFFT to the result in Step 2; (4) To down-sample the result in Step 3 to get The procedure to eliminate aliases for D, is similar, but D, will not be down-sampled. A copy of the uri-down-sampled A j will be used to reconstruct aj. The procedure to eliminate redundant images
when reconstructing d, is: (1) To apply FFT to reconstruction result of normal Mallat algorithm; (2) To set the amplitude to zero whose frej j+ quency is out of the interval [!./2 l , !./2 J; (3) To apply IFFT to the result in Step 2 to get d j •
2
Experimental Examination
2. 1
Controller design An experiment was performed to validate the effectiveness of the proposed anti-aliasing method. In an automatic gauge control (AGC) system, the sampled rolling force signal F can be described
as:
F=Fo+F1 +Fe+Fn
(4)
where, F o is the preset rolling force; F 1 is rolling force variation caused by variations of strip thickness; Fe is the variation of the rolling force caused by roll eccentricity; and F, IS the stochastic noise during sigllal sampling. In the system depicted in Fig. 3, the sampled rolling force and rolling speed are the input signals to the multi-resolution wavelet analyzer. Its outputs are F o+ F 1 , Fe' and F n, each in different frequency band. F o+ F 1 was applied to the AGC, Fe was applied to the eccentricity regulator, and F; was discarded. It should be noted that the control signal from the eccentricity regulator is applied to the hydraulic system in an inverse-phase manner for compensation.
2. 2
Experimental results The controlled plant In the experiment was a six-stand four-roll continuous hot-strip finishing mill in a steel factory. The diameter of the work roll was 800 rnm , and the diameter of the backup roll was 1 350 mm. The rolling speed setup of each stand was o. 65 m/s , 1. 29 m/s , 2.03 m/s , 2.90 m/s ,
Other signals Rolling force ignal Component of (Fo+F 1) Rolling spee signal AGC .--------, Multi-resolution Wavelet analyzer Eccentricity regulator Component of eccentricity
Fig. 3
AGC system with wavelet analyzer
Vol. 16
Journal of Iron and Steel Research, International
• 38 •
3.87 mis, and 4.77 m/s , respectively. The strip's entrance thickness is 50 mm and the finished strip thickness set-point is 3. 81 mm. The rolling force sampling frequency f. is 200 Hz in the experiment, and the sample length for the analyzer in Fig. 3 was 1 000 points. During the experiment, the rolling force data were renewed with newly sampled data in each control step. With the multi-resolution wavelet analyzer, Fa F] and Fe were separated for the AGC and the eccentricity regulator, respectively. A piece of sampled rolling force signal is shown in Fig. 4. The "sym5" wavelet was used in the anti-aliasing wavelet method, and the decomposition level N is equal to 6. Partial results of the anti-aliasing method are depicted in Fig. 5 to Fig. 7, where d, to d 3 are subbands of noise signal, and a comparison between the Mallat algorithm and the proposed method was made. It can be seen that the dominant frequencies were more pronounced because the redundant frequency components present in the Mallat algorithm were eliminated. The reconstructed detail of the rolling signal is
+
...
20.50 r - - - - - - - - - - - - - - - - ,
~::~
~o~l..
i r ~"----.- '- - - '-__ o
20
40 60 FrequencylHz
80
100
Fig. 6 Spectrum comparison of d 2 for Mallat algorithm (a) and antI-aliasing method (b)
t t
o
20
40 60 FrequencylHz
80
100
20.45
Fig. 7 Spectrum comparison of d 3 for Mallat algorithm (a) and antI-aliasing method (b)
20.40
~ 20.35 ~
.E
lil
20.25 20.20 20.15
Z
1:> .... ~ 'Cl QI
] ;:l
~
_ _~_ _~_ ___1 2.0 3.0 4.0 5.0 Timels
'--_~~_~
o
1.0
Fig. 4
Sampled rolling force signal
}"l
.~
r Ii 0
shown in Fig. 8, where d, to d, are sub-bands of roll eccentricity signal. The reconstructed Fa F], Fe' and F; are shown in Fig. 9. In the experiment, the thickness variation was reduced from ± 40 Jlm, using the ordinary AGC systern, to ± 15 Jlm, and the difference between actual and reference thickness is illustrated in Fig. 10. The
+
~ 20.30
20
40 60 FrequencylHz
80
I
100
Fig. 5 Spectrum comparison of d, for Mallat algorithm (a) and anti-aliasing method (b)
Fig. 8
Reconstructed detail of rolling force signal
Issue 2
Roll Eccentricity Compensation Based on Anti-Aliasing Wavelet Analysis Method
• 39 •
AGC systems, including those systems using traditional FFT methods and wavelet methods without the anti-aliasing algorithm. This is confirmed experimentally. The thickness variation is reduced from ±40 p'm to ± 15 p.m. The proposed multi-resolution wavelet analyzer can not only directly compensate roll eccentricity, but also indirectly improve the regulating quality of the AGC, because Fe has been separated from F o Fl. It should be pointed out that the calculation burden will increase when the FFT and IFFT are used during wavelet decomposition and reconstruction, but this disadvantage can be overcome by the improved hardware processors.
+
o
1.0
2.0
4.0
3.0
5.0
Timels
Fig.9
Reconstructed r,
+F
I ,
F., and F.
0.010 r - - - - - - - - - - - - - - - . ,
References : OJ)05
1
[IJ
-..:l
.~
-8
~ -0.005
[2J
~
~ -0.010
-0.015 '--_ _~_--""___~ 1.0 2.0 3.0 '" 0 Time/s
4.0
---' 5.0
[3J
[4J
Fig. 10
Difference between actual and reference thickness
r~ference thickness was 3.870 mm.
3
Conclusions [5J
The anti-aliasing ability of the proposed method is achieved by combining the Mallat algorithm with the FFT and IFFT. Experimental results demonstrate that redundant frequencies of each sub-signal and redundant images created during reconstruction are effectively eliminated. This guarantees the validity of the reconstructed sub-signals. This anti-aliasing multi-resolution wavelet method is valid to compensate for roll eccentricity and is superior to ordinary
[6J
[7J
[8J
Aistleitner K, Mattersdorfer L G, Haas W, et al. Neural Network for Identification of Roll Eccentricity in Rolling Mills [J]. Journal of Materials Processing Technology, 1996, 600-4): 387. WANG Wei-reno WANG Zheng-lin , SUN Yi-kang. Application of Multi-Resolution Wavelet Controller in Rolling Eccentricity Control [J]. Journal of University of Science and Technology Beijing, 2005, 27(6), 728 (in Chinese). LI Bo-qun, Application and Research of Synthetical AGC System in Hot Rolling Mill [D]. Beijing: University of Science and Technology Beijing, 2006 (in Chinese). HUANG Min, FANG Xiao-ke, WANG Jian-hui, et al. Roll Eccentricity Compensation Control for Strip Rolling Mills Based on Wavelet Packet De-Noising Theory [AJ. Institute of Electrical and Electronics Engineers Inc. eds, Proceedings of the 5th WCICA [CJ. Piscataway: Institute of Electrical and Electronics Engineers Inc. 2004. 3565. LI Yong, WANG Iun , HU Xian-shuo , et al. AGC System With Adaptive Threshold Method of Wavelet Transform [J]. Iron and Steel, 2007, 420): 39 (in Chinese). Mallat S G. A Theory for Multi-Resolution Signal Decomposition: The Wavelet Representation [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1989, 11(7): 674. YANG Iian-guo. Wavelet Analysis and Its Engineering Applications [MJ. Beijing, China Machine Press. 2005 (in Chinese) . YANG Jian-guo , Park S T. An Anti-Aliasing Algorithm for Discrete Wavelet Transform [J]. Mechanical System and Signal Processing, 2003. 17(5): 945.
..-"
-.;- ScienceDirect JOURNAL OF IRO!':'l AND STEEL RESEARCH, INTERNXfIONAL. 2009, 16(2): 35-39
Roll Eccentricity Compensation Based on Anti-Aliasing Wavelet Analysis Method CHEN Zhi-ming ,
LUO Fei,
XU Yu-ge ,
YU Wei
(College of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, Guangdong , China) Abstract: Roll eccentricity is an important factor causing thickness variations during hot strip rolling and might define the limit of strip thickness control accuracy. An improved multi-resolution wavelet transform algorithm was proposed to compensate for the roll eccentricity. The wavelet transform method had good localization characteristics in both the time and frequency domains for signal analysis; however, the wavelet method had a frequency-aliasing problem owing to the less than ideal cut-off frequency characteristics of wavelets. This made its component reconstruction of an inaccurate signal. To eliminate inherent frequency aliases in the wavelet transform, fast Fourier transform (FIT) and inverse fast Fourier transform (IFFT) were combined with the Mallat algqrithm, This synthesis was described in detail. Then, the roll eccentricity component was extracted from rolling force signal. An automatic gauge control (AGC) system added with a multi-resolution wavelet analyzer was designed. Experimental results showed that the anti-aliasing method could greatly restrain the inverse effect of eccentricity and the thickness control accuracy was improved from ± 40 I'm to ± 15 um, Key words: hot strip rolling; roll eccentricity compensation; wavelet analysis; frequency alias
During the hot strip rolling process, the strip passes through the roll gap and is cast into a thin sheet, as depicted in the upper part of Fig. 1. The shape of the roll, then, has a great influence on the processed thickness and shape of the strip. Because the demand for strip quality is becoming more critical, especially the thickness, much attention must be paid to roll eccentricity compensation. Roll eccentricity, which defines the strip thickness control accuracy, is usually caused by inexact roll grinding, nonuniformed thermal roll expansion, or by deviations between the axis of the roll barrel and the axis of the roll neck[I.2] , as shown in the bottom part of Fig. 1. Roll eccentricity is not a stationary signal. Its frequency and amplitude might vary with the change of rolling speed and with the thermal expansion of the rolls. Typically, roll eccentricity affects the measured strip thickness of a six-stand hot rolling mill ranging from 25 f-lm to 50 f-lm.
Ef: /~ Axis of roll neck
Fig. 1
Eccentricity
or roll
Foundation Item: Item Sponsored by National Natural Science Foundation of China (60774032); Provincial Natural Science Foundation of Guangdong Province of China (06025724); Key Project of Guangzhou Scientific Program of China (2007Z2-D0121); Special Research Fund of Ministry of Education of China for College Doctoral Subjects (20070561006) E-mail: [email protected]; Revised Date: June 16,2008 Biograpby:CHEN Zhi-ming(l981-), Male, Doctor Candidate,
• 36 •
Journal of Iron and Steel Research, International
Extensive research has been done on roll eccentricity compensation. Fast Fourier transform (FFT) is a traditional method in this research area. Shiozaki and Takahashi proposed the Fourier analyzer of roll eccentricity (FARE) method using the relationship between roll eccentricity and rolling force variatiorr". The FFT algorithm can separate desirable and noise components from a background signal. But the FFT has local contradictions between the time and frequency domains, and is generally only suitable for a stationary signal. Because the frequency and amplitude of roll eccentricity always vary with the change of rolling speed and with the abrasion and thermal expansion of the rolls, this restricts the FFT's ability to separate the eccentricity component from its background signal-the rolling force signal. The wavelet transform analysis method, which is developed from the Fourier transform, has good localization characteristics in both the time and frequency domains, and is usually applied to analyze the signal component in specified signal frequency band and time window. The use of wavelet transform for roll eccentricity compensation is an innovation with great potentiaP·4,5]. An adaptive threshold method of wavelet transform was proposed in Ref. [5 J, in which the decomposition level and the threshold values were determined adaptively , and a form of soft-threshold was used to reconstruct the eccentricity signal. Huang accomplished the eccentricity compensation based on wavelet packet de-noising theorl 4 ] . But it should be pointed out that frequency aliasing and redundant images are present during the wavelet decomposition and reconstruction process because of the less than ideal cut-off frequency of the used wavelets. This means that the decomposition and construction do not guarantee that the reconstructed roll eccentricity component reflects the real situation, and this leads to unsatisfied compensation. In this study, an improved multi-resolution wavelet method that combines the Mallat algorithm with the FFT and inverse fast Fourier transform (lFFT) is proposed. The algorithm eliminates frequency aliasing and yields accurate eccentricity compensation and based on it, a controller is designed.
1 1. 1
Vol. 16
(W"J)(a,b)=
lal- 1/ 2 [
:
r~bl dr
J(ryq;[
(1)
where, (W
Ao[J(t) J = f
l
Ct)
="fH (2t- k)A j-
A j [J(t) J
J
1
[J(t) J
Dj[J(t) =I(;(2t-k)A j- 1 [J(t) k
(2)
J
where, t is the discrete time sequence, t = 1 , 2, ... , N; f Ct) is thepriginal signal; j is the decomposition level, j = 1 , 2, ... , J, and ] = 10g2 N; Hand G are the quadrature mirror filters for decomposition in time domain, i. e. , H is the low-pass filter, and G is the high-pass filter; A j is the wavelet coefficient of the approximation component, i. e., the low-frequency component of J(t) at level is and D, is the wavelet coefficient of the detail component, i. e. , the high-frequency component of J ( t ) at level j. This procedure is depicted in Fig. 2, where J. is the sampling frequency. In this manner, an integrated signal is decomposed into a sequence of subsignals with different frequency bands. The reconstruction algorithm can be described as follows:
J
A j [J(t) =2}'h (t-2k)Aj + 1 [J(t) k
J+
2Ig (t- 2k)D j + 1 [J(t) J
(3)
k
Reconstruction is the reverse of decomposition. 1. 2
Frequency aliasing and anti-aliasing
In the decomposition of A j - 1 , for its approximation A j and detail D j , there are two key steps: con-
Anti-Aliasing Wavelet Method Mallat algorithm
The definition of the wavelet transform of a signal J(r) is as follows:
Fig.2
Decomposition of I(t) and its frequency distribution
Issue 2
• 37 •
Roll Eccentricity Compensation Based on Anti-Aliasing Wavelet Analysis Method
volution of A j - I with Hand G, respectively, with the results being A' j and D' j ; down-sampling of the convolution results, i. e. , one sample is kept out of two from A'j and tr.; with the results being A j and
u.. Similarly, there are two key steps during the reconstruction: up-sampling of A j and D j , i. e. , putting one zero between each sample of A j and D, ; and convolution of up-sampled A j and D, with hand g, respectively. For the Mallat algorithm, the down-sampling procedure will halve the sampling frequency, whereas the up-sampling procedure doubles it at each transform level. In this way, frequency alias and redundant images are created. If wavelet transform is used only for compression and reconstruction of original signal, the reconstructed signal will be exactly the same as the original, since the up- and down-samplings are opposite procedures. However, when the Mallat algorithm is used to reconstruct a sub-band signal, redundant frequencies and images are caused by the convolution and down-sampling and up-sampling procedures because of the less than ideal cutoff fq~quency characteristics of the wavelet filters[7.8J. It means that each sub-band includes redundant frequency components that should be in other sub-bands after the decomposition, and redundant images are created and remain in the reconstructed signal. This is detrimental while trying to extract a specific signal component from its complex integrated background signal. One solution to this problem is to eliminate the redundant frequency components and images by using the FFT and IFFT in the wavelet decomposition and reconstruction procedures'" .8J . The procedure of eliminating the aliases when calculating the approximation coefficient A j is as follows: (1) To apply FFT to the convolution result of A j - I with H; (2) To set the amplitude to zero if the frequency is out of the frequency band of A j (~!,/2j+1 here) ; (3) To apply IFFT to the result in Step 2; (4) To down-sample the result in Step 3 to get The procedure to eliminate aliases for D, is similar, but D, will not be down-sampled. A copy of the uri-down-sampled A j will be used to reconstruct aj. The procedure to eliminate redundant images
when reconstructing d, is: (1) To apply FFT to reconstruction result of normal Mallat algorithm; (2) To set the amplitude to zero whose frej j+ quency is out of the interval [!./2 l , !./2 J; (3) To apply IFFT to the result in Step 2 to get d j •
2
Experimental Examination
2. 1
Controller design An experiment was performed to validate the effectiveness of the proposed anti-aliasing method. In an automatic gauge control (AGC) system, the sampled rolling force signal F can be described
as:
F=Fo+F1 +Fe+Fn
(4)
where, F o is the preset rolling force; F 1 is rolling force variation caused by variations of strip thickness; Fe is the variation of the rolling force caused by roll eccentricity; and F, IS the stochastic noise during sigllal sampling. In the system depicted in Fig. 3, the sampled rolling force and rolling speed are the input signals to the multi-resolution wavelet analyzer. Its outputs are F o+ F 1 , Fe' and F n, each in different frequency band. F o+ F 1 was applied to the AGC, Fe was applied to the eccentricity regulator, and F; was discarded. It should be noted that the control signal from the eccentricity regulator is applied to the hydraulic system in an inverse-phase manner for compensation.
2. 2
Experimental results The controlled plant In the experiment was a six-stand four-roll continuous hot-strip finishing mill in a steel factory. The diameter of the work roll was 800 rnm , and the diameter of the backup roll was 1 350 mm. The rolling speed setup of each stand was o. 65 m/s , 1. 29 m/s , 2.03 m/s , 2.90 m/s ,
Other signals Rolling force ignal Component of (Fo+F 1) Rolling spee signal AGC .--------, Multi-resolution Wavelet analyzer Eccentricity regulator Component of eccentricity
Fig. 3
AGC system with wavelet analyzer
Vol. 16
Journal of Iron and Steel Research, International
• 38 •
3.87 mis, and 4.77 m/s , respectively. The strip's entrance thickness is 50 mm and the finished strip thickness set-point is 3. 81 mm. The rolling force sampling frequency f. is 200 Hz in the experiment, and the sample length for the analyzer in Fig. 3 was 1 000 points. During the experiment, the rolling force data were renewed with newly sampled data in each control step. With the multi-resolution wavelet analyzer, Fa F] and Fe were separated for the AGC and the eccentricity regulator, respectively. A piece of sampled rolling force signal is shown in Fig. 4. The "sym5" wavelet was used in the anti-aliasing wavelet method, and the decomposition level N is equal to 6. Partial results of the anti-aliasing method are depicted in Fig. 5 to Fig. 7, where d, to d 3 are subbands of noise signal, and a comparison between the Mallat algorithm and the proposed method was made. It can be seen that the dominant frequencies were more pronounced because the redundant frequency components present in the Mallat algorithm were eliminated. The reconstructed detail of the rolling signal is
+
...
20.50 r - - - - - - - - - - - - - - - - ,
~::~
~o~l..
i r ~"----.- '- - - '-__ o
20
40 60 FrequencylHz
80
100
Fig. 6 Spectrum comparison of d 2 for Mallat algorithm (a) and antI-aliasing method (b)
t t
o
20
40 60 FrequencylHz
80
100
20.45
Fig. 7 Spectrum comparison of d 3 for Mallat algorithm (a) and antI-aliasing method (b)
20.40
~ 20.35 ~
.E
lil
20.25 20.20 20.15
Z
1:> .... ~ 'Cl QI
] ;:l
~
_ _~_ _~_ ___1 2.0 3.0 4.0 5.0 Timels
'--_~~_~
o
1.0
Fig. 4
Sampled rolling force signal
}"l
.~
r Ii 0
shown in Fig. 8, where d, to d, are sub-bands of roll eccentricity signal. The reconstructed Fa F], Fe' and F; are shown in Fig. 9. In the experiment, the thickness variation was reduced from ± 40 Jlm, using the ordinary AGC systern, to ± 15 Jlm, and the difference between actual and reference thickness is illustrated in Fig. 10. The
+
~ 20.30
20
40 60 FrequencylHz
80
I
100
Fig. 5 Spectrum comparison of d, for Mallat algorithm (a) and anti-aliasing method (b)
Fig. 8
Reconstructed detail of rolling force signal
Issue 2
Roll Eccentricity Compensation Based on Anti-Aliasing Wavelet Analysis Method
• 39 •
AGC systems, including those systems using traditional FFT methods and wavelet methods without the anti-aliasing algorithm. This is confirmed experimentally. The thickness variation is reduced from ±40 p'm to ± 15 p.m. The proposed multi-resolution wavelet analyzer can not only directly compensate roll eccentricity, but also indirectly improve the regulating quality of the AGC, because Fe has been separated from F o Fl. It should be pointed out that the calculation burden will increase when the FFT and IFFT are used during wavelet decomposition and reconstruction, but this disadvantage can be overcome by the improved hardware processors.
+
o
1.0
2.0
4.0
3.0
5.0
Timels
Fig.9
Reconstructed r,
+F
I ,
F., and F.
0.010 r - - - - - - - - - - - - - - - . ,
References : OJ)05
1
[IJ
-..:l
.~
-8
~ -0.005
[2J
~
~ -0.010
-0.015 '--_ _~_--""___~ 1.0 2.0 3.0 '" 0 Time/s
4.0
---' 5.0
[3J
[4J
Fig. 10
Difference between actual and reference thickness
r~ference thickness was 3.870 mm.
3
Conclusions [5J
The anti-aliasing ability of the proposed method is achieved by combining the Mallat algorithm with the FFT and IFFT. Experimental results demonstrate that redundant frequencies of each sub-signal and redundant images created during reconstruction are effectively eliminated. This guarantees the validity of the reconstructed sub-signals. This anti-aliasing multi-resolution wavelet method is valid to compensate for roll eccentricity and is superior to ordinary
[6J
[7J
[8J
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