CN102540891A - Recursive extended least squares algorithm-based crystallizer ARMAX (Auto Regressive Moving Average Exogenous) model identification method - Google Patents

Recursive extended least squares algorithm-based crystallizer ARMAX (Auto Regressive Moving Average Exogenous) model identification method Download PDF

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CN102540891A
CN102540891A CN2012100162242A CN201210016224A CN102540891A CN 102540891 A CN102540891 A CN 102540891A CN 2012100162242 A CN2012100162242 A CN 2012100162242A CN 201210016224 A CN201210016224 A CN 201210016224A CN 102540891 A CN102540891 A CN 102540891A
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crystallizer
armax
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CN102540891B (en
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张华军
蔡炜
褚学征
陈方元
尉强
周登科
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Qidong Xianfeng High Pressure Pump Co Ltd
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Wisdri Engineering and Research Incorporation Ltd
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Abstract

The crystallizer armax identification Method based on recurrence extended least squares that the present invention relates to a kind of, specifically: being input u with crystallizer oil cylinder valve aperture, it is output y with crystallizer position, crystallizer armax model least square and target function are established on the basis of sampled data, variable u, y and e are constituted vector as white noise estimation value by the model parameter calculation residual error e being calculated using the last time
Figure 2012100162242100004DEST_PATH_IMAGE002
, according to least square method of recursion thought step by step calculation variable pk and lk, pass through pk and lk progressive alternate computation model unknown parameter. The present invention can accurately approach crystallizer armax unknown-model parameter global optimal solution, crystallizer armax model parameter can be carried out using present sample inputoutput data online to update, the update of crystallizer armax model parameter does not depend on and historical data, and accurate armax Model Distinguish parameter can be provided in time when model parameter changes.

Description

Crystallizer ARMAX identification Method based on recursion augmentation least square method
Technical field
The present invention relates to conticaster crystallizer Control System Design field in the iron and steel metallurgical industry; Relate in particular to a kind of based on recursion augmentation least square method (Recursive Extended Least Squares Algorithm, crystallizer ARMAX RELS) (Auto Regressive Moving Average eXogenous) identification Method.
Background technology
Mold oscillation to the strand demoulding and surface quality have directly, significant effects; In the actual casting cycle of sheet billet continuous casting; Pulling rate is normally along with the variation of working condition (like cast temperature) changes; For guaranteeing to obtain good strand stripping result and strand table quality, should guarantee that suitably adjust frequency, amplitude etc. vibrated basic parameter under the basicly stable prerequisite of vibratory process parameter.Yet; Obtain good frequency, amplitude control effect; Crystallizer control system that must be reasonable in design with fast, accurately tracking frequencies, amplitude set-point, and outstanding control system to be the basis with the model carry out systematic analysis and design, in view of the PID controller design method of present crystallizer control system based on experience; Be necessary at first crystallizer to be carried out Model Distinguish, on the rational model basis, carry out Control System Design again to obtain the better controlling effect.Because traditional least square method can only be carried out the ARX Model Distinguish; ARMAX model for having the coloured noise interference can't be realized parameter estimation; Therefore be necessary to utilize the augmentation least square method that crystallizer ARMAX model is carried out identification, obtain the model parameter under the coloured noise interference.
Summary of the invention
Technical matters to be solved by this invention is: a kind of crystallizer ARMAX identification Method based on recursion augmentation least square method is provided; This method can onlinely be carried out the Model Distinguish under the crystallizer coloured noise disturbed condition, and the crystallizer control system good for design performance provides scientific and rational mathematical model.
The present invention solves its technical matters and adopts following technical scheme:
Crystallizer ARMAX identification Method based on recursion augmentation least square method provided by the invention; Specifically: with crystallizer oil cylinder valve aperture is input u; With the crystallizer position is output y; On the sampled data basis, set up crystallizer ARMAX model least square and target function; Utilize the last model parameter that calculates to calculate residual error e as the white noise estimated value; Variable u, y and e are constituted vector
Figure BDA0000131524250000011
progressively calculate variable Pk and Lk, through Pk and Lk iterative computation model unknown parameter progressively according to least square method of recursion thought.
Above-mentioned crystallizer ARMAX identification Method provided by the invention based on recursion augmentation least square method, it may further comprise the steps:
(1) gathering inputoutput data, serves as input u (t) with crystallizer oil cylinder valve aperture, serves as that output y (t) gathers N to data sample Z with the crystallizer position N
(2) the ARX model that makes up under the interference of crystallizer white noise is:
A(q)y(t)＝B(q)u(t)+C(q)ε(t)，
Wherein: A (q)=1+a 1q -1+ a 2q -2+ ... + a Naq -na, B (q)=b 1q -1+ b 2q -2+ ... + b Nbq -nb, C (q)=1+c 1q -1+ c 2q -2+ ... + c Ncq -nc, q -1To mobile operator, q is that forward direction moves operator for afterwards, and na, nb, nc are arithmetic number, and ε (t) is a white Gaussian noise;
(3) make θ=[a 1a 2A Nab 1b 2B Nbc 1c 2C Nc] TFor the ARMAX model is treated identified parameters;
(4) the ARMAX model transferring being become ARX model
Figure BDA0000131524250000021
ε (t) is white Gaussian noise,
Figure BDA0000131524250000022
Figure BDA0000131524250000023
(5) make the objective function of the ARX Model Distinguish process that has white Gaussian noise be:
(6) utilizing the white Gaussian noise generating algorithm to generate 100 energy densities is 0.00000001 undesired signal ε ', and the ε (t) in the step (4) is replaced with ε ' (t);
(7) calculate
Figure BDA0000131524250000025
(8) calculate
(9) judge R (t 0) whether reversible, if irreversible then execution in step (7), otherwise carry out (10);
(10) make P (t 0)=R (t 0) -1, θ ^ ( t 0 ) = P ( t 0 ) * F ( t 0 ) ;
(11) calculate wherein 1≤i≤nc of residual error
Figure BDA0000131524250000028
;
(12) with obtaining
Figure BDA0000131524250000029
in ε (t-i) substitution (4) in (11)
(13) Order
Figure BDA00001315242500000210
Figure BDA00001315242500000211
(14) then the unknown parameter iterative formula is
Figure BDA00001315242500000212
(15) whether judgment data finishes, if do not finish then execution in step (11), otherwise the output result;
Through above-mentioned steps, realize crystallizer ARMAX Model Distinguish based on recursion augmentation least square method.
In the above-mentioned steps (6), when utilizing white Gaussian noise generating algorithm generted noise, the energy density of noise adopts 0.00000001, and is as far as possible little to guarantee the undesired signal amplitude.
In the above-mentioned steps (11), 1≤i≤nc.
The present invention compared with prior art has following main advantage:
One of which. can accurately approach crystallizer ARMAX model unknown parameter globally optimal solution;
They are two years old. and can onlinely utilize current sampling inputoutput data to carry out crystallizer ARMAX model parameter and upgrade;
They are three years old. and crystallizer ARMAX model parameter is upgraded and is not relied on and historical data;
They are four years old. the parameter of ARMAX Model Distinguish accurately can, model parameter in time be provided when changing.
Description of drawings
Fig. 1 is an ARMAX model structure schematic diagram;
Fig. 2 is a recursion augmentation least square method RELS process flow diagram;
Fig. 3 is that crystallizer ARMAX model is predicted the comparison diagram between output valve and the actual samples data based on the system model that the RELS algorithm obtains among the embodiment 1.
Embodiment
Crystallizer ARMAX identification Method based on recursion augmentation least square method provided by the invention; Specifically: with crystallizer oil cylinder valve aperture is input u; With the crystallizer position is output y; On the sampled data basis, set up crystallizer ARMAX model least square and target function; Utilize the last model parameter that calculates to calculate residual error e as the white noise estimated value; Variable u, y and e are constituted vector
Figure BDA0000131524250000031
progressively calculate variable Pk and Lk, through Pk and Lk iterative computation model unknown parameter progressively according to least square method of recursion thought.
Above-mentioned crystallizer ARMAX identification Method based on recursion augmentation least square method provided by the invention referring to Fig. 1 and Fig. 2, may further comprise the steps:
(1) gathering inputoutput data, serves as input u (t) with crystallizer oil cylinder valve aperture, serves as that output y (t) gathers N to data sample Z with the crystallizer position N
(2) the ARX model that makes up under the interference of crystallizer white noise is A (q) y (t)=B (q) u (t)+C (q) ε (t), wherein A (q)=1+a 1q -1+ a 2q -2+ ... + a Naq -na, B (q)=b 1q -1+ b 2q -2+ ... + b Nbq -nb, C (q)=1+c 1q -1+ c 2q -2+ ... + c Ncq -nc, q -1To mobile operator, q is that forward direction moves operator for afterwards, and na, nb, nc are arithmetic number, and ε (t) is a white Gaussian noise, and accompanying drawing 1 is ARMAX modular concept figure;
(3) make θ=[a 1a 2A Nab 1b 2B Nbc 1c 2C Nc] TFor the ARMAX model is treated identified parameters;
(4) the ARMAX model transferring being become ARX model
Figure BDA0000131524250000032
ε (t) is white Gaussian noise,
Figure BDA0000131524250000033
Figure BDA0000131524250000034
(5) order have white Gaussian noise the objective function of ARX Model Distinguish process for
Figure BDA0000131524250000035
(6) utilizing the white Gaussian noise generating algorithm to generate 100 energy densities is 0.00000001 undesired signal ε ', and the ε (t) in the step (4) is replaced with ε ' (t);
(7) calculate
Figure BDA0000131524250000036
(8) calculate
(9) judge R (t 0) whether reversible, if irreversible then execution in step (7), otherwise carry out (10);
(10) make P (t 0)=R (t 0) -1, θ ^ ( t 0 ) = P ( t 0 ) * F ( t 0 ) ;
(11) calculate wherein 1≤i≤nc of residual error
Figure BDA0000131524250000041
;
(12) with obtaining
Figure BDA0000131524250000042
in ε (t-i) substitution (4) in (11)
(13) make
Figure BDA0000131524250000043
Figure BDA0000131524250000044
(14) then the unknown parameter iterative formula is
Figure BDA0000131524250000045
(15) whether judgment data finishes, if do not finish then execution in step (11), otherwise the output result;
Through above-mentioned steps, realize crystallizer ARMAX Model Distinguish based on recursion augmentation least square method.
Below in conjunction with concrete application example the invention described above method is further specified, but do not limit the present invention.
Embodiment 1:
Certain steel mill's one slab caster mould sampled data is as shown in table 1, its sampling time interval Ts=0.003 second, number of data points N=250.
Select na=5, nb=3, the ARMAX crystallizer model of nc=2;
Make A (q)=1+a 1q -1+ a 2q -2+ a 3q -3+ a 4q -4+ a 5q -5, B (q)=b 1q -1+ b 2q -2+ b 3q -3, then system treats that identified parameters does θ ^ 0 = a 1 a 2 a 3 a 4 a 5 b 1 b 2 b 3 c 1 c 2 ;
Can get initiation parameter according to the invention described above method step (6)-(10) is:
θ ^ t 0 = 0.257 0.147 1.439 - 0.184 - 1.067 - 1.799 0.0517 - 0.097 - 0.025 - 0.022 ;
Can get final crystallizer ARMAX model parameter according to the invention described above method step (11)-(15) is:
θ t ^ = - 0.710082227544558 - 0.245984323673739 - 0.164608668068724 0.216539402002658 - 0.268073409150356 0.167959624343424 - 0.007927143946054 0.016238396845038 0.010429106721516 0.007791252485065 ;
Above-mentioned final crystallizer ARMAX model parameter is the parameter of said crystallizer ARMAX Model Distinguish based on recursion augmentation least square method.
The crystallizer na=5 of Fig. 3 for adopting the identification of RELS method to obtain; Nb=3; ARMAX model prediction output and the actual correlation curve of exporting between the sampled data during nc=2 can find that from Fig. 3 the RELS method can either accurately approach the crystallizer system characteristic, can onlinely calculate fast again.
Subordinate list
Crystallizer sample data among table 1 embodiment 1
Sequence number 1 2 3 4 5 6 7 8 9 10
Input 46.875 12.02619 11.91768 11.75492 12.22512 11.91406 11.90683 11.87066 11.54876 11.61386
Output 8.583912 8.715567 8.860388 8.91305 9.071036 9.189525 9.308015 9.466001 9.571325 9.663484
Sequence number 11 12 13 14 15 16 17 18 19 20
Input 11.71152 11.2594 11.13643 10.78559 10.61198 10.21412 10.04413 10.01157 9.939236 9.595631
Output 9.834635 9.953125 10.11111 10.2296 10.38759 10.50608 10.59823 10.70356 10.83521 10.96687
Sequence number 21 22 23 24 25 26 27 28 29 30
Input 9.255642 9.320747 8.915654 8.814381 8.977141 8.289931 8.289931 7.97526 7.722078 7.515914
Output 11.0327 11.17752 11.25651 11.30917 11.46716 11.53299 11.63831 11.73047 11.80946 11.88845
Sequence number 31 32 33 34 35 36 37 38 39 40
Input 7.309751 7.143374 6.71658 6.5068 6.072772 5.703848 5.645978 5.298756 4.774306 4.481337
Output 11.95428 12.05961 12.12543 12.21759 12.29659 12.33608 12.40191 12.49407 12.54673 12.62572
Sequence number 41 42 43 44 45 46 47 48 49 50
Input 4.000289 3.642216 3.096065 2.871817 2.654803 2.177373 1.884404 1.381655 0.907841 0.719763
Output 12.67839 12.75738 12.78371 12.81004 12.8627 12.88903 12.9417 12.98119 12.98119 13.00752
Sequence number 51 52 53 54 55 56 57 58 59 60
Input -0.24233 -1.54803 -3.35286 -5.12514 -6.89742 -8.87948 -10.5288 -12.5977 -14.5616 -16.2218
Output 12.99436 13.03385 13.04702 13.04702 13.06018 13.00752 13.00752 12.98119 12.9022 12.83637
Sequence number 61 62 63 64 65 66 67 68 69 70
Input -18.0194 -19.4734 -21.0576 -22.2186 -23.1156 -24.66 -25.6402 -26.3853 -27.2244 -27.4993
Output 12.71788 12.61256 12.44141 12.24392 12.1386 11.95428 11.74363 11.55932 11.29601 11.07219
Sequence number 71 72 73 74 75 76 77 78 79 80
Input -27.8827 -27.8501 -28.3095 -28.4252 -28.2661 -27.7416 -27.0906 -26.4685 -25.3328 -24.5913
Output 10.79572 10.6114 10.38759 10.15061 9.874132 9.610822 9.360677 9.071036 8.860388 8.583912
Sequence number 81 82 83 84 85 86 87 88 89 90
Input -23.2458 -21.8967 -20.3631 -18.6921 -16.6667 -14.7931 -13.2198 -10.8579 -8.65885 -6.30064
Output 8.333767 8.083623 7.833478 7.570168 7.346354 7.188368 6.951389 6.753906 6.556423 6.411603
Sequence number 91 92 93 94 95 96 97 98 99 100
Input -4.07624 -2.48119 -1.37442 -0.40509 0.596788 1.312934 1.902488 2.470341 2.933304 3.504774
Output 6.240451 6.121962 6.029803 5.937645 5.884983 5.858652 5.83232 5.83232 5.819155 5.819155
Sequence number 101 102 103 104 105 106 107 108 109 110
Input 3.978588 4.466869 5.038339 5.349392 5.78342 6.047454 6.394676 6.940828 7.143374 7.273582
Output 5.819155 5.819155 5.858652 5.871817 5.924479 5.963976 5.977141 6.042969 6.121962 6.174624
Sequence number 111 112 113 114 115 116 117 118 119 120
Input 7.606337 7.671441 8.036748 8.062066 8.445457 8.658854 8.626302 8.969907 9.1182 9.197772
Output 6.266782 6.319444 6.424768 6.47743 6.556423 6.674913 6.740741 6.832899 6.938223 6.990885
Sequence number 121 122 123 124 125 126 127 128 129 130
Input 9.693287 9.671586 9.982639 10.09115 10.01157 10.4239 10.44922 11.04601 11.09303 11.2377
Output 7.109375 7.188368 7.293692 7.425347 7.491175 7.609664 7.649161 7.76765 7.872974 7.951967
Sequence number 131 132 133 134 135 136 137 138 139 140
Input 11.54514 11.48727 11.79832 11.76939 11.95747 12.18533 12.01895 12.01895 12.0298 12.04789
Output 8.083623 8.162616 8.294271 8.386429 8.478588 8.623408 8.741898 8.860388 8.978877 9.110532
Sequence number 141 142 143 144 145 146 147 148 149 150
Input 11.93215 11.85619 11.78024 11.83449 11.38238 11.216 10.84708 10.93388 10.88686 10.60113
Output 9.229022 9.360677 9.466001 9.637153 9.768808 9.926794 10.01895 10.12428 10.2691 10.38759
Sequence number 151 152 153 154 155 156 157 158 159 160
Input 10.43475 10.05498 9.805411 9.733073 9.443721 9.42202 9.190538 8.846933 8.922888 8.470775
Output 10.53241 10.66406 10.75622 10.88788 10.96687 11.08536 11.20385 11.26968 11.40133 11.46716
Sequence number 161 162 163 164 165 166 167 168 169 170
Input 8.449074 8.098235 8.083767 7.826968 7.273582 7.24103 6.872106 6.720197 6.579138 6.061921
Output 11.58565 11.63831 11.73047 11.86212 11.91479 12.00694 12.07277 12.12543 12.23076 12.28342
Sequence number 171 172 173 174 175 176 177 178 179 180
Input 5.877459 5.389178 5.005787 4.680266 4.134115 3.870081 3.43967 3.002025 2.672888 2.267795
Output 12.37558 12.45457 12.50723 12.59939 12.65205 12.70472 12.77054 12.81004 12.8627 12.88903
Sequence number 181 182 183 184 185 186 187 188 189 190
Input 1.974826 1.548032 1.063368 0.831887 0.57147 -0.63657 -1.84823 -3.67115 -5.26982 -7.18678
Output 12.92853 12.98119 12.98119 12.99436 13.03385 13.00752 13.04702 13.03385 13.04702 13.04702
Sequence number 191 192 193 194 195 196 197 198 199 200
Input -9.08203 -10.7964 -12.8906 -14.5942 -16.4605 -17.9905 -19.7085 -21.3252 -22.3307 -23.6256
Output 13.00752 13.00752 12.9417 12.88903 12.78371 12.70472 12.59939 12.41507 12.27025 12.08594
Sequence number 201 202 203 204 205 206 207 208 209 210
Input -24.66 -25.293 -26.3346 -26.8012 -27.7742 -28.1648 -28.1829 -28.125 -28.0852 -27.8067
Output 11.86212 11.69097 11.45399 11.29601 11.07219 10.80888 10.54557 10.30859 10.05845 9.900463
Sequence number 211 212 213 214 215 216 217 218 219 220
Input -28.125 -27.2569 -26.5878 -25.4376 -24.3634 -22.9167 -21.394 -20.5548 -18.8368 -16.6522
Output 9.610822 9.360677 9.071036 8.807726 8.53125 8.254774 8.109954 7.859809 7.570168 7.346354
Sequence number 221 222 223 224 225 226 227 228 229 230
Input -14.7678 -12.5651 -10.4637 -8.0693 -6.29702 -3.94965 -2.27865 -1.24421 -0.18446 0.499132
Output 7.109375 6.911892 6.688079 6.582754 6.411603 6.227286 6.121962 6.016638 5.963976 5.911314
Sequence number 231 232 233 234 235 236 237 238 239 240
Input 1.240596 1.945891 2.452257 3.088831 3.653067 3.880932 4.58261 4.95515 5.414497 5.996817
Output 5.871817 5.858652 5.83232 5.819155 5.858652 5.83232 5.858652 5.871817 5.871817 5.924479
Sequence number 241 242 243 244 245 246 247 248 249 250
Input 6.268084 6.604456 6.77445 7.204861 7.526765 7.642506 8.018663 8.025897 8.098235 8.532263
Output 5.963976 6.029803 6.0693 6.121962 6.200955 6.253617 6.358941 6.451099 6.503761 6.609086

Claims (4)

1. crystallizer ARMAX identification Method based on recursion augmentation least square method; It is characterized in that with crystallizer oil cylinder valve aperture be input u; With the crystallizer position is output y; On the sampled data basis, set up crystallizer ARMAX model least square and target function; Utilize the last model parameter that calculates to calculate residual error e as the white noise estimated value; Variable u, y and e are constituted vector
Figure FDA0000131524240000011
progressively calculate variable Pk and Lk, through Pk and Lk iterative computation model unknown parameter progressively according to least square method of recursion thought.
2. crystallizer ARMAX identification Method according to claim 1 is characterized in that this method may further comprise the steps:
(1) gathering inputoutput data, serves as input u (t) with crystallizer oil cylinder valve aperture, serves as that output y (t) gathers N to data sample Z with the crystallizer position N
(2) the ARX model that makes up under the interference of crystallizer white noise is:
A(q)y(t)＝B(q)u(t)+C(q)ε(t)，
In the formula: A (q)=1+a 1q -1+ a 2q -2+ ... + a Naq -na, B (q)=b 1q -1+ b 2q -2+ ... + b Nbq -nb,
C (q)=1+c 1q -1+ c 2q -2+ ... + c Ncq -nc, q -1To mobile operator, q is that forward direction moves operator for afterwards, and na, nb, nc are arithmetic number, and ε (t) is a white Gaussian noise, and accompanying drawing 1 is ARMAX modular concept figure;
(3) make θ=[a 1a 2A Nab 1b 2B Nbc 1c 2C Nc] TFor the ARMAX model is treated identified parameters;
(4) the ARMAX model transferring is become in ARX model
Figure FDA0000131524240000012
formula: ε (t) is a white Gaussian noise,
Figure FDA0000131524240000014
(5) make the objective function of the ARX Model Distinguish process that has white Gaussian noise be:
Figure FDA0000131524240000015
(6) utilize the white Gaussian noise generating algorithm to generate one group of undesired signal ε ', the ε (t) in the step (4) is replaced with ε ' (t);
(7) calculate
Figure FDA0000131524240000016
(8) calculate
Figure FDA0000131524240000017
(9) judge R (t 0) whether reversible, if irreversible then execution in step (7), otherwise execution in step (10);
(10) make P (t 0)=R (t 0) -1, θ ^ ( t 0 ) = P ( t 0 ) * F ( t 0 ) ;
(11) calculate residual error
Figure FDA0000131524240000019
(12) with obtaining
Figure FDA00001315242400000110
in ε (t-i) substitution (4) in (11)
(13) Order
Figure FDA0000131524240000021
Figure FDA0000131524240000022
(14) then the unknown parameter iterative formula is
Figure FDA0000131524240000023
(15) whether judgment data finishes, if do not finish then execution in step (11), otherwise the output result;
Through above-mentioned steps, realize crystallizer ARMAX Model Distinguish based on recursion augmentation least square method.
3. crystallizer ARMAX identification Method according to claim 2 is characterized in that in the step (6), and when utilizing white Gaussian noise generating algorithm generted noise, the energy density of noise adopts 0.00000001, and is as far as possible little to guarantee the undesired signal amplitude.
4. crystallizer ARMAX identification Method according to claim 2 is characterized in that in the step (11) 1≤i≤nc.
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CN112130542A (en) * 2020-07-01 2020-12-25 浙江浙能台州第二发电有限责任公司 Control loop performance evaluation method based on normal operation data and system identification
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