已知数据根据微分方程拟合参数
RSCJ等效模型下，约瑟夫森微分方程 （二阶）：
i=sinφ+dφ/dt+B*d^2φ/dt^2.
已知数据是电压V和电流I。其中I=Ic*i，V=Ic*R*dφ/dt (Ic和R都为常数)。那么实测的I-V曲线也就相当于微分方程中的i-dφ/dt曲线。
问题是，怎么编程可以拟合出方程中的参数B?
不知道说清楚了没。
烦请大神指点！
非常感谢！
已知数据如下 （2000多点数据）：
电压V 电流I
2.843E-5 -0.03426
2.862E-5 -0.03424
3.04E-5 -0.03401
3.177E-5 -0.03415
3.211E-5 -0.03427
3.252E-5 -0.03433
3.366E-5 -0.03425
3.352E-5 -0.0343
3.508E-5 -0.03433
3.526E-5 -0.03412
3.62E-5 -0.03398
3.619E-5 -0.03428
3.698E-5 -0.0344
3.713E-5 -0.03435
3.749E-5 -0.0343
3.698E-5 -0.03427
3.647E-5 -0.0343
3.655E-5 -0.03408
4.536E-5 -0.09142
3.842E-5 -0.1531
3.155E-5 -0.21134
2.62E-5 -0.27255
2.14E-5 -0.33068
1.578E-5 -0.39221
1.05E-5 -0.44992
5.54E-6 -0.51152
6.3E-7 -0.56982
-5.33E-6 -0.63144
-1.094E-5 -0.68931
-1.666E-5 -0.75042
-2.383E-5 -0.80872
-2.886E-5 -0.86988
-3.547E-5 -0.92791
-4.186E-5 -0.98624
-4.772E-5 -1.04763
-5.521E-5 -1.10604
-6.328E-5 -1.16705
-6.972E-5 -1.22556
-7.739E-5 -1.28674
-8.554E-5 -1.34475
-9.417E-5 -1.4058
-9.901E-5 -1.43658
-1.0298E-4 -1.46427
-1.0631E-4 -1.49489
-1.1006E-4 -1.52544
-1.1483E-4 -1.55632
-1.1892E-4 -1.58374
-1.2459E-4 -1.61394
-1.2988E-4 -1.64465
-1.3545E-4 -1.67564
-1.405E-4 -1.70354
-1.458E-4 -1.73418
-1.5213E-4 -1.7648
-1.5772E-4 -1.79532
-1.6388E-4 -1.82203
-1.7219E-4 -1.85289
-1.7804E-4 -1.88384
-1.8457E-4 -1.91464
-1.9127E-4 -1.94216
-1.9825E-4 -1.97294
-2.0531E-4 -2.00376
-2.1291E-4 -2.03426
-2.2061E-4 -2.06128
-2.3031E-4 -2.09203
-2.4014E-4 -2.12309
-2.4901E-4 -2.15341
-2.5991E-4 -2.18149
-2.7017E-4 -2.21226
-2.8132E-4 -2.24261
-2.9137E-4 -2.27294
-3.0236E-4 -2.30072
-3.1248E-4 -2.33131
-3.2478E-4 -2.36186
-3.3488E-4 -2.39247
-3.4717E-4 -2.42013
-3.5995E-4 -2.45073
-3.7501E-4 -2.48167
-3.8859E-4 -2.51213
-4.0292E-4 -2.53929
-4.1732E-4 -2.57056
-4.3209E-4 -2.60118
-4.4787E-4 -2.6318
-4.6378E-4 -2.6589
-4.7996E-4 -2.69007
-4.9736E-4 -2.72055
-5.1381E-4 -2.7476
-5.3203E-4 -2.77826
-5.5044E-4 -2.8094
-5.6842E-4 -2.84002
-5.869E-4 -2.86749
-6.0572E-4 -2.89804
-6.2696E-4 -2.92901
-6.4829E-4 -2.95853
-6.6985E-4 -2.98622
-6.9259E-4 -3.01699
-7.1531E-4 -3.04772
-7.3785E-4 -3.07832
-7.593E-4 -3.10605
-7.8541E-4 -3.13686
-8.0963E-4 -3.16716
-8.3429E-4 -3.1978
-8.5901E-4 -3.22561
-8.9184E-4 -3.25636
-9.2303E-4 -3.28711
-9.5318E-4 -3.3179
-9.8179E-4 -3.34547
-0.00101 -3.37605
-0.00104 -3.40625
-0.00108 -3.43699
-0.00111 -3.46439
-0.00114 -3.49526
-0.00118 -3.52581
-0.00121 -3.55688
-0.00125 -3.58471
-0.00129 -3.61484
-0.00133 -3.64555
-0.00136 -3.67612
-0.0014 -3.7037
-0.00144 -3.73452
-0.00148 -3.76503
-0.00153 -3.79593
-0.00157 -3.8236
-0.00162 -3.85372
-0.00167 -3.88429
-0.00171 -3.91546
-0.00176 -3.94307
-0.00181 -3.97358
-0.00187 -4.00416
-0.00192 -4.03522
-0.00197 -4.06282
-0.00202 -4.09257
-0.00208 -4.1235
-0.00214 -4.15417
-0.00219 -4.18182
-0.00226 -4.21272
-0.00231 -4.24327
-0.00238 -4.27386
-0.00244 -4.30086
-0.00251 -4.33176
-0.00257 -4.36278
-0.00264 -4.39345
-0.00271 -4.42105
-0.00279 -4.45191
-0.00286 -4.48261
-0.00293 -4.51301
-0.003 -4.54017
-0.00308 -4.57092
-0.00316 -4.60168
-0.00324 -4.6296
-0.00333 -4.66014
-0.00341 -4.69041
-0.00349 -4.721
-0.00358 -4.74876
-0.00367 -4.77935
-0.00376 -4.81011
-0.00386 -4.84088
-0.00396 -4.8684
-0.00405 -4.8988
-0.00415 -4.92971
-0.00425 -4.96043
-0.00435 -4.98785
-0.00446 -5.01818
-0.00456 -5.04892
-0.00467 -5.07984
-0.00478 -5.10733
-0.0049 -5.13806
-0.00502 -5.16852
-0.00513 -5.19926
-0.00525 -5.22672
-0.00537 -5.25777
-0.00549 -5.28866
-0.00562 -5.31889
-0.00575 -5.34644
-0.00588 -5.37739
-0.00603 -5.40777
-0.00617 -5.43833
-0.00631 -5.46589
-0.00646 -5.49698
-0.00662 -5.52743
-0.00677 -5.55828
-0.00692 -5.5858
-0.00708 -5.61675
-0.00724 -5.64708
-0.0074 -5.67736
-0.00757 -5.70537
-0.00774 -5.73588
-0.00791 -5.76689
-0.00809 -5.79763
-0.00827 -5.82525
-0.00845 -5.85559
-0.00864 -5.88602
-0.00884 -5.9167
-0.00904 -5.9442
-0.00925 -5.97484
-0.00946 -6.00597
-0.00968 -6.03643
-0.0099 -6.06411
-0.01013 -6.09458
-0.01037 -6.12516
-0.01061 -6.1561
-0.01085 -6.18396
-0.01111 -6.21466
-0.01137 -6.24545
-0.01164 -6.27612
-0.0119 -6.30363
-0.01218 -6.33356
-0.01247 -6.36457
-0.01277 -6.39523
-0.01308 -6.42287
-0.0134 -6.45336
-0.01374 -6.48426
-0.01408 -6.51193
-0.01444 -6.54262
-0.0148 -6.57265
-0.01516 -6.60298
-0.01554 -6.63131
-0.01595 -6.66196
-0.01638 -6.69276
-0.01683 -6.72349
-0.01727 -6.75082
-0.01772 -6.78144
-0.01821 -6.81223
-0.01872 -6.84312
-0.01924 -6.87122
-0.01979 -6.90176
-0.02036 -6.93231
-0.02096 -6.96296
-0.02157 -6.99077
-0.02222 -7.0209
-0.02288 -7.05192
-0.02362 -7.08277
-0.02436 -7.11085
-0.02513 -7.14157
-0.02594 -7.17231
-0.02682 -7.20301
-0.02771 -7.23041
-0.02862 -7.26092
-0.02958 -7.29219
-0.03057 -7.32295
-0.03157 -7.35117
-0.03266 -7.38151
-0.03374 -7.41246
-0.03495 -7.44316
-0.03617 -7.47043
-0.03744 -7.5013
-0.03884 -7.53223
-0.04027 -7.56273
-0.04174 -7.59105
-0.04331 -7.6219
-0.04495 -7.65253
-0.0467 -7.68329
-0.04849 -7.71125
-0.0504 -7.74215
-0.05237 -7.77299
-0.05441 -7.80413
-0.05653 -7.83186
-0.0588 -7.86297
-0.06121 -7.89361
-0.06376 -7.92449
-0.06637 -7.95252
-0.06909 -7.9835
-0.07194 -8.01451
-0.07495 -8.04577
-0.07817 -8.07354
-0.0816 -8.1048
-0.08516 -8.13573
-0.0888 -8.16666
-0.09271 -8.19438
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-0.10121 -8.25706
-0.10584 -8.28851
-0.11069 -8.3167
-0.11579 -8.34803
-0.12136 -8.37931
-0.12699 -8.40748
-0.13276 -8.43929
-0.13916 -8.47076
-0.14582 -8.5024
-0.15284 -8.53115
-0.16031 -8.56259
-0.16807 -8.59395
-0.17611 -8.62529
-0.18441 -8.65404
-0.19331 -8.68631
-0.20283 -8.71804
-0.21237 -8.7502
-0.22233 -8.77941
-0.23268 -8.81111
-0.24328 -8.84256
-0.25414 -8.87425
-0.26498 -8.90359
-0.27613 -8.93633
-0.28697 -8.96813
-0.29757 -9
-0.30769 -9.02866
-0.31741 -9.06089
-0.32635 -9.09273
-0.33437 -9.1244
-0.34162 -9.15289
-0.34823 -9.18501
-0.35383 -9.21584
-0.35908 -9.24736
-0.36348 -9.27555
-0.36736 -9.30659
-0.37075 -9.33733
-0.37376 -9.36835
-0.3765 -9.39666
-0.37923 -9.42763
-0.38174 -9.45847
-0.38434 -9.48924
-0.38652 -9.51701
-0.38879 -9.54786
-0.39091 -9.5791
-0.39329 -9.61022
-0.39545 -9.63788
-0.39752 -9.66885
-0.39963 -9.69999
-0.40171 -9.73092
-0.4037 -9.75856
-0.40586 -9.78934
-0.40791 -9.82047
-0.40994 -9.85146
-0.41176 -9.87932
-0.41377 -9.91011
-0.41588 -9.94109
-0.41792 -9.97198
-0.41977 -9.99931
-0.42182 -10.0301
-0.42376 -10.06162
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我从纯数学的角度来分析，感觉还是少变量，你给的数据没有时间变量t，那么φ关于时间t的二阶偏导数没法求
假设实测出的任意两个数据点之间的时间间隔是0.01
迭代数: 20
计算用时(时:分:秒:微秒): 00:03:05:174
优化算法: 准牛顿法(BFGS) + 通用全局优化法
计算结束原因: 达到收敛判断标准
均方差(RMSE): 0.110099669633367
残差平方和(SSE): 25.140897863503
相关系数(R): 0.955259457355074
相关系数之平方(R^2): 0.91252063086631
决定系数(DC): 0.912484628250756
卡方系数(Chi-Square): -2.93073094393305
F统计(F-Statistic): 21613.5846186257
参数 最佳估算
---------- -------------
Ic -0.00290908945302359
R 24.0585117401893
C 697.878481125544------------------v积分之后和fei相差一个常数
B -0.131768084174069，
！！！假设实测出的任意两个数据点之间的时间间隔是0.01，强制楼上C为0，结果变化不大
迭代数: 55
计算用时(时:分:秒:微秒): 00:07:03:888
优化算法: 准牛顿法(BFGS) + 通用全局优化法
计算结束原因: 达到收敛判断标准
均方差(RMSE): 0.110102579429016
残差平方和(SSE): 25.142226765618
相关系数(R): 0.95525631311255
相关系数之平方(R^2): 0.912514623741383
决定系数(DC): 0.912480002347447
卡方系数(Chi-Square): -2.84566188814172
F统计(F-Statistic): 21611.9582637779
参数 最佳估算
---------- -------------
ic -0.00290998014913476
r 24.0584550429692
b -0.131743322619604
哦，这样啊。可惜当时没有导出时间数据。。。
非常感谢你的回复！
存在几个问题：
1.Ic和B应该是正值。拟合结果是负值的原因会不会与I是从0先增加到负向最大有关？
2.B值理论上是大于1的。
3.“v积分之后和φ相差一个常数”，不太懂。不知道你是怎么拟合的，电容C这个值也可以引入吗？你肯定是个行家。
4. 理论上C不可强制为0，否则曲线不会出现回滞.
5. 可否留下联系方式，想详细请教。
你的公式需要略微变形一下，至少要变成I和V的关系式，我也不是什么行家，至于联系方式，就在眼皮底下。