含分数阶的灰色模型及其在地基沉降预测中的应用
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摘要
基于灰色理论建立的传统沉降预测模型均为整数阶,存在不连续、与实测数据差异较大的缺陷。鉴于此,本文以灰色模型为研究对象,通过改变模型中的整数阶微分为分数阶微分来改进灰色模型的预测效果。本模型与传统模型的最大区别在于增加了分数阶阶次的识别,首先结合灰色理论得到无输入的常微分方程;接着对其引入输入项,并将常微分方程做变换得到含分数阶的微分方程;最后将该模型与实测数据进行对比,且与传统灰色理论沉降预测模型进行误差计算,发现本文建立的沉降预测模型可以较好地预测地基沉降。
The traditional settlement prediction models based on the gray theory are all of integer orders,and there are defects that are discontinuous and have large differences from the measured data. In this paper,an improved gray model,which uses fractional derivatives to replace the integer ones,is suggested. The biggest difference between this model and the traditional model is the increased recognition of fractional orders. First,combined with the gray theory,the ordinary differential equation with no input is obtained. Then,an input term is introduced to the equation,and the ordinary differential equation is transformed to obtain a differential equation with a fractional order. Finally,the model is compared with the measured data,and the error is calculated with the traditional gray theory settlement prediction model. The results show that the settlement prediction model established in this paper can provide better prediction for the foundation settlement.
The traditional settlement prediction models based on the gray theory are all of integer orders,and there are defects that are discontinuous and have large differences from the measured data. In this paper,an improved gray model,which uses fractional derivatives to replace the integer ones,is suggested. The biggest difference between this model and the traditional model is the increased recognition of fractional orders. First,combined with the gray theory,the ordinary differential equation with no input is obtained. Then,an input term is introduced to the equation,and the ordinary differential equation is transformed to obtain a differential equation with a fractional order. Finally,the model is compared with the measured data,and the error is calculated with the traditional gray theory settlement prediction model. The results show that the settlement prediction model established in this paper can provide better prediction for the foundation settlement.
引文
[1]甘友文,王志亮,郑华.地基沉降预测中的双曲线模型修正[J].水文地质工程地质,2004,31(1):98-100.[GAN Y W, WANG Z L, ZHENG H.Correction of hyperbolic model in foundation settlement prediction[J]. Hydrogeology&Engineering Geology,2004,31(1):98-100.(in Chinese)]
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[3]肖治宇,陈昌富,季永新.基于自适应神经-模糊推理系统的软土路基沉降预测模型[J].水文地质工程地质,2010,37(6):61-65.[XIAO Z Y,CHEN C F,JI Y X. Settlement prediction of soft clay roadbed based on adaptive Neuro-Fuzzy inference system model[J]. Hydrogeology&Engineering Geology,2010,37(6):61-65.(in Chinese)]
[4]唐利民.地基沉降预测模型的正则化算法[J].岩土力学,2010,31(12):3945-3948.[TANG L M.Regularization algorithm of foundation settlement prediction mode[J]. Rock and Soil Mechanics,2010,31(12):3945-3948.(in Chinese)]
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[6]曹文贵,印鹏,贺敏,等.考虑实测数据新旧程度的工后沉降单项模型预测方法[J].水文地质工程地质,2015,42(6):65-70.[CAO W G,YIN P,HE M, et al. A prediction method for postconstruction settlement of a single model with the consideration of the new or old degree of the measured data[J]. Hydrogeology&Engineering Geology,2015,42(6):65-70.(in Chinese)]
[7]刘国华,何勇兵,汪树玉.土石坝沉降预测中的多变量灰色预测模型[J].水利学报,2003,(12):84-88.[LIU G H, HE Y B, WANG S Y.Application of grey multi-variable forecasting model for the settlement of earth dam[J]. Journal of Hydraulic Engineering,2003,(12):84-88.(in Chinese)]
[8]李洪然,张阿根,叶为民.参数累积估计灰色模型及地面沉降预测[J].岩土力学,2008,29(12):3417-3421.[LI H R,ZHANG A G,YE W M.Accumulating method GM(1,1)model and prediction of land subsidence[J]. Rock and Soil Mechanics,2008,29(12):3417-3421.(in Chinese)]
[9]苗雨,谭显坤,杨慧琳,等.基于灰色模型的软土路基沉降预测及误差分析[J].地下空间与工程学报,2006,2(6):1049-1052.[MIAO Y,TAN X K,YANG H L,et al. Prediction and error analysis of post-construction settlement for soft soil embankment using the R. Usher grey model[J]. Chinese Journal of Underground Space and Engineering,2006,2(6):1049-1052.(in Chinese)]
[10]李篷,段淑清,王堃,等.灰色模型与曲线拟合沉降预测及相关性分析[J].地下空间与工程学报,2014,10(1):109-115.[LI P,D S Q,WANG K,et al. Settlement prediction by gray model and curve fitting and relevance analysis[J]. Chinese Journal of Underground Space and Engineering,2014,10(1):109-115.(in Chinese)]
[11] NAZARIAN P,HAERI M. Generalization of order distribution concept use in the fractional order system identification[J]. Signal Processing,2010,90(7):2243-2252.
[12] GABANO J,POINOT T. Fractional modeling and identification of thermal systems[J]. Signal Processing,2011,91(3):531-541.
[13] LEWANDOWSKI R, CHORAZYCZEWSKI B.Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models,used to modeling of viscoelastic dampers[J]. Computers and Structures,2010,88(1):1-17.
[14] MANSOURI R,BETTAYEB M,DJAMAH T. et al.Vector fitting fractional system identification using particle swarm optimization[J]. Applied Mathematics and Computation,2008,206(2):510-520.
[15] BENCHELLAL A, POINOT T, TRIGEASSOU C.Approximation and identification of diffusive interfaces by fractional models[J]. Signal Processing,2006,86(10):2712-2727.
[16] DJAMAH T,MANSOURI R,DJENNOUNE S,et al.Optimal low order model identification of fractional dynamic systems[J]. Applied Mathematics and Computation,2008,206(2):543-554.
[17]胡振南.双曲线法在路基沉降预测中的应用研究[J].公路工程,2011,36(3):145-148.[HU Z N. Study of the hyperbolic method in prediction of roadbed settlement[J]. Highway Engineering,2011,36(3):145-148.(in Chinese)]
[2]王志亮,黄景忠,李永池.沉降预测中的Asaoka法应用研究[J].岩土力学,2006,27(11):2025-2028.[Wang Zhiliang, Huang Jingzhong, Li Yongchi. Study on application of Asaoka’s method to settlement predication[J]. Rock and Soil Mechanics,2006,27(11):2025-2028.(in Chinese)]
[3]肖治宇,陈昌富,季永新.基于自适应神经-模糊推理系统的软土路基沉降预测模型[J].水文地质工程地质,2010,37(6):61-65.[XIAO Z Y,CHEN C F,JI Y X. Settlement prediction of soft clay roadbed based on adaptive Neuro-Fuzzy inference system model[J]. Hydrogeology&Engineering Geology,2010,37(6):61-65.(in Chinese)]
[4]唐利民.地基沉降预测模型的正则化算法[J].岩土力学,2010,31(12):3945-3948.[TANG L M.Regularization algorithm of foundation settlement prediction mode[J]. Rock and Soil Mechanics,2010,31(12):3945-3948.(in Chinese)]
[5]朱志铎,周礼红.软土路基全过程沉降预测的Logistic模型应用研究[J].岩土工程学报,2009,31(6):965-969.[ZHU Z D, ZHOU L H.Application of Logistic model in settlement prediction during complete process of embankment construction[J]. Chinese Journal of Geotechnical Engineering,2009,31(6):965-969.(in Chinese)]
[6]曹文贵,印鹏,贺敏,等.考虑实测数据新旧程度的工后沉降单项模型预测方法[J].水文地质工程地质,2015,42(6):65-70.[CAO W G,YIN P,HE M, et al. A prediction method for postconstruction settlement of a single model with the consideration of the new or old degree of the measured data[J]. Hydrogeology&Engineering Geology,2015,42(6):65-70.(in Chinese)]
[7]刘国华,何勇兵,汪树玉.土石坝沉降预测中的多变量灰色预测模型[J].水利学报,2003,(12):84-88.[LIU G H, HE Y B, WANG S Y.Application of grey multi-variable forecasting model for the settlement of earth dam[J]. Journal of Hydraulic Engineering,2003,(12):84-88.(in Chinese)]
[8]李洪然,张阿根,叶为民.参数累积估计灰色模型及地面沉降预测[J].岩土力学,2008,29(12):3417-3421.[LI H R,ZHANG A G,YE W M.Accumulating method GM(1,1)model and prediction of land subsidence[J]. Rock and Soil Mechanics,2008,29(12):3417-3421.(in Chinese)]
[9]苗雨,谭显坤,杨慧琳,等.基于灰色模型的软土路基沉降预测及误差分析[J].地下空间与工程学报,2006,2(6):1049-1052.[MIAO Y,TAN X K,YANG H L,et al. Prediction and error analysis of post-construction settlement for soft soil embankment using the R. Usher grey model[J]. Chinese Journal of Underground Space and Engineering,2006,2(6):1049-1052.(in Chinese)]
[10]李篷,段淑清,王堃,等.灰色模型与曲线拟合沉降预测及相关性分析[J].地下空间与工程学报,2014,10(1):109-115.[LI P,D S Q,WANG K,et al. Settlement prediction by gray model and curve fitting and relevance analysis[J]. Chinese Journal of Underground Space and Engineering,2014,10(1):109-115.(in Chinese)]
[11] NAZARIAN P,HAERI M. Generalization of order distribution concept use in the fractional order system identification[J]. Signal Processing,2010,90(7):2243-2252.
[12] GABANO J,POINOT T. Fractional modeling and identification of thermal systems[J]. Signal Processing,2011,91(3):531-541.
[13] LEWANDOWSKI R, CHORAZYCZEWSKI B.Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models,used to modeling of viscoelastic dampers[J]. Computers and Structures,2010,88(1):1-17.
[14] MANSOURI R,BETTAYEB M,DJAMAH T. et al.Vector fitting fractional system identification using particle swarm optimization[J]. Applied Mathematics and Computation,2008,206(2):510-520.
[15] BENCHELLAL A, POINOT T, TRIGEASSOU C.Approximation and identification of diffusive interfaces by fractional models[J]. Signal Processing,2006,86(10):2712-2727.
[16] DJAMAH T,MANSOURI R,DJENNOUNE S,et al.Optimal low order model identification of fractional dynamic systems[J]. Applied Mathematics and Computation,2008,206(2):543-554.
[17]胡振南.双曲线法在路基沉降预测中的应用研究[J].公路工程,2011,36(3):145-148.[HU Z N. Study of the hyperbolic method in prediction of roadbed settlement[J]. Highway Engineering,2011,36(3):145-148.(in Chinese)]