含分数阶的灰色模型及其在地基沉降预测中的应用

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。

Gray model with fractional order and its application to settlement prediction

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.