Recursive least square adjustment

In doubt ： The concept of recursive least squares adjustment should be the same as that of sequential adjustment , Because the concept of sequential adjustment is “ Sequential adjustment is also called successive correlation indirect adjustment , It divides observations into two or more groups , Make relevant indirect adjustment in the order of groups , So as to achieve the same result as the overall adjustment of the two-phase or multi-phase network .”

One 、 Preliminary knowledge

1、 Basic concepts

• Sequential adjustment is also called successive correlation indirect adjustment , It divides observations into two or more groups , Make relevant indirect adjustment in the order of groups , So as to achieve the same result as the overall adjustment of the two-phase or multi-phase network .
• Advantages of grouping adjustment ：1） When dealing with large adjustment problems , Because the order of the normal equation of group adjustment is low , It can overcome the difficulty of insufficient computer capacity when not grouping .2） Under certain circumstances , For example, zoning adjustment , Each group can calculate synchronously , Thus, the calculation cycle can be shortened .3） When a survey network has completed the adjustment calculation , When new observation data is added due to network expansion or encryption , Just take the original adjustment result as the relevant observation value and consider its weight inverse matrix , Adjust with the new observation , The same result as the overall adjustment of the new and old data can be obtained .

2、 Symbolic meaning

The symbols used are the same as those in the least squares adjustment

Least squares adjustment ：https://blog.csdn.net/Gou_Hailong/article/details/122188144

in addition , The meanings of several new symbols used in recursive least squares are as follows ：

Two 、 The principle of recursive least square adjustment

In general , More observations , As long as it is handled properly , The smaller the mean square error of the least squares estimation . The least square algorithm realized by batch processing , All measured values must be stored . If the number of measured values is very large , Then the computer must have huge storage capacity , This is obviously uneconomical . Recursive least square adjustment extracts the information of the estimated quantity from the measured value obtained each time , Used to correct the estimate obtained in the previous step . The more times you get measurements , The more times to correct , The higher the accuracy of estimation . The algorithm is described in detail below .

1、 The recursive formula

The starting term can be obtained by the least square .

2、 Recursive formula derivation