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Contents
1 Comparison of Online Recursive Least Squares to Online Recursive Square Root
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1.1 Conversion of Online Recursive Least Squares to formulation presented in Hsia
The starting update equations are^{[1]}
Eqn. 1 
and
Eqn. 2 
where
Eqn. 3 
From the RLS algorithm we have
Eqn. 4 
where
Eqn. 5 
leading to
Eqn. 6 
Comparing Eqns. 1 and 6 leads to the following conclusions
With these substitutions it is clear that the Kalman gain is
From the RLS algorithm we have
if . If for all iteration, then
After making the appropriate substitutions for , then becomes
If , then
In order to double check the result we start with the update equation for from the RLS algorithm
Making the correct substitutions we get
Eqn. 7 
This becomes
which is equal to Eqn. 7 is is symmetric. Since is originally defined as^{[2]}
it must be symmetric.
1.2 Conversion of Online Recursive Square Root to formulation presented in Hsia^{[3]}
The RSR update equations are^{[4]}
Eqn. 8 
where
Eqn. 9 
Eqn. 10 
Eqn. 11 
Eqn. 12 
 for
and
Eqn. 13 
 for
and
Eqn. 14 
and
Eqn. 15 
Just as the RLS algorithm was linked to the algorithm derived in Hsia^{[5]} the RSR algorithm from Zhao^{[6]} can also be shown to be equivalent be the correct substitutions. Again, we start with the error or innovation sequence defined as
for the RLS algorithm and
for the RSR algorithm. Teh resulting equalities fall out
Eqn. 16 
and
Eqn. 17 
.
Using Eqn. 17 in Eqn. 10
and Eqn. 11 becomes
If above is compared to . from Eqn. 3, then
with the following equality
Eqn. 18 
Then Eqn. 1 becomes
Use Eqn. 18 and get
Substitute in from Eqn. 3 and from Eqn. 16 to get
Note that is a scalar.
2 Conclusion
3 References
 Spradlin, Gabriel T. '"AN EXPLORATION OF PARAMETER IDENTIFICATION TECHNIQUES: CMG TEMPERATURE PREDICTION THEORY AND RESULTS." Master's Thesis, University of Houston, Houston, TX December 2005.
 Zhao, X. M., Shieh, L. S., Sunkel, J. W., and Yuan, Z. Z., "Selftuning Control of Attitude and Momentum Management of the Space Station," AIAA J. of Guidance, Control, and Dynamics, Vol. 15, No. 1, 1992, pp. 1727.
 Hsia, T. C., "System Identification (LeastSquares Methods)," 1977 D. C. Heath and Company, Lexington, MA. ISBN 0669996300