The year of this writing, 2009, is the 200th anniversary of Gauss’s (1809) Theory of the Motion of Heavenly Bodies Moving About the Sun in Conic Sections. One might suspect that after 200 years there is little to be learned about least squares, and that applications of the method are so routine as to be boring. While literature on new theory may be sparse, applications of the theory can be challenging. This author still spends a considerable fraction of his professional time trying to determine why least-squares and Kalman filter applications do not perform as expected. In fact, it is not unusual to spend more time trying to “fix” implementations than designing and developing them. This may suggest that insufficient time was spent in the design phase, but in most cases the data required to develop a better design were simply not available until the system was built and tested. Usually the problems are due to incorrect modeling assumptions, severe nonlinearities, poor system observability, or combinations of these.

This chapter discusses practical usage of least-squares techniques. Of particular interest are methods that allow analysis of solution validity, model errors, estimate confidence bounds, and selection of states to be estimated. Specific questions that are addressed (at least partially) in this chapter include:

1. Is the solution valid?

2. How unique is the solution?

3. What is the expected total ...

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