This paper demonstrates that order-recursive least squares (ORLS) algorithms based on orthogonal transformations and hyperbolic transformations can be systematically constructed in two steps. The first step is to determine the structure of the ORLS algorithm according to the property of the data vector in the LS estimation and the requirements to the output. The second step is to determine the proper implementation of building blocks of the ORLS structure using orthogonal or hyperbolic transformations. The canonical ORLS structure and some possible orthogonal/hyperbolic implementations of their building blocks are presented. It is also shown that some of the orthogonal transformations are only applicable to certain types of ORLS structures and not to others.
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