Estimation Problems and the Modulating Function Method: The Algebra of Modulating Functions
The modulating function method is an algebraic estimation framework that has been used for state and parameter estimation, as well as fault detection, of linear and some nonlinear systems. At the core of the method is the modulating function: a function that evaluates to 0 at the left or right boundaries up to a certain order of derivatives, being classified as left, right, or total modulating functions. Despite being an algebraic framework, there is no paper that analyzes the algebraic properties of the three types of modulating functions and the consequences of these properties. In light of this gap, this paper discusses the algebraic properties of modulating functions and, after formalizing their closedness and group properties, simple algorithms to construct new modulating functions are proposed, discussed, and illustrated. Moreover, the fact that total modulating functions form an algebra and a vector space is exploited to construct orthonormal modulating functions, which are then used to significantly improve the well-known matrix conditioning issue of the modulating function method, illustrated with the parameter estimation of a boat's nonlinear roll dynamics.