Hyperbolic Functions and the Modulating Function Method

The modulating function method is an integral transform method that has been used for state and parameter estimation of lumped systems, as well as estimation of distributed and fractional-order systems. The kernel of the modulation operator for input-output systems is called the modulating function, having the resulting filter characteristics dictated by this choice. Since this selection is crucial in the filter design process, different functions have been proposed over the years, until a different approach was introduced: obtaining the modulating function as a solution to an auxiliary problem. This approach is, in fact, common within mathematics, and well-known functions, such as hyperbolic sine and cosine, are obtained as the solution to a particular auxiliary system. Intriguingly, hyperbolic functions generate 4 new types of modulating functions in a simple manner. Moreover, 12 types of mixed hyperbolic modulating functions are obtained by using different hyperbolic functions to satisfy the boundary conditions. Lastly, an application example shows that hyperbolic modulating functions can display significantly lower bias than polynomial modulating functions.