Transfer function identification from impulse response via a new set of orthogonal hybrid functions (HF)

The present work proposes a new set of hybrid functions (HF) which evolved from the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). Traditional block pulse functions (BPF) still continue to be attractive to many researchers in the arena of control theory. Block pulse functions also gave birth to a few useful variants. Two such variants are SHF and TF. The former is efficient for analyzing sample-and-hold control systems, while triangular functions established their superiority in obtaining piecewise linear solution of various control problems. After developing the basic theory of HF, a few square integrable functions are approximated via this set in a piecewise linear manner. For such approximation, it is shown, the mean integral square error (MISE) is much less than block pulse function based approximation. The operational matrices for integration in HF domain are also derived. Finally, this new set is employed for solving identification problem from impulse response data. The results are compared with the solutions obtained via BPF, SHF, TF, etc. and many illustrations are presented.