ABSTRACT
This chapter discusses the history of orthogonal functions (OF), their classification and their developments. It describes the useful properties of block-pulse functions (BPF) and shifted Legendre polynomials (SLP). The chapter states the rationale for choosing BPFs and SLPs among all OFs, for the study of state estimation and optimal control problems. The OF approach became quite popular numerically and computationally as it converts calculus into algebra in the sense of least squares, that is dynamical equations of a system can be converted into a set of algebraic equations whose solution simply leads to the solution of dynamical equations. Depending upon their nature, OFs may be broadly classified in three categories: Piecewise Constant Orthogonal Functions, Orthogonal Polynomials, and Sine-Cosine Functions. The linear-quadratic-Gaussian control problem concerns linear systems disturbed by additive white Gaussian noise, incomplete state information and quadratic costs.
