Reconstruction of the Fourier expansion of inputs of linear time-varying systems

In this paper we propose a general method to estimate periodic unknown input signals of finite-dimensional linear time-varying systems. We present an infinite-dimensional observer that reconstructs the coefficients of the Fourier decomposition of such systems. Although the overall system is infinite dimensional, convergence of the observer can be proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces. Besides its low computational complexity and global convergence, this observer has the advantage of providing a simple asymptotic formula that is useful for tuning finite-dimensional filters. Two illustrative examples are presented.