Periodic inputs reconstruction of partially measured linear periodic systems

In this paper, the problem of input reconstruction for the general case of periodic linear systems driven by periodic inputs ẋ=A(t)x+A0(t)w(t),y=C(t)x is addressed where x(t)∈Rnx(t)∈Rn and A(t)A(t), A0(t)A0(t), and C(t)C(t) are T0T0-periodic matrices and ww is a periodic signal containing an infinite number of harmonics. The contribution of this paper is the design of a real-time observer of the periodic excitation w(t)w(t) using only partial measurement. The employed technique estimates the (infinite) Fourier decomposition of the signal. Although the overall system is infinite dimensional, convergence of the observer is proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces. This observer design relies on a simple asymptotic formula that is useful for tuning finite-dimensional filters. The presented result extends recent works where full-state measurement was assumed. Here, only partial measurement, through the matrix C(t)C(t), is considered.