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 x ̇ = A ( t ) x + A 0 ( t ) w ( t ) , y = C ( t ) x is addressed where x ( t ) ∈ R n and A ( t ) , A 0 ( t ) , and C ( t ) are T 0 -periodic matrices and w 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 ) 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 ) , is considered.