Window Observers for Linear Systems

Abstract

Given a linear system ẋ = Ax + Bu with output y = Cx and a window function w(t), i.e., ∀t, w(t) ∈ (0,1), and assuming that the window function is Lebesgue measurable, we refer to the following observer, x̂̇ = Ax̂ + Bu + w(t)LC(x - x̂) as a window observer. The stability issue is treated in this paper. It is proven that for linear time-invariant systems, the window observer can be stabilized by an appropriate design under a very mild condition on the window functions, albeit for linear time-varying system, some regularity of the window functions is required to achieve observer designs with the asymptotic stability. The corresponding design methods are developed. An example is included to illustrate the possible applications.