Uniqueness in time limited input reconstruction

Abstract

It is shown in this paper that an upper bound for the number of inputs that can be uniquely reconstructed from m outputs is rmax=mna-2+γna-1-1, where na is an upper bound on the number of time stations that support the inputs and γ=N+RankDc, with N being the system order and Dc the input feedthrough matrix (which may be zero). An alternative phrasing being that a lower bound for the number of sensors needed to reconstruct r inputs of limited duration is mmin=rna-1-γna-2-1. These results hold on the premise that the system is linear and observable, that the inputs are accurately reconstructed by a first-order hold and that the initial condition is known. At the limits, rmax ormmin, the problem is typically poorly conditioned, but the situation improves away from the bounds and the anticipated robustness can be readily checked. It is shown that for some conditions the number of inputs that can be robustly reconstructed can exceed the number of outputs, sometimes by a significant margin.