On l1 filtering and smoothing

The problems of optimal l₁ smoothing and suboptimal l₁ filtering are considered. It is shown that the optimal smoother is a finite-dimensional, noncausal system, and that it can be obtained by solving a fixed finite linear programming problem whose order is determined by the McMillan degree of the plant. In the case of l₁ filtering, it is shown that, given any ∈>0, there is a finite linear programming problem whose solution yields a finite-dimensional filter that achieves performance within ∈ of optimal. The order of this associated linear programming problem can be bounded, albeit conservatively, by an explicit function of ∈