Derivation of an optimal boundary layer width for the smooth variable structure filter

In this paper, an augmented form of the smooth variable structure filter (SVSF) is proposed. The SVSF is a state estimation strategy based on variable structure and sliding mode concepts. It uses a smoothing boundary to remove chattering (excessive switching along an estimated state trajectory). In its current form, the SVSF defines the boundary layer by an upper-bound on the uncertainties present in the estimation process (i.e., modeling errors, magnitude of noise, etc.). This is a conservative approach as one would be limiting the gain by assuming a larger smoothing boundary subspace than what is necessary. A more well-defined boundary layer will yield more accurate estimates. This paper derives a solution for an optimal boundary layer width by minimizing the trace of the a posteriori covariance matrix. The results of the derivation are simulated on a linear mechanical system for the purposes of control, and compared with the Kalman filter.