Inverse filtering and deconvolution

This paper studies the so-called inverse filtering and deconvolution problem from different angles. To start with, both exact and almost deconvolution problems are formulated, and the necessary and sufficient conditions for their solvability are investigated. Exact and almost deconvolution problems seek filters that can estimate the unknown inputs of the given plant or system either exactly or almostly whatever may be the unintended or disturbance inputs such as measurement noise, external disturbances, and model uncertainties that act on the system. As such they require strong solvability conditions. To alleviate this, several optimal and suboptimal deconvolution problems are formulated and studied. These problems seek filters that can estimate the unknown inputs of the given system either exactly, almostly or optimally in the absence of unintended (disturbance) inputs, and on the other hand, in the presence of unintended (disturbance) inputs, they seek that the influence of such disturbances on the estimation error be as small as possible in a certain norm (H2 or H8) sense. Both continuous- and discrete-time systems are considered. For discrete-time systems, the counter parts of all the above problems when an l step delay in estimation is present are introduced and studied. Next, we focus on the exact and almost deconvolution but this time when the uncertainties in plant dynamics can be structurally modeled by a ¿-block as a feedback element to the nominally known plant dynamics. This is done either in the presence or absence of external disturbances.