H2-Optimal Filter
The goal of optimal filtering is to design a filter that acts on the output of the generalized plant and optimizes the transfer matrix from to the filtered output.
Consider the continuous-time generalized LTI plant, with minimal state-space representation
where it is assumed that is Hurwitz. A continuous-time dynamic LTI filter with state-space representation
is designed to optimize the transfer function from to , which is given by
where
Optimal Filtering seeks to minimize the given norm of the transfer function There are two methods of synthesizing the H2-optimal filter.
Solve for and that minimize the objective function , subject to
Synthesis 2 is identical to Synthesis 1, with the exception of the final two matrix inequality constraints:
In both cases, if and then it is often simplest to choose in order to satisfy the equality constraint (above).
In both cases, the optimal H2 filter is recovered by the state-space matrices and
The problem of optimal filtering can alternatively be formulated as a special case of synthesizing a dynamic output "feedback" controller for the generalized plant given by
The synthesis methods presented in this page take advantage of the fact that the controller in this case is not a true feedback controller, as it only appears as a feedthrough term in the performance channel.
A list of references documenting and validating the LMI.