LMIs in Control/pages/H2-Optimal Filter

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.

The SystemEdit

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.

Synthesis 1Edit

Solve for  and   that minimize the objective function  , subject to

 

 

 

 

 

 

Synthesis 2Edit

Synthesis 2 is identical to Synthesis 1, with the exception of the final two matrix inequality constraints:

 

 

RemarkEdit

In both cases, if  and  then it is often simplest to choose  in order to satisfy the equality constraint (above).

ConclusionEdit

In both cases, the optimal H2 filter is recovered by the state-space matrices  and  

RemarkEdit

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.

External LinksEdit

A list of references documenting and validating the LMI.

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