Time and frequency domain design of functional filters

This paper proposes both time and frequency domain design of functional filters for linear time-invariant multivariable systems where all measurements are affected by disturbances. The order of this filter is equal to the dimension of the vector to be estimated. The time procedure design is based on the unbiasedness of the filter using a Sylvester equation; then the problem is expressed in a singular system one and is solved via Linear Matrix Inequalities (LMI) to find the optimal gain implemented in the observer design. The frequency procedure design is derived from time domain results by defining some useful Matrix Fractions Descriptions (MFDs) and mainly, establishing the useful and equivalent form of the connecting relationship that parameterizes the dynamic behavior between time and frequency domain, given by Hippe in the reduced-order case. A numerical example is given to illustrate our approach.