The goal of mixed
-optimal state estimation is to design an observer that minimizes the
norm of the closed-loop transfer matrix from
to
, while ensuring that the
norm of the closed-loop transfer matrix from
to
is below a specified bound.
Consider the continuous-time generalized plant
with state-space realization

where it is assumed that
is detectable.
The matrices needed as input are
.
The observer gain L is to be designed to minimize the
norm of the closed-loop transfer matrix
from the exogenous input
to the performance output
while ensuring the
norm of the closed-loop transfer matrix
from the exogenous input
to the performance output
is less than
, where

is minimized.
The form of the observer would be:

is to be designed, where
is the observer gain.
The LMI:
Optimal Observer
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The mixed
-optimal observer gain is synthesized by solving for
, and
that minimize
subject to
,

The mixed
-optimal observer gain is recovered by
, the
norm of
is less than
and the
norm of T(s) is less than
.
Link to the MATLAB code designing
- Optimal Observer
Code
Optimal Observer