# LMIs in Control/pages/MixedH2HinfinityOptimalobserver

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.

**The System**[edit]

Consider the continuous-time generalized plant with state-space realization

where it is assumed that is detectable.

**The Data**[edit]

The matrices needed as input are .

**The Optimization Problem**[edit]

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[edit]

The mixed -optimal observer gain is synthesized by solving for , and that minimize subject to ,

**Conclusion:**[edit]

The mixed -optimal observer gain is recovered by , the norm of is less than and the norm of T(s) is less than .

**Implementation**[edit]

Link to the MATLAB code designing - Optimal Observer

## External Links[edit]

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

## Related LMIs[edit]