# LMIs in Control/pages/HinfinityOptimalObserver

-Optimal observers yield robust estimates of some or all internal plant states by processing measurement data. Robust observers are increasingly demanded in industry as they may provide state and parameter estimates for monitoring and diagnosis purposes even in the presence of large disturbances such as noise etc. It is there where Kalman filters may tend to fail. State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system. The goal of -optimal state estimation is to design an observer that minimizes the norm of the closed-loop transfer matrix from w to z.

**The System**[edit]

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

**The Data**[edit]

The matrices needed as input are .

**The Optimization Problem**[edit]

The observer gain is to be designed such that the of the transfer matrix from w to z, given by

is minimized. The form of the observer would be:

**The LMI:** Optimal Observer[edit]

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

**Conclusion:**[edit]

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

**Implementation**[edit]

Link to the MATLAB code designing - Optimal Observer

https://github.com/Ricky-10/coding107/blob/master/HinfinityOptimalobserver

# 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.
- [1]
- [2]