# LMIs in Control/pages/Discrete-Time H2-Optimal Observer

LMIs in Control/pages/Discrete-Time H2-Optimal Observer

In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize the H2 norm, which conceptually is minimizing the average magnitude of error in the observer.

**The System**[edit | edit source]

where and is the state vector, and is the state matrix, and is the input matrix, and is the exogenous input, and is the output matrix, and is the feedthrough matrix, and is the output, and it is assumed that is detectable.

**The Data**[edit | edit source]

The matrices .

**The Optimization Problem**[edit | edit source]

An observer of the form:

is to be designed, where is the observer gain.

Defining the error state , the error dynamics are found to be

,

and the performance output is defined as

.

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

is minimized.

**The LMI:** Discrete-Time H2-Optimal Observer[edit | edit source]

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

where refers to the trace of a matrix.

**Conclusion:**[edit | edit source]

The -optimal observer gain is recovered by and the norm of is . The matrix is the observer gains that can be used to form the optimal observer:

**Implementation**[edit | edit source]

This implementation requires Yalmip and Sedumi.

**Related LMIs**[edit | edit source]

Mixed H2-Hinfinity discrete time observer

Discrete-Time_Hinfinity-Optimal_Observer

## External Links[edit | edit source]

This LMI comes from Ryan Caverly's text on LMI's (Section 5.1.2):

- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.

Other resources:

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.