# LMIs in Control/pages/H-2 filtering

LMIs in Control/pages/H-2 filtering

This is a WIP based on the template.

## Contents

**The System**[edit]

For the application of this LMI, we will look at linear systems that can be represented in state space as

where represent the state vector, the measured output vector, and the output vector of interest, respectively, is the disturbance vector, and and are the system matrices of appropriate dimension.

**The Data**[edit]

The data are (the disturbance vector), and and (the system matrices). Furthermore, the matrix is assumed to be stable

**The Optimization Problem**[edit]

We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:

where is the state vector, is the estimation vector, and are the coefficient matrices of appropriate dimensions.

Note that the combined complete system can be represented as

where is the estimation error,

is the state vector of the system, and are the coefficient matrices, defined as:

In other words, for the system defined above we need to find such that

where is a positive constant, and

**The LMI:** H-2 Filtering[edit]

For this LMI, the solution exists if one of the following sets of LMIs hold:

Matrices exist that obey the following LMIs:

or

Matrices exist that obey the following LMIs:

**Conclusion:**[edit]

To find the corresponding filter, use the optimized matrices from the first solution to find:

Or the second solution to find:

These matrices can then be used to produce to construct the filter described above, that will best eliminate the disturbances of the system.

**Implementation**[edit]

A link to CodeOcean or other online implementation of the LMI

**Related LMIs**[edit]

Links to other closely-related LMIs

## External Links[edit]

A list of references documenting and validating the LMI.

- 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.