# LMIs in Control/pages/LMI for System H2 Norm

-norm of System

The -norm is conceptually identical to the Frobenius (aka Euclidean) norm on a matrix. It can be used to determine whether the system **representation** can be reduced to its simplest form, thereby allowing its use in performing effective block-diagram algebra.

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

Suppose we define the **state-space** system if:

where , , , and for any . Then the -norm of the system can be determined as described below.

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

In order to determine the -norm of the system, we need the matrices , , and .

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

Suppose we wanted to to infer properties of the system behaviour (which is represented in the form **(A,B,C,D)**). Then it becomes necessary to ensure that the overall system forms an algebra, as the standard use of block-diagram algebra would otherwise be invalid. The only way this is possible is by calculating and/or -norms - both of which are signal norms that (in a certain sense) measure the size of the transfer function.

**The LMI:** The Norm[edit | edit source]

Assuming that , this means that the following are equivalent:

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

The LMI can be used to minimize the -norm of the system. It is worth noting that a finite -norm does not guarantee finite -norm, and that in order for the block diagram algebra to be valid, -norm must be finite.

**Implementation**[edit | edit source]

- Example Code - A GitHub link that contains code (titled "H2Norm.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

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

- -Filtering - LMI for -Filtering
- Discrete-Time Norm - LMI for -norm in the Discrete-Time case.

## External Links[edit | edit source]

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.