# LMIs in Control/Matrix and LMI Properties and Tools/Generalized H2 Norm

## Generalized $H_{2}$ Norm

The $H_{2}$  norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented $D$  matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, $G$  as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

## The System

Consider a continuous-time, linear, time-invariant system $G:L_{2e}\to L_{2e}$  with state space realization $(A,B,C,0)$  where $A\in \mathbb {R} ^{n\times n}$ , $B\in \mathbb {R} ^{n\times m}$ , $C\in \mathbb {R} ^{p\times n}$ , amd $A$  is Hurwitz. The generalized $H_{2}$  norm of $G$  is:

$\left\|G\right\|_{2,{\infty }}=\sup _{u\in L_{2},{u}\neq {\text{zero}}}$  ${\frac {\left\|Gu\right\|_{\infty }}{\ \left\|u\right\|_{2}}}$

## The Data

The transfer function $G$ , and system matrices $A$ , $B$ , $C$  are known and $A$  is Hurwitz.

## The LMI: Generalized $H_{2}$ Norm LMIs

The inequality $\left\|G\right\|_{2,{\infty }}<\mu$  holds under the following conditions:

1. There exists $P\in \mathbb {S} ^{n}$  and $\mu \in \mathbb {R} _{>0}$  where $P>0$  such that:

${\begin{bmatrix}A^{T}P+PA&PB\\*&-\mu 1\end{bmatrix}}<0$ .
${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$ .

2. There exists $Q\in \mathbb {S} ^{n}$  and $\mu \in \mathbb {R} _{>0}$  where $Q>0$  such that:

${\begin{bmatrix}QA^{T}+AQ&B\\*&-\mu 1\end{bmatrix}}<0$ .
${\begin{bmatrix}Q&QC^{T}\\*&\mu 1\end{bmatrix}}>0$ .

3. There exists $P\in \mathbb {S} ^{n},V\in \mathbb {R} ^{n\times n}$  and $\mu \in \mathbb {R} _{>0}$  where $P>0$  such that:

${\begin{bmatrix}-(V+V^{T})&V^{T}A+P&V^{T}B&V^{T}\\*&-P&0&0\\*&**-\mu 1&0\\*&*&*&-P\end{bmatrix}}<0$ .
${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$ .

## Conclusion:

The generalized $H_{2}$  norm of $G$  is the minimum value of $\mu \in \mathbb {R} _{>0}$  that satisfies the LMIs presented in this page.

## Implementation

This implementation requires Yalmip and Sedumi.

Generalized $H_{2}$  Norm - MATLAB code for Generalized $H_{2}$  Norm.