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

## Generalized ${\displaystyle H_{2}}$ Norm

The ${\displaystyle H_{2}}$  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 ${\displaystyle D}$  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, ${\displaystyle G}$  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 ${\displaystyle G:L_{2e}\to L_{2e}}$  with state space realization ${\displaystyle (A,B,C,0)}$  where ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ , ${\displaystyle B\in \mathbb {R} ^{n\times m}}$ , ${\displaystyle C\in \mathbb {R} ^{p\times n}}$ , amd ${\displaystyle A}$  is Hurwitz. The generalized ${\displaystyle H_{2}}$  norm of ${\displaystyle G}$  is:

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

## The Data

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

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

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

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

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

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

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

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

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

## Conclusion:

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

## Implementation

This implementation requires Yalmip and Sedumi.

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