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

Generalized NormEdit

The   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   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,   as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

The SystemEdit

Consider a continuous-time, linear, time-invariant system   with state space realization   where  ,  ,  , amd   is Hurwitz. The generalized   norm of   is:

   

The DataEdit

The transfer function  , and system matrices  ,  ,   are known and   is Hurwitz.

The LMI: Generalized Norm LMIsEdit

The inequality   holds under the following conditions:

1. There exists   and   where   such that:

 .
 .


2. There exists   and   where   such that:

 .
 .


3. There exists   and   where   such that:

 .
 .

Conclusion:Edit

The generalized   norm of   is the minimum value of   that satisfies the LMIs presented in this page.

ImplementationEdit

This implementation requires Yalmip and Sedumi.

Generalized   Norm - MATLAB code for Generalized   Norm.

Related LMIsEdit

LMI for System H_{2} Norm

External LinksEdit


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LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control