LMIs in Control/pages/H2 index

H2 Index Deduced LMI

Although there are ways to evaluate an upper bound on the H2, the verification of the bound on the H2-gain of the system can be done via the deduced condition.

The SystemEdit

We consider the generalized Continuous-Time LTI system with the state space realization of  

 

where  ,   and   are the system state, output, and the input vectors respectively.
The transfer function of such a system can be evaluated as:

 

The DataEdit

The system matrices   are known.

The Optimization ProblemEdit

For an arbitrary  (a given scalar), the transfer function satisfies

 

The H2-norm condition on Transfer function holds only when the matrix A is stable. And this can be conveniently converted to an LMI problem


if and only if 1. There exists a symmetric matrix   such that:
 ,  

2. There exists a symmetric matrix   such that:
 ,  

The LMI - Deduced Conditions for H2-norm Edit

These deduced condition can be derived from the above equations. According to this

For an arbitrary  (a given scalar), the transfer function satisfies

 

if and only if there exists symmetric matrices   and  ; and a matrix   such that
 
  
  

The above LMI can be combined with the bisection method to find minimum   to find the minimum upper bound on the H2 gain of  .

Conclusion:Edit

If there is a feasible solution to the aforementioned LMI, then the   upper bounds the norm of the system G(s).

ImplementationEdit

To solve the feasibility LMI, YALMIP toolbox is required for setting up the problem, and SeDuMi or MOSEK is required to solve the problem. The following link showcases an example of the problem:

https://github.com/yashgvd/ygovada

Related LMIsEdit

Bounded Real Lemma
Deduced LMIs for H-infinity index

External LinksEdit

A list of references documenting and validating the LMI.


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