LMIs in Control/pages/H2 index

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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 System[edit| edit source]

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 Data[edit| edit source]

The system matrices are known.

The Optimization Problem[edit| edit source]

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| edit source]

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| edit source]

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

Implementation[edit| edit source]

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 LMIs[edit| edit source]

Bounded Real Lemma
Deduced LMIs for H-infinity index

External Links[edit| edit source]

A list of references documenting and validating the LMI.


Return to Main Page:[edit| edit source]