LMIs in Control/pages/Detectability LMI

Detectability LMIEdit

Detectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair   is shown below.

The SystemEdit

 

where  ,  , at any  .

The DataEdit

The matrices necessary for this LMI are   and  . There is no restriction on the stability of  .

The LMI: Detectability LMIEdit

  is detectable if and only if there exists   such that

 .

Conclusion:Edit

If we are able to find   such that the above LMI holds it means the matrix pair   is detectable. In words, a system pair   is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input   and output  .

ImplementationEdit

This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Detectability_LMI.m

Related LMIsEdit

Stabilizability LMI

Hurwitz Stability LMI

Controllability Grammian LMI

Observability Grammian LMI

External LinksEdit

A list of references documenting and validating the LMI.


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