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



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  .


This implementation requires Yalmip and Sedumi.


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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