LMIs in Control/pages/Detectability LMI

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Detectability LMI[edit]

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 System[edit]

where , , at any .

The Data[edit]

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

The LMI: Detectability LMI[edit]

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 .

Implementation[edit]

This implementation requires Yalmip and Sedumi.

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

Related LMIs[edit]

Stabilizability LMI

Hurwitz Stability LMI

Controllability Grammian LMI

Observability Grammian LMI

External Links[edit]

A list of references documenting and validating the LMI.


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