# LMIs in Control/pages/Detectability LMI

**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]

**External Links**[edit]

A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.