# LMIs in Control/pages/Stabilizability LMI

Stabilizability LMI

A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of 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 A.

**The LMI:** Stabilizability LMIEdit

is stabilizable if and only if there exists such that

- ,

where the stabilizing controller is given by

- .

**Conclusion:**Edit

If we are able to find such that the above LMI holds it means the matrix pair is stabilizable. In words, a system pair is stabilizable if for any initial state an appropriate input can be found so that the state asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to *approach* as whereas controllability requires that the state must *reach* the origin in a *finite* time.

**Implementation**Edit

This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Stabilizability_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.