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 SystemEdit

 

where  ,  , at any  .

The DataEdit

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.

ImplementationEdit

This implementation requires Yalmip and Sedumi.

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

Related LMIsEdit

Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI

External LinksEdit

A list of references documenting and validating the LMI.


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