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



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.


This implementation requires Yalmip and Sedumi.


Related LMIsEdit

Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI

External LinksEdit

A list of references documenting and validating the LMI.

Return to Main Page:Edit