LMIs in Control/pages/Stabilizability LMI

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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 | edit source]

where , , at any .

The Data[edit | edit source]

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

The LMI: Stabilizability LMI[edit | edit source]

is stabilizable if and only if there exists such that

,

where the stabilizing controller is given by

.

Conclusion:[edit | edit source]

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 | edit source]

This implementation requires Yalmip and Sedumi.

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

Related LMIs[edit | edit source]

Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

Observability Grammian LMI

External Links[edit | edit source]

A list of references documenting and validating the LMI.


Return to Main Page:[edit | edit source]