LMIs in Control/pages/LMI for the Controllability Grammian

LMI to Find the Controllability Grammian

Being able to adjust a system in a desired manor using feedback and sensors is a very important part of control engineering. However, not all systems are able to be adjusted. This ability to be adjusted refers to the idea of a "controllable" system and motivates the necessity of determining the "controllability" of the system. Controllability refers to the ability to accurately and precisely manipulate the state of a system using inputs. Essentially if a system is controllable then it implies that there is a control law that will transfer a given initial state and transfer it to a desired final state . There are multiple ways to determine if a system is controllable, one of which is to compute the rank "controllability grammian". If the grammian is full rank, the system is controllable and a state transferring control law exists.

The SystemEdit


where  ,  , at any  .

The DataEdit

The matrices necessary for this LMI are   and  .   must be stable for the problem to be feasible.

The LMI: LMI to Determine the Controllability GrammianEdit

  is controllable if and only if   is the unique solution to


where   is the Controllability Grammian.


The LMI above finds the controllability grammian  of the system  . If the problem is feasible and a unique   can be found, then we also will be able to say the system is controllable. The controllability grammian of the system   can also be computed as:  , with control law   that will transfer the given initial state   to a desired final state  .


This implementation requires Yalmip and Sedumi.


Related LMIsEdit

Stabilizability LMI

Hurwitz Stability LMI

Detectability LMI

Observability Grammian LMI

External LinksEdit

A list of references documenting and validating the LMI.

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