LMIs in Control/pages/LMI for the Controllability Grammian

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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 System[edit | edit source]

where , , at any .

The Data[edit | edit source]

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

The LMI: LMI to Determine the Controllability Grammian[edit | edit source]

is controllable if and only if is the unique solution to

,

where is the Controllability Grammian.

Conclusion:[edit | edit source]

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 .

Implementation[edit | edit source]

This implementation requires Yalmip and Sedumi.

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

Related LMIs[edit | edit source]

Stabilizability LMI

Hurwitz Stability LMI

Detectability LMI

Observability Grammian LMI

External Links[edit | edit source]

A list of references documenting and validating the LMI.


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