# 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 System**Edit

where , , at any .

**The Data**Edit

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.

**Conclusion:**Edit

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

This implementation requires Yalmip and Sedumi.

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

**Related LMIs**Edit

**External Links**Edit

A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.