LMIs in Control/Matrix and LMI Properties and Tools/Schur Stabilizability

LMI for Schur Stabilizability

Schur Stabilization is one method of ensuring that a controller can be made to stabilize a system. The following LMI is one that determines whether or not a system is indeed Schur Stabilizable, or having the property of being able to be Schur Stabilized.

The SystemEdit

We consider the following system:

 

or the matrix pair (A,B). In both cases, the matrices  ,  ,  , and   are the state matrix, input matrix, state vector, and the input vector, respectively.

The DataEdit

The data required is both the matrices A and B as seen in the form above.

The Optimization ProblemEdit

The goal of the optimization is to find a valid symmetric P such that the following LMI is satisfied.

The LMI: LMI for Schur stabilizabilityEdit

The LMI problem is to find a symmetric matrix P and a matrix W satisfying:

 

Another LMI with the same result of finding Schur Stabilizability is to find a symmetric matrix P such that:

 

Conclusion:Edit

If the one of the above LMIs is found to be feasible, then the system is Schur Stabilizable and the Schur Stabilization LMI will always give a feasible result as well, in addition to a controller K that will Schur Stabilize the system.

ImplementationEdit

A link to Matlab codes for this problem in the Github repository:

https://github.com/maxwellpeterson99/MAE509Code

Related LMIsEdit

[1] - Schur Stabilization

External LinksEdit

[2] - LMI in Control Systems Analysis, Design and Applications

Return to Main PageEdit

[3] -Matrix and LMI Properties and Tools