# 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 System**Edit

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 Data**Edit

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

**The Optimization Problem**Edit

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.

**Implementation**Edit

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

**Related LMIs**Edit

[1] - Schur Stabilization

**External Links**Edit

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

**Return to Main Page**Edit

[3] -Matrix and LMI Properties and Tools