# LMIs in Control/pages/LMI for Schur Stabilization

LMIs in Control/pages/LMI for Schur Stabilization

**The System**[edit | edit source]

We consider the following system:

where , are the state vector and the input vector, respectively. Moreover, the state feedback control law is defined as follows:

Thus, the closed-loop system is given by:

**The Data**[edit | edit source]

**The Optimization Problem**[edit | edit source]

Find a matrix such that,

According to the definition of the spectral norms of matrices, this condition becomes equivalent to:

One can use the Lemma 1.2 in [1] page 14, the aforementioned inequality can be converted into:

**The LMI:** LMI for Schur stabilization[edit | edit source]

Title and mathematical description of the LMI formulation.

**Conclusion:**[edit | edit source]

This problem is a special case of Intensive Disk Region Design (page 230 in [1]). This problem may not have a solution even when the system is stabilizable. In other words, once there exists a solution, the solution is robust in the sense that when there are parameter perturbations, the closed-loop system's eigenvalues are not easy to go outside of a circle region within the unit circle [1].

**Implementation**[edit | edit source]

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

https://github.com/asalimil/LMI-for-Schur-Stability

**Related LMIs**[edit | edit source]

## External Links[edit | edit source]

A list of references documenting and validating the LMI.

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