LMIs in Control/pages/LMI for Schur Stabilization
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 Optimization ProblemEdit
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  page 14, the aforementioned inequality can be converted into:
The LMI: LMI for Schur stabilizationEdit
Title and mathematical description of the LMI formulation.
This problem is a special case of Intensive Disk Region Design (page 230 in ). 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 .
A link to Matlab codes for this problem in the Github repository:
A list of references documenting and validating the LMI.
-  - LMI in Control Systems Analysis, Design and Applications