# LMIs in Control/pages/Schur Stabilization

LMI for Schur Stabilization

Similar to the stability of continuous-time systems, one can analyze the stability of discrete-time systems. A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems and a linear time-invariant system with this property is called a Schur stable system.

**The System**Edit

We consider the following system:

where the matrices , , , and are the state matrix, input matrix, state vector, and the input vector, respectively.

Moreover, represents time in the discrete-time system and is the next time step.

The state feedback control law is defined as follows:

where is the controller gain. Thus, the closed-loop system is given by:

**The Data**Edit

The matrices and are given.

We define the scalar as with the range of .

**The Optimization Problem**Edit

The optimization problem is to find a matrix such that:

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

Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:

**The LMI:** LMI for Schur stabilizationEdit

The LMI for Schur stabilization can be written as minimization of the scalar, , in the following constraints:

**Conclusion:**Edit

After solving the LMI problem, we obtain the controller gain and the minimized parameter . 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

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

**Related LMIs**Edit

**External Links**Edit

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