# LMIs in Control/pages/Quadratic Schur Stabilization

**LMI for Quadratic Schur Stabilization**

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 with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.

**The System**Edit

Consider discrete time system

where , , at any .

The system consist of uncertainties of the following form

where , , and

**The Data**Edit

The matrices necessary for this LMI are , , and

**The LMI:**Edit

There exists some X > 0 and Z such that

**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:

**Conclusion:**Edit

The Controller gain matrix is extracted as

It follows that the trajectories of the closed-loop system (A+BK) are stable for any

**Implementation**Edit

https://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m

**Related LMIs**Edit

**External Links **Edit

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMI in Control Systems Analysis, Design and Applications