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 SystemEdit

Consider discrete time system


where  ,  , at any  .
The system consist of uncertainties of the following form


where  , ,  and  

The DataEdit

The matrices necessary for this LMI are  ,  ,  and  

The LMI:Edit

There exists some X > 0 and Z such that


The Optimization ProblemEdit

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 Controller gain matrix is extracted as  


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



Related LMIsEdit

Schur Complement
Schur Stabilization

External Links Edit