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:

 

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  

ImplementationEdit

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

Related LMIsEdit

Schur Complement
Schur Stabilization

External Links Edit