LMIs in Control/pages/LMI for Schur Stabilization

LMIs in Control/pages/LMI for Schur Stabilization

The SystemEdit

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 DataEdit

 

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 [1] page 14, the aforementioned inequality can be converted into:

 

The LMI: LMI for Schur stabilizationEdit

Title and mathematical description of the LMI formulation.

 

Conclusion:Edit

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].

ImplementationEdit

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

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIsEdit

LMI for Hurwitz stability

External LinksEdit

A list of references documenting and validating the LMI.

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

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