LMIs in Control/pages/LMI for Schur Stabilization

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LMIs in Control/pages/LMI for Schur Stabilization

The System[edit | edit source]

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 Data[edit | edit source]

The Optimization Problem[edit | edit source]

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 stabilization[edit | edit source]

Title and mathematical description of the LMI formulation.

Conclusion:[edit | edit source]

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 | edit source]

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


Related LMIs[edit | edit source]

LMI for Hurwitz stability

External Links[edit | edit source]

A list of references documenting and validating the LMI.

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

Return to Main Page:[edit | edit source]