LMIs in Control/pages/Schur Stabilization

LMI for Schur Stabilization

Similar to the stability of continuous-time systems, one can analyze the stability of discrete-time systems. 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 and a linear time-invariant system with this property is called a Schur stable system.

The SystemEdit

We consider the following system:

 

where the matrices  ,  ,  , and   are the state matrix, input matrix, state vector, and the input vector, respectively.

Moreover,   represents time in the discrete-time system and   is the next time step.

The state feedback control law is defined as follows:

 

where   is the controller gain. Thus, the closed-loop system is given by:

 

The DataEdit

The matrices   and   are given.

We define the scalar as   with the range of  .

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 LMI: LMI for Schur stabilizationEdit

The LMI for Schur stabilization can be written as minimization of the scalar,  , in the following constraints:

 

Conclusion:Edit

After solving the LMI problem, we obtain the controller gain   and the minimized parameter  . 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

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

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