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 System 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 Data The matrices and are given. We define the scalar as with the range of . The Optimization Problem 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 stabilization The LMI for Schur stabilization can be written as minimization of the scalar, , in the following constraints: Conclusion: 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]. Implementation A link to Matlab codes for this problem in the Github repository: https://github.com/asalimil/LMI-for-Schur-Stability Related LMIs LMI for Hurwitz stability External Links - LMI in Control Systems Analysis, Design and Applications Return to Main Page LMIs in Control/Tools