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.
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 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:
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 ). 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 .
A link to Matlab codes for this problem in the Github repository:
-  - LMI in Control Systems Analysis, Design and Applications