LMIs in Control/Matrix and LMI Properties and Tools/D-Stability Settling Time Poles

LMI for Settling Time Poles

The following LMI allows for the verification that poles of a system will fall within a settling time constraint. This can also be used to place poles for settling time when the system matrix includes a controller, such as in the form A+BK.

The SystemEdit

We consider the following system:


or the matrix  , which is the state matrix.

The DataEdit

The data required is the matrix A and the settling time   you wish to verify.

The Optimization ProblemEdit

To begin, the constraint of the pole locations is as follows:  , where z is a complex pole of A. We define  . The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.

The LMI: LMI for Settling Time PolesEdit

The LMI problem is to find a matrix P > 0 satisfying:



If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the settling time specification of  , and the poles of A satisfy the previously defined constraint.


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


Related LMIsEdit

[1] - D-stabilization

[2] - D-stability Controller

[3] - D-stability Observer

External LinksEdit

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

[5] - A course on LMIs in Control by Matthew Peet

Return to Main PageEdit

[6] -Matrix and LMI Properties and Tools