LMIs in Control/Matrix and LMI Properties and Tools/D-Stability Rise Time Poles
LMI for Rise Time Poles
The following LMI allows for the verification that poles of a system will fall within a rise time constraint. This can also be used to place poles for rise time when the system matrix includes a controller, such as in the form A+BK.
We consider the following system:
or the matrix , which is the state matrix.
The data required is the matrix A and the rise 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 Rise 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 rise 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:
 - D-stabilization
 - D-stability Controller
 - D-stability Observer
 - LMI in Control Systems Analysis, Design and Applications
 - A course on LMIs in Control by Matthew Peet
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