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 System[edit | edit source]
We consider the following system:
or the matrix , which is the state matrix.
The Data[edit | edit source]
The data required is the matrix A and the settling time you wish to verify.
The Optimization Problem[edit | edit source]
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 Poles[edit | edit source]
The LMI problem is to find a matrix P > 0 satisfying:
Conclusion:[edit | edit source]
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.
Implementation[edit | edit source]
A link to Matlab codes for this problem in the Github repository:
Related LMIs[edit | edit source]
 - D-stabilization
 - D-stability Controller
 - D-stability Observer
External Links[edit | edit source]
 - LMI in Control Systems Analysis, Design and Applications
 - A course on LMIs in Control by Matthew Peet
Return to Main Page[edit | edit source]
 -Matrix and LMI Properties and Tools