LMIs in Control/Matrix and LMI Properties and Tools/D-Stability Max Percent Overshoot Poles
LMI for Max Percent Overshoot Poles
The following LMI allows for the verification that poles of a system will within a maximum percent overshoot constraint. This can also be used to place poles for max percent overshoot 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 max percent overshoot 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. The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.
The LMI: LMI for Max Percent Overshoot Poles[edit | edit source]
The LMI problem is to find a matrix P 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 max percent overshoot 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:
https://github.com/maxwellpeterson99/MAE509Code
Related LMIs[edit | edit source]
[1] - D-stabilization
[2] - D-stability Controller
[3] - D-stability Observer
External Links[edit | edit source]
[4] - LMI in Control Systems Analysis, Design and Applications
[5] - A course on LMIs in Control by Matthew Peet
Return to Main Page[edit | edit source]
[6] -Matrix and LMI Properties and Tools