# 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

We consider the following system:

or the matrix , which is the state matrix.

**The Data**Edit

The data required is the matrix A and the max percent overshoot you wish to verify.

**The Optimization Problem**Edit

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 PolesEdit

The LMI problem is to find a matrix P satisfying:

**Conclusion:**Edit

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

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

**Related LMIs**Edit

[1] - D-stabilization

[2] - D-stability Controller

[3] - D-stability Observer

**External Links**Edit

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

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

**Return to Main Page**Edit

[6] -Matrix and LMI Properties and Tools