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

We consider the following system:

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax\end{aligned}}}

or the matrix ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ , which is the state matrix.

The Data

The data required is the matrix A and the max percent overshoot ${\displaystyle M_{p}}$  you wish to verify.

The Optimization Problem

To begin, the constraint of the pole locations is as follows: ${\displaystyle z-z^{*}+{{\pi } \over ln({M_{p}})}|z+z^{*}|{\leq }0}$ , 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

The LMI problem is to find a matrix P satisfying:

{\displaystyle {\begin{aligned}{\begin{bmatrix}\pi (AP+(AP)^{T})&lnM_{p}(AP-(AP)^{T})\\lnM_{p}(AP-(AP)^{T})^{T}&\pi (AP+(AP)^{T})\end{bmatrix}}<0\\\end{aligned}}}

Conclusion:

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 ${\displaystyle z-z^{*}+{{\pi } \over ln({M_{p}})}|z+z^{*}|{\leq }0}$ , and the poles of A satisfy the previously defined constraint.

Implementation

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

Related LMIs

[1] - D-stabilization

[2] - D-stability Controller

[3] - D-stability Observer