User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/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:

$$ \begin{align} \dot x(t)&=Ax \end{align}$$

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

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

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

$$\begin{align}\begin{bmatrix} \pi(AP+(AP)^{T}) & ln M_p(AP-(AP)^{T})\\ ln M_p(AP-(AP)^{T})^{T} & \pi(AP+(AP)^{T})\end{bmatrix} < 0\\ \end{align}$$

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

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs
- D-stabilization

- D-stability Controller

- D-stability Observer

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