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LMI for Rise Time Poles

The following LMI allows for the verification that poles of a system will fall within a rise time constraint. This can also be used to place poles for rise time 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 rise time $$t_r$$ you wish to verify.

The Optimization Problem
To begin, the constraint of the pole locations is as follows: $$z^{*} z-{1.8^2 \over {t_r}^2}{\leq}0$$, where z is a complex pole of A. We define $$r^2{\geq}z^{*} z$$. The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.

The LMI: LMI for Rise Time Poles
The LMI problem is to find a matrix P> 0 satisfying:

$$\begin{align} \begin{bmatrix} -rP & AP\\ (AP)^{T} & -rP\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the rise time specification of $$z^{*} z-{1.8^2 \over {t_r}^2} {\leq} 0$$, 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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