# LMIs in Control/Matrix and LMI Properties and Tools/D-Stability Rise Time Poles

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:

{\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 rise time ${\displaystyle t_{r}}$  you wish to verify.

## The Optimization Problem

To begin, the constraint of the pole locations is as follows: ${\displaystyle z^{*}z-{1.8^{2} \over {t_{r}}^{2}}{\leq }0}$ , where z is a complex pole of A. We define ${\displaystyle r^{2}{\geq }z^{*}z}$ . 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:

{\displaystyle {\begin{aligned}{\begin{bmatrix}-rP&AP\\(AP)^{T}&-rP\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 rise time specification of ${\displaystyle z^{*}z-{1.8^{2} \over {t_{r}}^{2}}{\leq }0}$ , and the poles of A satisfy the previously defined constraint.

## Implementation

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

## Related LMIs

[1] - D-stabilization

[2] - D-stability Controller

[3] - D-stability Observer