# 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:

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

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{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 $z^{*}z-{1.8^{2} \over {t_{r}}^{2}}{\leq }0$ , 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

 - D-stabilization

 - D-stability Controller

 - D-stability Observer