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

LMI for Settling Time Poles

The following LMI allows for the verification that poles of a system will fall within a settling time constraint. This can also be used to place poles for settling 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 settling time ${\displaystyle t_{s}}$  you wish to verify.

## The Optimization Problem

To begin, the constraint of the pole locations is as follows: ${\displaystyle {(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0}$ , where z is a complex pole of A. We define ${\displaystyle 2Re(z){\leq }-\alpha }$ . The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.

## The LMI: LMI for Settling Time Poles

The LMI problem is to find a matrix P > 0 satisfying:

{\displaystyle {\begin{aligned}AP+(AP)^{T}+\alpha P&<0\\\end{aligned}}}

## Conclusion:

If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the settling time specification of ${\displaystyle {(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0}$ , 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