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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:

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

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 settling time $t_{s}$ you wish to verify.

The Optimization Problem

To begin, the constraint of the pole locations is as follows: ${(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0$, where z is a complex pole of A. We define $2Re(z){\leq }-\alpha$. 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:

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

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 ${(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0$, and the poles of A satisfy the previously defined constraint.

Implementation

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs

[1] - D-stabilization

[2] - D-stability Controller

[3] - D-stability Observer

External Links

[4] - LMI in Control Systems Analysis, Design and Applications

[5] - A course on LMIs in Control by Matthew Peet

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[6] -Matrix and LMI Properties and Tools