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

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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[edit | edit source]

We consider the following system:

or the matrix , which is the state matrix.

The Data[edit | edit source]

The data required is the matrix A and the settling time you wish to verify.

The Optimization Problem[edit | edit source]

To begin, the constraint of the pole locations is as follows: , where z is a complex pole of A. We define . 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[edit | edit source]

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

Conclusion:[edit | edit source]

If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the settling time specification of , and the poles of A satisfy the previously defined constraint.

Implementation[edit | edit source]

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

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs[edit | edit source]

[1] - D-stabilization

[2] - D-stability Controller

[3] - D-stability Observer

External Links[edit | edit source]

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

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

Return to Main Page[edit | edit source]

[6] -Matrix and LMI Properties and Tools