Solving LMI problem with LME (linear matrix equality)

Hi. Can't seem to find this information anywhere.

I need to solve an LMI problem for a state space system in descriptor form. The issue is I can't seem to find how to handle an equality constraint like this:

E'*P = P*E > 0

in the LMI toolbox.

My intuiton is telling me I need to add slack variables to turn this into another LMI but I can't seem to find this in any literature.

Would something like this work?

min c*x + t

E'*P-P*E<t*I, t>0

Answer 1

[Amine Dehak]

hello here is your answer

[Guilherme Barbosa]

This is very usefull! But I have a question, do you have the source of this image?

[Francisco-Ronay López-Estrada]

Somebody found the source of such Lemma?

[Amine Dehak]

Yes sorry. the source is the book "Control of Singular Systems with Random Abrupt Changes"

https://books.google.fr/books?id=66Te6BnZnMIC&pg=PA67&lpg=PA67&dq=%22to+overcome+this+we+can+use+the+following+LMI+condition+that+may+approximate%22&source=bl&ots=w6r5liIU9F&sig=ACfU3U1ZGaHOaH8BF6UO0rRzA9Qxe5iC6Q&hl=en&sa=X&ved=2ahUKEwiK4OWj0rHsAhUP8BQKHSUsDg0Q6AEwAHoECAEQAg#v=onepage&q=%22to%20overcome%20this%20we%20can%20use%20the%20following%20LMI%20condition%20that%20may%20approximate%22&f=false