Convert Riccati matrix inequalities into LMI

[stevens]

Hey,

i am quite new in the topic LMIs... That's why I would like to ask, if someone can help me, solving these riccati matrix inequalities....

Can I form them into LMIs? 

Thank you very much!

Best regards,

Steven

[Johan Löfberg]

yes, schur complements

[stevens]

Thank you very much  for that hint! 

With the schur complement I got these three LMIs. Is that correct?

so P1 and P2 should be symmetric?

Is it possible to define an equality (==) for the third equation in yalmip? 

Best regards,

Steven

[Johan Löfberg]

You still have quadratic terms in P so you will have to apply a Schur complements once again

However, the equality is nonlinear and nonconvex, and cannot be linearized, so you have a pretty bad problem. Equalities are just == in YALMIP, but this is a nonlinear equality and hence an LMI solver will not be able to deal with this

[stevens]

Thank you very much Johan Löfberg!

[stevens]

Sorry, but I have a further question to a similar topic. I try to solve these three LMIs (22), (26a) and (26b) with the unkown matrices: X, Y, T, Wb, Wc, W_.

Unfortunately, I don't know how to handle the inequality W_ >= T*Y*T' (highlighted), because it is nonlinear.

Is there a possibility to implement this inequality?

Thank you for your support!

Best regards,

Steven

[Johan Löfberg]

W>=TYT is not possible to represent if you have all those variables as decision variables. Hence, I suspect you misunderstand some notation , and 22 is  unclear what they mean.  As the expression is non-symmetric, you would have to introduce a new variable Z, and add the constraints Z == [E 0;0 E]*[T I;I X], Z>=0