I think you really have to have a good look at your model. It is probably an extremely bad model numerically (bad scaling/almost uncontrollable etc). With SeDuMi, I get a pole at +0.28, with SDPT3 at 1e-6. Such a difference in the optimal solution suggest the old law "crap in crap out". Use the results with extreme caution.

By aiming for a strictly feasible solution
solvesdp(F,obj,sdpsettings('solver','sedumi','shift',1e-4));

I can manage to get a strictly feasible solution, but Acl is still not Hurwitz. That suggests that either your recovery code is wrong, or you are doing something very ill-conditioned during the computation of the controller recovery.

I will have another look at the nonlinear model and check whether there are some choices I have made wrong when I linearized it. And also the recovery part.
Thanks very much for helpful info

Tore