Check the (in)feasibility of a LMI problem

[Luo ArChen]

Hi, Johan,

  

  I am trying to check the (in)feasibility of a LMI problem, the problem is described as follow [1]:

and the corresponding problem is

The authors of [1] said that this system is asymptotically stable when $h$ belongs to [0.200,2.04].

I am trying to check this results by the code as the attachment. However, no matter how I change $h$, e.g., $h=1000$ or $h=0.1$, the problem is always feasible, i.e., 'Successfully solved (MOSEK)'. Also, I can find that for most cases, the eigenvalues of matrix $R$ are not all positive, e.g. ( 1.0e-08*-0.2098,  1.0e-08* -0.0691) when $h=1000$. I am wondering if this is a numerical problem, or I just misunderstand something.

Since I am new to LMI, please forgive me if this is a foolish question.

Thank you!

[1] A. Seuret, and F. Gouaisbaut. "Wirtinger-based integral inequality: Application to time-delay systems." Automatica 49.9(2013):2860-2866.

[Johan Löfberg]

There is no such thing as strict inequalities in practice, and YALMIP is telling you that loud and clear through warnings.

0 is a feasible solution

>> assign([S(:);R(:);P(:)],0);

>> check(constraints)

 

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   ID|          Constraint|   Primal residual|   Dual residual|

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   #1|   Matrix inequality|                 0|      1.5351e-14|

|   #2|   Matrix inequality|                 0|        0.020144|

|   #3|   Matrix inequality|                 0|          1.6622|

|   #4|   Matrix inequality|                 0|      5.2068e-10|

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

If x is a feasible solution, then so is t*x for any t>=0. Hence, you should dehomogenize the problem and when doing so you can add a strict condition, by adding P>=eye(2'N). With tht, it is infeasible

[Johan Löfberg]

and in the paper, they seem to have identity matrices (I) whereas you create matrices full of ones.

[Luo ArChen]

Thank you for you quick reply!

According to your suggestions, I have dehomogenized this problem you mentioned and it works! For the identity matrices, I have corrected it by myself.

Similar results to the paper are now gotten by simulations. Many thanks to your kind and valuable help!!

[Johan Löfberg]

You should look at your solutions. When you have h=4 and selfshift=1, P is already on the order of 10^5, i.e. the problem appears to be ill-posed, i.e., the optimal solution tends to infinity, or at least it is numerically huge

with h=5, mosek has numerical issues and returns UNKNOWN status. The solution is feasible as seen by check, but some of the solutions are insane, such as xi=77732352068.1663

You have a badly posed problem.