Issues with SOS and SDP decomposition

[David Webster]

Hello,

I am a fairly new YALMIP user and am running into some trouble.

I want to obtain the v(x)'*Q*v(x) == p(x) for the following system:

x1' = -x1+2*x2^2

x2' = x1-x2-x1*x2-x2^3

I know that if the system were one variable, e.g. x' = x+x^2, this decomposition is trivial and explained in the basic YALMIP tutorials on SOS, but because of the multiple variables YALMIP is not taking the input.

Code below:

x1 = sdpvar(1,1); x2 = sdpvar(1,1);

p = [-x1+2*x2^2; x1-x2-x1*x2-x2^3;];

F = sos(p);

solvesos(F)

My only goal is to obtain the polynomial Lyapunov function for the above system.

Any suggestions on syntax changes?

Thanks,

David

[Johan Löfberg]

Well, p(1) and p(2) are trivially not sos. If you want to do Lyapunov analysis, you have to begin by defining a Lyapunov function, and I see no such thing in your code

I would advice you to first read up on how you actually define a stability problem using SOS. It is explained here (and many other papers)

http://www.control.isy.liu.se/~johanl/2009_TAC_SOS.pdf

Then you define a parameterized SOS problem (the parameters being the coefficients of the Lyapunov function)

http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.SumOfSquares

http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Examples.MoreSOS

[Johan Löfberg]

Here is an old unfinished example
http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Examples.NonlinearStability

Make sure you read the stuff above first though so you know what you are doing