Help on Sos

Q. In my course on linear control, we learned that asymptotic convergence is global, but in nonlinear control we learned about the " domain of attraction. " The instructor mentioned that it can be hard to figure out what the region of attraction (ROA) is, but that there is something called " SOS " that can be used. I know that SOS stands for sum of squares, but other than that I don't know anything about it. Is there anyone at IEEE Control Systems Magazine who can explain SOS? Andy: I'm happy to try to help, with the assistance of my colleagues Ufuk Topcu, Pete Seiler, and Gary Balas. It's important to note that we're users of SOS methods, not experts, but I think we can answer your question or at least point you in the right direction. Your question leads with " what is SOS? " so let's begin there. Once that's out of the way, just a few steps lead to optimizations whose feasible solutions yield certified , quantitative inner estimates of the region of attraction. In its basic form, SOS applies to polynomials in several real variables. A polynomial is a finite linear combination of monomials. For example, the polynomial q(x 1 , x 2) J x 1 2 1 2x 1 4 1 2x 1 3 x 2 2 x 1 2 x 2 2 1 5x 2 4 (1) is a linear combination of five mo-nomials in two variables. Quadratic polynomials, such as x T Qx, where Q is a symmetric matrix, appear frequently in control theory. This form can be generalized to polynomials of higher degree, namely, if p (x) is a polynomial of degree less than or equal to 2d, then a Gram matrix representation is p (x) 5 z T (x) Qz (x) , where z (x) is a vector of monomials of degree less than or equal to d, and Q is a symmetric matrix. For example, the polynomial q(x 1 , x 2) can be represented as z T (x) Qz (x) , where z (x) J ≥ x 1 x 1 2 x 1 x 2 x 2 2 ¥ , Q J ≥ 1 0 0 0 0 2 1 20.5 0 1 0 0 0 20.5 0 5 ¥. The Gram matrix Q is not unique due to the dependencies among the mo-nomials in z. In this …