Based on recent results on Sum of Squares (SOS) to polynomial theory, we address exact relaxation to quadratic stability arisen from stability analysis and controller synthesis in the fuzzy control systems represented by TS models. In this paper we show how to characterize the exact solution to the quadratic stability and by approximating the cone of copositive matrices via SOS that is solvable by SOSTOOLS. Certificate of convergence of such copositive relaxation is provided. Examples from current results are provided to illustrate our approach. It is shown that SOS relaxations can characterize exactness of the decision problems.