In this paper, we will study the state feedback control problem of polynomial nonlinear systems using fractional Lyapunov functions. By adding constraints to bound the variation rate of each state, the general difficulty of calculating derivative of nonquadratic Lyapunov function is effectively overcome. As a result, the state feedback conditions are simplified as a set of Linear Matrix Inequalities (LMIs) with polynomial entries. Computationally tractable solution is obtained by Sum-of-Squares (SOS) decomposition. And it turns out that both of the Lyapunov matrix and the state feedback gain are state dependent fractional matrix functions, where the numerator as well as the denominator can be polynomials with flexible forms and higher nonlinearities involved in. Same idea is extended to a class of output dependent nonlinear systems and the stabilizing output feedback controller is specified as polynomial of output. Synthesis conditions are similarly derived as using constant Lyapunov function except that all entries in LMIs are polynomials of output with derivative of output involved in. By bounding the variation rate of output and gridding on the bounded interval, the LMIs are solvable by SOS decomposition. Finally, two examples are used to materialize the design scheme and clarify the various choices on state boundaries.
