This paper deals with H_{\infty} control problem for nonlinear conformable fractional order systems. We first derive new sufficient condition for exponential stability of nonlinear conformable fractional order systems based on Lyapunov-like function method for conformable fractional order systems and linear matrix inequalities (LMIs) approach. Then, by introducing a new concepts of H_{\infty} control problem for nonlinear conformable fractional order systems, we study H_{\infty} performance analysis and H_{\infty} state feedback controller design problems for the considered systems. In terms of LMIs, a sufficient condition is proposed to ensure the nonlinear conformable fractional order systems are not only exponentially stable, but also satisfy H_{\infty} performance. An explicit expression for state feedback controllers is also designed to make the closed-loop system is exponentially stable with H_{\infty} performance. Finally, numerical examples are given to illustrate the validity and effectiveness of the proposed results