The nonlinear observers are proposed for a nonlinear system. To improve the characteristics such as stability, convergence, and $H^{$infty$}$</TEX> filter performance criterion, we utilize an $H^{$infty$}$</TEX> filter Riccati equation or a modified $H^{$infty$}$</TEX> filter Riccati equation with a freedom parameter. Using the Lyapunov function method, the characteristics of the observers are analyzed. Then the in-flight alignment for a strapdown inertial navigation system(SDINS) is designed using the proposed observer. And the additive quaternion error model is especially used to reduce the uncertainty of the SDINS error model. Simulation results show that the observer with the modified $H^{$infty$}$</TEX> filter Riccati equation effectively improves the performance of the in-flight alignment.