We present an analytical model of a single natural guide star astronomical adaptive optics system, in closed loop mode. The model is used to simulate the long exposure system point spread function, using the spatial frequency (or Fourier) approach, and complement an initial open loop model. Applications range from system design, science case analysis and AO data reduction. All the classical phase errors have been included: deformable mirror fitting error, wavefront sensor spatial aliasing, wavefront sensor noise, and the correlated anisoplanatic and servo-lag error. The model includes the deformable mirror spatial transfer function, and the actuator array geometry can be different from the wavefront sensor lenslet array geometry. We also include the dispersion between the sensing and the correction wavelengths. Illustrative examples are given at the end of the paper.