Technological advances and the ever-growing human quest for improving the resolution of telescope observations are motivating the design of larger and larger ground telescopes: indeed, the larger is the telescope lens diameter, the better is the diffraction limited resolution of the telescope. Unfortunately, the terrestrial atmospheric turbulence, if not properly compensated, negatively affects the telescope observations, limiting its real resolution. Adaptive Optics (AO) systems are used in large ground telescopes in order to compensate the effect of the atmosphere, and hence to make the real telescope resolution be determined by the diffraction properties of the lens. AO systems exploit the measurements of wavefront sensors to estimate the current values of the atmospheric turbulence, and compensate its effect by properly adapting the shape of a set of deformable mirrors. As the size of the telescope lenses is increasing, then the size of the AO system (e.g. the number of deformable mirror actuators and the size of the wavefront sensor) is increasing as well. This causes the increase of the computational burden needed to compute a proper compensation of the effect of the atmosphere. Consequently, as the potential telescope resolution increases, the task of the AO systems becomes more challenging. Motivated by the need of providing AO solutions useful for the next generations of ground telescopes, then a number of efficient algorithms have been recently considered in the literature to solve the problems related to the AO system. This paper considers the combination of a recently proposed very efficient phase reconstruction method, namely the CuRe, with a properly defined Kalman filter in order to obtain a dynamic compensation of the atmospheric turbulence. The performance of the proposed approach is investigated in some simulations.