Optimal control of tokamak plasmas requires efficient and accurate prediction of heat and matter transport. Growing from kinetic resonant instabilities, turbulence saturates by involving many scales, from the small vortex up to the back-reaction on the density and temperature profiles. Self-organisation processes are of particular interest, encompassing spontaneous zonal flow genera- tion and transport by avalanche. First principle numerical simulation codes like GYSELA allow studying the gyro-kinetic evolution of the particle distribution function. The large model size and cost prompts the need for reduction. Removing velocity dimensions is the so-called collisionless closure problem for fluid equations. Earlier approaches are extended and generalised by calling to the dynamical systems and optimal control litterature. In particular, we apply the balanced truncation and rational interpolation to the one-dimensional linear VlasovPoisson problem. The interpolation method features a cheap and versatile formulation, opening the door to wider use for more complex phenomena. Quasi-linear theory is the reference model for turbulent effects. The GYSELA three-dimensional output is analysed to estimate the robustness of linear properties in turbulent filaments. Key quasi-linear quantities carry over to the non-linear regime. Effective velocities and shape of turbulent structures are computed, and match expected group velocities and linear eigenmode. Nevertheless, the turbulent potential spectrum must be specified externally to quasi- linear models. This results in radially travelling unstable linear solutions that share many properties of turbulent avalanches seen in numerical simulations.