Parameters uncertainty in the model of power electronic converters e.g., cascaded grid connected multilevel inverters with LCL coupling can lead to the stability and robustness issues of closed-loop systems in a real system. Hence, closed-loop stabilization of the power electronic converters in a wide range of changes is a challenging task. Also, it can result in the spoiling of THD value in a grid-connected inverter beyond the acceptable and standard ranges. To cope with these issues, in this paper, an adaptive nonlinear controller is developed based on the Lyapunov stability criteria for a multilevel shunt active filter with three cascaded H-bridges. Despite the complex model of the mentioned system, it is shown that the designed adaptive controller can improve the performance of the closed-loop system by estimating the uncertain values of inductors in the LCL coupling network. The stability and robustness of the controller are proved using the Lyapunov stability criteria in the whole operational range of the system. The simulation results show that the proposed controller is stable and robust in a wide range of load current changes. Moreover, to stabilize the DC link voltages of the grid-connected inverter, a novel approach based on the mean minimum value of DC-link voltage errors is used which has led to the robust and fast stabilization of these voltages. It should be noted that all simulations have been performed by using the EMTDC/PSCAD toolbox. Finally, a small-scale prototype has been implemented for experimental verification of the system.