The presented research introduces a control strategy for a three-phase grid-tied LCL-filtered quasi-Z-source inverter (qZSI) using a Lyapunov-function-based method and cascaded proportional-resonant (PR) controllers. The suggested control strategy ensures the overall stability of the closed-loop system and eliminates any steady-state inaccuracy in the grid current. The inverter current and capacitor voltage reference values of qZSI are created by the utilization of cascaded coupled proportional-resonant (PR) controllers. By utilizing synchronous reference frame and Lyapunov function- based control, the requirement to perform derivative operations and anticipate inductance and capacitance are avoided, resulting in achieving the goal of zero steady-state error in the grid current. The qZSI can accomplish shoot-through control by utilizing a simple boost control method. Computer simulations demonstrate that the suggested control strategy effectively achieves the desired control objectives, both in terms of steady-state and dynamic performance.