In this paper, double-line frequency ($2\omega$ ) ripple suppression and SMC with time-invariant (fixed) switching frequency methods are proposed for single-phase grid-connected three-level neutral-point-clamped quasi-Z-source inverters. The $2\omega $ ripple suppression method is based on the 180 phase difference existing between the $2\omega $ ripple components of the capacitor and inductor voltages in the dc-side. Hence, when these components are added in the closed-loop, a phase cancellation occurs so that the inductor current reference can be generated without $2\omega $ ripple component. In this case, the actual inductor current, which is forced to track its reference, has no $2\omega $ ripple component. In addition, the grid current control is achieved via sliding mode control (SMC). Unlike the existing SMC methods, the proposed SMC achieves fixed switching frequency which is made possible by eliminating the discontinuities in the sliding surface function using a boundary layer. The proposed ripple suppression method together with the SMC method offers many advantages such as fast dynamic response, zero grid current error, simple implementation, robustness to parameter variations and fixed switching frequency. The effectiveness of the proposed control method is verified experimentally under steady-state and transient conditions.