An integrated Z-source inverter for the single-phase single-stage grid-connected photovoltaic system is proposed in this paper. The inverter integrates three functional blocks including maximum-power-point-tracking, step-up/down DC-side voltage and output grid-connected current. According to the non-minimum-phase characteristic presented in DC-side and the functional demands of the system, two constant-frequency sliding-mode controllers with integral compensation are proposed to guarantee the system robustness. By using two controllers, the effects caused by the non-minimum-phase characteristic are mitigated. Under the circumstance of that the input voltage or the grid-connected current changes suddenly, the notches/protrusions following the over-shoot/ under-shoot of the DC-bus voltage are eliminated. The quality of grid-connected current is ensured. Also, a small-signal modelling method is employed to analyze the close-loop system. A 300W prototype is built in the laboratory. A solar-array simulator (SAS) is used to verify the systematic responses in the experiment. The correctness and validity of the inverter and proposed control algorithm are proved by simulation and experimental results.