Most PV systems are usually controlled by a Maximum Power Point Tracking (MPPT) algorithm to maximize the generated electrical power. However, the maximum power is often unstable and depends on the solar irradiance and temperature. This makes it difficult to control the power grid supply-demand balance due to fluctuations caused by the increase of renewable and variable PV systems. This paper proposes a new control algorithm for a PV-connected inverter called Specified Power Point Tracking (SPPT) control in addition to the conventional Maximum Power Point Tracking (MPPT) control. The PV system is controlled to generate the maximum power or a specified power depending on the electricity transactions comes from the electricity trading system. A high-speed FPGA-based digital adaptive hysteresis current control method, which has fast and stable response and simple structure comparing with the popular Sine-triangle Pulse Width Modulation (SPWM) method, is proposed to implement the MPPT and SPPT control. The adaptive hysteresis current band is calculated adaptively to improve a disadvantage of the classical fixed band hysteresis current control on the varying switching frequency. A reference current used in the adaptive hysteresis current control is calculated such that the output power of the PV-connected inverter is maximized in the MPPT control or is maintained at a given value in the SPPT control. The experimental and simulation results show that the PV-connected inverter under the proposed control algorithm generates the desired power almost exactly and yields stable and fast response despite the varying irradiance.