A low-capacitance static compensator (LC-StatCom) based on a cascaded H-bridge (CHB) multilevel converter can be an efficient and low-cost solution for reactive power compensation in power grids. However, the reduced capacitance, and, therefore, faster capacitor voltage dynamics, makes it challenging to control it. Classical cascade control schemes, in which the bandwidths associated with the inner current loop and the outer capacitor voltage loop need to be separated enough, are the most reported approaches. Nevertheless, for the LC-StatCom case, the voltage loop can be almost as fast as that one of the current loop. For this reason, this article proposes a control approach that does not separate voltage and current dynamics. Concretely, we propose a multi-input linear time-variant control law, based on incremental passivity theory for CHB multilevel converters. The proposed control guarantees stability despite saturation in the control signal. In addition, it provides capacitor voltage balance without the need to use a specific balancing stage. This article, first, reviews the average model of the converter and, then, analyses the steady-state dynamic behavior to determine the desired and coherent reference signals. Then, it applies an incremental passivity control approach to drive the LC-StatCom state variables to the desired references. After introducing the proposed control law, experimental results on a seven-level 1-kVA CHB LC-StatCom are shown to demonstrate its excellent performance in steady state and transients.