This paper presents a new direct tilt control (DTC) design to improve the lateral stability and the driving comfort of narrow tilting vehicles. To this end, a conceptual model is constructed from the vehicle dynamics, a simplified model of the driving environment, and the vehicle perceived acceleration. This latter is considered as the main performance variable of the related H2 control problem. The conceptual model is then represented in a polytopic linear parameter varying (LPV) form for control purposes. To avoid the use of costly vehicle sensors while favoring the simplest control structure for real-time implementation, a new LPV static output feedback (SOF) control method is proposed. Thanks to Lyapunov stability arguments, physical constraints on both system states and DTC actuator are explicitly considered in the design procedure to improve the safety and the comfort of passengers. Moreover, a parameter-dependent Lyapunov function is exploited for theoretical developments. In this way, the finite bounds on vehicle speed and acceleration are effectively taken into account in the control design to reduce the conservatism. The H2 control design is recast as an LMI-based optimization which can be easily solved with numerical solvers. The resulting robust DTC controller is evaluated with realistic driving scenarios.