A convertible sofa of the perpendicular-sleeping type, said sofa comprising: a fixed structure supporting a foldable structure which comprises three panels, namely, a central panel having two opposite sides articulated on a front panel and a rear panel, respectively, along axes of articulation parallel to a transverse direction, where the foldable structure can be folded/unfolded moving in a longitudinal direction between: a sofa configuration in which the front panel forms a seat, the central panel forms - a backrest, and the rear panel is folded over the central panel; and a sleeping configuration in which the three panels are substantially coplanar, so as to form a bed base; and said sofa further comprising a system of actuation of folding/unfolding of the foldable structure between the sofa configuration and the sleeping configuration, and vice versa; and where the system of actuation comprises at least one motorized actuator comprising a body within which a rod is movable in translation, where the body is articulated on one of the fixed or foldable structures and the end of the rod is articulated on the other of the fixed or foldable structures, the movement of the rod causing the foldable structure to fold/unfold said sofa being characterized in that the rear panel is articulated on the fixed structure along an axis of articulation parallel to the axes of articulation, and the body of the actuator is articulated on the fixed structure or on the rear panel, and the end of the rod is, conversely, articulated on the rear panel or on the fixed structure. a fixed structure supporting a foldable structure which comprises three panels, namely, a central panel having two opposite sides articulated on a front panel and a rear panel, respectively, along axes of articulation parallel to a transverse direction, where the foldable structure can be folded/unfolded moving in a longitudinal direction between: a sofa configuration in which the front panel forms a seat, the central panel forms - a backrest, and the rear panel is folded over the central panel; and a sleeping configuration in which the three panels are substantially coplanar, so as to form a bed base; and said sofa further comprising a system of actuation of folding/unfolding of the foldable structure between the sofa configuration and the sleeping configuration, and vice versa; and where the system of actuation comprises at least one motorized actuator comprising a body within which a rod is movable in translation, where the body is articulated on one of the fixed or foldable structures and the end of the rod is articulated on the other of the fixed or foldable structures, the movement of the rod causing the foldable structure to fold/unfold said sofa being characterized in that the rear panel is articulated on the fixed structure along an axis of articulation parallel to the axes of articulation, and the body of the actuator is articulated on the fixed structure or on the rear panel, and the end of the rod is, conversely, articulated on the rear panel or on the fixed structure. a sofa configuration in which the front panel forms a seat, the central panel forms - a backrest, and the rear panel is folded over the central panel; and a sleeping configuration in which the three panels are substantially coplanar, so as to form a bed base; and said sofa further comprising a system of actuation of folding/unfolding of the foldable structure between the sofa configuration and the sleeping configuration, and vice versa; and where the system of actuation comprises at least one motorized actuator comprising a body within which a rod is movable in translation, where the body is articulated on one of the fixed or foldable structures and the end of the rod is articulated on the other of the fixed or foldable structures, the movement of the rod causing the foldable structure to fold/unfold said sofa being characterized in that the rear panel is articulated on the fixed structure along an axis of articulation parallel to the axes of articulation, and the body of the actuator is articulated on the fixed structure or on the rear panel, and the end of the rod is, conversely, articulated on the rear panel or on the fixed structure.