1. A damping force controller which controls damping force generated by a damping force generation apparatus against externally input vibration by operating an actuator, comprising: control input calculation means for calculating a control input u by use of a state feedback controller K designed such that an L 2 gain of a closed loop control system S including a generalized plant G and the state feedback controller K for the generalized plant G becomes less than a previously set positive constant γ, the generalized plant G being designed by making use of a delay-approximated model M which is a control model in which a delay element R representing operation delay of the actuator and a delay compensation element R* cancelling out the delay act on a mechanical motion model of the damping force generation apparatus, which is a bilinear system, and which is designed to approximate a control model designed on the basis of the mechanical motion model of the damping force generation apparatus and in consideration of the delay, when a change amount of the speed dr/dt-dy/dt of the externally input vibration is equal to or less than a previously set very small amount; and operation control means for controlling the operation of the actuator on the basis of the control input u calculated by the control input calculation means.