In this article, a general and systematical quantitative robust linear parameter varying control method is proposed for active vibration control of linear parameter varying flexible structures such that a complete set of control objectives can be considered, especially the reduction of necessarily required control energy and the control input. To achieve this goal, the phase and gain control policies are employed in linear parameter varying H∞ control designs for suitable selection of weighting functions. The designed parameter-dependent H∞ controller allows us to explicitly consider the time-varying parameters of the dynamical models for saving the control energy and achieving other control objectives such as the specification of vibration reduction and qualitative robustness properties to both parametric and dynamic uncertainties. Then, various reliable robustness analyses are conducted to quantitatively verify the robustness properties in both deterministic and probabilistic senses. The design processes and the effectiveness of the proposed control method are illustrated by active vibration control of a non-collocated piezoelectric cantilever beam excited by an external position-varying force which is the disturbance to be rejected. This plant has typical parameter (force position)-dependent dynamics and is modeled as a linear parameter varying system whose time-varying parameter is the actual position of the disturbance. The numerical simulations demonstrate that, compared to the classical H∞ control and the acceleration feedback control, the proposed control method allows us to compute a quantitatively robust force-position-dependent H∞ controller whose benefit is requiring less control energy and smaller control input, while satisfying the same control objectives in the frequency and time domains.