A method of designing linear-parameter varying (LPV) control systems based on the parameterisation of all quadratically stabilising controllers is presented. Conceptions of doubly coprime factorisation and Youla parameterisation of LTI systems are extended to LPV systems with respect to quadratic stability using a state-space expression. The parameterisation of closed-loop systems, which are affine with any quadratically stable Q-parameter, is then described. This description enables the application of the Q-parameter approach to a variety of LPV control-system designs. Above all, a systematic H∞ strategy is focused on and a necessary and sufficient condition and also a design scheme of Q to obtain L2-gain performance, are clarified.