This paper is concerned with the gain-scheduled controller synthesis problems for linear parameter varying (LPV) systems. In the case where the state space matrices of the LPV system depend affinely on the time-varying parameters, the standard linear matrix inequalities (LMI's) are helpful in dealing with these synthesis problems provided that we accept the notion of quadratic stability. However, in the case of rational parameter dependence, the standard LMI's carry some deficiency and do not lead us to numerically tractable conditions in a straightforward fashion. This paper clarifies that dilated LMI's are effective in overcoming the difficulties and deriving numerically tractable conditions. The dilated LMI's also make possible to employ a parameter-dependent Lyapunov variable to achieve control objectives, which is promising to alleviate the conservatism stemming from a quadratic (parameter-independent) Lyapunov variable.