Linear parameter-varying (LPV) modelling and control of a nonlinear partial differential equation (PDE) is considered in this article. The one-dimensional viscous Burgers' equation is discretised using a finite difference scheme; the boundary conditions are taken as control inputs and the velocities at two grid points are assumed to be measurable. A nonlinear high-order state space model is generated and proper orthogonal decomposition is used for model order reduction. After assessing the accuracy of the reduced model, a low-order functional observer is designed to estimate the reduced states which are linear combinations of the velocities at all grid points. A discrete-time quasi-LPV model that is affine in scheduling parameters is derived based on the reduced model. A polytopic LPV controller is synthesised based on a generalised plant containing the LPV model and the functional observer. More generally, the proposed method can be used to design an LPV controller for a quasi-LPV system with non-measurable scheduling parameters. Simulation results demonstrate the high tracking performance and disturbance and measurement noise rejection capabilities of the designed LPV controller compared with a linear quadratic Gaussian (LQG) controller based on a linearised model.