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In this technical note we examine the relationship between linear hybrid automata (LHA) and piecewise affine (PWA) systems. While a LHA is an autonomous non-deterministic model, a PWA is a deterministic model with inputs. Through the key idea of modeling the uncertainty associated with LHA transitions as input disturbances in a PWA model, by extending continuous-time PWA models to include the dynamics of discrete states and resets we show in a constructive way that a LHA can be equivalently represented as a PWA system, where equivalent means that the two systems generate the same trajectories. Besides filling in a missing theoretical link between the LHA modelling framework and the PWA modelling framework, the result has the practical advantage of enabling the use of several existing control theoretical tools developed for PWA models to a wider class of hybrid systems.