This paper investigates low-order observer design for bilinear systems with input constraints. A bilinear Luenberger-type observer with an H-infinity performance measure is formulated and the resulting synthesis problem is posed as a matrix inequality optimization for a linear parameter varying system. The resulting (nonconvex) bilinear matrix inequality problem is then solved with an LMI-based algorithm to find low-order nominal and robust quadratically stable observers. The performance of these observers are compared with that of a Kalman filter. In addition to alleviating the need to know the noise spectrum and its lower real-time computational burden, the H-infinity filter is shown to be robust to model uncertainties. The online radiation force estimation problem for a wave energy converter with bilinear dynamics is considered as an example.