Output feedback controllers for disturbance attenuation of linear systems with constraints on inputs are studied. The constraints are limited actuator capacity as well as limits on the actuator rates. The main features include guaranteed performance levels that depend on both actuator magnitude and rate limits, and controllers that operate at, or close to, maximum actuator capacity. The controllers have a structure similar to those used in the linear parameter varying (LPV) approach. Sufficient conditions for finding appropriate nonlinear control laws are expressed in terms of a convex search.