This article focuses on dynamic output feedback and robust control of quasi linear parabolic partial differential equations (PDE) systems with time-varying uncertain variables. Especially processes that are described by dissipative PDEs are considered. The states of the process required for designing controllers are dynamically estimated from limited number of noisy process measurements employing an Extended Kalman filter. The issue of utilizing these estimated states in a robust controller to achieve the desired process objective is investigated. The controller design needs to address both model uncertainty and sensor noise. The methodology is employed on an representative example wherein the desired objective is to stabilize an unstable operating point in a catalytic rod, where an exothermic reaction occurs. A finite dimensional robust controller, utilizing dynamically estimated states, is used to successfully stabilize the process to an open-loop unstable steady-state.