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This paper deals with the use of feedback control to prevent hysteresis and surge in axial-flow compressors. We present a dynamic output-feedback controller that semiglobally stabilizes every rotating stall equilibrium, and a range of axisymmetric equilibria of the Moore-Greitzer model for axial-flow compressors. The dynamic controller combines a two-state-variable-feedback backstepping controller from the literature with a nonlinear, reduced order, high-gain observer that estimates the mass flow through the compressor from measurements of the pressure rise across it. Given an equilibrium and a compact inner bound on the domain of attraction, we use Lyapunov techniques to compute an explicit lower bound on the observer gain such that the specified equilibrium is asymptotically stable for the closed-loop system, with a domain of attraction that contains the specified inner bound. We use a numerical example to illustrate how the inner bound on the domain of attraction can be specified so that the closed-loop compressor does not exhibit hysteresis and surge oscillations even in response to changes in the throttle setting that are dictated by large and sudden changes in the desired operating point. Simulation results are used to demonstrate the absence of hysteresis and surge in the closed-loop compressor dynamics.