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This paper investigates the use of nonlinear direct output feedback in the control of linear time-invariant systems. Inparticular, it is concerned with those nonlinear controllers which result in a closed loop system which is quadratically stable. That is, the closed loop nonlinear system is asymptotically stable and furthermore, a quadratic Lyapunov function can be used to establish this stability. The main result of this paper can be stated roughly as follows: Any linear system which can be made quadratically stable using nonlinear direct output feedback control can also be stabilized using linear direct output feedback control.