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We present a general formulation of the problem of multi-variable anti-windup bumpless transfer (AWBT) controller synthesis which provides sufficient conditions for stability covering a large class of typical input nonlinearities such as saturation, deadzone, and switching/override nonlinearities. The resulting synthesis method demonstrates graceful performance degradation whenever input nonlinearities are active through minimization of a weighted L ₂ gain. The main contribution of the paper is that the AWBT controller synthesis, using static compensation, is cast as a convex optimization over linear matrix inequalities.