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.