Best estimated inverse versus inverse of the best estimator

The construction of a feed-forward controller frequently requires the estimation of an inverse function. Two possible methods to achieve this are: (i) learning the best estimated inverse (BEI), a method that is sometimes referred to as direct inverse learning and (ii) learning the inverse of the best estimator (IBE), a method that is sometimes referred to as indirect inverse learning. We analyze a general control problem, in the presence of noise, and show analytically that these two methods are asymptotically significantly different, even for simple linear non-redundant systems. We further demonstrate that the IBE method is typically superior for control purposes.