The "Model Reference Adaptive Control (MRAC)" is a popular approach from the early nineties to our days. Its basic idea is the application of proper feedback that makes the behavior of the controlled system identical to that of the "reference model" that normally is simple enough to control. The idea has many particular variants with the common feature that they are designed by the use of Lyapunov's 2nd ("direct") method that normally applies a quadratic Lyapunov function constructed of the tracking error and further additional terms. Though this approach normally guarantees global asymptotic stability, its use can entail complicated tuning that may have disadvantages whenever very fast applications are needed. In this paper an alternative problem tackling, the application of "Robust Fixed Point Transformations (RFPT)" in the MRAC technique is recommended. This approach applies strongly saturated, multiplicative nonlinear terms causing a kind of "deformation" of the input of the available imprecise system model. Instead parameter tuning that is typical in the traditional MRAC it operates with a simple convergence guaranteed only within a local basin of attraction. This technique can well compensate the simultaneous consequences of modeling errors and external disturbances that normally can "fob" the more traditional, tuning based approaches. As a potential application paradigm the novel MRAC control of a "cart-beam-hamper" system is considered. The conclusions of the paper are illustrated by simulation results.