Crossover Model

The crossover model of compensatory operation grows out of the observation that pilots develop the necessary dynamics to produce in the pilot-airplane combination a particular transfer function in the crossover region of frequencies (McRuer, 1988). The pilot-airplane open-loop transfer function developed has the remarkably simple form of an integrated time delay, or rnc (e-Ts)/s, where the open-loop gain is rnc, т is the delay, and s is the Laplace operator. The open-loop gain rnc is called the crossover frequency, the frequency at which the open-loop amplitude response crosses the 1.0- or 0-db line.

The closed-loop frequency response, or ratio of output to input for the crossover model, is flat at 1.0 at low frequencies, meaning that the output follows exactly the input. As input frequency is raised, the frequency at which the output drops 3 db lower than the input, or to only 70 percent of the input, is considered a cutoff for all practical purposes. This frequency defines the closed-loop system bandwidth. For the crossover model, the frequency that defines closed-loop system bandwidth is also the frequency rnc for which the open-loop system has a gain of 1.0.

The crossover model time delay т is actually a low-frequency approximation, valid at crossover frequencies, for numerous pilot and control system delays and higher order lag terms. That part of т due to the pilot becomes greater as the lead contributed by the pilot increases, a cost of additional pilot effort (McRuer, 1988). This reduces the available crossover frequency for other system lags.