# Crossover Pilot Model

Represent crossover pilot model

Pilot Models

## Description

The Crossover Pilot Model block represents the pilot model described in Mathematical Models of Human Pilot Behavior. (For more information, see [1]). This pilot model is a single input, single output (SISO) model that represents some aspects of human behavior when controlling aircraft. When modeling human pilot models, use this block for more accuracy than that provided by the Tustin Pilot Model block. This block is also less accurate than the Precision Pilot Model block.

The Crossover Model takes into account the combined dynamics of the human pilot and the aircraft, using the following form around the crossover frequency:

`${Y}_{p}{Y}_{c}=\frac{{\omega }_{c}{e}^{-\tau s}}{s}.$`

In this equation:

VariableDescription
YpPilot transfer function.
YcAircraft transfer function.
ωcCrossover frequency.
τTransport delay time caused by the pilot neuromuscular system.

If the dynamics of the aircraft (Yc) change, Yp changes correspondingly. From the options provided in the Type of control parameter, specify the dynamics of the aircraft. The preceding table lists the possible types of control that you can select for the aircraft.

 Note:   This block is valid only around the crossover frequency. It is not valid for discrete inputs such as a step.

This block has non-linear behavior. If you want to linearize the block (for example, with one of the Simulink® `linmod` functions), you might need to change the Pade approximation order. The Crossover Pilot Model block implementation incorporates the Simulink Transport Delay block with the Pade order (for linearization) parameter set to `2` by default. To change this value, use the `set_param` function, for example:

`set_param(gcb,'pade','3')`

## Parameters

Type of control

From the list, select one of the following options to specify the type of dynamics control that you want the pilot to have over for the aircraft.

Option (Controlled Element Transfer Function)Transfer Function of Controlled Element (Yc)Transfer Function of Pilot (Yp)YcYpNotes
`Proportional`
`${K}_{c}$`
`$\frac{{K}_{p}{e}^{-\tau s}}{s}$`
`$\frac{{K}_{c}{K}_{p}{e}^{-\tau s}}{s}$`

`Rate or velocity`
`$\frac{{K}_{c}}{s}$`
`${K}_{p}{e}^{-\tau s}$`
`$\frac{{K}_{c}{K}_{p}{e}^{-\tau s}}{s}$`

`Spiral divergence`
`$\frac{{K}_{c}}{{T}_{I}s-1}$`
`${K}_{p}{e}^{-\tau s}$`
`$\frac{{K}_{c}{K}_{p}{e}^{-\tau s}}{\left({T}_{I}s-1\right)}$`

`Second order - Short period`
`$\frac{{K}_{c}{\omega }_{n}{}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}$`
`$\frac{{K}_{p}{e}^{-\tau s}}{{T}_{I}s+1}$`
`$\begin{array}{l}\frac{{K}_{c}{\omega }_{n}{}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}×\\ \frac{{K}_{p}{e}^{-\tau s}}{{T}_{I}s+1}\end{array}$`
Short
period,
with ${\omega }_{n}>1/\tau$
`Acceleration (*)`
`$\frac{{K}_{c}}{{s}^{2}}$`
`${K}_{p}s{e}^{-\tau s}$`
`$\frac{{K}_{c}{K}_{p}{e}^{-\tau s}}{s}$`

`Roll attitude (*)`
`$\frac{{K}_{c}}{s\left({T}_{I}s+1\right)}$`
`${K}_{p}\left({T}_{L}s+1\right){e}^{-\tau s}$`
`$\frac{{K}_{c}{K}_{p}{e}^{-\tau s}}{s}$`
With
TL ≈ TI
`Unstable short period(*)`
`$\frac{{K}_{c}}{\left({T}_{I1}s+1\right)\left({T}_{I2}s-1\right)}$`
`${K}_{p}\left({T}_{L}s+1\right){e}^{-\tau s}$`
`$\frac{{K}_{c}{K}_{p}{e}^{-\tau s}}{\left({T}_{I2}s-1\right)}$`
With
TL≈ TI1
`Second order - Phugoid(*)`
`$\frac{{K}_{c}{\omega }_{n}{}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}$`
`${K}_{p}\left({T}_{L}s+1\right){e}^{-\tau s}$`
`$\frac{{K}_{c}{K}_{p}{\omega }_{n}^{2}{e}^{-\tau s}}{s}$`
Phugoid,
with $\begin{array}{l}{\omega }_{n}\ll 1/\tau ,\\ 1/{T}_{L}\approx \zeta {\omega }_{n}\end{array}$

* Indicates that the pilot model includes a Derivative block, which produces a numerical derivative. For this reason, do not send discontinuous (such as a step) or noisy input to the Crossover Pilot Model block. Such inputs can cause large outputs that might render the system unstable.

VariableDescription
KcAircraft gain.
KpPilot gain.
τPilot time delay.
TILag constant.
ζDamping ratio for the aircraft.
ωnNatural frequency of the aircraft.

Calculated value

From the list, select one of the following options to specify which value the block is to calculate:

• `Crossover frequency` — The block calculates the crossover frequency value. Selecting this option disables the Crossover frequency (rad/s) parameter.

• `Pilot gain` — The block calculates the pilot gain value. Selecting this option disables the Pilot gain parameter.

Controlled element gain

Specifies the gain of the aircraft controlled dynamics.

Pilot gain

Specifies the pilot gain.

Specifies a crossover frequency value, rad/s. This value ranges from 1 to 10 rad/s.

Pilot time delay(s)

Specifies the total pilot time delay, in seconds. This value typically ranges from 0.1 s to 0.2 s.

Pilot lag constant

Specifies the pilot lag constant.

## Inputs and Outputs

InputDimension TypeDescription

First

1-by-1 Contains the command for the signal that the pilot model controls.

Second

1-by-1 Contains the signal that the pilot model controls.

OutputDimension TypeDescription

First

1-by-1 Contains the command for the aircraft.

## References

[1] McRuer, D. T., Krendel, E., Mathematical Models of Human Pilot Behavior. Advisory Group on Aerospace Research and Development AGARDograph 188, Jan. 1974.