View source for Ducted Fan

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The '''Ducted Fan problem''' is a classical nonlinear benchmark in optimal control with multiple input and state constraints. This description is taken from [[#OCPjl | [1]]].

It models the planar motion of a ducted fan aircraft, described by its horizontal and vertical positions <math>( x_1 , x_2 )</math> , the angle <math>\alpha</math> with respect to the vertical, and their velocities <math>( v_1 , v_2 , v_\alpha )</math> .
The inputs are the body-fixed thrust components <math>( u_1 , u_2 )</math> , generated by moving flaps at the end of the duct.

The objective is to steer the fan from the origin to a horizontal position of <math>1 \mathrm{m}</math> at altitude <math>0</math>, with zero final velocities and attitude, in a free final time <math>t_f</math>, while minimising a trade-off between control effort and transition time.

== Mathematical formulation ==
We summarize the states as <math>x := (x_1, v_1, x_2, v_2, \alpha, v_\alpha)</math>.

<p>
<math>
 \begin{array}{lll}
 \displaystyle \min_{u, t_\mathrm{f}} && \frac{1}{t_\mathrm{f}} \int_0^{t_\mathrm{f}} \left( 2 u_1^2(t) + u_2^2(t) \right) \mathrm{d}t + \mu \, t_\mathrm{f} \\
 \text{subject to} \\
\quad \dot{x_1}(t) & = & v_1(t),\\
\quad \dot{v_1}(t) & = & \frac{1}{m} \left( u_1 \cos \alpha - u_2 \sin \alpha \right), \\
\quad \dot{x_2}(t) & = & v_2(t), \\
\quad \dot{v_2}(t) & = & \frac{1}{m} \left( -\mathrm{mg} + u_1 \sin \alpha + u_2 \cos \alpha \right), \\
\quad \dot{\alpha} & = & v_\alpha, \\
\quad \dot{v}_\alpha & = & \frac{r}{J} u_1, \\
\quad x(0) &=& (0, 0, 0, 0, 0, 0)^T, \\
\quad x(t_\mathrm{f}) &=& (1, 0, 0, 0, 0, 0)^T, \\
\quad u_1(t) & \in & [-5, 5] \ \quad \forall t \in [0,t_\mathrm{f}], \\
\quad u_2(t) & \in & [0, 17] \ \quad \forall t \in [0,t_\mathrm{f}], \\
\quad \alpha(t) & \in & [-30, 30] \ \quad \forall t \in [0,t_\mathrm{f}]
  \end{array}
</math>
</p>

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value
|-
| align=center | <math>m</math> || align=right | 2.2 <math>\mathrm{kg}</math> 
|- 
| align=center | <math>J</math> || align=right | 0.05 <math>\mathrm{kg \cdot m^2}</math>
|-
| align=center | <math>r</math> || align=right | 0.2 <math>\mathrm{m}</math>
|- 
| align=center | <math>\mathrm{mg}</math> || align=right | 4 <math>\mathrm{N}</math>
|- 
| align=center | <math>\mu</math> || align=right | 1
|}

The weight <math>\mu</math> balances control effort and transition time.

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
 Image:Double_Oscillator.png| States and discretized control for a local optimum.
</gallery>

== Miscellaneous and Further Reading ==
This formulation and a detailed description can be found in [[#OCPjl|[1]]].

== References ==
<span id="OCPjl">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/double_oscillator.md<br>


[[Category:MIOCP]]
[[Category:ODE model]]