Ducted Fan

Ducted Fan
State dimension:	1
Differential states:	4
Discrete control functions:	1

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

It models the planar motion of a ducted fan aircraft, described by its horizontal and vertical positions $(x_1,x_2)$ , the angle $\alpha$ with respect to the vertical, and their velocities $(v_1,v_2,v_{\alpha })$ . The inputs are the body-fixed thrust components $(u_1,u_2)$ , 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 $1\mathrm{m}$ at altitude $0$, with zero final velocities and attitude, in a free final time $t_f$, while minimising a trade-off between control effort and transition time.

Mathematical formulation

We summarize the states as $x:=(x_1,v_1,x_2,v_2,\alpha ,v_{\alpha })$.

$\begin{array}{lll}\underset{u,t_{\mathrm{f}}}{\min }& & \frac{1}{t_{\mathrm{f}}}\underset{0}{\overset{t_{\mathrm{f}}}{\int }}(2u_1^2(t)+u_2^2(t))\mathrm{d}t+\mu t_{\mathrm{f}}\\ \text{subject to}\\ \dot{x_1}(t)& =& v_1(t),\\ \dot{v_1}(t)& =& \frac{1}{m}(u_1\cos \alpha -u_2\sin \alpha ),\\ \dot{x_2}(t)& =& v_2(t),\\ \dot{v_2}(t)& =& \frac{1}{m}(-\mathrm{m}\mathrm{g}+u_1\sin \alpha +u_2\cos \alpha ),\\ \dot{\alpha }& =& v_{\alpha },\\ {\dot{v}}_{\alpha }& =& \frac{r}{J}{u}_1,\\ x(0)& =& (0,0,0,0,0,0{)}^T,\\ x(t_{\mathrm{f}})& =& (1,0,0,0,0,0{)}^T,\\ u_1(t)& \in & [-5,5]\mathrm{\forall }t\in [0,t_{\mathrm{f}}],\\ u_2(t)& \in & [0,17]\mathrm{\forall }t\in [0,t_{\mathrm{f}}],\\ \alpha (t)& \in & [-30,30]\mathrm{\forall }t\in [0,t_{\mathrm{f}}]\end{array}$

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$m_1$	100 $\mathrm{k}\mathrm{g}$	First mass directly affected by $F(t)$
$m_2$	2 $\mathrm{k}\mathrm{g}$	Second mass influenced by damping control
$k_1$	100 $\mathrm{N}/\mathrm{m}$	Spring connecting first mass to reference
$k_2$	3 $\mathrm{N}/\mathrm{m}$	Coupling spring between the two masses
$c$	0.5 $\mathrm{N}\mathrm{s}/\mathrm{m}$	Damping affecting second mass
$T$	$2\pi$	Duration of the motion
$u$	-	Modulates the damping of the second mass

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and discretized control for a local optimum.

Miscellaneous and Further Reading

This formulation and a detailed description can be found in [1].

References

[1] 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