Ducted Fan: Difference between revisions

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

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Revision as of 10:12, 24 November 2025

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 $α$ with respect to the vertical, and their velocities $(v_{1}, v_{2}, v_{α})$ . 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 $1 m$ 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 $x : = (x_{1}, v_{1}, x_{2}, v_{2}, α, v_{α})$ .

$\begin{array}{lll} \min_{u, t_{f}} & \frac{1}{t_{f}} \int_{0}^{t_{f}} (2 u_{1}^{2} (t) + u_{2}^{2} (t)) d t + μ t_{f} \\ subject to \\ \dot{x_{1}} (t) & = & v_{1} (t), \\ \dot{v_{1}} (t) & = & \frac{1}{m} (u_{1} \cos α - u_{2} \sin α), \dot{x_{2}} (t) & = & v_{2} (t), \\ \dot{v_{2}} (t) & = & \frac{1}{m} (- m g + u_{1} \sin α + u_{2} \cos α), \\ \dot{α} & = & v_{α}, \\ {\dot{v}}_{α} & = & \frac{r}{J} u_{1}, \\ x (0) & = & (0, 0, 0, 0, 0, 0)^{T}, \\ x (t_{f}) & = & (1, 0, 0, 0, 0, 0)^{T}, \\ u_{1} (t) & \in & [- 5, 5] \forall t \in [0, t_{f}], \\ u_{2} (t) & \in & [0, 17] \forall t \in [0, t_{f}], \\ α (t) & \in & [- 30, 30] \forall t \in [0, t_{f}] \end{array}$

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$m_{1}$	100 $k g$	First mass directly affected by $F (t)$
$m_{2}$	2 $k g$	Second mass influenced by damping control
$k_{1}$	100 $N / m$	Spring connecting first mass to reference
$k_{2}$	3 $N / m$	Coupling spring between the two masses
$c$	0.5 $N s / m$	Damping affecting second mass
$T$	$2 π$	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

@@ Line 13: / Line 13: @@
 == 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} && \frac{1}{2}\int_0^T \left( x_0(t)^2 + x_1(t)^2 + u(t)^2 \right) \\
+  \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_0}(t) & = & x_2(t),\\
+\quad \dot{x_1}(t) & = & v_1(t),\\
-\quad \dot{x_1}(t) & = & x_3(t), \\
+\quad \dot{v_1}(t) & = & \frac{1}{m} \left( u_1 \cos \alpha - u_2 \sin \alpha \right),
-\quad \dot{x_2}(t) & = & - \frac{k_1 + k_2}{m_1} \cdot x_0 + \frac{k_2}{m_1} \cdot x_1 + \frac{1}{m_1} \sin\left(\frac{2\pi}{T} \cdot t\right), \\
+\quad \dot{x_2}(t) & = & v_2(t), \\
-\quad \dot{x_3}(t) & = &  \frac{k_2}{m_2} x_0(t) - \frac{k_2}{m_2} x_1(t) - \frac{c(1-u)}{m_2} x_3(t), \\
+\quad \dot{v_2}(t) & = & \frac{1}{m} \left( -\mathrm{mg} + u_1 \sin \alpha + u_2 \cos \alpha \right), \\
-\quad x_1(0) &=& 0, \\
+\quad \dot{\alpha} & = & v_\alpha, \\
-\quad x_2(0) &=& 0, \\
+\quad \dot{v}_\alpha & = & \frac{r}{J} u_1, \\
-\quad u(t) & \in & [-1, 1] \ \quad \forall t \in [0,T]
+\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>