Ducted Fan: Difference between revisions

Ducted Fan
State dimension:	1
Differential states:	6
Discrete control functions:	3

Latest revision as of 10:43, 28 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
$m$	2.2 $k g$
$J$	0.05 $k g \cdot m^{2}$
$r$	0.2 $m$
$m g$	4 $N$
$μ$	1

The weight $μ$ balances control effort and transition time.

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/ducted_fan.md

@@ Line 1: / Line 1: @@
 {{Dimensions
 |nd        = 1
-|nx        = 4
+|nx        = 6
-|nw        = 1
+|nw        = 3
 }}
@@ Line 10: / Line 10: @@
 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 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.
+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} && \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] \ & \forall t \in [0,t_\mathrm{f}], \\
+\quad u_2(t) & \in & [0, 17] \ & \forall t \in [0,t_\mathrm{f}], \\
+\quad \alpha(t) & \in & [-30, 30] \ & \forall t \in [0,t_\mathrm{f}]
    \end{array}
 </math>
@@ Line 34: / Line 40: @@
 {| border="1" align="center" cellpadding="5" cellspacing="0"
 |- bgcolor=#c7c7c7
-! Symbol !! Value !! Description
+! Symbol !! Value
 |-
-| align=center | <math>m_1</math> || align=right | 100 <math>\mathrm{kg}</math> || First mass directly affected by <math>F( t )</math>
+| align=center | <math>m</math> || align=right | 2.2 <math>\mathrm{kg}</math>
 |-
-| align=center | <math>m_2</math> || align=right | 2 <math>\mathrm{kg}</math>||  Second mass influenced by damping control
+| align=center | <math>J</math> || align=right | 0.05 <math>\mathrm{kg \cdot m^2}</math>
 |-
-| align=center | <math>k_1</math> || align=right | 100 <math>\mathrm{N/m}</math>|| Spring connecting first mass to reference
+| align=center | <math>r</math> || align=right | 0.2 <math>\mathrm{m}</math>
-|-
-| align=center | <math>k_2</math> || align=right | 3 <math>\mathrm{N/m}</math>|| Coupling spring between the two masses
-|-
-| align=center | <math>c</math> || align=right | 0.5 <math>\mathrm{Ns/m}</math>|| Damping affecting second mass
 |-
-| align=center | <math>T</math> || align=right | <math>2 \pi</math> || Duration of the motion
+| align=center | <math>\mathrm{mg}</math> || align=right | 4 <math>\mathrm{N}</math>
 |-
-| align=center | <math>u</math> || align=right | - || Modulates the damping of the second mass
+| align=center | <math>\mu</math> || align=right | 1
 |}
+The weight <math>\mu</math> balances control effort and transition time.
 == Reference Solutions ==
@@ Line 55: / Line 59: @@
 Here is one local solution to the above control problem.
-<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
+<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
-  Image:Double_Oscillator.png| States and discretized control for a local optimum.
+  Image:Ducted_Fan.png| States and discretized control for a local optimum.
 </gallery>
@@ Line 63: / Line 67: @@
 == 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>
+<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/ducted_fan.md<br>
 [[Category:MIOCP]]
 [[Category:ODE model]]