Hang Glider

Hang Glider
State dimension:	1
Differential states:	4
Discrete control functions:	2

The Hang Glider problem is a classical benchmark in optimal control. This description is taken from [1].

It consists of steering a hang glider from an initial horizontal position and altitude to a target altitude while maximising the horizontal distance travelled. The glider dynamics incorporate lift, drag, gravity, and the effect of a thermal updraft. The control variable is the lift coefficient $c_L$, which modulates the aerodynamic lift and influences the trajectory through the thermal region.

Mathematical formulation

$\begin{array}{lll}\underset{u}{\min }& & -x(t_f)\\ \text{subject to}\\ \dot{x}(t)& =& v_x(t),\\ \dot{y}(t)& =& v_y(t),\\ \dot{v_x}(t)& =& -\frac{L(t)\cdot w(t)+D(t)\cdot v_x(t)}{mv(t)},\\ \dot{v_y}(t)& =& \frac{L(t)\cdot v_x(t)-D(t)\cdot w}{mv(t)}-g,\\ x_1(t)& \ge & 0& \mathrm{\forall }t\in [0,T],\\ x(0)& =& (x_0,y_0,v_{x,0},v_{y,0}{)}^T,\\ y(t_f)& =& y_f\\ v_x(t_f)& =& v_{x,f}\\ v_y(t_f)& =& v_{y,f}\\ \end{array}$

with the auxiliary equations: Failed to parse (syntax error): {\displaystyle v = \sqrt{v_x^2 + w^2}, \quad \\ w = v_y - U_\text{updraft}(x), \\ L = \frac{1}{2} \rho S c_L v^2, \\ D = \frac{1}{2} \rho S (c_0 + c_1 c_L^2) v^2, \\ U_\text{updraft}(x) = u_c\, (1 - r) e^{-r}, \\ r = \left( \frac{x}{r_0} - 2.5 \right)^2, }

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$\alpha$	3	Weight on state
$\beta$	0	Weight on squared state
$\gamma$	0.5	Weight on squared control
$T$	10	Final 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/robbins.md