Difference between revisions of "Goddart's rocket problem"

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(→‎Mathematical formulation)
(→‎Mathematical formulation)
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  & m(0) &=& m_0, \\
 
  & m(0) &=& m_0, \\
 
  & r(T) &=& r_T, \\
 
  & r(T) &=& r_T, \\
−
  & D(r(\cdot),v(\cdot))&\le& C \\
+
  & D(r(t),v(t))&\le& C \\
 
& T free
 
& T free
 
\end{array}  
 
\end{array}  

Revision as of 13:13, 19 January 2016

In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.

Variables

The state variables r,v,m describe the altitude(radius), speed and mass.

The drag is given by

D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).
All units are renormalized.

Mathematical formulation

\begin{array}{llcll} \displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex] \mbox{s.t.} & \dot{r}(t) & = & v, \\ & \dot{v}(t) & = & -\frac{1}{r(t)^2} + \frac{1}{m(t)} (T_{max}u(t)-D(r,v)) \\[1.5ex] & \dot{m}(t) & = & -b T_{max} u(t), \\ & u(t) &\in& [0,1] \\ & r(0) &=& r_0, \\ & v(0) &=& v_0, \\ & m(0) &=& m_0, \\ & r(T) &=& r_T, \\ & D(r(t),v(t))&\le& C \\ & T free \end{array}

Parameters

\begin{array}{rcl} r_0 &=& 1 \\ v_0 &=& 0 \\ m_0 &=& 1 \\ r_T &=& 1.01 \\ b &=& 7 \\ T_{max} &=& 3.5 \\ A &=& 310 \\ k &=& 500 \\ C &=& 0.6 \end{array}

Reference Solution

The following reference solution was generated using BOCOP. The optimal value of the objective function is -0.63389.


  • Reference solution plots

  • Control u over time.

  • Position r over time.

  • Speed v over time.

  • Mass m over time.

References

The Problem can be found in the BOCOP User Guide.

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