Difference between revisions of "Goddart's rocket problem"

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

Latest revision as of 17:08, 22 February 2016

Goddart's rocket problem
State dimension: 1
Differential states: 3
Continuous control functions: 1
Path constraints: 1
Interior point equalities: 4


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 respectively.

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} & = & v, \\ & \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex] & \dot{m} & = & -b T_{max} u, \\ & u(t) &\in& [0,1] \\ & r(0) &=& r_0, \\ & v(0) &=& v_0, \\ & m(0) &=& m_0, \\ & r(T) &=& r_T, \\ & D(r,v)&\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.

Source Code

Model descriptions are available in:

References

The Problem can be found in the BOCOP User Guide.

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