Difference between revisions of "Particle steering problem"
From mintOC
FelixMueller (Talk | contribs) (→Mathematical formulation) |
FelixMueller (Talk | contribs) (→Mathematical formulation) |
||
| Line 21: | Line 21: | ||
& \ddot{x}_1 & = & a \cos (u), \\ | & \ddot{x}_1 & = & a \cos (u), \\ | ||
& \ddot{x}_2 & = & a \sin (u), \\ | & \ddot{x}_2 & = & a \sin (u), \\ | ||
| − | & x( | + | & x(0) &=& (0, 0)^T, \\ |
& \dot{x}(0) &=& (0, 0)^T, \\ | & \dot{x}(0) &=& (0, 0)^T, \\ | ||
& x_2 (t_f) &=& 5, \\ | & x_2 (t_f) &=& 5, \\ | ||
Revision as of 19:18, 5 May 2016
| Particle steering problem | |
|---|---|
| State dimension: | 1 |
| Differential states: | 2 |
| Discrete control functions: | 1 |
| Interior point equalities: | 7 |
The Particle steering problem minimizes "the time taken for a particle, acted upon by a thrust of constant magnitude, to achieve a given altitude and terminal velocity." (Cite and problem taken from the COPS library)
Mathematical formulation
The problem is given by
![\begin{array}{llcl}
\displaystyle \min_{x, u, t_f} & t_f \\[1.5ex]
\mbox{s.t.}
& \ddot{x}_1 & = & a \cos (u), \\
& \ddot{x}_2 & = & a \sin (u), \\
& x(0) &=& (0, 0)^T, \\
& \dot{x}(0) &=& (0, 0)^T, \\
& x_2 (t_f) &=& 5, \\
& \dot{x}(t_f) &=& (45, 0)^T, \\
& u(t) &\in& [-\frac{\pi}{2},\frac{\pi}{2}].
\end{array}](https://mintoc.de/images/math/6/8/0/680aac214a5aba1bf8e9049939671655.png)
where 
is the position of the particle, 
is the control angle and 
is the constant magnitude of thrust.
Source Code
Model descriptions are available in
