Difference between revisions of "Particle steering problem"
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| + | where <math> (x_1, x_2) </math> is the position of the particle, <math> u </math> is the control angle and <math> a </math> is the constant magnitude of thrust. | ||
== Source Code == | == Source Code == | ||
Revision as of 19:17, 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(t_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/b/9/3/b93ac5698f674646a14328d8b261a025.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
