Difference between revisions of "Hanging chain problem"
From mintOC
FelixMueller (Talk | contribs) (→Parameters) |
FelixMueller (Talk | contribs) (→Parameters) |
||
| Line 37: | Line 37: | ||
\begin{array}{rcl} | \begin{array}{rcl} | ||
[t_0, t_f] &=& [0, 1],\\ | [t_0, t_f] &=& [0, 1],\\ | ||
| − | (a,b) &=& ( | + | (a,b) &=& (1,3),\\ |
Lp &=& 4. | Lp &=& 4. | ||
\end{array} | \end{array} | ||
Revision as of 18:42, 5 May 2016
| Hanging chain problem | |
|---|---|
| State dimension: | 1 |
| Differential states: | 2 |
| Discrete control functions: | 1 |
| Interior point equalities: | 2 |
The Hanging chain problem is concerned with finding a chain (of uniform density) of length 
suspendend between two points 
with minimal potential energy. (Problem taken from the COPS library)
Mathematical formulation
The problem is given by
![\begin{array}{llcl}
\displaystyle \min_{x, u} & x_2(t_f) \\[1.5ex]
\mbox{s.t.} & \dot{x}_1 & = & u, \\
& \dot{x}_2 & = & x_1 (1+u^2)^{1/2}, \\
& \dot{x}_3 & = & (1+u^2)^{1/2}, \\
& x(t_0) &=& (a,0,0)^T, \\
& x_1(t_f) &=& b, \\
& x_3(t_f) &=& Lp, \\
& x(t) &\in& [0,10], \\
& u(t) &\in& [-10,20].
\end{array}](https://mintoc.de/images/math/b/c/7/bc780667f14aed98c5f25693189fe3bb.png)
Parameters
In this model the parameters used are
![\begin{array}{rcl}
[t_0, t_f] &=& [0, 1],\\
(a,b) &=& (1,3),\\
Lp &=& 4.
\end{array}](https://mintoc.de/images/math/e/e/0/ee0e9c5601a4e38d2c026c83b4756eb8.png)
Source Code
Model descriptions are available in
