Hanging chain problem: Difference between revisions
FelixMueller (talk | contribs) Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 |nre = 2 }}<!-- Do not insert line break here or Dimensions Box moves up in the layout... -->The Hanging chain..." |
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<math> | <math> | ||
\begin{array}{llcl} | \begin{array}{llcl} | ||
\displaystyle \min_{x, u} &x_2(t_f) \\[1.5ex] | \displaystyle \min_{x, u} & x_2(t_f) \\[1.5ex] | ||
\mbox{s.t.} & \dot{x}_1 & = & u, \\ | \mbox{s.t.} & \dot{x}_1 & = & u, \\ | ||
& \dot{x}_2 & = & x_1 (1+u^2)^{1/2}, \\ | & \dot{x}_2 & = & x_1 (1+u^2)^{1/2}, \\ | ||
& \dot{x}_3 & = & (1+u^2)^{1/2), \\ | & \dot{x}_3 & = & (1+u^2)^{1/2), \\ | ||
& x(t_0) &=& (a,0,0)^T, \\ | & x(t_0) &=& (a,0,0)^T, \\ | ||
\end{array} | \end{array} | ||
</math> | </math> | ||
Revision as of 17:41, 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
Failed to parse (unknown function "\begin{array}"): {\displaystyle \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, \\ \end{array} }
Parameters
In this model the parameters used are \begin{array}{rcl} [t_0, t_f] &=& [0, 1],\\ (a,b) &=& (0.4, 0.2),\\ Lp &=& 4. \end{array}
Source Code
Model descriptions are available in