Difference between revisions of "Hanging chain problem"
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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 18:41, 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 (syntax error): \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
