Hanging chain problem: Difference between revisions
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{{Dimensions | {{Dimensions | ||
|nd = 1 | |nd = 1 | ||
|nx = | |nx = 3 | ||
| | |nu = 1 | ||
|nre = | |nc = 4 | ||
|nre = 5 | |||
}}<!-- Do not insert line break here or Dimensions Box moves up in the layout... | }}<!-- Do not insert line break here or Dimensions Box moves up in the layout... | ||
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\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_1(t_f) &=& b, \\ | |||
& x_3(t_f) &=& Lp, \\ | |||
& x_i(t) &\in& [0,10], \quad i=1,2,3, \\ | |||
& u(t) &\in& [-10,20]. | |||
\end{array} | \end{array} | ||
</math> | </math> | ||
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In this model the parameters used are | In this model the parameters used are | ||
<math> | |||
\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} | ||
</math> | |||
== Source Code == | == Source Code == | ||
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* [[:Category:AMPL/TACO | AMPL/TACO code]] at [[Hanging chain problem (TACO)]] | * [[:Category:AMPL/TACO | AMPL/TACO code]] at [[Hanging chain problem (TACO)]] | ||
* [[:Category:Gekko | GEKKO Python code]] at [[Hanging chain problem (GEKKO)]] | |||
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here --> | <!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here --> | ||
[[Category:MIOCP]] | [[Category:MIOCP]] | ||
[[Category:ODE model]] | [[Category:ODE model]] | ||
[[Category: | [[Category:Minimum energy]] | ||
Latest revision as of 16:04, 22 February 2026
| Hanging chain problem | |
|---|---|
| State dimension: | 1 |
| Differential states: | 3 |
| Continuous control functions: | 1 |
| Path constraints: | 4 |
| Interior point equalities: | 5 |
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
Parameters
In this model the parameters used are
Source Code
Model descriptions are available in