Difference between revisions of "Fuller's initial value multimode problem"
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| − | --> This site describes a Fuller's problem variant with no terminal constraints and additional Mayer term for penalizing deviation from given reference values. Furthermore, this variant comprises four binary controls instead of only one control. | + | -->This site describes a Fuller's problem variant with no terminal constraints and additional Mayer term for penalizing deviation from given reference values. Furthermore, this variant comprises four binary controls instead of only one control. |
== Mathematical formulation == | == Mathematical formulation == | ||
Latest revision as of 23:56, 8 January 2018
| Fuller's initial value multimode problem | |
|---|---|
| State dimension: | 1 |
| Differential states: | 2 |
| Discrete control functions: | 4 |
| Interior point equalities: | 2 |
This site describes a Fuller's problem variant with no terminal constraints and additional Mayer term for penalizing deviation from given reference values. Furthermore, this variant comprises four binary controls instead of only one control.
Mathematical formulation
For ![t \in [t_0, t_f]](https://mintoc.de/images/math/5/5/8/55823791d9100bcb5461801aff4f6edd.png)
almost everywhere the mixed-integer optimal control problem is given by
![\begin{array}{llcl}
\displaystyle \min_{x, w} & \int_{t_0}^{t_f} x_0^2 & \; \mathrm{d} t & + (x(t_f)-x_T)^2 \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 & = & x_1+ \sum\limits_{i=1}^{4} c_{0,i} \omega_i, \\
& \dot{x}_1 & = & 1 + \sum\limits_{i=1}^{4} c_{1,i} \omega_i, \\[1.5ex]
& 1 &=& \sum\limits_{i=1}^{4}w_i(t), \\
& x(0) &=& x_S, \\
& w(t) &\in& \{0, 1\}.
\end{array}](https://mintoc.de/images/math/1/9/c/19cd2b7156ddbe83f4ad2b666cead397.png)
Parameters
We use 
together with:
![\begin{array}{rcl}
[t_0, t_f] &=& [0, 1],\\
(c_{0,1}, c_{1,1}) &=& (0, -2),\\
(c_{0,2}, c_{1,2}) &=& (0, -0.5),\\
(c_{0,3}, c_{1,3}) &=& (0, -3),\\
(c_{0,4}, c_{1,4}) &=& (0, 0).
\end{array}](https://mintoc.de/images/math/7/f/2/7f2a2b8ecd6fa85d2e48098cf5349e4d.png)
Reference Solutions
If the problem is relaxed, i.e., we demand that 
be in the continuous interval ![[0, 1]](https://mintoc.de/images/math/c/c/f/ccfcd347d0bf65dc77afe01a3306a96b.png)
instead of the binary choice 
, the optimal solution can be determined by means of direct optimal control.
The optimal objective value of the relaxed problem with 
is 
. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 
.
- Reference solution plots
Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and
.
Optimal relaxed controls.
Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and
. The relaxed controls were approximated by Combinatorial Integral Approximation.
Optimal binary controls.
