Fuller's initial value multimode problem - Revision history

ClemensZeile at 22:56, 8 January 2018

2018-01-08T22:56:11Z

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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.		-->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.


	Furthermore, this variant comprises four binary controls instead of only one control.

	== Mathematical formulation ==		== Mathematical formulation ==

ClemensZeile at 22:55, 8 January 2018

2018-01-08T22:55:56Z

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	}}<!-- 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...

	--> 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 ==

ClemensZeile: /* Reference Solutions */

2018-01-08T22:53:54Z

Reference Solutions

← Older revision		Revision as of 22:53, 8 January 2018
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	If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.		If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

	The optimal objective value of the relaxed problem with <math> n_t=6000, \, n_u=~~100~~ </math> is <math>1.~~10988197e~~-05</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>3.~~94891656e-05~~</math>.		The optimal objective value of the relaxed problem with <math> n_t=6000, \, n_u=60 </math> is <math>1.08947605e-05</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>0.000422127329</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
	Image:~~FullerRelaxed~~ 6000 40 1.png\| Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=~~100~~</math>.		Image:MmfullerRelaxed 6000 100 1.png\| Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=60</math>.
	Image:~~FullerRelaxed~~ 6000 40 2.png\| Optimal relaxed controls.		Image:MmfullerRelaxed 6000 100 2.png\| Optimal relaxed controls.
	Image:~~FullerCIA~~ 6000 40 1.png\| Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=~~100~~</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:MmfullerCIA 6000 100 1.png\| Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=60</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	Image:~~FullerCIA~~ 6000 40 2.png\| Optimal binary controls.		Image:MmfullerCIA 6000 100 2.png\| Optimal binary controls.
	</gallery>		</gallery>

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2018-01-08T22:34:43Z

Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 4 |nre = 2 }} This site descri..."

New page

{{Dimensions
|nd = 1
|nx = 2
|nw = 4
|nre = 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 <math>t \in [t_0, t_f]</math> almost everywhere the mixed-integer optimal control problem is given by

<p>
<math>
\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}
</math>
</p>

== Parameters ==

We use <math>x_S = x_T = (0.01, 0)^T</math> together with:

<math>
\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}
</math>

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with <math> n_t=6000, \, n_u=100 </math> is <math>1.10988197e-05</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>3.94891656e-05</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
Image:FullerRelaxed 6000 40 1.png| Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=100</math>.
Image:FullerRelaxed 6000 40 2.png| Optimal relaxed controls.
Image:FullerCIA 6000 40 1.png| Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
Image:FullerCIA 6000 40 2.png| Optimal binary controls.
</gallery>


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]