Difference between revisions of "Fuller's initial value problem"

From mintOC
Jump to: navigation, search
(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... -->This site describ...")
 
(→‎Parameters)
 
(4 intermediate revisions by the same user not shown)
Line 15: Line 15:
 
<math>
 
<math>
 
\begin{array}{llcl}
 
\begin{array}{llcl}
−
  \displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 & \; \mathrm{d} t + (x(t_f)-x_T)^2 \\[1.5ex]
+
  \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, \\
 
  \mbox{s.t.} & \dot{x}_0 & = & x_1, \\
 
  & \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
 
  & \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
Line 23: Line 23:
 
</math>
 
</math>
 
</p>
 
</p>
−
 
  
 
== Parameters ==
 
== Parameters ==
  
−
We use <math>x_S = x_T = (0.01, 0)^T</math>.
+
We use <math>x_S = x_T = (0.01, 0)^T</math> and <math>(t_0,t_f) = (0,1)</math>.
−
 
+
  
 
== Reference Solutions ==
 
== Reference Solutions ==
Line 34: Line 32:
 
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=12000, \, n_u=400 </math> is <math>x_2(t_f) =1.82875272</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.82878681</math>.   
+
The optimal objective value of the relaxed problem with  <math> n_t=6000, \, n_u=150 </math> is <math>1.45412214e-05</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>2.40273813e-05</math>.   
  
−
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
+
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
−
  Image:MmlotkaRelaxed_12000_30_1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=400</math>.
+
  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=150</math>.
−
  Image:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
+
  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=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
 +
Image:FullerCIA 6000 40 2.png| Optimal binary controls.
 
</gallery>
 
</gallery>
  
Line 54: Line 54:
 
[[Category:Chattering]]
 
[[Category:Chattering]]
 
[[Category:Sensitivity-seeking arcs]]
 
[[Category:Sensitivity-seeking arcs]]
−
[[Category:Population dynamics]]
+
 
  
  

Latest revision as of 23:26, 8 January 2018

Fuller's initial value problem
State dimension: 1
Differential states: 2
Discrete control functions: 1
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.

Mathematical formulation

For t \in [t_0, t_f] 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, \\ & \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex] & x(0) &=& x_S, \\ & w(t) &\in& \{0, 1\}. \end{array}

Parameters

We use x_S = x_T = (0.01, 0)^T and (t_0,t_f) = (0,1).

Reference Solutions

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

The optimal objective value of the relaxed problem with n_t=6000, \, n_u=150 is 1.45412214e-05. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 2.40273813e-05.

  • Reference solution plots

  • Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and n_t=6000, \, n_u=150.

  • Optimal relaxed controls.

  • Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and n_t=6000, \, n_u=150. The relaxed controls were approximated by Combinatorial Integral Approximation.

  • Optimal binary controls.

Retrieved from "https://mintoc.de/index.php?title=Fuller%27s_initial_value_problem&oldid=2183"