Difference between revisions of "Three Tank multimode problem"

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== Reference Solutions ==
 
== Reference Solutions ==
  
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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.  
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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 and the CIA decomposition. We denote the relaxed control values with <math>a(t)\in[0, 1]</math>.
  
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The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=100  </math> is <math>2.59106823</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>2.59121008</math>.   
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The optimal objective value of the relaxed problem with  <math> n_t=100, \, n_u=100  </math> is <math>8.775979</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>8.789487</math>.   
  
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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  Image:MmdoubletankRelaxed 12000 120 1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=100</math>.
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  Image:three_tank_relaxed_solution.png| Optimal relaxed controls and states determined by an direct approach with python 3.6 and CasADi, applied Multiple Shooting, 4th order Runge Kutta scheme and <math>n_u=100</math>.
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  Image:MmdoubletankCIA_12000_120_1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
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  Image:three_tank_binary_solution.png| According optimal binary controls and states determined by the direct approach. The relaxed controls were approximated by Combinatorial Integral Approximation.
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Image:three_tank_rounding_solution.png| Binary and relaxed control values as part of the Combinatorial Integral Approximation problem
 
</gallery>
 
</gallery>
  

Revision as of 10:12, 14 March 2020

Three Tank multimode problem
State dimension: 1
Differential states: 3
Discrete control functions: 3
Interior point equalities: 2

This site describes a Double tank problem variant with three binary controls instead of only one control and three tanks, i.e., three differential states representing different compartments.

Mathematical formulation

The mixed-integer optimal control problem is given by

\begin{array}{llll} \displaystyle \min_{x,w} & \displaystyle \int_{0}^{T} & k_1(x_2-k_2)^2 + k_3(x_3-k_4)^2 \; \text{d}t\\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = -\sqrt{x_1}+c_1 w_1 + c_2 w_2 - w_3 \sqrt{c_3 x_3}, \\[1.5ex] & \dot{x}_2 & = \sqrt{x_1}-\sqrt{x_2}, \\[1.5ex] & \dot{x}_3 & = \sqrt{x_2}-\sqrt{x_3}+w_3 \sqrt{c_3 x_3}, \\[1.5ex] & x(0) & = (2,2,2)^T, \\[1.5ex] & 1 & = \sum\limits_{i=1}^{3}w_i(t), \\ & w_i(t) &\in \{0, 1\}, \quad i=1\ldots 3. \end{array}


Parameters

These fixed values are used within the model.

T=12, c_1=1, c_2=2, c_3=0.8, k_1=2, k_2=3, k_3 = 1, k_4 = 3.

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 and the CIA decomposition. We denote the relaxed control values with a(t)\in[0, 1].

The optimal objective value of the relaxed problem with n_t=100, \, n_u=100 is 8.775979. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 8.789487.

  • Reference solution plots

  • Optimal relaxed controls and states determined by an direct approach with python 3.6 and CasADi, applied Multiple Shooting, 4th order Runge Kutta scheme and n_u=100.

  • According optimal binary controls and states determined by the direct approach. The relaxed controls were approximated by Combinatorial Integral Approximation.

  • Binary and relaxed control values as part of the Combinatorial Integral Approximation problem

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