Difference between revisions of "Double Tank multimode problem"

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  Image:MmlotkaRelaxed_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: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>.
 
  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.
 
  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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Revision as of 16:29, 10 January 2018

Double 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.

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 \; \text{d}t\\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = \sum\limits_{i=1}^{3} c_{i}\; w_i,-\sqrt{x_1}, \\[1.5ex] & \dot{x}_2 & = \sqrt{x_1}-\sqrt{x_2}, \\[1.5ex] & x(0) & = (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=10, c_1=1, c_2=0.5, c_3=2, k_1=2, k_2=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.

The optimal objective value of the relaxed problem with n_t=12000, \, n_u=100 is 2.59106823. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 2.59121008.

  • Reference solution plots

  • Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and n_t=12000, \, n_u=100.

  • Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and n_t=12000, \, n_u=100. The relaxed controls were approximated by Combinatorial Integral Approximation.

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