Three Tank multimode problem

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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

 \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.


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.

Source Code

Model description is available in (using pycombina for solving the (CIA) rounding problem step):