Three Tank multimode problem

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

$\begin{array}{lll}\underset{x,w}{\min }& \underset{0}{\overset{T}{\int }}& k_1(x_2-k_2{)}^2+k_3(x_3-k_4{)}^2\text{d}t\\ \text{s.t.}& {\dot{x}}_1& =-\sqrt{x_1}+c_1{w}_1+c_2{w}_2-w_3\sqrt{c_3{x}_3},\\ & {\dot{x}}_2& =\sqrt{x_1}-\sqrt{x_2},\\ & {\dot{x}}_3& =\sqrt{x_2}-\sqrt{x_3}+w_3\sqrt{c_3{x}_3},\\ & x(0)& =(2,2,2{)}^T,\\ & 1& =\sum _{i=1}^3{w}_i(t),\\ & w_i(t)& \in \{0,1\},i=1\dots 3.\end{array}$

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 100 discretization intervals.
• 
  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

Source Code

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

• Casadi code at Three Tank multimode problem (python/casadi)