Double Tank

Double Tank
State dimension:	1
Differential states:	2
Continuous control functions:	0
Discrete control functions:	1
Interior point equalities:	2

The double tank problem is a basic example for a switching system. It contains the dynamics of an upper and a lower tank, connected to each other with a pipe. The goal is to minimize the deviation of a certain fluid level $k_2$ in the lower tank. The problem was introduced and discussed in a variety of publications for the optimal control of constrained switched systems, e.g. Henrion et al. and Vasudevan et al.

Mathematical formulation

$\begin{array}{lll} \displaystyle \min_{\sigma} & \displaystyle \int_{0}^{T}k_1(x_2-k_2)^2 \; \text{d}t &\\[1.5ex] \mbox{s.t.} & \displaystyle \dot{x}_1(t) = c_{\sigma(t)}-\sqrt{x_1(t)} \qquad &\text{for } t\in[0,T], \\[1.5ex] & \displaystyle \dot{x}_2(t) = \sqrt{x_2(t)}-\sqrt{x_2(t)} \qquad &\text{for } t\in[0,T], \\[1.5ex] & \displaystyle x_i(0)=x_{i0} \qquad &\text{for } i=1,2, \\[1.5ex] & \displaystyle \sigma(t) \in \{1,2\} \qquad &\text{for } t\in[0,T],\\[1.5ex] \end{array}$

The two states of the system correspond to the fluid levels of an upper and a lower tank. The output of the upper tank flows into the lower tank, the output of the lower tank exits the system, and the flow into the upper tank is restricted to either $c_1$ [lt/s] or $c_2$ [lt/s]. The dynamics in each mode are then derived using Torricelli’s law, as shown in constraints 1 and 2. The objective of the optimization is to have the fluid level in the lower tank equal to $k_2$ [m], as reflected in the cost function.

Parameters

In an exemplary test, the parameters were chosen to be:

State variables

Symbol	Initial value ($x_{i0}$)
$x_1(t)$	$2$
$x_2(t)$	$2$

Parameters

Symbol	Value
$k_1$	$2$
$k_2$	$3$
$c_1$	$1$
$c_2$	$2$
$T$	$10$

Reference Solution

By introducing the lifts $l_i=\sqrt{x_i}$, algebraically constrained as $l_i^2=x_i, \; l_i\geq 0,$ the problem is recast with polynomial data. In this way way switch in connection with GloptiPoly3(Verlinkung),(Erklärung) can be applied.

Source Code

With the parameters above, the optimal control problem was tested using the following code.