Two-Tank System

Two tanks are connected in series. The inflow $q_{\mathrm{in}}$ of tank 1 is controlable by a pump with the voltage $u_{\mathrm{A}}$ , whereby the inflow $q_{1,2}$ and the outflow $q_{\mathrm{out}}$ of tank 2 are adjustable by valves. Moreover, the height $h_2$ of tank 2 is measured and the height $h_1$ of tank 1 is unknown. Both tanks have the same overall height $H$ and the same base area $A_{\mathrm{T}}$ .

Figure made with TikZ

For the mathematical represenation both tanks are considered seperatly.

For the first tank:

$A_{\mathrm{T}} \dot{h}_1(t) & = q_{\mathrm{in}} - q_{1,2}$

assuming the inflow

$q_{\mathrm{in}} & = K u_{\mathrm{A}}$

with the proportional factor $K$ and the flow between the tanks with Torricelli’s law

$q_{1,2} & = A_{\mathrm{out},1} \sign(h_1 - h_2)\sqrt{ 2 g \left|h_1 - h_2\right|}$

For the second tank:

$A_{\mathrm{T}} \dot{h}_2(t) & = q_{1,2} - q_{\mathrm{out}}$

assuming the linear resistance to flow

$q_{\mathrm{out}} & = A_{\mathrm{out},2} \sqrt{ 2 g h_2}$

The complete nonlinear system with the system vector

$\boldsymbol{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} h_1 \\ h_2 \end{pmatrix}$

is given by

$\dot{\boldsymbol{x}} = \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} \frac{K}{A_{\mathrm{T}}} u_{\mathrm{A}} - \frac{A_{\mathrm{out},1}}{A_{\mathrm{T}}} \sign(x_1 - x_2)\sqrt{ 2 g \left|x_1 - x_2\right|} \\ \frac{A_{\mathrm{out},1}}{A_{\mathrm{T}}} \sign(x_1 - x_2)\sqrt{ 2 g \left|x_1 - x_2\right|} - \frac{A_{\mathrm{out},2}}{A_{\mathrm{T}}} \sqrt{ 2 g x_2} \end{pmatrix}.$

Violations of the model’s boundary conditions are the water levels of both tanks exceed the maximal height $H$

$x_1 > H, \quad x_2 > H.$

The height of tank 2

$y & = x_2 = h_2$

is chosen as output.

The example comes with two controllers. The CppPIDController implemenents a PID controller in C++ and uses pybind11 as binding between C++ and python. The py:class:CppStateController linearizes the nonlinear model in a chosen steady state and applies static state feedback. The state feedback is implemented in C++ and uses pybind11 as binding between C++ and Python.

The example comes with one observers. The py:class:CppHighGainObserver implements a High-Gain observer for the nonlinear system in C++, which uses pybind11 as binding between C++ and python.

A 3D visualizer isn’t implemented, but a 2D visualization can be used instead.

An external py:data:settings.py file contains all parameters. All implemented classes import their initial values from here.

At program start, the main loads two regimes from the file default.sreg. controller-PID is a setting using a PID controller to stabilize the water level of tank 2 at a specific height. controller-State is a settings using a nonlinear observer and a linearized state feedback to stabilize the water level of tank 2 at a specific height.

The structure of __main__.py allows starting the example without navigating to the directory and using an __init__.py file to outsource the import commands for additionl files.