Placeholder for the InvertedPendulum example model, available on the Wolfram SystemModeler website.

Wolfram Language




Library Dependency

This is a placeholder model. It requires the ModelPlug and PlanarMechanics libraries.

  • ModelPlug is a free Modelica library that allows you to link your simulation models to the real world by the use of an Arduino Board or any other board following the Firmata protocol). ModelPlug opens up a wide variety of ways to interact with your models, for example by using buttons, switches, input sensor information or even actuators such as motors and servos.
  • The free PlanarMechanics library was created especially for modeling multibody systems with two-dimensional mechanical components. Compared to the MultiBody library, currently available in the Modelica Standard Library, it is simpler to use and it is more optimized to planar modeling. Planar models of mechanical systems are useful in many different applications, for example, in contact problems that are more easily modeled in 2D than in 3D.

The following documentation is taken from the main example of the downloadable model. Once you have downloaded all the model dependencies, the model can be downloaded here.

LQR-controlled inverted pendulum system



This model studies how a linear-quadratic regulator (LQR) can be used to stabilize an inverted pendulum.

The model requires the free PlanarMechanics library, availiable from the SystemModeler Library Store.



The inverted pendulum model consists of a pendulum and a cart, with the axis of rotation of the pendulum being located at the center of the cart. The pendulum is initialized with its center of mass above the rotation axis. When located directly above the cart, the pendulum will be in steady state and will stay there until disturbed.

While a pendulum that hangs straight down will be in a stable position, an inverted pendulum is unstable. That means that any small disturbance will cause the pendulum to tip over and never return to its original position.



Control System

A control system can be used to stabilize the unstable pendulum. Here, a linear-quadratic regulator is used. First, the regulator measures the states of the system, namely the cart position and velocity, the pendulum angle and the angular velocity. The regulator then calculates a force that should be applied on the cart in order for all the states to become zero. In other words, the regulator aims to have the pendulum pointing straight up (defined here as 0 degrees) and the cart return to the origin.




By simulating the model, you can try different values of the parameters and see how the control system responds.

To simulate the model and see the generated 3D animation, follow the steps below:

  • Click the Simulate button simulate.
  • When the simulation is finished, click the Animate button animate in Simulation Center.
  • Use your mouse or trackpad to drag the animation into a good angle and zoom in with your scroll wheel or by using the trackpad. Then click the Play button simPlay to play the animation.

You can change the magnitude of the force that is applied to the pendulum by changing the pendulumDisturbanceForce parameter.