# Unstable systems

## Inverted pendulum

- Details
- Last Updated on Friday, 04 October 2013 12:45
- Published on Sunday, 20 May 2012 18:45
- Written by Jaroslav Kolařík
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**System description**

The system consists of a cart which can be moved along a metal guiding bar. An aluminium rod with a cylindrical weight is fixed to the cart by an axis. This system is unstable and non-linear with one input and two outputs. Input signal is control voltage of a DC motor which can change position of the cart. The outputs are cart position and angle of the pendulum rod. Both outputs are measured by incremental encoders.

1 - Servo amplifier, 2 - Motor, 3 - Drive wheel, 4 - Transmission belt, 5 - Metal guiding bar

6 - Cart, 7 - Pendulum weight, 8 - Guide roll, 9 - Pendulum rod

**Parameters of the real system AMIRA PS600:**

All the used constants were either taken from the producer (Amira, 2000) or identified by experiments (Chalupa & Bobál, 2008; Marholt, Gazdoš & Dostál, 2011):

Parameter |
Symbol |
Value & Unit |
Parameter |
Symbol |
Value & Unit |

Cart weight | mc | 4 kg | Inertia moment | Θ | 0.08433 kg.m2 |

Pendulum weight | mp | 0.36 kg | Cart friction | Fr | 6.5 kg/s |

Total weight | m | 4.36 kg | Pendulum friction | C | 0.00652 kg.m2/s |

Pendulum length | l | 0.42 m | Rate constant | ka | 7.5 N/V |

**Mathematical model**

The system can be described by the following nonlinear differential equations (Amira, 2000):

where *F* represents input signal, which is the force produced by the DC motor. Output signals are *r* - cart position (*r*' - denotes cart speed) and *φ* - pendulum angle (*φ*' - denotes pendulum angular speed). Symbol *g* is the gravity acceleration constant. All other constants and symbols are clearly defined in the presented table above.

The nonlinear differential equations were linearized in the operating point *φ* = 0 (top unstable position of the pendulum rod). A transfer function of the pendulum angle for this operating point then takes the following form (Marholt, Gazdoš & Dostál, 2011):

**Simulation files**

Downloading files is possible only for registered users.

**Information sources**

AMIRA. *PS600 Laboratory Experiment Inverted Pendulum*. Duisburg : Amira GmbH, 2000, 351p.