# Unstable systems

## Inverted pendulum

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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

Information sources

AMIRAPS600 Laboratory Experiment Inverted Pendulum. Duisburg : Amira GmbH, 2000, 351p.

CHALUPA, P., BOBÁL, V. Modelling and predictive control of inverted pendulum, Proceedings of 22nd European Conf. Modelling and Simulation, pp. 531-537, ISBN 978-0-9553018-5-8, Nicosia-Cyprus, June 2008, Digitaldruck Pirrot GmbH Sbr.-Dudweiler.

MARHOLT, J., GAZDOŠ, F., DOSTÁL, P. Control of the Unstable System of the Inverted Pendulum Using the Polynomial Approach. Cybernetic Letters. 2011, p. 1-5. ISSN 1802-3525.

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