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; Hits: 2883

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):