About Fuzzy Control

Swing-up Inverted Pendulum

It seems natural to assume that most of people have a common experience of playing with an inverted stick on top of one’s palm at childhood. If you may remember, the know-how in stabilizing the stick lies on your technique to move your arm by judging the movement of the stick head with your eyes. It is a human rule of thumb that even a child can subconsciously go through the experiences of feedback control without knowledge of control theory.

In this Fuzzy control training kit, a mechanical arm and motor take places of the human arm and muscles. Instead of the human eyes, the potentiometers attached at the end and center of the arm provide the tilting status of the pendulum and the position data of the arm.

There are several reasons why the inverted pendulum is employed as a control object. Comparative low cost for fabrication is one reason. It is the other reason to have an advantage of easy check by a glance whether the control works well or not since the inverted pendulum is unstable and needs a fulltime control to prevent from toppling down.

As the typical means of automatic control to achieve such tasks, there are Modern control theory and Fuzzy control theory.

Modern Control Theory
In Modern linear control theory, it is required to exactly introduce a mathematical model (state equation) which represents the characteristics of the control object and then design the controller based on it. Its steps can be briefly shown as follows: step-1 to seek the equations of motion (horizontal, vertical and rotary directions) for the arm and pendulum, step-2 to convert the sought out nonlinear differential equations into linear ones by assuming its tilting angle as extremely small and step-3 to determine the mathematical model (state equation) by adding the time functions to the model which are normally used for the linear control theory as well as deforming them.
According to the determined model, the controller must be designed in consideration of such characteristics as stability / controllability of the first and second order lag models, stability of feedback control, feedback-gain design by pole placement, parameter identification, etc.

Fuzzy Control Theory
In Fuzzy control theory, it is not required to understand or use the knowledge of dynamics, Modern control theory and mathematics. It adopts certain rules which is the translation of the human experience, sixth-sense and know-how.
With regard to the inverted pendulum, most people already have some experiences to orally express the control method without need of further experience anew.
In case of Fuzzy control, it is simply required to translate the contents of the oral expressions into the membership functions so that the inverted pendulum can be stabilized comparatively easy. The contents explained in Modern control theory above can be controlled by translating the human rule of thumb acquired at childhood into 5 rules of the membership functions. In this "Pendulum Fuzzy Control Course', three steps of control to swing-up the tilted pendulum, to stop and stabilize it at around its top-dead-point (TDP) are realized by creating 11 rules. Upon designing the kit (selection of motor), however, it is required to compute and select the devices suitable to the purpose.

Upon creating the Fuzzy rules for a swing-up inverted pendulum, the procedure requires such three steps of creation and tuning of the rules as: to optimally stabilize the pendulum at around its top-dead-center, to swing up the tilted pendulum toward its TDP at a proper motor speed and to securely stop the pendulum at the proximity of its TDP.
The above procedure was actually applied upon developing this training kit since the control of the inverted pendulum was the original plan and the swing-up motion was just added lately.
Even though 11 rules are applied here, there actually exist 6 rules (11 / 2 = 5.5) since clockwise and counterclockwise motions are similar.

1.Stabilizing Inverted Pendulum at TDP

Rule No. 1 - 3

Move the arm toward the pendulum tilting direction

Rule No. 4 - 5

Move the arm toward the pendulum tilting direction faster so as to stop the arm not to reach to the mechanical stopper

2.Swinging-up Tilted Pendulum toward TDP

Rule No. 5, 6

Move the arm opposite to the pendulum tilting direction

3.Stopping Swing-up Pendulum at TDP

Rule No. 9, 10

Move the arm away from the approaching pendulum since the reverse motion causes rotation

Rule No. 8, 11

Move the arm further away from the approaching pendulum to kill its momentum unless the pendulum stops by the above rule No. 9, 10

 (C) 2004 ADWIN Corp.