Analysis of a Pendulum Problem.ppt

7/30/2019 Analysis of a Pendulum Problem.ppt

1/17

Analysis of a Pendulum

Problem

after Jan Jantzen

http://www.erudit.de/erudit/demos/cartball/index.htm


2/17

Inverted pendulum

Balancing an inverted pendulum is a good demonstrationproblem, because it is difficult, swift, and spectacular.

It is a standard problem used in many classrooms andcommercial software packages.

This version is not the usual pole balancer, but rather a

steel ball rolling on a pair of arched tracks. The objective of the demo is to present the basic concepts

of fuzzy control, in an easily accessible manner.

The ball can be balanced using conventional techniques for

comparison. Fuzzy control is different in the sense that the control

strategy is a set of rules rather than mathematicalequations.


3/17

The cart moves on a pair of tracks horizontally mounted on

a heavy support.

The control objective is to balance the ball on the top of the

arc and at the same time place the cart in a desired position.

We will analyze the ball and cart separately and apply the

basic physical equations related to the vertical reaction force

Y and the horizontal reaction force K.

Friction forces are neglected.

The problem


4/17


5/17


6/17

They are nonlinear due to the trigonometric functions,

and they are coupled such that occurs on the left side

of (A-6) and on the right side of (A-7); the situation isthe reverse in the case of .

y


7/17


8/17

The model can be linearized around the origin. In order to avoid

errors we will linearize (A-6)-(A-7) rather than the nonlinear

state-space equations. Introduce the following approximations tothe trigonometric functions


9/17


10/17

With the data in Table 1 the

actual values of the

constants are:

a = -1.34

b = 0.301

c = 14.3

D = -0.386


11/17

State feedback control

Notice that the control signal is now the voltage U rather than

the force F, for convenience.

The block diagram shows how the four

states are fed back into the controller,which combines them linearly.


12/17

This is a state-space form as well, but of the closed-loop system.

Stability is guaranteed if none of the eigenvalues of the closed-loop system

matrix A+BKare in the right half of the complex plane (all ks must be

positive).

Jorgensen found (in 1974) by trial and error the following values satisfactory:

K= [5,5,120,8]

Using optimization techniques (Linear Quadratic RegulatorMatlab

Toolbox, will give a fast and stable controller with little overshoot from

K= [24,24,162,44]


13/17

Cascade Control

It is quite intuitive to divide the system into twosubsystems, one for the ball, another for the cart;

it makes it more manageable.

Theball seems to require faster control reactionthan the positioning of the cart,

and it is standard practice to have a fast inner loop,

in this case a PD controller reacting on the ball anglemakes it reach its reference ,

which takes commands from a slower outer loop,

in this case a PD controller reacting on the cart position

r


14/17

System Block Diagram


15/17

Fuzzy control of a pendulum problem

Fuzzy control Demo
http://localhost/var/Program%20Files/ERUDIT/Pendulum%20-%20Fuzzy%20Controller/Pendulum.exehttp://localhost/var/Program%20Files/ERUDIT/Pendulum%20-%20Fuzzy%20Controller/Pendulum.exe


16/17

The default membership

functions are triangular.

Examples of membership

functions are

MVL (moves left),

SST (stands still), and

MVR (moves right).


17/17

Graph

Show Charts

When enabled the following Plots show up after starting a new simulation:

- cart positiony and cart control signal U1 against time

- cart phase plot,g1*y againstg2*dy

- ball angle and ball control signal U2 against time.

- ball phase plot,g3* againstg4*

- ball control signal U1, cart control signal U2, and U1+U2 against time

d

Documents

Analysis of a Pendulum Problem.ppt