Transcript Analysis of a Pendulum Problem

Analysis of a Pendulum
Problem
after Jan Jantzen
http://www.erudit.de/erudit/demos/cartball/index.htm
Inverted pendulum
• Balancing an inverted pendulum is a good demonstration
problem, because it is difficult, swift, and spectacular.
• It is a standard problem used in many classrooms and
commercial 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 mathematical
equations.
The problem
• 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.
They are nonlinear due to the trigonometric functions,
and they are coupled such that y occurs on the left side
of (A-6) and on the right side of (A-7); the situation is
the reverse in the case of  .


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 to
the trigonometric functions
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
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.
• 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+BK are in the right half of the complex plane (all k’s 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 Regulator – Matlab
Toolbox, will give a fast and stable controller with little overshoot from
K= [24,24,162,44]
Cascade Control
• It is quite intuitive to divide the system into two
subsystems, one for the ball, another for the cart;
– it makes it more manageable.
• The ball seems to require faster control reaction
than 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 angle 
makes it reach its reference  r ,
– which takes commands from a slower outer loop,
• in this case a PD controller reacting on the cart position
System Block Diagram
Fuzzy control of a pendulum problem
Fuzzy control Demo
The default membership
functions are triangular.
Examples of membership
functions are
• MVL (moves left),
• SST (stands still), and
• MVR (moves right).
Graph
Show Charts
When enabled the following Plots show up after starting a new simulation:
- cart position y and cart control signal U1 against time
- cart phase plot, g1*y against g2*dy
- ball angle  and ball control signal U2 against time.
- ball phase plot, g3*  against g4*d
- ball control signal U1, cart control signal U2, and U1+U2 against time