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 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

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 is the reverse in the case of .

y

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 ,

– which takes commands from a slower outer loop, • in this case a PD controller reacting on the cart position

r

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*

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

d