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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