In this chapter, a particular chaotic system is presented for which the control surface has been obtained. Specifically, a ball bouncing on an oscillating table is considered. The frequency of oscillation of the table is adjusted so that the ball bounces to a constant height. The control variable, the desired frequency of oscillation,ω, is a function of two state variables: (1) the velocity of the ball when it leaves the table, v, and (2) the phase angle of the table at impact, θ.

The control surface of the chaotic bouncing ball system is very complex. However, despite its complexity, this surface can be defined accurately with fuzzy production rules. A GA is used to specify the fuzzy membership functions because the FC must be extremely accurate to suitably control the chaotic system. Unfortunately, just using GAs to define the membership functions does not guarantee the necessary precision. The FC must utilize a large number of membership function forms, and it must also be provided with great flexibility in defining the consequent for the fuzzy rules. Triangular, trapezoidal, sinusoidal, and exponential membership functions are necessary. The flexibility in defining the membership functions for the consequent of the rules is obtained by employing a real-valued consequent.

Details are provided in this chapter for an FC that is then used to control a computer simulation of the chaotic bouncing ball system. The results demonstrate the effectiveness of using an FC to control a chaotic system. These results are important for two reasons. First, controlling chaotic systems is very difficult, and second, fuzzy logic provides a viable alternative to some of the highly mathematical techniques that have been reported. The successful use of fuzzy logic to control a chaotic system demonstrates that inexact reasoning can be used to obtain very precise control.

16.3 The Chaotic System

The bouncing ball system was selected simply to demonstrate the effectiveness of using a particular control strategy to manipulate chaotic systems. It was chosen from the many possible chaotic systems because it is easy for most people to gain a “feel” for the response of this system, and because it is relatively simple.

A ball is dropped from an arbitrary height, and bounces on a table that moves sinusoidally up and down as shown in Figure 16.1. While the gravitational force of the system and the coefficient of restitution of the ball are constant, the height to which the ball bounces is determined strictly by the velocity of the ball relative to the table at impact, and by the phase angle of the table when an impact occurs. If the ball impacts the table when the table is moving upward, the relative velocity is large and the ball bounces higher than it would on a stationary table. If the ball impacts the table when the table is moving downward, the relative velocity is small and the ball bounces lower than it would on a stationary table. The net effect is that the height to which the ball bounces is deterministic, yet it has no apparent long-term pattern. Furthermore, the time-history of the ball (its position at a particular time) is extremely sensitive to the initial height from which it is dropped — the system is chaotic.

Figure 16.1  The chaotic bouncing ball system possesses the complex attributes characteristic of chaotic systems yet is simple enough for most people to get a feel for it.

Poincaré maps are used to determine whether a system is chaotic. A Poincaré map for the bouncing ball system appears in Figure 16.2. In this plot, the polar coordinate r, the distance from the origin, represents the velocity of the ball at impact. The angular coordinate θ represents the phase angle of the table when an impact occurs. Thus, a point on the Poincaré map completely describes the state of the bouncing ball system at the time of each collision. The time of the next impact, and the state of the bouncing ball at the time of the next impact can be computed from this information. Since the Poincaré map does not consist of either a finite set of points forming a pattern (indicating the system is in some way repetitive) or a closed orbit (indicating the system is almost periodic or quasiperiodic), the motion is chaotic. The ball bounces to a seemingly random height each time it contacts the table. However, this does not have to be the case.

Figure 16.2  A Poincaré map can be used to determine whether a system is chaotic. In this plot, the polar coordinate r, the distance from the origin, represents the velocity of the ball at impact. The angular coordinate θ represents the phase angle of the table when an impact occurs.

The frequency at which the table oscillates is adjustable, and changing this frequency dramatically affects the height to which the ball bounces. In fact, the frequency of oscillation can be altered to force the ball to bounce to a constant height, i.e., to force the ball to always strike the table with the same relative velocity at the same phase angle. Thus, this becomes the objective of a control problem: to adjust the table’s frequency of oscillation in such a way as to cause the ball to rise to the same maximum height after each bounce. Moreover, a Poincaré map provides a convenient mechanism for evaluating the effectiveness of a controller. An effective control system should produce a Poincaré map that consists of a single point, because the response of the system will be wholly repetitive.

The bouncing ball system presents an especially appealing problem environment because a control surface can be described analytically using the equations of motion of the system, and thus there is an exact solution to which the control surface developed by the GA-FC can be compared. Second, the details of the GA-FC are described. This description includes the necessary information for utilizing a GA to tune a FC. Third, results are presented that portray the effectiveness of using an FC to manipulate the chaotic bouncing ball system. These results are important because they demonstrate the effectiveness of using fuzzy logic to obtain a precise solution.