16.6 Results

A GA-FC was developed that can efficiently manipulate the bouncing ball system. This controller alters the frequency of oscillation of the table, based solely on the velocity with which the ball strikes and on the phase angle of the table at the time of impact to ensure the ball bounces to a constant height. As pointed out earlier, the effectiveness of such a controller can be judged from a Poincaré map. Such a Poincaré map for the bouncing ball system wherein a trained GA-FC has been put into operation was created. However, this map included only five points in the space. Thus, the figure was not very clear. There are so few points because the trained controller is put into operation after four bounces. The resulting figure is far different from Figure 16.2, which shows a Poincaré map for the bouncing ball system with no control. The GA-FC has instilled order in an otherwise chaotic system.

For many, a Poincaré map is a natural vehicle for judging the performance of a controller in the realm of chaos. For others, however, it is difficult to use this type of map. To assist interpretation, Figure 16.5 shows the maximum height to which the ball bounces as a function of time. Recall that the objective of the controller was to adjust the frequency of oscillation of the table in such a manner as to force the ball to always bounce to a constant height. Figure 16.5 leaves no doubt as to the effectiveness of the GA-FC. Figure 16.5 shows the same simulation as does the Poincaré map depicted described above. In both, the controller is put into operation after four bounces.

For the simulation depicted in Figure 16.5, the controller is able to adjust the oscillatory frequency of the table so as to immediately cause the ball to bounce to the target height. However, this is certainly not always the case. A limit placed on the range of acceptable values for ω that reflects the limit on the hardware that could be used if the system were to be built. This limitation restricts the controller by restricting the amount of energy that can be imparted to or dissipated from the bouncing ball. As an example, if the ball is bouncing well above the target height, the maximum energy dissipation will occur when the ball strikes the table when the table has its maximum downward velocity. Since the value of ω is restricted, so too is the energy that can be dissipated. Likewise, if the ball is bouncing to a height well below the target height, the maximum energy that can be imparted to the ball will occur when the ball strikes the table when the table has its maximum upward velocity. Since the value of ω is restricted, so also is the energy that can be imparted to the ball.

Figure 16.5  The above plot of maximum ball height as a function of time indicates that the GA-FC effectively manipulates the chaotic system.

The GA-FC produces a control surface that can be compared with the analytical control surface shown in Figure 16.3. The surface produced by the GA-FC is shown in Figure 16.6. Notice that the GA-FC control surface is characterized by all of the complexities contained in the analytical control surface. However, simply comparing these two surfaces in such an inexact fashion is by no means adequate. Therefore, Figure 16.7 shows an error surface, which is simply the absolute value of the difference between the control actions prescribed by the analytical controller and by the GA-FC. Note that there is excellent agreement between the two controllers for most of the space. However, there are a few places in which the difference between the two controllers is substantial. These regions are characterized by discontinuities in the control surface. The GA-FC is sometimes unable to represent these discontinuities adequately because of the fact that fuzzy logic requires a discretization of the search space. Certainly this discretization is more flexible than the discretization possible with conventional set theory. At this point, the authors yield to the classical trade-off made in virtually all computational work; that of accuracy versus computational time. The magnitude of the error surface could well be reduced with greater discretization (the definition of more fuzzy sets), but this adds to the complexity of the controller and increases the time the controller needs to prescribe a control action.

Figure 16.6  The control surface produced by the GA-FC is remarkably similar to the analytical control surface.

Figure 16.7  The above error surface indicates the effectiveness of the GA-FC for controlling the chaotic system.

For most people, the information depicted by Figures 16.5 and 16.7 should be adequate to demonstrate the effectiveness of the GA-FC. However, there are some who prefer statistical analysis over graphical comparisons. For these people, the following analysis is included. For the purposes of this analysis, error, E, is defined as:

where ω_analytical is the oscillatory frequency of the table prescribed by the analytical controller and ω_GA-FC is the oscillatory frequency of the table prescribed by the GA-FC. One hundred thousand points were sampled to compute the average error, E_avg⁼0.4673%, and the standard deviation, σ=3.141. Moreover, 98.54% of the points sampled had an average error of less than 1.0%. The GA-FC produces an accurate depiction of the analytical control surface.

The above discussion demonstrates the accuracy of the GA-FC for the bouncing ball system. As with most chaotic systems, a controller for the bouncing ball system must be more than just accurate; it must also be fast. It is quite important to note that the analytical controller described earlier in this chapter takes on the order of one second to compute its control action, whereas the GA-FC takes on the order of milliseconds to compute its control action. This requirement for fast and accurate controllers becomes even more important in some other chaotic systems. An example of a chaotic system in which speed is at a premium appears in the steel industry in the form of an electric arc furnace (Ochs and Hartman, 1988).