16.4 Surface-Fitting and the Analytical Solution

As stated above, the problem of controlling the bouncing ball system is a surface-fitting problem: for any possible value of the ball’s velocity at impact, v, and any possible value of the phase angle of the table, 0 ≤ θ ≤ 2π, the frequency of oscillation of the table, ω, can be adjusted to ensure that the ball bounces to the desired height, y_desired. The equations of motion can be used to compute the control surface. Although the equations of motion for the system are rather simple, obtaining the solution to these equations is tedious. The solution is the following equation, which includes both y_desired and the value of ω necessary to ensure the ball bounces to a maximum height of y_desired:

where g is the acceleration due to gravity, A is a constant that indicates the magnitude of the oscillation, e is the coefficient of restitution, ν₀ is the initial velocity of the ball, and t_c is the time at which the next collision occurs where θ = ωt_c.

Equation (16.1) gives the appropriate value of ω for any given values of v_o, θ, e, A, and y_desired. In this study, the following values were used: e=0.9, A=1.0, and y_desired⁼20.0. With these values of the parameters, an appropriate control space consisted of 10.0 ≤ ν_ball ≤ 40.0 and 0 ≤ θ≤ 2π. For values of ν_ball and θ in this range, equation (5) must be solved numerically. The process of solving this equation, although not difficult using a traditional root-finding method, is quite tedious. There are multiple roots to the equation, and great care must be taken to ensure that the correct root is found and used.

Equation (16.1) was evaluated at numerous points using a combination of a Newton method and a bisection method. This approach allowed for the location and selection of the appropriate roots. The results yield the analytical control surface shown in Figure 16.3. Note the complex nature of this surface. This complexity must be fully described with fuzzy logic, because the bouncing ball system is very sensitive to alterations in ω.

Figure 16.3  The “analytical control surface” shown above depicts the complex control surface for this chaotic system. Further, since this is a chaotic system, the controller being developed must fit this surface precisely.

16.5 GA-Designed Fuzzy Logic Controller

The procedure for developing a GA-FC for the bouncing ball system is the same as presented in the early chapters of this book. Thus, only the details of the application are presented here. As presented here, the chaotic system does not require an adaptive controller because there are no changing system parameters that are not condition variables. An adaptive controller would be required if a parameter such as the coefficient of restitution were changing. However, for our current GA-based approach to work the changes in e would have to occur infrequently relative to the time constant of the system so that the GA would have time to discover new control strategies.

The fuzzy controller takes two input variables, v and θ, and one output variable, ω. Because of the sensitive nature of the chaotic system, numerous linguistic terms are required to adequately represent each of the pertinent controlled variables, v and θ. Twenty-nine linguistic terms were used to represent both v and θ. The manipulated variable, ω, was represented using triangles only, and an infinite number of linguistic terms were used since the real-valued consequent approach was employed. As normal, these linguistic terms are defined by membership functions. The analytical control surface shown in Figure 16.3 depicts the intricate detail of the terrain that must be represented with a fuzzy rule base. Notice that the contours of the control surface seem to be linear in some places, sinusoidal in other places, and exponential in still other places. Those sections of the surface that appear to be linear are accurately represented by triangular membership functions (trapezoidal functions could have also been used). However, to accurately represent the other regions, more complex membership function forms were needed. Intuitively, one might try to represent the sinusoidal regions with sinusoidal membership functions, and the exponential regions with exponential membership functions. Both sinusoidal and exponential membership functions can be represented with appropriate general equations. The sinusoidal membership functions are described with an equation of the following form:

where c is the center (location of maximum height) and w is the width. The exponential membership functions are described with an equation of the following form:

where c is the center (location of maximum height), w is the width, and k is a constant defining the curvature of the exponential function. This happens to be a case where intuition is well served. It turns out that the above three membership function forms (triangular, sinusoidal, and exponential) can represent the control surface. These membership function forms are shown in Figure 16.4.

Figure 16.4  The above membership function forms were necessary and adequate to represent the complex control surface of the chaotic system.

A GA can be used to design both the rules and membership functions for the chaotic FC. Although the problem is larger (29 membership functions for each of the two condition variables) than those in previous chapters, the approach to using the GA is the same. A portion of a bit string is used to represent each of the membership functions while other pieces of the bit string are used to represent rules. Because of the vast size of the search space (there are 29*29=841 rules), the problem was tackled in pieces. The search problem was solved piecewise; only small portions of the search space were solved at a time. In this way, the GA was presented with a manageable problem.