Certainly, this rule cannot actually be applied in a meaningful way until the linguistic terms shown in italics are provided with exact meaning using fuzzy membership functions. Thus, both fuzzy controllers and fuzzy models depict a relationship between “input” and “output” variables using linguistic rules and membership functions that define the linguistic terms appearing in the rules.

Figure 12.1  The tasks of developing both control systems and computer models are to define optimum surfaces. In the case of a controller, the correct control action is a function of the state of the system. In the case of a modeling system, the correct future state depends on the current state and any control actions taken on the system.

Once it is understood that computer models define a particular surface just as controllers do, then the steps necessary to develop a fuzzy computer model become apparent. In fact, the steps used for developing a fuzzy controller apply with only slight modification. The following steps can be used to develop a fuzzy model of a physical system:

1)  Determine the state variables and the forcing functions (those things that effect changes in the environment to be modeled) to be considered. These are the variables that represent the current state of the system and the forcings that are applied to the system to elicit a change in state.

2)  Determine the new state variables to be computed. These are the variables that the model is predicting.

3)  Describe the fuzzy sets for the current state variables, the forcings, and the new state variables to be predicted.

4)  Establish fuzzy production rules that cover all of the possible conditions that exist in the problem environment.

5)  Define the fuzzy membership functions for all descriptive terms to be used in the fuzzy rules.

6)  Apply the fuzzy production rules by taking a weighted average of the actions prescribed by all of the rules. The result is a single crisp numeric value for each of the state variables being predicted.

This procedure is almost identical to the procedure set forth in Chapter 2 for developing a fuzzy control system. And, in the next sections, we hope to show that these steps can be used to develop efficient computer models.

12.3 Fuzzy Model of A Grinding Circuit

In Chapter 11 we developed a neural network model of a grinding circuit. At that time, we pointed out some of the highlights of grinding, and some of the rationale for modeling grinding. Since we have already covered these points in some detail, we choose simply to provide a review at this point.

Grinding is a necessary component in the separation of a number of minerals, and improvements in grinding could provide substantial cost savings for the minerals separation industry. Because of the complex physics associated with grinding, it is a particularly difficult unit process to model. Thus, researchers have begun to turn their attention away from the development of first-principle models and toward alternative mechanisms of modeling.

The problem of modeling grinding using fuzzy mathematics is identical to the problem definition set forth in the neural network modeling application. Specifically, the chore is to predict four performance measures: (1) the fineness of the ground product, (2) energy costs associated with the process, (3) distribution modulus that is the packing density at minimal viscosity, and (4) the apparent viscosity of the ground product. Each of these four indicators are important in varying industrial situations. Although there are a number of parameters that can alter the performance of a grinding circuit, there are four variables that are especially important: (1) x_s, the percent solids by weight, (2) x_B, the maximum ball size, (3) x_M, mill speed, and (4) x_D, dispersant addition. Thus, the intent of the current effort is to develop a linguistic model in which x_s, x_B, x_M, and x_D are used to predict the fineness of the product, the energy consumed in a grinding circuit, the distribution modulus, and the apparent viscosity of the ground product. Further, only data describing values of these eight variables is to be used in the development of the model. Figure 12.2 shows a schematic of the linguistic computer model.

Figure 12.2  The linguistic model receives as inputs: (1) x_s, the percent solids by weight, (2) x_B, the maximum ball size, (3) x_M, mill speed, and (4) x_D, dispersant addition. It uses values of these parameters to estimate values of: (1) the fineness of the ground product, (2) energy costs associated with the process, (3) distribution modulus that is the packing density at minimal viscosity, and (4) the apparent viscosity of the ground product.

Fuzzy linguistic models describe a relationship between input and output variables in linguistic terms. Like conventional rule-based systems (commonly called expert systems), fuzzy linguistic models use a set of production rules that are of the form

The left-hand side of the rules consists of combinations of the input variables (the variables x_s, x_B, x_M, and x_D). The right-hand side of the rules consists of combinations of the output variables (the variables f, e, dm, and v). Unlike conventional expert systems, fuzzy linguistic models discussed here employ rules that utilize fuzzy terms like those appearing in human rules-of-thumb, and like those appearing in fuzzy controllers. For example, a reasonable rule for a fuzzy linguistic model used for a grinding circuit might be: