Granular Synthesis of Rule-Based Models and Function Approximation Using Rough Sets

Abstract

This chapter suggests a new method to develop rule-based models using concepts about rough sets. The rules encapsulate relations among variables and give a mechanism to link granular descriptions of the models with their computational procedures. An estimation procedure is suggested to compute values from granular representations encoded by rule sets. The method is useful to develop granular models of static and dynamic nonlinear systems and processes. Numerical examples illustrate the main features and the usefulness of the method.

Rough Controller Synthesis

“…In this example the linear function was associated with the function of a proportional-integral controller. These results can also be confronted with those obtained in the work referenced in Pinheiro et al, 2010. In Example 5 a practical context of adaptive gains is synthesized through a rough controller in the control of a nonlinear system.…”

“…As an example of the calculation of the polynomial coefficient functions, using the decision rule in the form (12) with x 1 (k) = 0, x 1 (m) = 1, x 2 (k) = 0, x 2 (m) = 1, y (k) = 0 and y (m) = 2, where using (13) we have y = ((2 -0)/2)((x 1 -0)/(1 -0) + (x 2 -0)/(1 -0)) = x 1 + x 2 which defines the coefficients of (16). Other examples of fuzzy models obtained with this methodology are illustrated in Pinheiro et al, 2010. (17)…”

Time series forecasting through rule-based models obtained via rough sets

“…Section 2 brings a quick review on the concepts of rough sets and describes the method proposed in Pinheiro et al (2009), it presents an extension of how the method may be applied in time series forecasting. Section 3 presents a brief review about artificial neural networks and how they may be applied in predictions.…”

“…An extension of the method described in Pinheiro et al (2009) will be used, which shows how rough set-based models may be used as function approximators.…”