A three-dimensional fuzzy control methodology for a class of distributed parameter systems

The traditional fuzzy set is two-dimensional (2-D) with one dimension for the universe of discourse of the variable and the other for its membership degree. This 2-D fuzzy set is not able to handle the spatial information. The traditional fuzzy logic controller (FLD) developed from this 2-D fuzzy set should not be able to control the distributed parameter system that has the tempo-spatial nature. A three-dimensional (3-D) fuzzy set is defined to be made of a traditional fuzzy set and an extra dimension for spatial information. Based on concept of the 3-D fuzzy set, a new fuzzy control methodology is proposed to control the distributed parameter system. Similar to the traditional FLC, it still consists of fuzzification, rule inference, and defuzzification operations. Different to the traditional FLC, it uses multiple sensors to provide 3-D fuzzy inputs and possesses the inference mechanism with 3-D nature that can fuse these inputs into a so called "spatial membership function." Thus, a simple 2-D rule base can still be used for two obvious advantages. One is that rules will not increase as sensors increase for the spatial measurement; the other is that computation of this 3-D fuzzy inference can be significantly reduced for real world applications. Using only a few more sensors, the proposed FLC is able to process the distributed parameter system with little complexity increased from the traditional FLC. The 3-D FLC is successfully applied to a catalytic packed-bed reactor and compared with the traditional FLC. The results demonstrate its effectiveness to the nonlinear unknown distributed parameter process and its potential to a wide range of engineering applications.