CHAPTER 6

TAKAGI–SUGENO FUZZY SYSTEMS

Takagi–Sugeno (T–S) fuzzy systems are more general than the Mamdani fuzzy systems discussed in Chapters 3 and 4. In fact, it can be shown that Mamdani systems are special cases of T–S fuzzy systems. The T–S systems are important because they enable a kind of control called parallel distributed control, they facilitate fuzzy identification of dynamic systems and adaptive fuzzy control, and they enable stability proofs for certain closed-loop systems involving fuzzy controllers. Their drawback is that they are less intuitive than Mamdani systems.

In T–S systems, the consequents of the rules do not involve fuzzy sets as do Mamdani systems, but instead are mathematical expressions. The mathematical expressions can be any linear functions of any variables. In this book, we consider only consequents that are either memoryless affine functions of the fuzzy system’s inputs, or one of the linear dynamic system model forms discussed in Sections 5.1 or 5.2. In the former case, the T–S fuzzy system performs an interpolation between memoryless functions. In the latter case, the T–S system performs an interpolation between dynamic systems. The latter case is useful for fuzzy identification and control.

6.1 TAKAGI–SUGENO FUZZY SYSTEMS AS INTERPOLATORS BETWEEN MEMORYLESS FUNCTIONS

Consider a T–S fuzzy system with R rules of the form:

(6.1)

The input universes ...