MODELING OF DYNAMIC PLANTS AS FUZZY SYSTEMS
The estimation methods of Chapter 8 can be used to find fuzzy models for nonlinear dynamic systems. The determination of a model for an unknown system is known as identification. Usually, the object of doing so is to utilize the model in a parallel distributed control scheme (see Chapter 7). Design of a parallel distributed controller is relatively straightforward because the consequents in the fuzzy model are linear systems.
In Section 9.1, we present a method of deriving a fuzzy model for a nonlinear system whose equations of motion are known. This enables parallel distributed control of the nonlinear system when derivation of the controller directly from the nonlinear equations of motion may be impossible.
If the controller is to be capable of adjusting itself in real time (as in adaptive fuzzy control), the method used for identification must be recursive. In Chapter 8, we give two methods of recursive identification: recursive least squares (RLS) and gradient. Therefore, we will concentrate on these methods in this chapter.
9.1 MODELING KNOWN PLANTS AS T–S FUZZY SYSTEMS 
Many nonlinear systems with known mathematical models may be exactly modeled on a bounded domain in the state space with Takagi–Sugeno (T–S) fuzzy systems. Consider a nonlinear time-invariant system for which x = 0 is the equilibrium point. The basic idea of modeling it with a T–S fuzzy system is to express it as a series of linear dynamic systems, ...