Takagi-Sugeno (TS, for short) fuzzy controllers have been used and treated as black-box controllers, and there exists no explicit structure of any TS fuzzy controller in the literature. In this paper, we investigate the analytical structure of the TS fuzzy controllers that use our newly introduced simplified TS control rule scheme. Our scheme requires that all the rule consequent employ a common linear function and be proportional to one another. The scheme drastically reduces the number of adjustable parameters required by the original TS rule scheme. Product fuzzy logic AND was employed in the rule antecedents. We theoretically proved that the TS controllers using the simplified rule scheme were nonlinear variable gain controllers. The proportionality of the rule consequent parameterized the gain variation characteristics. As special cases, the TS controllers with three or two input variables and the simplified linear rule consequent were nonlinear variable gain PID, PI or PD controllers. As an illustration, we mathematically analyzed the nonlinear variable gain PI controllers, and revealed that the gain variation empowered the controllers to outperform their linear counterparts. Stability of the nonlinear PI control systems was also addressed. A practical example is given to show how to achieve the desired characteristics of the gain variation by properly designing the simplified rules.