TSK Observers for Discrete Type-1 and Type-2 Fuzzy Systems

This paper proposes a new observer design methodology for type-1 and type-2 Takagi-Sugeno-Kang fuzzy systems. The design methodology does not require a common Lyapunov function and is, therefore, applicable to systems with nondetectable consequents. Our observers include fuzzy observers as well as constant gain observers. The observers are exponentially stable, and the designer can specify the rate of exponential convergence. The observer designs can be tested using linear matrix inequalities. The observers can be combined with a fuzzy controller for observer state feedback. The observer design methodology is demonstrated using the TORA system, and the observer state feedback is applied to the control of the inverted pendulum. The results show that the observer design provides good estimates of the state and can be used in a control system design even if the system has nondetectable consequents. Comparison to results from the literature shows that the new design can provide stable observer state feedback, where other observer designs fail.