In this paper, the design problem of a delayed output feedback control scheme using two-layer interval fuzzy observers for a class of nonlinear systems with state and output delays is investigated. The Takagi-Sugeno-type fuzzy linear model with an online update law is used to approximate the nonlinear system. Based on the fuzzy model, a two-layer interval fuzzy observer is used to reconstruct the system states according to equal interval output time delay slices. Subsequently, a delayed output feedback adaptive fuzzy controller is developed to overcome the nonlinearities, time delays, and external disturbances such that H∞tracking performance is achieved. The linguistic information is developed by setting the membership functions of the fuzzy logic system and the adaptation parameters to estimate the model uncertainties directly using linear analytical results instead of estimating nonlinear system functions. The tracking error dynamics are designed to enable our adaptive controller to avoid the filtering of the basis vectors whose dimension is much larger than that of the state vector of the controlled system. This is achieved by not imposing the strictly positive real condition. Based on the Lyapunov stability criterion and linear matrix inequalities, some sufficient conditions are derived so that all the states of the system are uniformly ultimately bounded. Therefore, the effect of the external disturbances on the tracking error can be attenuated to any prescribed level, and H∞tracking control is achieved. Building on our previous work in this area, the proposed control scheme is extended to handle a class of uncertain nonlinear systems with state and output delays and external disturbances. This is achieved through the use of a robust variable structure scheme and H∞control techniques. Finally, a numerical example of a two-link robot manipulator is studied to illustrate the effectiveness of the proposed control scheme.

Haptic interface devices are machines that are controlled by the human arm contact forces. These devices are necessary elements of virtual reality machines. These devices may be programmed to give the human arm the sensation of forces associated with various arbitrary maneuvers. As examples, these devices can give the human the sensation that he/she is maneuvering a mass, or pushing onto a spring or a damper. In general, these devices may be programmed for any trajectory-dependent force. To illustrate and verify the analysis of these machines, a two-degree-of-freedom electrically-powered haptic interface device was designed and built at the Human Engineering Laboratory (HEL) of the University of California-Berkeley.>