Abstract
An observer design for a class of nonlinear systems with unknown inputs is considered. Takagi–Sugeno fuzzy bilinear systems represent a wide class of nonlinear systems, and these systems with unknown inputs are an ideal model for many physical systems. For such systems, a design method for obtaining an observer that estimates the state of the system is proposed. A parallel distributed observer (PDO), which is constructed with local linear observers and the appropriate grade of the membership functions, is a conventional observer for Takagi–Sugeno fuzzy bilinear systems. However, it is known that its design conditions have conservativeness. In this paper, to reduce the conservatism in the design conditions, non-PDO with new design conditions is proposed. Our design conditions are derived from a multiple Lyapunov function, which depends on the membership function with time-delay in the premise variables. This method eventually reduces the conservatism and enables us to construct an observer for a wide class of nonlinear systems. When the premise variables are the state variables that are not measurable, Takagi–Sugeno fuzzy bilinear systems can represent a wider class of nonlinear systems. Hence, an observer design for fuzzy bilinear systems with unmeasurable premise variables is also proposed. Finally, numerical examples are given to illustrate our design methods.
Highlights
It is well known that the Takagi–Sugeno fuzzy system has a great potential to describe a wide class of nonlinear systems [1]
A bilinear system is a class of nonlinear systems, and its analysis and synthesis are of importance
Both the fuzzy bilinear system and standard fuzzy system can describe nonlinear systems, a fuzzy bilinear system has the advantage of its representation with a lesser number of subsystems, which significantly reduces the conservatism in analysis and synthesis [7]
Summary
It is well known that the Takagi–Sugeno fuzzy system has a great potential to describe a wide class of nonlinear systems [1]. The recent papers [17,18] proposed the control design method based on a new multiple Lyapunov function approach Since their Lyapunov function has an integral of the membership functions, any information on the derivatives of the membership functions is not necessary. Followed by [17,18], a class of multiple Lyapunov functions that contain an integral of the membership function of fuzzy systems is adopted This approach requires no information on the derivative of the membership function and is shown to reduce the conservatism in control design conditions. Our multiple Lyapunov function eventually does not require the upper bound of the derivative of the membership function Based on such a multiple Lyapunov function, an observer design method of fuzzy bilinear systems is proposed. Numerical examples are shown to illustrate our observer design method and to show the effectiveness of our approach
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