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This article presents a new observer design framework for a class of nonlinear descriptor systems with unknown but bounded inputs. In the presence of unmeasured nonlinearities, that is, premise variables, designing nonlinear observers is known as particularly challenging. To solve this problem, we rewrite the nonlinear descriptor system in the form of a Takagi-Sugeno (TS) fuzzy model with nonlinear consequents. This model reformulation enables an effective use of the differential mean value theorem to deal with the mismatching terms involved in the estimation error dynamics. These nonlinear terms, issued from the unmeasured nonlinearities of the descriptor system, cause a major technical difficulty for TS fuzzy-model-based observer design. The descriptor form is treated through a singular redundancy representation. For observer design, we introduce into the Luenberger-like observer structure a virtual variable aiming at estimating the one-step ahead state. This variable introduction allows for free-structure decision variables involved in the observer design to further reduce the conservatism. Using Lyapunov-based arguments, the observer design is reformulated as an optimization problem under linear matrix inequalities with a single line search parameter. The estimation error bounds of both the state and the unknown input can be minimized by means of a guaranteed 𝓁 ∞ -gain performance level. The interests of the new 𝓁 ∞ TS fuzzy observer design are clearly illustrated with two physically motivated examples.

This paper presents a new approach to design static output feedback (SOF) controllers for constrained Takagi-Sugeno fuzzy systems with nonlinear consequents. The proposed SOF fuzzy control framework is established via the absolute stability theory with appropriate sector-bounded properties of the local state and input nonlinearities. Moreover, both state and input constraints are explicitly taken into account in the control design using set-invariance arguments. Especially, we include the local sector-bounded nonlinearities of the fuzzy systems in the construction of both the nonlinear controller and the nonquadratic Lyapunov function. Within the considered local control context, the new class of nonquadratic Lyapunov functions provides an effective solution to estimate the closed-loop domain of attraction, which can be nonconvex and even disconnected. The convexification procedure is performed using specific congruence transformations in accordance with the special structures of the proposed SOF controllers and nonquadratic Lyapunov functions. Consequently, the fuzzy SOF control design can be reformulated as an optimization problem under strict LMI constraints with a linear search parameter. Compared to existing fuzzy SOF control schemes, the new structures of the control law and the Lyapunov function are more general and offer additional degrees of freedom for the control design. Both theoretical arguments and numerical illustrations are provided to demonstrate the effectiveness of the proposed approach in reducing the design conservatism.

This paper investigates the design of unknown input (UI) observers for a large class of nonlinear systems using Takagi-Sugeno (TS) fuzzy modeling. To avoid the well-known issue on the unmeasured premise variables in fuzzy observer design, we reformulate the nonlinear systems in a TS fuzzy form with local nonlinear models. A particular feature of these so-called N-TS fuzzy models is that all the unmeasured nonlinearities are isolated in a nonlinear consequent. Together with a judicious use of the differential mean value theorem, the N-TS fuzzy reformulation enables an effective framework to design fuzzy UI observers. Based on an UI decoupling technique, no specific information on the UI is required for fuzzy observer design. The asymptotic estimations of both the state and the UI are guaranteed with fuzzy Lyapunov arguments. The observer gains can be effectively computed following an LMI-based design procedure. Numerical illustrations are given to demonstrate the interests of the proposed method over related existing results.