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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.

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The problems of observer‐based feedback control and stability analysis for a T‐S fuzzy system are investigated in this paper. Based on system output variable, a membership function dependent observer and a controller subject to observer errors are constructed to estimate the unknown system state and establish the closed‐loop feedback control system. The derived stability criteria, which is not numerically solvable, is converted to a solvable optimization problem. An improved membership function dependent approach is proposed to reduce the conservativeness. The approximated values of membership functions and their derivatives are calculated based on the proposed piecewise‐function method. Then, a piecewise decaying rate setting technique is presented to adjust the error convergent speed. A three‐rule fuzzy system stability analysis example is given to show the conservativeness reduction effects, while an inverted pendulum cart control process is used to show the effectiveness of the proposed observer‐based control scheme.

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