In this paper, based on the nonmonotonic Lyapunov functions, a new less conservative state feedback controller synthesis method is proposed for a class of discrete time nonlinear systems represented by Takagi-Sugeno (T-S) fuzzy systems. Parallel distributed compensation (PDC) state feedback is employed as the controller structure. Also, a T-S fuzzy observer is designed in a manner similar to state feedback controller design. The observer and the controller can be obtained separately and then combined together to form an output feedback controller by means of the Separation theorem. Both observer and controller are obtained via solving a sequence of linear matrix inequalities. Nonmonotonic Lyapunov method allows the design of controllers for the aforementioned systems where other methods fail. Illustrative examples are presented which show how the proposed method outperforms other methods such as common quadratic, piecewise or non quadratic Lyapunov functions.

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This paper handles the identification of nonlinear systems through linear time-varying (LTV) approximation. The mathematical form of the nonlinear system is unknown and regenerated through an experiment followed by LTV and linear parameter-varying (LPV) estimation and integration. By employing a well-designed experiment the linearized model of the nonlinear system around a time-varying trajectory is obtained. The result is an LTV approximation of the nonlinear system around that trajectory. Having estimated the LTV model, an LPV model is identified. It is shown that the parameter-varying (PV) coefficients of this LPV model are partial derivatives of the nonlinear system evaluated at the trajectory. In this paper, we will show that there exists a relation between the LPV coefficients. This structural relation in the LPV model ensures the integrability of PV coefficients for nonlinear reconstruction. Indeed, the vector of the LPV coefficients is the gradient of the nonlinear system evaluated at the trajectory. Then, the nonlinear system is reconstructed through symbolic integration of the coefficients. The proposed method is a data-driven scheme that can reconstruct an estimate of the nonlinear system and its mathematical form using input-output measurements. Finally, the use of the proposed method is illustrated via a simulation example.