Fuzzy Parabolic Partial Differential Equation Research Articles

Fuzzy Boundary Sampled-Data Control for Nonlinear Parabolic DPSs.

For a nonlinear parabolic distributed parameter system (DPS), a fuzzy boundary sampled-data (SD) control method is introduced in this article, where distributed SD measurement and boundary SD measurement are respected. Initially, this nonlinear parabolic DPS is represented precisely by a Takagi-Sugeno (T-S) fuzzy parabolic partial differential equation (PDE) model. Subsequently, under distributed SD measurement and boundary SD measurement, a fuzzy boundary SD control design is obtained via linear matrix inequalities (LMIs) on the basis of the T-S fuzzy parabolic PDE model to guarantee exponential stability for closed-loop parabolic DPS by using inequality techniques and a LF. Furthermore, respecting the property of membership functions, we present some LMI-based fuzzy boundary SD control design conditions. Finally, the effectiveness of the designed fuzzy boundary SD controller is demonstrated via two simulation examples.

Finite‐time dissipative control of fuzzy distributed parameter CPSs with quantization under cyber attacks

AbstractThe finite‐time dissipative control for a class of nonlinear distributed parameter Cyber‐Physical Systems (CPSs) is studied in this article. By replacing the nonlinear term with the Takagi‐Sugeno (T‐S) fuzzy model, the nonlinear distributed parameter CPSs are represented by a fuzzy parabolic partial differential equation (PDE). To reduce the transmission burden of communication channels, the state information is considered to be dynamically quantized before transmission. Then, a fuzzy state feedback controller under cyber attacks is designed. Sufficient design criteria for the fuzzy controller and the adjusting parameters of dynamic quantizer are developed to obtain the finite‐time dissipative performance of the fuzzy closed‐loop system by the free‐matrix inequality approach. Finally, a simulation study in a cascaded of two cylindrical lithium batteries is explored to validate the efficiency of the control strategy.

Quantized-Output-Based Exponential Stabilization of Fuzzy DPSs With Time-Varying Delay and Nonlinearities

The exponential stabilization of nonlinear distributed parameter systems (DPSs) with time-varying delay is studied in this paper. With the employment of the Takagi-Sugeno (T-S) fuzzy model, the nonlinear DPSs are described as a fuzzy parabolic partial differential equation (PDE). Then, with the consideration of the measurement output and control input information quantized by a kind of dynamic quantizers, fuzzy static and dynamic output feedback controllers are proposed. The time-delayed feedback controllers are also provided in the forms of static and dynamic. On the basis of the derived T-S fuzzy PDE model, Lyapunov-based control design strategies are proposed with respect to the effect of quantization. Moreover, by constructing appropriate Lyapunov functionals, sufficient conditions for designing the output feedback control gains and the quantizers' adjusting parameters which guarantee the exponential stability of the fuzzy closed-loop system are provided in terms of standard linear matrix inequalities (LMIs). Finally, we explore the output feedback controllers into the FitzHugh-Nagumo (FHN) equation and Fisher equation as applications. The efficiency of the design strategies is indicated through the simulation results.

Formula omitted] fuzzy intermittent boundary control for nonlinear parabolic distributed parameter systems

For nonlinear space-varying parabolic distributed parameter systems, this paper introduces an H∞ fuzzy intermittent boundary control, where the output measurements are only available at some specified boundary position (i.e., boundary measurements). Initially, a Takagi–Sugeno fuzzy parabolic partial differential equation is used to precisely describe the nonlinear space-varying parabolic distributed parameter system. Then, under boundary measurements, an H∞ fuzzy intermittent boundary control design based on the Takagi–Sugeno fuzzy parabolic partial differential equation model ensuring the exponential stability with an H∞ performance for closed-loop space-varying distributed parameter system is subsequently obtained via spatial linear matrix inequalities by employing inequality techniques and piecewise switching-time-dependent Lyapunov function. Furthermore, in order to solve the H∞ fuzzy intermittent boundary controller design of nonlinear space-varying parabolic distributed parameter systems under boundary measurements, we express spatial linear matrix inequalities as linear matrix inequalities and further present some linear matrix inequality based fuzzy intermittent boundary control design conditions respecting the property of membership functions. Finally, two simulation examples are offered to demonstrate the effectiveness of the proposed H∞ fuzzy intermittent boundary control method.

AN IMPLICIT METHOD FOR FUZZY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we consider an implicit finite difference method for solving fuzzy partial differential equations (FPDEs). We present stability of this method and solve the parabolic equation with this scheme.