Static Collocated Piecewise Fuzzy Control Design of Quasi-Linear Parabolic PDE Systems Subject to Periodic Boundary Conditions

This paper presents a Lyapunov and partial differential equation (PDE)-based methodology to solve static collocated piecewise fuzzy control design of quasi-linear parabolic PDE systems subject to periodic boundary conditions. Two types of piecewise control, i.e., globally piecewise control and local...

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Bibliographic Details
Published inIEEE transactions on fuzzy systems Vol. 27; no. 7; pp. 1479 - 1492
Main Authors Wang, Jun-Wei, Li, Han-Xiong
Format Journal Article
LanguageEnglish
Published New York IEEE 01.07.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects
Actuators
Boundary conditions
Computer simulation
Control design
Control stability
Control systems
Controllers
Design methodology
Feedback control
Fuzzy control
Fuzzy systems
Mathematical analysis
Mathematical model
Mathematical models
Matrix methods
Nonlinearity
Parabolic differential equations
Partial differential equations
Piecewise control
Poincaré–Wirtinger inequality
quasi-linear partial differential equation (PDE)
Takagi–Sugeno (T–S) fuzzy model
Wirtinger inequality
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Summary:This paper presents a Lyapunov and partial differential equation (PDE)-based methodology to solve static collocated piecewise fuzzy control design of quasi-linear parabolic PDE systems subject to periodic boundary conditions. Two types of piecewise control, i.e., globally piecewise control and locally piecewise control are considered, respectively. A Takagi-Sugeno (T-S) fuzzy PDE model that is constructed via local sector nonlinearity method is first employed to accurately describe spatiotemporal dynamics of quasi-linear PDEs. Based on the T-S fuzzy PDE model, a static collocated piecewise fuzzy feedback controller is constructed to guarantee the locally exponential stability of the resulting closed-loop system. Sufficient conditions for the existence of such fuzzy controller are developed by applying vector-valued Poincaré-Wirtinger inequality and its variants and a linear matrix inequality (LMI) relaxation technique. These sufficient conditions are presented in terms of standard LMIs. Finally, the performance of the suggested fuzzy controller is illustrated by numerical simulation results of a nonlinear PDE system described by quasi-linear FitzHugh-Nagumo equation with periodic boundary conditions.
ISSN:1063-6706
1941-0034
DOI:10.1109/TFUZZ.2018.2881667