Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semi-linear Parabolic PDE Systems

TY - JOUR

T1 - Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semi-linear Parabolic PDE Systems

AU - Wang, Jun Wei

AU - Tsai, Shun Hung

AU - Li, Han Xiong

AU - Lam, Hak Keung

PY - 2018/2/24

Y1 - 2018/2/24

N2 - This paper employs a Takagi-Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial $L^\infty$ norm $\|\cdot\|_\infty$ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe the complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov--Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincar\'{e} inequality in 1D spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\|\cdot\|_\infty$, if the sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance the exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.

AB - This paper employs a Takagi-Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial $L^\infty$ norm $\|\cdot\|_\infty$ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe the complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov--Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincar\'{e} inequality in 1D spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\|\cdot\|_\infty$, if the sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance the exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.

KW - Agmon's inequality

KW - distributed parameter systems

KW - exponential stability

KW - Sampled-data control

KW - Takagi-Sugeno fuzzy partial differential equation model

UR - http://www.scopus.com/inward/record.url?scp=85042704838&partnerID=8YFLogxK

U2 - 10.1109/TFUZZ.2018.2809686

DO - 10.1109/TFUZZ.2018.2809686

M3 - Article

AN - SCOPUS:85042704838

SN - 1063-6706

JO - IEEE Transactions on Fuzzy Systems

JF - IEEE Transactions on Fuzzy Systems

ER -