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Research Papers

# Stabilization of a Class of Nonlinear Coupled Ordinary Differential Equation–Partial Differential Equation Systems Via Sampled-Data H∞ Fuzzy Controller

[+] Author and Article Information
S. Dharani

Department of Mathematics,
Bharathiar University,
e-mail: sdharanimails@gmail.com

R. Rakkiyappan

Department of Mathematics,
Bharathiar University,
e-mail: rakkigru@gmail.com

S. Lakshmanan

Research Center for Wind Energy Systems,
Kunsan National University,
558 Daehak-ro,
Gunsan-si 54150, South Korea
e-mail: lakshm85@gmail.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 27, 2017; final manuscript received July 21, 2017; published online November 10, 2017. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 140(4), 041007 (Nov 10, 2017) (12 pages) Paper No: DS-17-1052; doi: 10.1115/1.4037651 History: Received January 27, 2017; Revised July 21, 2017

## Abstract

The main intention of this study is to develop a sampled-data $H∞$ fuzzy controller design to analyze the stability of coupled ordinary differential equation (ODE)–partial differential equation (PDE) systems, where the nonlinear coupled system is expressed by Takagi–Sugeno (T–S) fuzzy models. The coupled ODE–PDE system in this paper constitutes an n–dimensional nonlinear subsystem of ODEs and a scalar linear parabolic subsystem of PDE. Then, in regard to the T–S model representation, Lyapunov technique is utilized to model a sampled-data $H∞$ fuzzy controller to stabilize the contemplated coupled systems and to attain a desired $H∞$ disturbance attenuation performance. The formulated time-dependent Lyapunov functional makes full use of the accessible information about the actual sampling pattern. The outcome of the sampled-data $H∞$ fuzzy control problem is expressed as linear matrix inequality (LMI) optimization problem which can be solved effectively by using any of the available softwares. Finally, hypersonic rocket car model is furnished with simulation results to exhibit the efficacy of the proposed theoretical results.

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## Figures

Fig. 1

State responses of closed-loop coupled ODE–PDE system without disturbances

Fig. 2

State responses of closed-loop coupled ODE–PDE system with disturbances

Fig. 3

State responses of closed-loop coupled ODE–PDE system without disturbances under state feedback controller

Fig. 4

State responses of closed-loop coupled ODE–PDE system with disturbances under state feedback controller

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