Research Papers

Delay-Dependent Exponential Stabilization for Linear Distributed Parameter Systems With Time-Varying Delay

[+] Author and Article Information
Jun-Wei Wang

Key Laboratory of Knowledge Automation for
Industrial Processes of Ministry of Education,
School of Automation and Electrical Engineering,
University of Science and Technology Beijing,
Beijing 100083, China
e-mails: junweiwang@ustb.edu.cn;

Chang-Yin Sun

School of Automation,
Southeast University,
Nanjing 210096, China

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 24, 2016; final manuscript received October 9, 2017; published online December 19, 2017. Assoc. Editor: Mazen Farhood.

J. Dyn. Sys., Meas., Control 140(5), 051003 (Dec 19, 2017) Paper No: DS-16-1570; doi: 10.1115/1.4038374 History: Received November 24, 2016; Revised October 09, 2017

This paper extends the framework of Lyapunov–Krasovskii functional to address the problem of exponential stabilization for a class of linearly distributed parameter systems (DPSs) with continuous differentiable time-varying delay and a spatiotemporal control input, where the system model is described by parabolic partial differential-difference equations (PDdEs) subject to homogeneous Neumann or Dirichlet boundary conditions. By constructing an appropriate Lyapunov–Krasovskii functional candidate and using some inequality techniques (e.g., spatial integral form of Jensen's inequalities and vector-valued Wirtinger's inequalities), some delay-dependent exponential stabilization conditions are derived, and presented in terms of standard linear matrix inequalities (LMIs). These stabilization conditions are applicable to both slow-varying and fast-varying time delay cases. The detailed and rigorous proof of the closed-loop exponential stability is also provided in this paper. Moreover, the main results of this paper are reduced to the constant time delay case and extended to the stochastic time-varying delay case, and also extended to address the problem of exponential stabilization for linear parabolic PDdE systems with a temporal control input. The numerical simulation results of two examples show the effectiveness and merit of the main results.

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Grahic Jump Location
Fig. 1

Open-loop trajectory of ‖y(⋅,t)‖2, closed-loop trajectory of ‖y(⋅,t)‖2, and trajectory of ‖u(⋅,t)‖2 via Corollary 1

Grahic Jump Location
Fig. 2

Open-loop trajectories of ‖y1(⋅,t)‖2 and ‖y2(⋅,t)‖2 for the semilinear systems (49)(51)

Grahic Jump Location
Fig. 3

Closed-loop trajectories of ‖y1(⋅,t)‖2, ‖y2(⋅,t)‖2 and ‖u(⋅,t)‖2 via Theorem 1

Grahic Jump Location
Fig. 4

Closed-loop trajectories of ‖y1(⋅,t)‖2, ‖y2(⋅,t)‖2 and |u(t)| via Theorem 7

Grahic Jump Location
Fig. 5

Closed-loop trajectories of ‖y1(⋅,t)‖2 and ‖y2(⋅,t)‖2 as well as ‖u(⋅,t)‖2 via Theorem 5

Grahic Jump Location
Fig. 6

Stochastic time-varying delay h(t)



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