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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;
wangjunwei-1984@163.com

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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References

Nilsson, J. , 1998, “Real-Time Control Systems With Delays,” Ph.D. dissertation, Lund Institute of Technology, Lund, Sweden.
Gu, K. , Kharitonov, V. L. , and Chen, J. , 2003, Stability of Time-Delay Systems, Birkhauser, Boston, MA. [CrossRef]
Zhong, Q.-C. , 2006, Robust Control of Time-Delay Systems, Springer-Verlag, London.
Hespanha, J. P. , Naghshtabrizi, P. , and Xu, Y. , 2007, “ A Survey of Recent Results in Networked Control Systems,” Proc. IEEE, 95(1), pp. 138–162. [CrossRef]
Wu, M. , He, Y. , She, J.-H. , and Liu, G.-P. , 2004, “ Delay-Dependent Criteria for Robust Stability of Time-Varying Delay Systems,” Automatica, 40(8), pp. 1435–1439. [CrossRef]
He, Y. , Wang, Q.-G. , Xie, L. H. , and Lin, C. , 2007, “ Further Improvement of Free-Weighting Matrices Technique for Systems With Time-Varying Delay,” IEEE Trans. Autom. Control, 52(2), pp. 293–299. [CrossRef]
Gouaisbaut, F. , and Peaucelle, D. , 2006, “ Delay-Dependent Robust Stability of Time Delay Systems,” Fifth IFAC Symposium on Robust Control Design, Toulouse, France, ▪.
Gouaisbaut, F. , and Peaucelle, D. , 2006, “ Delay-Dependent Stability Analysis of Linear Time Delay Systems,” Sixth IFAC Workshop on Time-Delay Systems, L'Aquila, Italy, ▪.
Han, Q.-L. , 2005, “ Absolute Stability of Time-Delay Systems With Sector-Bounded Nonlinearity,” Automatica, 41(12), pp. 2171–2176. [CrossRef]
Liu, K. , Suplin, V. , and Fridman, E. , 2010, “ Stability of Linear Systems With General Sawtooth Delay,” IMA J. Math. Control Inf., 27(4), pp. 419–436. [CrossRef]
Seuret, A. , and Gouaisbaut, F. , 2013, “ Jensen's and Wirtinger's Inequalities for Time-Delay Systems,” 11th IFAC Workshop on Time-Delay Systems, Grenoble, France, ▪.
Seuret, A. , and Gouaisbaut, F. , 2013, “ Wirtinger-Based Integral Inequality: Application to Time-Delay Systems,” Automatica, 49(9), pp. 2860–2866. [CrossRef]
Wu, J. , 1996, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York. [CrossRef]
Allegretto, W. , and Papini, D. , 2007, “ Stability for Delayed Reaction-Diffusion Neural Networks,” Phys. Lett. A, 360(6), pp. 669–680. [CrossRef]
Gaffney, E. A. , and Monk, N. A. M. , 2006, “ Gene Expression Time Delays and Turing Pattern Formation Systems,” Bull. Math. Biol., 68(1), pp. 99–130. [CrossRef] [PubMed]
Forys, U. , and Marciniak-Czochra, A. , 2003, “ Logistic Equations in Tumor Growth Modeling,” Int. J. Appl. Math. Comput. Sci., 13(3), pp. 317–325.
Wang, P. , 1963, “ Asymptotic Stability of a Time-Delayed Diffusion System,” ASME J. Appl. Mech., 30(4), pp. 500–504. [CrossRef]
Wang, P. , 1964, “ Optimum Control of Distributed Parameter Systems With Time Delays,” IEEE Trans. Autom. Control, 9(1), pp. 13–22. [CrossRef]
Kim, M. , 1974, “ Synthesis of Linear Optimum Distributed Parameter Systems With Time Delays,” Proc. IEEE, 62(8), pp. 1177–1179. [CrossRef]
Wang, T. , 1994, “ Stability in Abstract Functional-Differential Equations—I: General Theorems,” J. Math. Anal. Appl., 186(2), pp. 534–558. [CrossRef]
Wang, T. , 1994, “ Stability in Abstract Functional-Differential Equations—II: Applications,” J. Math. Anal. Appl., 186(3), pp. 835–861. [CrossRef]
Nicaise, S. , and Pignotti, C. , 2006, “ Stability and Instability Results of the Wave Equation With a Delay Term in the Boundary or Internal Feedback,” SIAM J. Control Optim., 45(5), pp. 1561–1585. [CrossRef]
Gurevich, S. V. , 2013, “ Dynamics of Localized Structures in Reaction-Diffusion Systems Induced by Delayed Feedback,” Phys. Rev. E, 87, p. 052922. [CrossRef]
Fridman, E. , and Orlov, Y. , 2009, “ Exponential Stability of Linear Distributed Parameter Systems With Time-Varying Delays,” Automatica, 45(1), pp. 194–201. [CrossRef]
Luo, Y.-P. , Xia, W.-H. , Liu, G.-R. , and Deng, F.-Q. , 2009, “ LMI Approach to Exponential Stabilization of Distributed Parameter Control Systems With Delay,” Acta Autom. Sin., 35(3), pp. 299–304. [CrossRef]
Wang, J.-W. , Sun, C.-Y. , Xin, X. , and Mu, C.-X. , 2014, “ Sufficient Conditions for Exponential Stabilization of Linear Distributed Parameter Systems With Time Delays,” IFAC Proc. Vol., 47(3), pp. 6062–6067. [CrossRef]
Solomon, O. , and Fridman, E. , 2015, “ Stability and Passivity Analysis of Semilinear Diffusion PDEs With Time-Delays,” Int. J. Control, 88(1), pp. 180–192. [CrossRef]
Wang, J.-W. , and Wu, H.-N. , 2015, “ Some Extended Wirtinger's Inequalities and Distributed Proportional-Spatial Integral Control of Distributed Parameter Systems With Multi-Time Delays,” J. Franklin Inst., 352(10), pp. 4423–4445. [CrossRef]
Gahinet, P. , Nemirovskii, A. , Laub, A. J. , and Chilali, M. , 1995, “LMI Control Toolbox for Use With MATLAB,” MathWorks, Natick, MA.
Liu, Z. , and Zheng, S. , 1999, Semigroups Associated With Dissipative Systems, Chapman and Hall/CRC, Boca Raton, FL.
Curtain, R. F. , and Zwart, H. J. , 1995, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York. [CrossRef]
Fridman, E. , and Blighovsky, A. , 2012, “ Robust Sampled-Data Control of a Class of Semilinear Parabolic Systems,” Automatica, 48(5), pp. 826–836. [CrossRef]
Am, B. N. , and Fridman, E. , 2014, “ Network-Based Distributed H Filtering of Parabolic Systems,” Automatica, 50(12), pp. 3139–3146. [CrossRef]
Wang, J.-W. , Li, H.-X. , and Wu, H.-N. , 2014, “ Distributed Proportional Plus Second-Order Spatial Derivative Control for Distributed Parameter Systems Subject to Spatiotemporal Uncertainties,” Nonlinear Dyn., 76(4), pp. 2041–2058. [CrossRef]
Bode, M. , and Purwins, H.-G. , 1995, “ Pattern Formation in Reaction-Diffusion Systems-Dissipative Solitons in Physical Systems,” Physica D, 86(1–2), pp. 53–63. [CrossRef]
Mao, X. , Koroleva, N. , and Rodkina, A. , 1998, “ Robust Stability of Uncertain Stochastic Differential Delay Equations,” Syst. Control Lett., 35(5), pp. 325–336. [CrossRef]

Figures

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. 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)

Grahic Jump Location
Fig. 2

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

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