A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Advanced Search
Volume 2 Issue 4
Oct.  2015

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 15.3, Top 1 (SCI Q1)
    CiteScore: 23.5, Top 2% (Q1)
    Google Scholar h5-index: 77， TOP 5
Turn off MathJax
Article Contents
Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2015  >  2(4): 353-357
Fudong Ge, Yangquan Chen and Chunhai Kou, "Cyber-physical Systems as General Distributed Parameter Systems: Three Types of Fractional Order Models and Emerging Research Opportunities," IEEE/CAA J. of Autom. Sinica, vol. 2, no. 4, pp. 353-357, 2015.
Citation: Fudong Ge, Yangquan Chen and Chunhai Kou, "Cyber-physical Systems as General Distributed Parameter Systems: Three Types of Fractional Order Models and Emerging Research Opportunities," IEEE/CAA J. of Autom. Sinica, vol. 2, no. 4, pp. 353-357, 2015.
shu

Cyber-physical Systems as General Distributed Parameter Systems: Three Types of Fractional Order Models and Emerging Research Opportunities

  • 1. the College of Information Science and Technology, Donghua University, Shanghai 201620, China;

  • 2. the Mechatronics, Embedded Systems and Automation Lab, University of California, Merced, CA 95343, USA;

  • 3. the Department of Applied Mathematics, Donghua University, Shanghai 201620, China

Funds:

This work was supported by Chinese Universities Scientific Fund (CUSF-DHD-2014061) and Natural Science Foundation of Shanghai (15ZR1400800).

    Abstract
  • Cyber-physical systems (CPSs) are man-made complex systems coupled with natural processes that, as a whole, should be described by distributed parameter systems (DPSs) in general forms. This paper presents three such general models for generalized DPSs that can be used to characterize complex CPSs. These three different types of fractional operators based DPS models are: fractional Laplacian operator, fractional power of operator or fractional derivative. This research investigation is motivated by many fractional order models describing natural, physical, and anomalous phenomena, such as sub-diffusion process or super-diffusion process. The relationships among these three different operators are explored and explained. Several potential future research opportunities are then articulated followed by some conclusions and remarks.

     

    • FullText(HTML)
  • loading
    • References(42)
  • [1]
    Lee E A. Cyber physical systems: design challenges. In: Proceedings of the 2008 11th IEEE International Symposium on Object Oriented Real-Time Distributed Computing (ISORC). Orlando, FL: IEEE, 2008. 363-369
    [2]
    Song Z, Chen Y Q, Sastry C R, Tas N C. Optimal Observation for Cyber-physical Systems: a Fisher-Information-Matrix-Based Approach. London: Springer, 2009.
    [3]
    Tricaud C, Chen Y Q. Optimal Mobile Sensing and Actuation Policies in Cyber-physical Systems. London: Springer, 2012.
    [4]
    El Jai A, Pritchard A J. Sensors and Controls in the Analysis of Distributed Systems. New York: Halsted Press, 1988.
    [5]
    Podlubny I. Fractional Differential Equations, Vol. 198. New York: Academic Press, 1999.
    [6]
    Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier Science, 2006.
    [7]
    Klimek M. On Solutions of Linear Fractional Differential Equations of a Variational Type. Czestochowa: Publishing Office of Czestochowa University of Technology, 2009.
    [8]
    Torvik P J, Bagley R L. On the appearance of the fractional derivative in the behavior of real materials. Journal of Applied Mechanics, 1984, 51(2): 294-298
    [9]
    Mandelbrot B B. The Fractal Geometry of Nature, Vol. 173. San Francisco: W. H. Freeman, 1983.
    [10]
    Lv G Y, Duan J Q, He J C. Nonlocal elliptic equations involving measures. Journal of Mathematical Analysis and Applications, 2015, 432(2): 1106-1118
    [11]
    Lv G Y, Duan J Q. Martingale and weak solutions for a stochastic nonlocal Burgers equation on bounded intervals. arXiv preprint arXiv: 1410.7691 [math. P R] 28 Oct. 2014.
    [12]
    He J C, Duan J Q, Gao H J. Global solutions for a nonlocal Ginzberg-Landau equation and a nonlocal Fokker-Plank equation. arXiv preprint arXiv: 1312. 5836 [math. A P] 20 Dec. 2013.
    [13]
    Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bulletin des Sciences Mathématiques, 2012, 136(5): 521-573
    [14]
    Du Q, Gunzburger M, Lehoucq R B, Zhou K. A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Mathematical Models and Methods in Applied Sciences, 2013, 23(3): 493-540
    [15]
    Applebaum D. Lévy Processes and Stochastic Calculus (Second edition). Cambridge: Cambridge University Press, 2009.
    [16]
    Du Q, Gunzburger M, Lehoucq R B, Zhou K. Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Review, 2012, 54(4): 667-696
    [17]
    Choi W, Kim S, Lee K A. Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian. Journal of Functional Analysis, 2014, 266(11): 6531-6598
    [18]
    Barrios B, Colorado E, De Pablo A, Sánchez U. On some critical problems for the fractional Laplacian operator. Journal of Differential Equations, 2012, 252(11): 6133-6162
    [19]
    Micu S, Zuazua E. On the controllability of a fractional order parabolic equation. SIAM Journal on Control and Optimization, 2006, 44(6): 1950-1972
    [20]
    Fattorini H O, Russell D L. Exact controllability theorems for linear parabolic equations in one space dimension. Archive for Rational Mechanics and Analysis, 1971, 43(4): 272-292
    [21]
    Kwaśnicki M. Eigenvalues of the fractional Laplace operator in the interval. Journal of Functional Analysis, 2012, 262(5): 2379-2402
    [22]
    Ros-Oton X, Serra J. The dirichlet problem for the fractional Laplacian: regularity up to the boundary. Journal de Mathématiques Pures et Appliquées, 2014, 101(3): 275-302
    [23]
    Gorenflo R, Mainardi F. Random walk models for space-fractional diffusion processes. Fractional Calculus and Applied Analysis, 1998, 1(2): 167-191
    [24]
    Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives: Theory and Applications. New York: Gordon and Breach Science Publishers, 1993.
    [25]
    Landkof N S, Doohovskoy A P. Foundations of Modern Potential Theory (Grundlehren der mathematischenWissenschaften). New York: Springer, 1972.
    [26]
    Luchko Y, Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica, 1999, 24(2): 207-233
    [27]
    Jiang H, Liu F, Turner I, Burrage K. Analytical solutions for the multiterm time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. Journal of Mathematical Analysis and Applications, 2012, 389(2): 1117-1127
    [28]
    Yang Q, Liu F, Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Applied Mathematical Modelling, 2010, 34(1): 200-218
    [29]
    Liu F, Anh V, Turner I. Numerical solution of the space fractional Fokker-Planck equation. Journal of Computational and Applied Mathematics, 2004, 166(1): 209-219
    [30]
    Ashyralyev A. A note on fractional derivatives and fractional powers of operators. Journal of Mathematical Analysis and Applications, 2009, 357(1): 232-236
    [31]
    Komatsu H. Fractional powers of operators. Pacific Journal of Mathematics, 1966, 19(2): 285-346
    [32]
    Komatsu H. Fractional powers of operators, II. Interpolation spaces. Pacific Journal of Mathematics, 1967, 21(1): 89-111
    [33]
    Komatsu H. Fractional powers of operators, III. Negative powers. Journal of the Mathematical Society of Japan, 1969, 21(2): 205-220
    [34]
    Carracedo C M, Alix M S. The Theory of Fractional Powers of Operators. Amsterdam: Elsevier, 2001.
    [35]
    Balakrishnan A V. Fractional powers of closed operators and the semigroups generated by them. Pacific Journal of Mathematics, 1960, 10(2): 419-439
    [36]
    Umarov S. Introduction to Fractional and Pseudo-differential Equations with Singular Symbols, Vol. 41. Switzerland: Springer, 2015.
    [37]
    Hahn M, Kobayashi K, Umarov S. SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations. Journal of Theoretical Probability, 2012, 25(1): 262-279
    [38]
    Umarov S. Algebra of pseudo-differential operators with variable analytic symbols and propriety of the corresponding equations. Differ. Equ., 1991, 27(6): 753-759
    [39]
    Hahn M, Umarov S. Fractional Fokker-Planck-Kolmogorov type equations and their associated stochastic differential equations. Fractional Calculus and Applied Analysis, 2011, 14(1): 56-79
    [40]
    Afifi L, Chafia A, El Jai A. Regionally efficient and strategic actuators. International Journal of Systems Science, 2002, 33(1): 1-12
    [41]
    Ge F D, Chen Y Q, Kou C H. Regional controllability of anomalous diffusion generated by the time fractional diffusion equations. In: ASME IDETC/CIE 2015, Boston, Aug. 2-5, 2015, DETC2015-46697. See also: arXiv preprint arXiv: 1508. 00047
    [42]
    Chen Y Q, Moore K L, Song Z. Diffusion boundary determination and zone control via mobile actuator-sensor networks (MAS-net): challenges and opportunities. In: Proc. SPIE 5421, Intelligent Computing: Theory and Applications II. Orlando, FL: SPIE, 2004. 102-113
    • Related Articles
    • Supplements(0)

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索
    Get Citation
    PDF
    XML

    Article Metrics

    Article views (1305) PDF downloads(17) Cited by()

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return