Applied and Computational Mathematics

| Peer-Reviewed |

Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics

Received: 20 August 2014    Accepted: 10 September 2014    Published: 20 September 2014
Views:       Downloads:

Share This Article

Abstract

This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.

DOI 10.11648/j.acm.20140305.14
Published in Applied and Computational Mathematics (Volume 3, Issue 5, October 2014)
Page(s) 217-224
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Small Gain Theorem, Distributed Control, Robust Control, nD Transfer Function

References
[1] H. K. Khalil, Nonlinear Systems Second Edition, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1996.
[2] A. W. Naylor and G. R. Sell, Linear Operator Theory in Engineering and Science, Springer-Verlag New York Inc., New York, 1982.
[3] J. Doyle, B. Francis, A. Tannenbaum, Feedback Control Theory, Macmillan, New York, 1992.
[4] K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1996.
[5] B.-S. Hong, P.-J. Su, C.-Y. Chou, C.-I. Hung, Realization of non-Fourier phenomena in heat transfer with 2D transfer function, Appl. Math. Model. 35 (8) (2011) 4031-4043.
[6] C.-Y. Chou, System Identification and Feedback Control of Non-Fourier Heat Transfer with 2D Transfer Function, Doctoral Dissertation, National Chung Cheng University, Taiwan, 2012.
[7] B.-S. Hong, C.-Y. Chou, T.-Y. Lin, 2D transfer function modeling of thermoacoustic vibration engines with boundary heat-flux control, Asian J. Control (to appear).
[8] A. Preumont, A. François, P. De Man, N. Loix, K. Henrioulle, Distributed sensors with piezoelectric films in design of spatial filters for structural control, J. Sound Vibr. 282 (3-5) (2005) 701-712.
[9] A.W. Brown, B.G. Colpitts, K. Brown, Distributed sensor based on dark-pulse Brillouin scattering, IEEE Photon. Technol. Lett. 17 (7) (2005) 1501-1503.
[10] T.V. karnaukhova, E.V. Pyatetskaya, Basic relations of the theory of thermoviscoelastic plates with distributed sensors, Int. Appl. Mech. 45 (6) (2009) 660-669.
[11] T.V. Karnaukhova, E.V. Pyatetskaya, Basic equations for termoviscoelastic plates with distributed actuators under monoharmonic loading, Int. Appl. Mech. 45 (2) (2009) 200-214.
[12] C. Menon, F. Carpi, D.D. Rossi, Concept design of novel bio-inspired distributed actuators for space applications, Acta Astronaut. 65 (5-6) (2009) 825-833.
[13] R. Rabenstein, L. Trautmann, Multidimensional transfer function models, IEEE Trans. Circuits Syst. Fund. Theor. Appl. 49 (6) (2002) 852-861.
[14] R. Rabenstein, L. Trautmann, Digital sound synthesis of string instruments with the functional transformation method, Signal Process. 83 (8) (2003) 1673-1688.
[15] B.-S. Hong, Construction of 2D isomorphism for 2D H1 -control of Sturm-Liouville Systems, Asian J. Control 12 (2) (2010) 187-199.
[16] B. Bamieh, F. Paganini, M.A. Dahleh, Distributed control of spatially invariant systems, IEEE Trans. Autom. Control 47 (7) (2002) 1091-1107.
[17] D.M. Gorinevsky, G. Stein, Structured uncertainty analysis of robust stability for multidimensional array systems, IEEE Trans. Autom. Control 48 (9) (2003) 1557-1568.
[18] G.E. Stewart, D.M. Gorinevsky, G.A. Dumont, Two-dimensional loop shaping, Automatica 39 (5) (2003) 779-792.
[19] G.E Stewart, D.M. Gorinevsky, G.A Dumont, Feedback controller design for a spatially distributed system: the paper machine problem, IEEE Trans. Control. Syst. Technol. 11 (5) (2003) 612-628.
[20] D.M. Gorinevsky, S. Boyd, G. Stein, Design of low-bandwidth spatially distributed feedback, IEEE Trans. Autom. Control 53 (1) (2008) 257 - 272.
[21] F.M. Callier, C.A. Desoer, An algebra of transfer functions for distributed linear time-invariant systems, IEEE Trans. Circuits Syst. 25 (9) (1978) 651-662.
[22] F.M. Callier, J. Winkin, LQ-optimal control of infinite-dimensional systems by spectral factorization, Automatica 28 (4) (1992) 757-770.
[23] F.M. Callier, L. Dumortier, J. Winkin, On the nonnegative self-adjoint solutions of the operator Riccati equation for infinite dimensional systems, Integr. Equ. Oper. Theory 22 (2) (1995) 162–195.
[24] P. Grabowski, F.M. Callier, Boundary control systems in factor form: transfer functions and input-output maps, Integr. Equ. Oper. Theory 41 (1) (2001) 1–37.
[25] I. Podlubny, Fractional-order systems and -controllers, IEEE Trans. Autom. Control 44 (1) (1999) 208-214.
[26] D. Valerio, J.S. da Costa, Tuning of fractional Controllers Minimising H2 and H1 Norms, Acta Polytech. Hung. 3 (4) 2006 55-70.
[27] B.M. Vinagre, V. Feliu, Optimal fractional controllers for rational order system: a special case of the Wiener-Hopf spectral factorization method, IEEE Trans. Autom. Control 52 (12) (2007) 2385-2389.
[28] C.A. Monje, B.M. Vinagre, V. Feliu, Y.-Q. Chen, Tuning and auto-tuning of fractional order controllers for industry applications, Control Eng. Practice 16 (2008) 798-812.
[29] F. Padula, A. Visioli, Tuning rules for optimal PID and fractional-order PID controllers, J. Process Control 21 (2011) 69-81.
[30] M.E. Valcher, On the internal stability and asymptotic behavior of 2-D positive systems, IEEE Trans. Circuits Syst. 44 (7) (1997) 602-613.
[31] M.E. Valcher, State-space descriptions and observability properties of spatiotemporal finite-dimensional autonomous behaviors, Syst. Control Lett. 44 (2) (2001) 91-102.
[32] E. Fornasini, M.E. Valcher, Controllability and reachability of 2-D positive system: A Graph Theoretic Approach, IEEE Trans. Circuits Syst. 52 (3) (2005) 576-585.
[33] P. Lancaster, I. Zaballa, Diagonalizable quadratic eigenvalue problems, Mech. Syst. Signal Pr. 23 (4) (2009) 1134-1144.
[34] J. Carlos, Z. Anaya, Diagonalization of quadratic matrix polynomials, Syst. Control Lett. 59 (2) (2010) 105-113.
[35] Z. Shu, J. Lam, H. Gao, B. Du, L. Wu, Positive observers and dynamic output-feedback controllers for interval positive linear systems, IEEE T. Circuits.-I 55 (10) (2008) 3209-3222.
Author Information
  • Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi 62012, Taiwan

Cite This Article
  • APA Style

    Boe-Shong Hong. (2014). Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Applied and Computational Mathematics, 3(5), 217-224. https://doi.org/10.11648/j.acm.20140305.14

    Copy | Download

    ACS Style

    Boe-Shong Hong. Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Appl. Comput. Math. 2014, 3(5), 217-224. doi: 10.11648/j.acm.20140305.14

    Copy | Download

    AMA Style

    Boe-Shong Hong. Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Appl Comput Math. 2014;3(5):217-224. doi: 10.11648/j.acm.20140305.14

    Copy | Download

  • @article{10.11648/j.acm.20140305.14,
      author = {Boe-Shong Hong},
      title = {Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {5},
      pages = {217-224},
      doi = {10.11648/j.acm.20140305.14},
      url = {https://doi.org/10.11648/j.acm.20140305.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20140305.14},
      abstract = {This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.},
     year = {2014}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics
    AU  - Boe-Shong Hong
    Y1  - 2014/09/20
    PY  - 2014
    N1  - https://doi.org/10.11648/j.acm.20140305.14
    DO  - 10.11648/j.acm.20140305.14
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 217
    EP  - 224
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20140305.14
    AB  - This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.
    VL  - 3
    IS  - 5
    ER  - 

    Copy | Download

  • Sections