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Authors
Perdon, Anna Maria; Anderlucci, Maria

Digital Object Identifier (DOI)
10.3182/20090901-3-RO-4009.00039

Page Numbers:
242-247

Index Terms
systems with time-delays,singular systems,disturbance decoupling,geometric approach

Abstract
The Disturbance Decoupling Problem by state feedback for linear generalized state systems with a finite number of commensurable point delays is formulated and investigated using as models linear singular systems with coeefficients in a ring. Necessary and sufficient conditions for its solution are given in geometric terms, as well as algorithmic procedures to test them. Examples are worked out in details.

References

[1] Jean Assan, Jean François Lafay, and A. M. Perdon.
    Computation of maximal pre-controllability submodules
    over a noetherian ring. Systems & Control Letters, 37
    (3):153-161, July 1999a.
[2] Jean Assan, Jean François Lafay, A. M. Perdon, and
    Jean Jacques Loiseau. Effective computation of maximal
    controlled invariant submodules over a principal ring. In
    Proceedings of the 38th IEEE Conference on Decision
    and Control, volume 4, pages 4216-4221, Phoenix, AZ,
    1999b.
[3] G. Basile and G. Marro. Controlled and conditioned
    invariant subspace in linear system theory. Journal
    Optimization Theory and Applications, 3, 1969.
[4] G. Basile and G. Marro. Controlled and Conditioned
    Invariants in Linear System Theory. Prentice Hall,
    Englewood Cliffs, New Jersey 07632, 1992. ISBN 0-13-
    172974-8.
[5] D. Cobb. State feedback impulse elimination for singular
    systems over a hermite domain. SIAM Journal on
    Control and Optimization, 44(6):2189-2209, January
    2006.
[6] G. Conte and A. M. Perdon. Current Trends in Nonlinear
    Systems and Control, chapter Unknown Input Observers
    and Residual Generators for Linear Time Delay Systems.
    Birkhäuser, 2005.
[7] G. Conte and A. M. Perdon. Systems over rings: Geometric
    theory and applications. Annual Review in Control, 24
    (1):pp 113-124, 2000.
[8] G. Conte and A. M. Perdon. The geometric approach for
    systems over a ring. In Proceedings of the 37th IEEE
    Conference on Decision and Control, volume 3, pages
    3557-3562, 1998.
[9] G. Conte and A. M. Perdon. Stability issues in the
    disturbance decoupling problem for systems over rings.
    In Ed. Libreria Progetto, editor, A tribute to Antonio
    Lepshyl, G. Picci e M.E. Valcher Eds., Padova, Italy,
    2007.
[10] G. Conte and A. M. Perdon. Systems over principal ideal
     domains. a polynomial model approach. SIAM Journal
     on Control and Optimization, 20, 1982.
[11] G. Conte and A. M. Perdon. The disturbance decoupling
     problem for systems over a ring. SIAM Journal on
     Control and Optimization, 33, 1995.
[12] L. R. Fletcher and A. Aasaraai. On disturbance decoupling
     in descriptor systems. SIAM Journal on Control and
     Optimization, 27(6):1319-1332, November 1989.
[13] Emilia Fridman. A lyapunov-based approach to stability
     of descriptor systems with delay. In Proceedings of
     IEEE International Conference on Decision and Control,
     volume 3, pages 2850-2855, Orlando, Florida, USA,
     December 2001.
[14] Feliks Ruvimovic Gantmacher. Théorie des matrices.
     Tome II: questions spéciales et applications. Dunod,
     1966.
[15] M. L. J. Hautus. Feedback Control of Linear and Non-linear
      Systems (Proc. joint workshop Bielefeld/Roma),
     volume 18 of Lecture Notes in Control ad Information
     Sciences, chapter Controlled Invariance in Systems over
     Rings, pages 107-122. Springer-Verlag, Berlin, 1982.
[16] E. W. Kamen. Mathematical System Theory: The influence
     of R. E. Kalman, A.C. Antoulas Ed., chapter
     Linear system over rings: from R. E. Kalman to the
     present. Springer-Verlag, Berlin, 1991.
[17] Frank L. Lewis. Geometric design techniques for observers
     in singular systems. Automatica, 26(2):411-415, March
     1990.
[18] Yongqing Liu and Xiangsheng Xie. On problem of stabilizaiion
     by output feedback for linear singular systems
     with time delay: A new approach. In Proceedings of
     The IEEE International Conference on Industrial Technology,
     pages 561-564, 1996.
[19] S. Ma and Z. Cheng. Observer design for discrete time-delay
      singular systems with unknown inputs. In Proceedings
     of the American Control Conference, volume 6,
     pages 4215-4219, Portland, OR, USA, June 2005.
[20] M. Malabre. Generalized linear systems: Geometric and
     structural approach. Linear Algebra and Its Applications,
     122/123/124:591-621, 1989.
[21] A. M. Perdon and M. Anderlucci. Impulse elimination by
     derivative feedback for singular systems with delay. In
     submitted to the 48th CDC, 2009.
[22] A. M. Perdon and M. Anderlucci. An unknown input
     observer for singular time-delay systems. In Proc.
     of the 14th Mediterranean Conference on Control and
     Automation (MED'06), Ancona, Italy, June 28-30 2006.
[23] A. M. Perdon and M. Anderlucci. A residual generator
     for singular time-delay systems. In Proc. of the 15th
     Mediterranean Conference on Control and Automation
     (MED'07), Athens, Greece, June 27-29 2007.
[24] A. M. Perdon and M. Anderlucci. Disturbance decoupling
     problem for neutral systems with delay: A geometric approach.
     In Proc. of the 16th Mediterranean Conference
     on Control and Automation (MED'08), Ajaccio, France,
     June 25-27 2008.
[25] A. M. Perdon, M. Anderlucci, and M. Caboara. Efficient
     algorithms for geometric control of systems over rings.
     International Journal of Control - Special Issue on The
     Use of Computer Algebra Systems for Computer Aided
     Control System Design, 79(11):1401-1417, November
     2006.
[26] J. Lam S. Xu and Y. Zou. Simplified descriptor system
     approach to delay-dependent stability and performance
     analyses for time-delay systems. In IEE Proceedings
     on Control Theory and Applications, volume 152, pages
     147-151, 2005.
[27] CoCoATeam. CoCoA: a system for doing Computations
     in Commutative Algebra. Freely
     available at http://cocoa.dima.unige.it
     as well as from the mirror site in America
     ftp://ftp.reed.edu/mirrors/cocoa.
[28] W. M. Wonham. Linear Multivariable Control: a Geometric
     Approach. Springer-Verlag, Berlin, 3rd edition, 1985.
     ISBN 0-38796-071-6.
[29] Zheng-Guang Wu and Wu-Neng Zhou. Delay-dependent
     robust stabilization for uncertain singular systems with
     state delay. Acta Automatica Sinica, 33(7):714-718,
     Julay 2007.
[30] Shengyuan Xu, Paul Van Dooren, Radu Stefan, and James
     Lam. Robust stability and stabilization for singular
     systems with state delay and parameter uncertainty.
     IEEE Transactions on Automatic Control, 47(7):1122-
     1128, July 2002.
[31] S. Zhu Y. Wang and Z. Cheng. Disturbance decoupling for
     singular systems. In 43rd IEEE Conference on Decision
     and Control, Atlantis, Bahamas, 2004.