Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Publisher: Elsevier
Languages: English
Types: Unknown
Subjects: QA
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials".
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [1] N. Karcanias, S. Fatouros, M. Mitrouli and G. Halikias, Approximate greatest common divisor of many polynomials, generalised resultants, and strength of approximation, Comput & Maths with Applications, vol. 51(12), 2006, pp 1817-1830.
    • [2] N. Karcanias, C. Giannakopoulos and M. Hubbard, Almost zeros of a set of polynomials of R[s], Int. J. Control, vol. 38(6), 1983, pp 1213-1238.
    • [3] M. Marcus, Finite dimensional multilinear algebra (in two parts), Marcel Deker, New York, 1973.
    • [4] N. Karcanias and C. Giannakopoulos, On Grassmann invariants and almost zeros of linear systems and the determinantal zero, pole assignment problem, Int. J. Control, vol 40, 1984.
    • [5] N. Karcanias and C. Giannakopoulos, Necessary and Sufficient Conditions for Zero Assignment by Constant Squaring Down, Linear Algebra and Its Applications, Special Issue on Control Theory, vol. 122/123/124, pp 415- 446, 1989.
    • [6] N. Karcanias, Multivariable Poles and Zeros, in Control Systems, Robotics and Automation, Encyclopedia of Life Support Systems (EOLSS), UNESCO, Eolss Publishers, UK, 2002 [http://www.eolss.net].
    • [7] B. Kouvaritakis and A.G.F. McFarlane, Geometric approach to analysis & synthesis of system zeros. Part II: Non-Square Systems, Int. J. Control, vol. 23, pp 167-181, 1976.
    • [8] H.H. Rosenbrock, State Space and Multivariable Theory, Nelson, London, 1970.
    • [9] T. Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, NJ, 1980.
    • [10] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover Publ., New York, 1969.
    • [11] G.D. Forney, Minimal Bases of Rational Vector Spaces with Applications to Multivariable Systems, SIAM J. Control, vol. 13, pp 493-520, 1975.
    • [12] N. Karcanias and J. Leventides, Grassman Invariants, Matrix Pencils and Linear System Properties, Linear Algebra & Its Applications, vol. 241-243, pp 705-731, 1996.
    • [13] J. Leventides and N. Karcanias N, Structured Squaring Down and Zero Assignment, Int. J. Control, vol. 81, pp 294-306, 2008.
    • [14] S. Barnett, Matrices Methods and Applications, Clarendon Press, Oxford, 1990.
    • [15] S. Fatouros and N. Karcanias, Resultant Properties of GCD of polynom and a Factorisation Represent of GCD, Int. J. of Control, vol. 76, pp 1666-1683, 2003.
    • [16] N. Karcanias and M. Mitrouli, Approximate algebraic computations of algebraic invariants, Symbolic methods in control systems analysis and design, IEE Control Engin. Series, vol. 56, pp 135-168, 1990.
    • [17] N. Karcanias, On the basis matrix characterisation of controllability subspaces, Int. J. Control, vol. 29, pp 767-786, 1979.
    • [18] S. Jaffe and N. Karcanias, Matrix pencil characterisation of almost (A, B)- invariant subspaces: A classification of geometric concepts, Int. J. Control, vol. 33, pp 51-93, 1981.
    • [19] N. Karcanias and P. Macbean, Structural invariants and canonical forms of linear multivariable systems, 3rd IMA Intern Conf. On Control Theory, Academic Press, pp 257-282, 1981.
    • [20] D.L. Boley and W.S. Lu, Measuring how far a controllable system is from an uncontrollable one, IEEE Tran. Auto. Contr., vol AC-31(3), pp 249-251, 1986.
    • [21] L.Elsner and C. He, An algorithm for computing the distance to uncontrollability, Syst. Contr. Lett., vol 4(5), pp 263-264, 1984.
    • [22] M. Wicks and R.A. DeCarlo, Computing the distance to an uncontrollable system, IEEE Tran. Auto. Contr., vol AC-36(1), pp 39-49, 1991.
    • [23] H.H. Rosenbrock, Structural properties of linear dynamical systems. International Journal of Control, Vol 20, pp 177-189, 1994.
    • [24] C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM , Philadelphia, 2000.
    • [25] J. Chen, M.K.H. Fan and C.N. Nett, Structured singular values and stability analysis of uncertain polynomials, Part 1: the generalized μ , Systems and Control Letters, 23:53-65, 1994.
    • [26] G. Halikias, G. Galanis, N. Karcanias and E. Milonidis, Nearest common root of polynomials, approximate Greatest Common Divisor and the structured singular value, to be published in IMAMCI, 2012.
    • [27] N. Karcanias, Invariance properties and characterisation of the greatest common divisor of a set of polynomials, Int. J. Control, 46, 1751-1760, 1987.
    • [28] M. Mitrouli, N. Karcanias, Computation of the GCD of polynomials using Gaussian transformation and shifting, Int. Journ. Control, 58, 211- 228, 1993.
    • [29] D. Christou, N. Karcanias, M. Mitrouli, The ERES method for computing the approximate GCD of several polynomials, Applied Numerical Mathematics, 60, 94-114, 2010.
    • [30] N. Karcanias, M. Mitrouli, A matrix pencil based numerical method for the computation of the GCD of polynomials, IEEE Trans. Autom. Cont., 39, 977-981, 1994.
  • No related research data.
  • No similar publications.

Share - Bookmark

Cite this article