On differential structures of polynomial spaces in control theory

Abstract

A valuable number of works has been published about Hurwitz and Schur polynomials in order to known better their properties. For example it is known that the sets of Hurwitz and Schur polynomials are open and no convex sets. Besides, the set of monic Schur polynomials is contractible. Now we study this set using ideas from differential topology, and we prove that the space of Schur complex polynomials with positive leading coefficient, and the space of Hurwitz complex polynomials which leading coefficient having positive real part, have structure of trivial vector bundle, and each space of (Schur complex and real, Hurwitz complex) polynomials has a differential structure diffeomorphic to some known spaces.

Keywords

Schur polynomials / Hurwitz complex polynomials / trivial vector bundle

References

[1] Ackermann, J. (2002). Robust Control: The Parameter Space Approach. Springer Verlag

[2] Aguirre B., Suárez R.. Algebraic test for the Hurwitz stability of a given segment of polynomials. boletin de la sociedad matematica mexicana, 2006, 3(122): 261-275.

[3] Aguirre-Hernández B., Frías-Armenta M.E., Verduzco F.. Smooth trivial vector bundle structure of the space of Hurwitz polynomials. AUTOMATICA, 2009, 45: 2864-2868.

[4] Barmish, B.R. (1994). New Tools for Robustness of Linear Systems. McMillan

[5] Bhattacharyya, S.P., Chapellat, H. & Keel, L.H. (1995). Robust Control: The Parametric Approach. Prentice-Hall

[6] Bialas S.A.. A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices. Bull. Polish. Acad. Sci. Tech. Sci., 1985, 33(9-10): 473-480.

[7] Blanchini F., Tempo R., Dabbene F.. Computation of the minimum destabilizing volume for interval and affine families of polynomials. IEEE Transactions on Automatic Control, 1998, 43(18): 1159-1163.

[8] Bose N.K.. Digital Filters: Theory and Application, 1993, New York: Elsevier Science North-Holland, Krieger Publishing Co

[9] Bröcker, T.H. & Jänich, K. (1982). Introduction to Differential Topology. Cambridge University Press

[10] Dabbene F., Polyak B.T., Tempo R.. On the complete instability of interval polynomials. Systems and Control Letters, 2007, 56(6): 431-438.

[11] Duan G.R., Wang M.Z.. Properties of the entire set of Hurwitz polynomials and stability analysis of polynomial families. IEEE Transactions on Automatic Control, 1994, 39(12): 2490-2494.

[12] Fam A.T., Meditch J.S.. A canonical parameter space for linear system design. IEEE Transactions on Automatic Control, 1978, 23(3): 454-458.

[13] Gantmacher, F.R. (1959). Matrix Theory, Vol 2. AMS Chelsea Publishing

[14] Garcia, F.R., Aguirre B. & Suarez, R. (2008). Stabilization of linear sampled-data systems by a time-delay feedback control. Mathematical Problems in Engineering, Article ID 270518, doi:10.1155/2008/270518

[15] Hermite C.. Sur le nombre des recines dune equation algebrique comprise entre des limites donnes. Journal fur die reine und angewandte mathematik, 1856, 52: 39-51.

[16] Hinrichsen D., Pritchard J.. Mathematical Systems Theory I, modeling, state space analysis, stability and robustness, 2005, Berlin: Springer-Verlag

[17] Hirsch, M.W. (1997). Differential topology. In: Mathematics, 33. Springer

[18] Hurwitz, A. Uber die Bedingungen, unter welchen eine Gleichung nur Wurzlen mit negativen reellen Teilen besitzt. Mathematische Annalen, 46: 273–284

[19] Jury, E.I. Sampled-data Control Systems. John Wiley & Sons, Inc., New York

[20] Jury, E.I. Inners and Stability of Dynamic Systems. Krieger Publishing Co., New York

[21] Kharitonov V.L.. Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Dif. Urav., 1978, 14: 2086-2088.

[22] Lancaster, P. & Tismenetsky, M. (1985). The Theory of Matrices with Applications. Academic Press

[23] Willems J.C., Tempo R.. The Kharitonov theorem with degree drop. IEEE Transactions on Automatic Control, 1999, 44(11): 2218-2220.