Extension of The Cayley-Hamilton Theorem to Continuous-time Linear Systems With Delays

• Autorzy: Kaczorek T.
• Języki publikacji: EN

Abstrakty

EN

The classical Cayley-Hamilton theorem is extended to continuous-time linear systems with delays. The matrices $A_0, A_1, dots, A_h in R^{n times n}$ of the system with $h$ delays $dot xleft(t right) = A_0 xleft(t right) + sum_{i = 1}^h {A_i xleft( {t - hi} right) + Buleft( t right)}$ satisfy $nh + 1$ algebraic matrix equations with coefficients of the characteristic polynomial $pleft( {s,w}right) = det left[ {I_n s - A_0 - A_1 w - cdots - A_h w^h }right]$, $w = e^{- hs}$.

Słowa kluczowe

EN

Cayley-Hamilton theorem; continuous time; linear system; delay; extension

PL

twierdzenie Cayleya-Hamiltona; czas ciągły; system liniowy; opóźnienie; rozszerzenie

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2005
• Tom: Vol. 15, no 2
• Strony: 231--234
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Kaczorek T.

• Institute of Control and Industrial Electronics Warsaw University of Technology Koszykowa 75, 00-662 Warsaw, Poland,

Bibliografia

• [1] Busłowicz M. and Kaczorek T. (2004): Reachability and minimum energy control of positive linear discrete-time systems with one delay. — Proc. 12-th Mediterranean Conf. Control and Automation, Kasadasi, Turkey: Izmir (on CDROM).
• [2] Chang F.R. and Chan C.N. (1992): The generalized Cayley-Hamilton theorem for standard pencis. — Syst. Contr. Lett., Vol. 18, No. 192, pp. 179–182.
• [3] Gałkowski K. (1996): Matrix description of multivariable polynomials. — Lin. Alg. and Its Applic., Vol. 234, No. 2, pp. 209–226.
• [4] Gantmacher F.R. (1974): The Theory of Matrices. — Vol. 2. — Chelsea: New York.
• [5] Kaczorek T. (1992/1993): Linear Control Systems.—Vols. I, II, Tauton: Research Studies Press.
• [6] Kaczorek T. (1994): Extensions of the Cayley-Hamilton theorem for 2D continuous-discrete linear systems. — Appl. Math. Comput. Sci., Vol. 4, No. 4, pp. 507–515.
• [7] Kaczorek T. (1995a): An existence of the Cayley-Hamilton theorem for singular 2D linear systems with non-square matrices.— Bull. Pol. Acad. Techn. Sci., Vol. 43, No. 1, pp. 39–48.
• [8] Kaczorek T. (1995b): An existence of the Cayley-Hamilton theorem for nonsquare block matrices and computation of the left and right inverses of matrices. — Bull. Pol. Acad. Techn. Sci., Vol. 43, No. 1, pp. 49–56.
• [9] Kaczorek T. (1995c): Generalization of the Cayley-Hamilton theorem for nonsquare matrices. — Proc. Int. Conf. Fundamentals of Electrotechnics and Circuit Theory XVIIISPETO, Ustron-Gliwice, Poland, pp. 77–83.
• [10] Kaczorek T. (1998): An extension of the Cayley-Hamilton theorem for a standard pair of block matrices. — Appl. Math. Comput. Sci., Vol. 8, No. 3, pp. 511–516.
• [11] Kaczorek T. (2005): Generalization of Cayley-Hamilton theorem for n-D polynomial matrices. — IEEE Trans. Automat. Contr., No. 5, (in press).
• [12] Lancaster P. (1969): Theory of Matrices. — New York, Academic, Press.
• [13] Lewis F.L. (1982): Cayley-Hamilton theorem and Fadeev’s method for the matrix pencil. [sE-A] — Proc. 22nd IEEE Conf. Decision and Control, San Diego, USA, pp. 1282–1288.
• [14] Lewis F.L. (1986): Further remarks on the Cayley-Hamilton theorem and Fadeev’s method for the matrix pencil [sE-A]. — IEEE Trans. Automat. Contr., Vol. 31, No. 7, pp. 869–870.
• [15] Mertizios B.G and Christodoulous M.A. (1986): On the generalized Cayley-Hamilton theorem.—IEEE Trans. Automat. Contr., Vol. 31, No. 1, pp. 156–157.
• [16] Smart N.M. and Barnett S. (1989): The algebra of matrices in n–dimensional systems.—Math. Contr. Inf., Vol. 6, No. 1, pp. 121–133.
• [17] Theodoru N.J. (1989): M-dimensional Cayley-Hamilton theorem. — IEEE Trans. Automat. Contr., Vol. AC-34, No. 5, pp. 563-565.
• [18] Victoria J. (1982): A block Cayley-Hamilton theorem. — Bull. Math. Soc. Sci. Math. Roum, Vol. 26, No. 1, pp. 93–97.