About the Band Formula for the Solution of a Generalized Krylov Problem for Affine Dynamic System

The article describes the band formula that is obtained for the solution of a generalized Krylov problem by using the theorem and Cayley-Hamilton identity, definitions of right and left maximum rank zero divisors for the given matrix. It consists in finding the characteristic polynomial coefficients for nonlinear affine dynamic system. Numerical example of analytical calculation of characteristic polynomial coefficients is given for the control problem of longitudinal spacecraft ’s motion with use of slewings on entering the Earth atmosphere. In this case it is a non-linear affine system of the third order. The given calculation is made both for open- and closed-loop feedback control systems.

References

[1] Gantmakher F.R. Teoriya matrits [Theory of matrices]. Moscow, Nauka Publ., 1988.

[2] Kaczorek T. Vectors and matrices in automation and electrotechnics. Warsaw, Polish Scientific Publisher, 1988.

[3] Kaczorek T. Generalization of the Cayley-Hamilton theorem for nonsquare matrices. Proc. Int. Conf. "Fundamentals of Electrotechnics and Circuit Theory XVIII-SPETO", 1988, pp. 77-83.

[4] Victoria J. A block-Cayley-Hamilton theorem. Bull. Math. Soc. Sci. Math. Roum, 1982, vol. 26, no. 1, pp. 93-97.

[5] Kaczorek T. An extension of the Cayley-Hamilton theorem for non-square block matrices and computation of the left and right inverses of matrices. Bull. Pol. Acad. Techn. Sci., 1995, vol. 43, no. 1, pp. 39-56.

[6] Chang F.R., Chen C.N. The generalized Cayley-Hamilton theorem for standard pencils. Systems and Control Lett., 1992, no. 18, pp. 179-182.

[7] Lewis F.L. Further remarks on the Cayley-Hamilton theorem and Fadeev’s method for the matrix pencil. IEEE Trans. Automat. Control, 1986, vol. 31, pp. 869-870.

[8] Smart N.M., Barnett S. The algebra of matrices in n-dimensional systems. Math. Control Inform., 1989, vol. 6, pp. 121-133.

[9] Kaczorek T. Generalization of the Cayley-Hamilton theorem for n-d polynomial matrices. IEEE Trans. Automat. Control, 2005, vol. 50, pp. 671-674.

[10] Kaczorek T. An extension of the Cayley-Hamilton theorem for nonlinear time-varying system. Int. J. Appl. Math. Comput. Sci., 2006, vol. 16, no. 1, pp. 141-146.

[11] Misrikhanov M.Sh., Ryabchenko V.N. Analysis and synthesis of linear dynamical systems based on the band formulas. Vestn. Ivanovskiy Gos Energ. Univ. (IGEU) [Bull. of the Ivanovo State Power Eng. Uni.], 2005, iss. 5, pp. 243-248 (in Russ.).

[12] Misrikhanov M.Sh., Ryabchenko V.N. Algebraic and matrix methods in the theory of linear MIMO-systems. Vestn. Ivanovskiy Gos Energ. Univ. (IGEU) [Bull. of the Ivanovo State Power Eng. Uni.], 2005, iss. 5, pp. 196-240 (in Russ.).

[13] Krasnoshchechenko V.I., Krishchenko A.P. Nelineynye sistemy: geometricheskie metody analiza i sinteza [Nonlinear systems: geometric methods for analysis and synthesis]. Moscow, MGTU im N.E. Baumana Publ., 2005. 520 p.

[14] Podchukaev V.A. Analiticheskie metody teorii avtomaticheskogo upravleniya [Analytical methods of control theory]. Moscow, Fizmatlit Publ., 2002. 448 p.

[15] Yaroshevskiy V.A. Vkhod v atmosferu kosmicheskikh letatel’nykh apparatov [Entrance to the atmosphere of spacecraft]. Moscow, Nauka Publ., 1988. 336 p.

[16] Zubov N.E., Mikrin E.A., Ryabchenko V.N. Synthesis of control law for longitudinal motion of the spacecraft in the Earth’s atmosphere during landing. Jelektr. nauchno-tehn. Izd. "Inzhenernyj zhurnal: nauka i innovacii" MGTU im. N.E. Baumana [El. Sc.-Techn. Publ. "Eng. J.: Science and Innovation" of Bauman MSTU], 2013, no. 10 (22) (in Russ.).