Abstract
A new formula for solving the Krylov problem on the basis of band matrices is proposed. Properties of the obtained formula are studied for problems of analysis and synthesis of linear controlled systems.
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Faddeev, D.K. and Faddeeva, V.N., Vychislitel’nye metody lineinoi algebry (Computational Methods of Linear Algebra), St. Petersburg: Lan’, 2002.
Andreev, Yu.N., Upravlenie konechnomernymi lineinymi ob"ektami (Control of Finite-Dimensional Linear Objects), Moscow: Nauka, 1976.
Demmel, J.W., Applied Numerical Linear Algebra, Philadelphia: SIAM, 1997. Translated under the title Vychislitel’naya lineinaya algebra. Teoriya i prilozheniya, Moscow: Mir, 2001.
Bernstein, D.S., Matrix Mathematics, Princeton: Princeton Univ. Press, 2005.
Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley, 1972. Translated under the title Lineinye optimal’nye sistemy upravleniya, Moscow: Mir, 1977.
Wonham, W.M., Linear Multivariable Control: A Geometric Approach, New York: Springer-Verlag, 1979. Translated under the title Lineinye mnogomernye sistemy upravleniya: geometricheskii podkhod, Moscow: Nauka, 1980.
Kailath, T., Linear Systems, Englewood Cliffs: Prentice Hall, 1980.
Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
Misrikhanov, M.Sh., Invariantnoe upravlenie mnogomernymi sistemami. Algebraicheskii podkhod (Invariant Control of Multidimensional Systems: An Algebraic Approach), Moscow: Nauka, 2007.
Dorf, R.C. and Bishop, R.H., Modern Control Systems, Englewood Cliffs: Prentice Hall, 2004. Translated under the title Sovremennye sistemy upravleniya, Moscow: Laboratoriya Bazovykh Znanii, 2004.
Krylov, A.N., On the Numerical Solution to the Equation that Defines Frequencies of Small Oscilations of Material Systems in Technical Questions, Izv. Akad. Nauk SSSR, 7 Ser. Otdel. Mat. Estestv. Nauk, 1931, no. 4, pp. 491–539.
Voevodin, V.V. and Kuznetsov, Yu.A., Matritsy i vychisleniya (Matrices and Computations), Moscow: Nauka, 1984.
Gantmakher, F.R., Teoriya matrits, Moscow: Nauka, 1988. Translated under the title The Theory of Matrices, New York: Chelsea, 1959.
Kalman, R., Contributions to the Theory of Optimal Control, Bull. Socied. Mat. Mexic., 1960, vol. 5, pp. 102–119.
Kalman, R.E., Falb, P.L., and Arbib, M.A., Topics in Mathematical Systems Theory, New York: McGraw-Hill, 1969. Translated under the title Ocherki po matematicheskoi teorii sistem, Moscow: Mir, 1971.
Misrikhanov, M.Sh., Band Controllability and Observability, Avtom. Telemekh., 2005, no. 12, pp. 93–104.
Zybin, E.Yu., Misrikhanov, M.Sh., and Ryabchenko, V.N., Minimal Parameterization of Solutions to Linear Matrix Equations, in Sovremennye metody upravleniya mnogosvyaznymi dinamicheskimi sistemami (Modern Control of Dynamic Systems), Krasovskii, A.A., Ed., Moscow: Energoatomizdat, 2003, pp. 191–202.
Commault, C., Lafay, J.F., and Malabre, M., Structure of Linear Systems. Geometric and Transfer Matrix Approaches, Kybernetika, 1991, vol. 27, no. 3, pp. 170–185.
Taufer, J., Lösing der Randwertprobleme für Systeme von linearen Differentialgleichungen, Rozpravy CSAV, Rada Mat. Prir. Ved, Praha, 1973, vol. 83, no. 5. Translated under the title Reshenie granichnykh zadach dlya sistem lineinykh differentsial’nykh uravnenii, Moscow: Nauka, 1981.
Cohn, P.M., Free Rings and Their Relations, London: Academic, 1971. Translated under the title Svobodnye kol’tsa i ikh svyazi, Moscow: Mir, 1975.
Misrikhanov, M.Sh. and Ryabchenko, V.N., Algebraic Systems with Locally Varying Matrices for Control in Electrical Energetics, Avtom. Telemekh., 2006, no. 5, pp. 133–141.
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Original Russian Text © M.Sh. Misrikhanov, V.N. Ryabchenko, 2007, published in Avtomatika i Telemekhanika, 2007, No. 12, pp. 53–69.