Abstract
Approaches and numerical algorithms for reducing the order of mathematical models of multidimensional dynamical systems that are based on the Krylov’s subspaces method are described. To calculate matrices representing the reduced models in the state space, Lanczos’ and Arnoldi’s methods are used. Practical examples of the reduction of large-scale systems are presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. N. Krylov, “On numerical solution of equation governing frequencies of small oscillations of material systems,” Izv. Akad. Nauk SSSR, Ser. OMEN., No. 4, 491–539 (1931).
Yu. A. Andreev, Control of Finite-Dimensional Linear Objects (Nauka, Moscow, 1976) [in Russian].
A. A. Voronov, Introduction to Dynamics of Complex Controlled Systems (Nauka, Moscow, 1985) [in Russian].
M. Sh. Misrikhanov, Invariant Control of Multidimensional Systems: An Algebraic Approach (Nauka, Moscow, 2007) [in Russian].
M. Sh. Misrikhanov, Classical and New Methods for Analysis of Dynamical Multidimensional Systems (Energoatomizdat, Moscow, 2004) [in Russian].
M. Sh. Misrikhanov, “Band controllability and observability tests,” Autom. Remote Control, No. 12, 1953–1963 (2005).
M. Sh. Misrikhanov and V. N. Ryabchenko, “Algebraic and matrix methods in the theory of linear MIMO systems,” Vestn. IGEU, No. 5, 196–240 (2005).
M. Sh. Misrikhanov and V. N. Ryabchenko, “The quadratic eigenvalue problem in electric power systems,” Autom. Remote Control, No. 5, 698–720 (2006).
M. Sh. Misrikhanov and V. N. Ryabchenko, “The band formula for A.N. Krylov’s problem,” No. 12, 2142–2157 (2007).
M. Sh. Misrikhanov and V. N. Ryabchenko, “Reduction of the Rosenbrock matrix in analysis of invariant zeros of the linear MIMO-system,” Autom. Remote Control, No. 12, 1675–1691 (2008).
A. V. Bogachev, E. A. Vorob’eva, N. E. Zubov, E. A. Mikrin, M. Sh. Misrikhanov, V. N. Ryabchenko, and S. N. Timakov, “Unloading angular momentum for inertial actuators of a spacecraft in the pitch channel,” J. Comput. Syst. Sci. Int. 50, 483–490 (2011).
Essentials of Robust Control, Ed. by K. Zhou (Prentice Hall, Upper Saddle River, New Jersey, 1998).
T. Kailath, Linear Systems (Prentice Hall, Upper Saddle River, New Jersey, 1990).
N. E. Zubov, E. A. Mikrin, M. Sh. Misrikhanov, and V. N. Ryabchenko, “Solution of linear matrix equations and Lyapunov’s inequalities by the Krylov subspaces method,” Vestn. MGTU Im. N.E. Baumana, No. 2, 103–119 (2014).
Handbook of Automatic Control Theory, Ed. by A. A. Krasovskii (Nauka, Moscow, 1987) [in Russian].
G. H. Golub and C. F. van Loan, Matrix Computations (John Hopkins Univ. Press, 1989; Mir, Moscow, 1999).
M. Yu. Balandin and E. P. Shurina, Methods for Solving Systems of Linear Algebraic Equations of High Dimension (Izd-vo NGTU, Novosibirsk, 2000) [in Russian].
Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, Philadelphia, 2003).
V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Calculations (Nauka, Moscow, 1984) [in Russian].
D. S. Bernstein, Matrix Mathematics (Princeton Univ. Press, Princeton, 2005).
A. C. Antoulas, Approximation of Large-Scale Dynamical Systems (SIAM, Philadelphia, 2005).
C. Lanczos, “An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators,” J. Res. Nat. Bur. Stand. 45, 255–282 (1950).
W. E. Arnoldi, “The Principle of Minimized Iteration in the Solution of the Matrix Eigenvalue Problem,” Quart. Appl. Math. 9, 17–29 (1951).
D. Boley and G. Golub, “The Nonsymmetric Lanczos Algorithm and Controllability,” Syst. Control Lett. 16, 97–105 (1991).
D. S. Watkins, Fundamentals of Matrix Computations (Wiley, 2002; BINOM. Laboratoriya znanii, Moscow, 2006).
Gantmakher, F.R., Theory of Matrices (Nauka, Moscow, 1988).
E. Grimme, D. Sorensen, and P. van Dooren, “Model reduction of state-space systems via an implicitly restarted Lanczos method,” Numerical Algorithms 12, 1–31 (1995).
A. C. Antoulas, “A new result on positive real interpolation and model reduction,” Systems Control Lett. 54, 361–374 (2005).
R. W. Freund, “Model reduction methods based on Krylov subspaces,” Acta Numerica 12, 267–319 (2003).
S. Gugercin and A. C. Antoulas, “An H2 error expression for the Lanczos procedure,” Proc. of the 42nd IEEE Conf. on Decision Procedure, 2003.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N.E. Zubov, E.A. Mikrin, M.Sh. Misrikhanov, A.V. Proletarskii, V.N. Ryabchenko, 2015, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2015, No. 2, pp. 3–21.
Rights and permissions
About this article
Cite this article
Zubov, N.E., Mikrin, E.A., Misrikhanov, M.S. et al. Reduction of large-scale dynamical systems by the Krylov subspaces method: Analysis of approaches. J. Comput. Syst. Sci. Int. 54, 165–183 (2015). https://doi.org/10.1134/S1064230715020148
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064230715020148