Band Formulas for Analysis and Synthesis of Controlled Dynamic MIMO Systems

The application of A.N. Krylov’s method in the control theory for solving various problems of analysis and synthesis of linear dynamic multi-input multi-output (MIMO) systems is discussed. These problems include the calculation of a balanced implementation of the linear MIMO system transfer matrix in the state space; reduction and decomposition of a model of this system in the state space; the definition of controlled and observed subspaces; stabilization using state elements feedback; synthesis of the control that provides the system invariance to the external perturbation. An approach is presented for analysis and synthesis of linear dynamical MIMO-systems on the basis of band formulas of controllability analysis. Using transformations of controllability band matrices, the band formula, connecting the MIMO-system parameters and the characteristic polynomial coefficients, is found.

References

[1] Andreev Yu.N. Upravlenie konechnomernymi linejnymi ob’ektami [Control of finitedimensional linear plants]. Moscow, Nauka Publ., 1976. 424 p.

[2] Wonham W.M. Linear Multivariable Control: A Geometric Approach. 3rd ed. New York, Springer-Verlag, 1985. 334 p. (Russ. Ed.: Linejnye mnogomernye sistemy upravlenija: geometricheskij podhod. Moscow, Nauka Publ., 1980. 376 p.).

[3] Kailath T. Linear Systems. Prentice-Hall, Inc., Englewood Cliffs, N.J.: 1980. 682 p.

[4] Dorf R.S., Bishop R.H. Modern Control Systems. 12th ed. Prentice-Hall, 2011. 1034 p. (Russ. Ed.: Sovremennye sistemy upravlenija. Moscow, Laboratorija Bazovyh Znanij Publ., 2004. 831 p.).

[5] Misrikhanov M.Sh. Invariantnoe upravlenie mnogomernymi sistemami. Algebraicheskij podhod. [Invariant control of multivariable systems. Algebraic approach]. Moscow, Nauka Publ., 2007. 284 p.

[6] Voevodin V.V., Kuznetsov Yu.A. Matricy i vychislenija [Matrices and computations]. Moscow, Nauka Publ., 1984. 386 p.

[7] Demmel J.W. Applied Numerical Linear Algebra. SIAM, 1997. 184 p. (Russ. Ed.: Vychislitel’naja linejnaja algebra. Teorija i prilozhenija. Moscow, Mir Publ., 2001. 153 p.).

[8] Misrikhanov M.Sh., Ryabchenko V.N. Analysis and Synthesis of Linear Dynamic Systems Based on Banded Formulas. Vestn. Ivanovskiy Gos Energ. Univ. (IGEU) [Herald of the Ivanovo State Power Eng. Un.], 2005, iss. 5, pp. 243-248 (in Russ.).

[9] Misrikhanov M.Sh., Ryabchenko V.N. The band formula for A.N. Krylov’s problem. Avtom. Telemekh. [Automation and Remote Control, vol. 68, no. 12, pp. 2142-2157], 2007, no. 12, pp. 53-69 (in Russ.).

[10] Nordstrom K., Norlander H. On the multi input pole placement control problem. Proc. 36th IEEE Conf. Decision and Control., 1997, vol. 5, pp. 4288-4293.

[11] Misrikhanov M.Sh., Ryabchenko V.N. Algebraic and Matrix Methods in the Theory of linear MIMO Systems. Vestn. Ivanovskiy Gos Energ. Univ. (IGEU) [Herald of the Ivanovo State Power Eng. Un.], 2005, iss. 5, pp. 196-240 (in Russ.).