Geodesic
The technique of splitting operators in perturbation control theory
Kybernetika, Tome 41 (2005) no. 1, pp. 15-32 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The paper presents the technique of splitting operators, intended for perturbation analysis of control problems involving unitary matrices. Combined with the technique of Lyapunov majorants and the application of the Banach or Schauder fixed point principles, it allows to obtain rigorous non-local perturbation bounds for a set of sensitivity analysis problems. Among them are the reduction of linear systems into orthogonal canonical forms, the general feedback synthesis problem, and the pole assignment problem in particular, as well as other basic problems in control theory and linear algebra.
The paper presents the technique of splitting operators, intended for perturbation analysis of control problems involving unitary matrices. Combined with the technique of Lyapunov majorants and the application of the Banach or Schauder fixed point principles, it allows to obtain rigorous non-local perturbation bounds for a set of sensitivity analysis problems. Among them are the reduction of linear systems into orthogonal canonical forms, the general feedback synthesis problem, and the pole assignment problem in particular, as well as other basic problems in control theory and linear algebra.
Classification : 93B10, 93B28, 93B35, 93B50, 93C05
Keywords: perturbation analysis; canonical forms; feedback synthesis 
@article{KYB_2005_41_1_a1,
     author = {Konstantinov, Mihail M. and Petkov, Petko Hr. and Christov, Nicolai D.},
     title = {The technique of splitting operators in perturbation control theory},
     journal = {Kybernetika},
     pages = {15--32},
     year = {2005},
     volume = {41},
     number = {1},
     mrnumber = {2130482},
     zbl = {1249.93057},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2005_41_1_a1/}
}
TY  - JOUR
AU  - Konstantinov, Mihail M.
AU  - Petkov, Petko Hr.
AU  - Christov, Nicolai D.
TI  - The technique of splitting operators in perturbation control theory
JO  - Kybernetika
PY  - 2005
SP  - 15
EP  - 32
VL  - 41
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/KYB_2005_41_1_a1/
LA  - en
ID  - KYB_2005_41_1_a1
ER  - 
%0 Journal Article
%A Konstantinov, Mihail M.
%A Petkov, Petko Hr.
%A Christov, Nicolai D.
%T The technique of splitting operators in perturbation control theory
%J Kybernetika
%D 2005
%P 15-32
%V 41
%N 1
%U http://geodesic.mathdoc.fr/item/KYB_2005_41_1_a1/
%G en
%F KYB_2005_41_1_a1
Konstantinov, Mihail M.; Petkov, Petko Hr.; Christov, Nicolai D. The technique of splitting operators in perturbation control theory. Kybernetika, Tome 41 (2005) no. 1, pp. 15-32. http://geodesic.mathdoc.fr/item/KYB_2005_41_1_a1/

[1] Grebenikov E. A., Ryabov, Yu. A.: Constructive Methods for Analysis of Nonlinear Systems (in Russian). Nauka, Moscow 1979 | MR

[2] Hermann R., Martin C. L.: Application of algebraic geometry to systems theory. Part I. IEEE Trans. Automatic Control 22 (1977), 19–25 | DOI | MR

[3] Higham N. J.: Optimization by direct search in matrix computations. SIAM J. Matrix Anal. Appl. 14 (1993), 317–333 | DOI | MR | Zbl

[4] Kantorovich L. V., Akilov G. P.: Functional Analysis in Normed Spaces. Pergamon, New York 1964 | MR | Zbl

[5] Konstantinov M., Mehrmann, V., Petkov P.: Perturbation Analysis for the Hamiltonian Schur Form. Technical Report 98-17, Fakultät für Mathematik, TU-Chemnitz, Chemnitz 1998

[6] Konstantinov M. M., Petkov, P. Hr., Christov N. D.: Invariants and canonical forms for linear multivariable systems under the action of orthogonal transformation groups. Kybernetika 17 (1981), 413–424 | MR | Zbl

[7] Konstantinov M. M., Petkov, P. Hr., Christov N. D.: Nonlocal perturbation analysis of the Schur system of a matrix. SIAM J. Matrix Anal. Appl. 15 (1994), 383–392 | DOI | MR | Zbl

[8] Konstantinov M. M., Petkov, P. Hr., Christov N. D.: Sensitivity analysis of the feedback synthesis problem. IEEE Trans. Automatic Control 42 (1997), 568–573 | DOI | MR | Zbl

[9] Konstantinov M. M., Petkov P. Hr., Christov N. D., Gu D. W., Mehrmann V.: Sensitivity of Lyapunov equations. In: Advances in Intelligent Systems and Computer Science (N. E. Mastorakis, ed.). WSES Press, 1999, pp. 289–292

[10] Konstantinov M. M., Petkov P. Hr., Gu D. W., Postlethwaite I.: Perturbation Analysis in Finite Dimensional Spaces. Tech. Rep. 96–18, Engineering Department, Leicester University, Leicester 1996

[11] Konstantinov M. M., Petkov P. Hr., Gu D. W., Postlethwaite I.: Perturbation analysis of orthogonal canonical forms. Linear Algebra Appl. 251 (1997), 267–291 | MR | Zbl

[12] Ortega J., Rheinboldt W.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York 1970 | MR | Zbl

[13] Petkov P. Hr., Christov N. D., Konstantinov M. M.: A new approach to the perturbation analysis of linear control problems. Preprints 11th IFAC World Congr., Tallin 1990, pp. 311–316 | MR

[14] Petkov P. Hr., Christov N. D., Konstantinov M. M.: Computational Methods for Linear Control Systems. Prentice–Hall, New York 1991 | Zbl

[15] Petkov P. Hr., Christov N. D., Konstantinov M. M.: Sensitivity of orthogonal canonical forms for single-input systems. In: Proc. 22nd Spring Conference of UBM, Sofia 1992, pp. 66–73

[16] Petkov P. Hr., Christov N. D., Konstantinov M. M.: Perturbation analysis of orthogonal canonical forms and pole assignment for single-input systems. In: Proc. 2nd European Control Conference, Groningen 1993, pp. 1397–1400

[17] Petkov P. Hr., Christov N. D., Konstantinov M. M.: Perturbation controllability analysis of linear multivariable systems. Preprints 12th IFAC World Congress, Sydney 1993, pp. 491–494

[18] Sun J.-G.: On perturbation bounds for the QR factorization. Linear Algebra Appl. 215 (1995), 95–111 | MR | Zbl

[19] Sun J.-G.: Perturbation bounds for the generalized Schur decomposition. SIAM J. Matrix Anal. Appl. 16 (1995), 1328–1340 | DOI | MR | Zbl