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In this paper we have factorized matrix polynomials into a complete set of spectral factors using a new design algorithm, and with some systematic procedures a complete set of block roots (solvents) have been obtained. The newly developed procedure is just an extension of the (scalar) Horner method to its block form for use in the computation of the block roots of matrix polynomial, the block-Horner method bringing a local iterative nature, faster convergence, nested programmable scheme, needless of any prior knowledge of the matrix polynomial, with the only one inconvenience, which is the strong dependence on the initial guess. In order to avoid this trap, we proposed a combination of two computational procedures, for which the complete program starts with the right block-Q.D. algorithm. It is then followed by a refinement of the right factor by block-Horner’s algorithm. This results in the global nature of the program, which is faster in execution, has well defined initial conditions, and good convergence in much less time.
Czasopismo
Rocznik
Tom
Strony
41--76
Opis fizyczny
Bibliogr. 42 poz., rys.
Twórcy
autor
- Institute of Aeronautics and Space Studies, Saad Dahlab University, Blida, Algeria
autor
- Mechanical Engineering, Materials and Structures Laboratory, Institute of Science and Technology, Tissemsilt University Center, Tissemsilt, Algeria
autor
- Department of Control Systems, Electronics and Electrotechnics Institute, University of Boumerdes (Ex:INELEC), Algeria
autor
- Department of Control Systems, Electronics and Electrotechnics Institute, University of Boumerdes (Ex:INELEC), Algeria
autor
- Western Macedonia University of Applied Sciences, Kozani, Greece
Bibliografia
- Ahn, S. M. (1982) Stability of a Matrix Polynomial in Discrete Systems. IEEE Trans. on Auto. Contr., AC-27, 1122-1124.
- Barnett, S. (1971) Matrices in Control Theory, Van Nostrand Reinhold, New York.
- Bekhiti, B., Dahimene, A., Nail, B., Hariche, K. and Hamadouche, A. (2015) On Block Roots of Matrix Polynomials Based MIMO Control System Design. 4th International Conference on Electrical Engineering ICEE Boumerdes.
- Burden, R. L. and Faires, J. D. (2005) Numerical Analysis, 8th Edition. Thomson Brooks/Cole, Belmont, CA, USA.
- Chen, C. T. (1984) Linear System Theory and Design. Holt, Reinhart and Winston.
- Dahimene, A. (2009) Incomplete matrix partial fraction expansion. Control and Cybernetics, 38, 3.
- DiStefano, J. J., Stubberud, A. R. and Williams, I.J. (1967) Theory and Problems of Feedback and Control Systems. Mc Graw Hill.
- Denman, E. D. (1977) Matrix polynomials, roots, and spectral factors. A &. Math. Comput. 3:359-368.
- Denman, E. D. and Beavers, A. N. (1976) The matrix sign function and computations in systems. Appl. Math. Comput. 2:63-94.
- Dennis, J. E., Traub, J. F. and Weber, R. P. (1976) The algebraic theory of matrix polynomials. J. Numer. Anal. 13:831-845.
- Dennis, J. E., Traub, J. F. and Weber, R. P. (1978) Algorithms for solvents of matrix polynomials. J. Numer. Anal. 15:523-533.
- Gohberg, I., Kaashoek, M. A. and Rodman, L. (1978) Spectral analysis of operator polynomial and a generalized Vandermonde matrix, 1. The finite-dimensional case. In: Topics in Functional Analysis. Academic Press, 91-128.
- Gohberg, I., Lancaster P. and Rodman, L. (1982) Matrix Polynomials. Academic Press.
- Hariche, K. (1987) Interpolation Theory in the Structural Analysis of λ-matrices. Chapter 3, Ph. D. Dissertation, University of Houston.
- Hariche, K. and Denman, E. D. (1988) On Solvents and Lagrange Interpolating λ-Matrices. Applied Mathematics and Computation 25321–332.
- Hariche, K. and Denman, E. D. (1989) Interpolation Theory and λ-Matrices. Journal of Mathematical Analysis and Applications 143, 53 and 547.
- Henrici, P. (1958) The Quotient-Difference Algorithm. Nat. Bur. Standards Appl. Math. Series, 49, 23-46.
- Higham, N. J. (2008) Functions of Matrices: Theory and Computation. SIAM. Kailath, T. and Li, W. (1980) Linear Systems. Prentice Hall.
- Kucera, V. (1979) Discrete Linear Control: The Polynomial Equation Approach. John Wiley.
- Meyer, C. D. (2000) Matrix Analysis and Applied Linear Algebra. SIAM.
- Parlett, B. N. (1967) The LU and QR Algorithms. In: A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers, 2. John Wiley.
- Pereira, E. (2003) On solvents of matrix polynomials. Appl. Numer. Math., 47, 197–208,
- Pereira, E. (2003) Block eigenvalues and solution of differential matrix equation. Mathematical Notes, Miskolc, 4, 1, 45–51.
- Pathan, A. and Collyer, T. (2003) The wonder of Horner’s method. Mathematical Gazette, 87, 509, 230-242.
- Resende, P. and Kaskurewicz, E. (1989) A Sufficient Condition for the Stability of Matrix Polynomials. IEEE Trans. on. Auto. Contr., AC-34, 539-541.
- Shieh, L. S. and Sacheti, S. (1978) A Matrix in the Block Schwarz Form and the Stability of Matrix Polynomials. Int. J. Control, 27, 245-259.
- Shieh, L. S. and Chahin, N. (1981) A computer-aided method for the factorization of matrix polynomials. Internat. Systems Sci., 12:1303-1316.
- Shieh, L. S., Tsay, Y. T. and Coleman, N. I. (1981) Algorithms for solvents and spectral factors of matrix polynomials. Internat. J. Control, 12:1303-1316 .
- Shieh, L. S. and Tsay, Y. T. (1981) Transformation of solvent and spectral factors of matrix polynomial, and their applications. Internat. J. Control, 34:813-823.
- Shieh, L. S. and Tsay, Y. T. (1982a) Transformation of a class of multivariable control systems to block companion forms. IEEE Trans. Automat. Control, 27:199-203.
- Shieh, L. S. and Tsay, Y. T. (1982b) Block modal matrices and their applications to multivariable control systems. IEE Proc. D Control Theory Appl., 2:41-48.
- Shieh, L. S. and Tsay, Y. T. (1984) Algebraic-geometric approach for the modal reduction of large-scale multivariable systems. IEE Proc. D Control Theory Appl. 131(1):23-26.
- Shieh, L. S., Chang, F. R. and McInnis, B. C. (1986) The block partial fraction expansion of a matrix fraction description with repeated block poles. IEEE Trans. Automut. Control. 31:23-36.
- Shieh, L. S., Chang, F. R. and McInnis, B. C. (1986) The block partial fraction expansion of a matrix fraction description with repeated blocl poles. IEEE Trans. Auto. Cont. AC-31, 236-239.
- Solak, M. K. (1987) Divisors of Polynomial Matrices: Theory and Applications. IEEE Trans. on Auto. Contr., AC-32, 916-919.
- Tsai, J. S. H., Shieh, I. S. and Shen, T. T. C. (1988) Block power method for computing solvents and spectral factors of matrix polynomials. Internut. Computers and Math. Appl., 16:683-699.
- Tsay, Y. T. and Shieh, L. S. (1983) Block decompositions and block modal controls of multivariable control systems. Automatica, 19, 1, 29-40.
- Tsai, J. S. H., Chen C. M. and Shieh, L. S. (1992) A Computer-Aided Method for Solvents and Spectral Factors of Matrix Polynomials. Applied Mathematics and Computation, 47:211-235.
- Yaici, M., Hariche, K. and Clark, T. (2014) Contribution to the Polynomial Eigenvalue Problem. International Journal of Mathematics, Computational, Natural and Physical Engineering, 8(10).
- Yaici, M. and Hariche, K. (2014a) On eigenstructure assignment using block poles placement. European Journal of Control, 20(5), 217–226.
- Yaici, M. and Hariche, K. (2014b) On Solvents of Matrix Polynomials. International Journal of Modeling and Optimization, 4, 4.
- Zabot, H and Hariche, K. (1997) On solvents-based model reduction of MIMO systems. International Journal of Systems Science, 28, 5, 499-505.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
Identyfikator YADDA
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