Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment. (English) Zbl 0734.65037
The authors propose a numerical method for solving a Sylvester-type equation \(AX-XH=G,\) where, with given \(N\times N\) matrix A (considered here as large and sparse) and \(N\times m\) matrix G, they construct an \(m\times m\) Hessenberg matrix H with a preassigned spectrum, and an \(N\times m\) orthonormal matrix X.
Reviewer: A.Pil’kevich (Kiev)
MSC:
| 65F30 | Other matrix algorithms (MSC2010) |
| 65F50 | Computational methods for sparse matrices |
| 65K10 | Numerical optimization and variational techniques |
| 15A24 | Matrix equations and identities |
| 93B55 | Pole and zero placement problems |
Keywords:
partial spectrum assignment; Sylvester-type matrix equation; Arnoldi method; Luenberger observer; Hessenberg-Schur method; partial pole- assignment problem; large matrices; Hessenberg matrix with preassigned spectumReferences:
| [1] | Balas, M. J., Trends in large space structure control theory: Fondest dreams, wildest hopes, IEEE Trans. Automat. Control, AC-2, 522-535 (1982) · Zbl 0496.93007 |
| [2] | Bhattacharyya, S. P.; Desouza, E., Pole assignment via Sylvester equation, Systems Control Lett., 1, 4, 261-263 (1982) · Zbl 0473.93037 |
| [3] | Boley, D.; Golub, G. H., The Lanczos-Arnoldi algorithm and controllability, Systems Control Lett., 4, 317-324 (1987) · Zbl 0547.93021 |
| [4] | Chen, C. T., Linear System Theory and Design (1984), New York |
| [5] | Datta, B. N., An algorithm to assign eigenvalues in a Hessenberg matrix: Single-input case, IEEE Trans. Automat. Control, AC-32, 414-417 (1987) · Zbl 0612.93020 |
| [6] | Gear, W. C.; Saad, Y., Iterative solution of linear equations in ODE codes, SIAM J. Sci. Statist. Comput., 4, 583-601 (1983) · Zbl 0541.65051 |
| [7] | Golub, G. H.; Nash, S.; Van Loan, C., A Hessenberg-Schur method for the problem \(AX + XB =C\), IEEE Trans. Automat. Control, AC-24, 909-910 (1979) · Zbl 0421.65022 |
| [8] | Golub, G. H.; Van Loan, C., Matrix Computations (1984), The Johns Hopkins U.P |
| [9] | Konstantinov, M.; Petkov, P.; Christov, N., Synthesis of linear systems with desired equivalent form, J. Assoc. Comput. Mach., 6, 27-35 (1980) · Zbl 0439.93020 |
| [10] | Luenberger, D., Observing the state of a linear system, IEEE Trans. Mil. Electron., MIL-8, 74-80 (1964) |
| [11] | Miminis, G.; Paige, C. C., An algorithm for pole assignment of time invariant linear systems, Internat. J. Control, 35, 130-138 (1981) |
| [12] | Nichols, N. K., Robustness in Partial Pole Placement, (Technical Report NA/1/86 (1986), Univ. of Reading: Univ. of Reading England) · Zbl 0626.93024 |
| [13] | Patel, R. V.; Misra, P., Numerical Algorithms for eigenvalue assignment by state feedback, Proc. IEEE, 17, 1755-1764 (1984) |
| [14] | Saad, Y., A projection method for partial pole assignment in linear state feedback, IEEE Trans. Automat. Control, 33, 3, 290-297 (1988) · Zbl 0641.93031 |
| [15] | Saad, Y., Projection methods for solving large sparse eigenvalue problems, (Kagstrom, B.; Ruhe, A., Matrix Pencils, Proceedings, Pitea Havsbad, Univ. of Umea, Sweden. Matrix Pencils, Proceedings, Pitea Havsbad, Univ. of Umea, Sweden, Lecture Notes in Math, 973 (1982), Springer-Verlag: Springer-Verlag Berlin), 121-144 · Zbl 0501.65014 |
| [16] | Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7, 856-869 (1986) · Zbl 0599.65018 |
| [17] | Saad, Y., Practical use of some Krylov subspace methods for solving indefinite and nonsymmetric linear systems, SIAM J. Numer. Anal., 19, 784-811 (1984) |
| [18] | Siljak, D. D., Large Scale Dynamic Systems: Stability and Structure (1977), North Holland: North Holland New York · Zbl 0358.34044 |
| [19] | Van Dooren, P., Reduced order observers: A new algorithm and proof, Systems Control Lett., 4, 243-251 (1984) · Zbl 0542.93018 |
| [20] | Van Dooren, P.; Verhaegen, M., On the use of unitary state-space transformations, (Linear Algebra and its Role in Systems Theory (1985), Amer. Math Soc), 447-463, Contemp. Math. · Zbl 0571.93015 |
| [21] | Walker, H., Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Statist. Comput., 9, 152-163 (1988) · Zbl 0698.65021 |
| [22] | West-Vukovich, George; Davison, Edward, The decentralized control of large flexible space structures, IEEE Trans. Automat. Control, AC-29, 10, 866-879 (1984) · Zbl 0549.93005 |
| [23] | Wilkinson, J. H., The Algebraic Eigenvalue Problem (1965), Oxford U.P: Oxford U.P Oxford · Zbl 0258.65037 |
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