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A direct algorithm for pole assignment of time-invariant multi-input linear systems using state feedback. (English) Zbl 0652.93015

A numerically stable algorithm is presented for the computation of the state feedback matrix for multi-input linear systems such that the closed-loop sysem matrix has desired eigenvalues. Good numerical behaviour in the algorithm is obtained by the use of orthogonal transformations. The numerical stability of the algorithm is proved doing a backward rounding error analysis. The performance of the algorithm is illustrated by two numerical examples.
Reviewer: T.Kaczorek

MSC:

93B40 Computational methods in systems theory (MSC2010)
93B55 Pole and zero placement problems
93C05 Linear systems in control theory
65G50 Roundoff error
65K10 Numerical optimization and variational techniques
93B17 Transformations

Software:

Matlab
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Full Text: DOI

References:

[1] Cavin, R. K.; Bhattacharyya, S. P., Robust and well conditioned eigenstructure assignment via Sylvester’s equation, (Proc. 1982 American Control Conf.. Proc. 1982 American Control Conf., Arlington, VA (1982)) · Zbl 0512.93035
[2] Davison, E. J., On pole assignment in multi-variable linear systems, IEEE Trans. Aut. Control, AC-13, 747-748 (1968)
[3] Eising, R., Pole assignment, a new proof and algorithms, (Memorandum COROR 81-10 (1981), Eindhoven University of Technology, Department of Mathematics and Computing Science, Statistics and Operations Research Group: Eindhoven University of Technology, Department of Mathematics and Computing Science, Statistics and Operations Research Group Eindhoven, The Netherlands) · Zbl 0497.93017
[4] Fahmy, M. M.; O’Reilly, J., On eigenstructure assignment in linear multivariable systems, IEEE Trans. Aut. Control, AC-27, 690-693 (1982) · Zbl 0484.93046
[5] Francis, G. F., The QR transformation, A unitary analogue to the LR transformation—Part I, II. A unitary analogue to the LR transformation—Part I, II, Computer J., 332-345 (1961), 1962 · Zbl 0104.34304
[6] Golub, G. H.; Nash, S.; Van Loan, C., A Hessenberg Schur method for the problem \(AX + XB = C\), IEEE Trans. Aut. Control, AC-24, 909-913 (1979) · Zbl 0421.65022
[7] Heymann, M., On pole assignment in multi-input controllable linear systems, IEEE Trans. Aut. Control, AC-13, 748-749 (1968)
[8] Kautsky, J.; Nichols, N. K.; Van Dooren, P., Robust pole assignment in linear state feedback, Int. J. Control, 41, 1129-1155 (1985) · Zbl 0567.93036
[9] Kublanovskaya, V. N., On some algorithms for the solution of the complete eigenvalue problem, USSR Comput. Math. Math. Phys., 3, 635-657 (1961) · Zbl 0128.11702
[10] Luenberger, D. G., Canonical forms for linear multivariable systems, IEEE Trans. Aut. Control, AC-12, 290-293 (1967)
[11] Moler, C., (MATLAB User’s Guide (1982), Department of Computer Science, University of New Mexico)
[12] Miminis, G. S., Numerical algorithms for control-lability and eigenvalue allocation, (M.Sc. Thesis (1981), McGill University, School of Computer Science: McGill University, School of Computer Science Montreal, Quebec, Canada)
[13] Miminis, G. S., Numerical algorithms for the pole placement problem, (Ph.D. dissertation (1985), School of Computer Science, McGill University: School of Computer Science, McGill University Montreal, Quebec, Canada) · Zbl 0478.93022
[14] Miminis, G. S., An algorithm for the computation of the distance of a linear dynamic system from the nearest uncontrollable one (1988), In preparation
[15] Miminis, G. S.; Paige, C. C., An algorithm for pole assignment of time invariant linear systems, Int. J. Control, 35, 341-354 (1982) · Zbl 0478.93022
[16] Miminis, G. S.; Paige, C. C., Algorithm for pole assignment of time invariant multi-input linear systems, (Proc. 21st IEEE Conf. on Decision and Control, Orlando, Florida, Vol. 1 (1982)), 62-67 · Zbl 0478.93022
[17] Paige, C. C., Properties of numerical algorithms related to computing controllability, IEEE Trans. Aut. Control, AC-26, 130-138 (1981) · Zbl 0463.93024
[18] Patel, R. V.; Misra, P., Numerical algorithms for eigenvalue assignment by state feedback, (Proc. IEEE, 72 (1984)), 1755-1764
[19] Petkov, P. Hr.; Christov, N. D.; Konstantinov, M. M., A computational algorithm for pole assignment of linear input systems, 23rd IEEE CDC, 3, 1770-1773 (1984) · Zbl 0546.93024
[20] Retallack, P. G.; MacFarlane, A. G.J., Pole-shifting techniques for multivariable feedback systems, (Proc. Inst. Elec. Eng., 117 (1970)), 1037-1038
[21] Stewart, G. W., (Introduction to Matrix Computations (1973), Academic Press: Academic Press New York) · Zbl 0302.65021
[22] Van Dooren, P. M., The generalized eigenstructure problem in linear system theory, IEEE Trans. Aut. Control, AC-26, 111-129 (1981) · Zbl 0462.93013
[23] Van Dooren, P. M., Deadbeat control: a special inverse eigenvalue problem, ((1984), BIT), 681-699 · Zbl 0549.93019
[24] Varga, A., A Schur method for pole assignment, IEEE Trans. Aut. Control, AC-26, 517-519 (1981) · Zbl 0475.93040
[25] Wilkinson, J. H., (The Algebraic Eigenvalue Problem (1965), Oxford University Press: Oxford University Press Oxford) · Zbl 0258.65037
[26] Wilkinson, J. H., Modern error analysis, SIAM Rev., 13, 548-568 (1971) · Zbl 0243.65018
[27] Wonham, W. M., On pole assignment in multi-input controllable linear systems, IEEE Trans. Aut. Control, AC-12, 660-665 (1967)
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