“A Schur Method for Designing LQ-optimal Systems with Prescribed Eigenvalues”
Authors: David Di Ruscio and Jens G. Balchen,Affiliation: NTNU, Department of Engineering Cybernetics
Reference: 1990, Vol 11, No 1, pp. 55-72.
Keywords: Linear optimal regulator, pole placement, multivariable control systems, control system design, computer-aided design, robustness
Abstract: In this paper a new algorithm for solving the LQ-optimal pole placement problem is presented. The method studied is a variant of the classical eigenvector approach and instead uses a set of Schur vectors, thereby gaining substantial numerical advantages. An important task in this method is the LQ-optimal pole placement problem for a second order (sub) system. The paper presents a detailed analytical solution to this problem. This part is not only important for solving the general n-dimensional problem but also provides an understanding of the behaviour of an optimal system: The paper shows that in some cases it is an infinite number; in others a finite number, and in still others, non state weighting matrices Q that give the system a set of prescribed eigenvalues. Equations are presented that uniquely determine these state weight matrices as a function of the new prescribed eigcnvalues. From this result we have been able to derive the maximum possible imaginary part of the eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen.
PDF (1986 Kb)
DOI: 10.4173/mic.1990.1.5DOI forward links to this article:
| [1] David Di Ruscio (1991), doi:10.4173/mic.1991.3.5 |
| [2] David Di Ruscio (1992), doi:10.4173/mic.1992.1.2 |
| [3] David Di Ruscio (1990), doi:10.4173/mic.1990.2.5 |
| [4] D. Di Ruscio (1991), doi:10.1109/CDC.1991.261580 |
| [5] D. Di Ruscio (1992), doi:10.1109/CDC.1992.371469 |
[1] AMIN, M.A. (1985). Optimal pole shifting for continuous multivariable systems, Int. J. Control, 41, 701-707 doi:10.1080/0020718508961157
[2] AMIN, M.A., HASSAN, M.M. (1985). Comments on ´On optimal pole assignment in a specified region´, Int. J. Control, 41, 1631-1632 doi:10.1080/0020718508961222
[3] ANDERSON, B.D.O., MOORE, J.B. (1969). Linear system optimisation with prescribed degree of stability, I.E.E.E. Trans. on Automatic Control, 116, 2083-2087.
[4] BAR-NESS, Y. (1978). Optimal closed-loop poles assignment, Int. J. Control, 27, 421-430 doi:10.1080/00207177808922380
[5] DICKMAN, A. (1987). On the robustness of multivariable linear feedback systems in state-space representation, I.E.E.E. Trans. on Automatic Control, 32, 407-410 doi:10.1109/TAC.1987.1104611
[6] DOYLE, J.C., WALL, J.E., STEIN, G. (1982). Performance and robustness analysis for structured uncertainty, Proceedings 21st I.E.E.E. Conf. Decision and Control, Orlando, Florida.I.E.E.E., New York, pp. 629-636.
[7] GULUB, G.H., VAN LOAN, C.F. (1983). Matrix Computations, North Oxford Academic Publishers Ltd, England ISBN 0-946536-05-8.
[8] GRAUPE., D. (1972). Derivation of weighting matrices towards satisfying eigenvalue requirements, Int. J. Control, 16, 881-888 doi:10.1080/00207177208932318
[9] HEGER, F., FRANK, P.M. (1984). Linear regulators with prescribed eigenvalues for a family of linear systems, In S.G. Tzatestas (ed.), Multivariable Control (D. Reidel Publishing Company), pp. 281-292.
[10] JUANG, J.C., LEE, T.T. (1984). On optimal pole assignment in a specified region, Int. J. Control, 40, 65-79 doi:10.1080/00207178408933257
[11] KALMAN, R.E. (1964). When is a linear system optimal?, Trans. A.S.E.M. 86 D, pp. 51-60.
[12] KAWASAKI, M., SHIMEMURA, E. (1983). Determining quadratic weighting matrices to locate poles in a Specified region, Automatica, 19, 557-560 doi:10.1016/0005-1098(83)90011-0
[13] KAWASAKI, M., SHIMEMURA, E. (1988). Pole placement in a specified region based on a linear quadratic regulator, Int. J. Control, 48, 225-240 doi:10.1080/00207178808906171
[14] LAUB, A.J. (1979). A Schur method for solving algebraic Riccati equations, IEEE Trans. on Automatic Control, 24, 913-921 doi:10.1109/TAC.1979.1102178
[15] MEDANIC, J., THARP, H.S., PERKINS, W.R. (1988). Pole placement by performance criterion modification, IEEE Trans. on Automatic Control, 33,469-472 doi:10.1109/9.1229
[16] MOLINIARI, B.P. (1977). The time-invariant linear-quadratic optimal control problem, Automatica, 13, 347-357 doi:10.1016/0005-1098(77)90017-6
[17] SMITH, B.T. et al. (1976). Matrix Eigensystems Routines - Eispack Guide, 2nd ed, (Lecture notes in Computer Science), vol. 6 (Springer-Verlag, New York).
[18] SAIF, M. (1989). Optimal linear regulator pole-placement by weight selection, Int. J. Control, 50, 399-414 doi:10.1080/00207178908953369
[19] SHIEH, L.S., DIB, H.M. McINNIS, B.C. (1986). Linear quadratic regulators with eigenvalue placement in a vertical strip, IEEE Trans. on Automatic Control, 31, 241-243 doi:10.1109/TAC.1986.1104233
[20] SHIEH, L.S., DIB, H.M., GANESAN, B. (1988). Linear quadratic regulators with eigenvalue placement in a specified region, Automatica, 24, 819-823 doi:10.1016/0005-1098(88)90058-1
[21] SOLHEIM, A.O. (1972). Design of optimal control systems with prescribed eigenvalues, Int. J. Control, 15, 143-160 doi:10.1080/00207177208932136
[22] STEWART, G.W. (1976). Algorithm 506 HQR3 and EXCHNG: Fortran subroutines for calculating and ordering the eigenvalues of a Real Upper Hessenberg Matrix, ACM Trans. Math. Soft., 2, 275-280 doi:10.1145/355694.355700
[23] THARP, H.S. (1989). Optimal pole-placement in discrete systems, Proceedings of the American Control Conference, Pittsburgh, Pennsylvania, pp. 1286-1291, published by IEEE Service Center.
[24] VARGA, A. (1981). A Schur method for pole assignment, IEEE Trans. on Automatic Control, 26, 517-519 doi:10.1109/TAC.1981.1102605
BibTeX:
@article{MIC-1990-1-5,
title={{A Schur Method for Designing LQ-optimal Systems with Prescribed Eigenvalues}},
author={Di Ruscio, David and Balchen, Jens G.},
journal={Modeling, Identification and Control},
volume={11},
number={1},
pages={55--72},
year={1990},
doi={10.4173/mic.1990.1.5},
publisher={Norwegian Society of Automatic Control}
};