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Truncated balanced realization of a stable non-minimal state-space system. (English) Zbl 0642.93015
The authors present a numerically reliable algorithm to compute the balanced realization of a state-space system that may be arbitrarily close to being unobservable and/or uncontrollable. The resulting realization, which is known to be a good approximation of the original system, must be minimal and therefore may contain a reduced number of states. In addition to real matrix multiplications, the algorithm only requires the solution of two Lyapunov equations and one singular value decomposition of an upper-triangular matrix.
The authors also present two examples of the application of the truncated balanced realization algorithm considered in the paper. The first example is intended to be reproducible, based on the system considered by K. Glover [cf., ibid. 39, 1115-1193 (1984; Zbl 0543.93036)]. The second arises from the design of a controller for an unstable helicopter, using \(H^{\infty}\) optimization.
Reviewer: N.Osetinski

MSC:
93B20 Minimal systems representations
93B40 Computational methods in systems theory (MSC2010)
93C05 Linear systems in control theory
15A18 Eigenvalues, singular values, and eigenvectors
15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
93B15 Realizations from input-output data
Software:
LINPACK; NAG
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References:
[1] DOI: 10.1016/0167-6911(86)90037-X · Zbl 0595.93024
[2] DONGARRA J. J., UNPACK Users’ Guide (1979)
[3] DOI: 10.1080/00207178408933239 · Zbl 0543.93036
[4] DOI: 10.1093/imanum/2.3.303 · Zbl 0492.65017
[5] DOI: 10.1016/0167-6911(86)90110-6 · Zbl 0589.93009
[6] DOI: 10.1007/BFb0043756
[7] DOI: 10.1016/0167-6911(85)90014-3 · Zbl 0559.93036
[8] LAUB A. J., Proc. Joint Automatic Control Conf. (1980)
[9] DOI: 10.1109/TAC.1981.1102568 · Zbl 0464.93022
[10] NAG Mark II Library Manual (1984)
[11] DOI: 10.1109/TAC.1982.1102945 · Zbl 0482.93024
[12] POSTLETHWAITE , I. , Gu , D.W. , and O’YOUNG , S. D. , 1986 a , Workshop on Computer Aided Control System Design , Salford , 2 – 4 July ; 1986 b, Oxford University Engineering Department Report, OUEL 1648/86 .
[13] POSTLETHWAITE I., Proc. I.E.E.E. Conf. on Decision and Control (1986)
[14] CTRL-C Users’ Guide. (1986)
[15] TOMBS , M. S. , 1987 , Robust control system design with application to high performance helicopters . D.Phil, thesis ( in preparation ), University of Oxford .
[16] WILLIAMS T. W. C., Proc. Insm elect. Engrs 133 pp 73– (1986)
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