Diffeomorphisms between Sets of Linear Systems

  • R. Ober
  • P. A. Fuhrmann
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 62)


Diffeomorphisms are given between different subsets of linear systems of fixed McMillan degree. The sets considered are the set of all systems of fixed McMillan degree, the subset of stable systems, the subset of bounded real systems, the subset of positive real systems, the subset of stable systems with Hankel singular values bounded by one. State space techniques are used in the proofs.


Linear System Canonical Form Stable System Riccati Equation Antistable System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    B. D. O. Anderson. On the computation of the Cauchy index. Quarterly of applied mathematics, pages 577–582, 1972.Google Scholar
  2. [2]
    R. W. Brockett. Some geometric questions in the theory of linear systems. IEEE Transactions on Automatic Control, 21:449–454, 1976.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    R. Bucy. The Riccati equation and its bounds. Journal Computer Systems Science, 6:343–353, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    D. Delchamps. A Note on the Analyticity of the Riccati Metric, volume 18, pages 37–42. AMS, Providence, RI, 1980. C.I. Byrnes and C.F. Martin, eds.Google Scholar
  5. [5]
    D. Delchamps. Analytic stabilization and the algebraic Riccati equation. In Proceedings 22nd Conference on Decision and Control, San Antonio, Texas, USA, December 1983.Google Scholar
  6. [6]
    D. Delchamps. Analytic feedback control and the algebraic Riccati equation. IEEE Transactions on Automatic Control, 29(11), 1984.Google Scholar
  7. [7]
    U. B. Desai and D. Pal. A transformation approach to stochastic model reduction. IEEE Transactions on Automatic Control, 29:1097–1100, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    P. Fuhrmann and R. Ober. A functional approach to LQG-balanced realizations. International Journal of Control, 57: 627–741, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    K. Glover. Some geometrical properties of linear systems with implications in identification. In Proceedings IFAC World Congress, Boston, 1975.Google Scholar
  10. [10]
    K. Glover and D. McFarlane. Robust stabilization of normalized coprime factors: An explicit H solution. In Proceedings Automatic Control Conference, Atlanta, Georgia, USA, 1988.Google Scholar
  11. [11]
    M. Green. Balanced stochastic realizations. Linear Algebra and Its Applications, 98:211–247, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    B. Hanzon. Identifiability, Recursive Identification and Spaces of Linear Systems. PhD thesis, Erasmus University Rotterdam, 1986.Google Scholar
  13. [13]
    B. Hanzon. On the differentiable manifold of fixed order stable linear systems. Systems and Control Letters, 13:345–352, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    P. Harshavardhana, E. Jonckheere, and L. Silverman. Stochastic balancing and approximation — stability and minimality. IEEE Transactions on Automatic Control, 29(8):744–746, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    U. Helmke. A global parametrization of asymptotically stable linear systems. Systems and Control Letters, 13:383–389, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    E. A. Jonckheere and L. M. Silverman. A new set of invariants for linear systems — application to reduced order compensator design. IEEE Transactions on Automatic Control, 28:953–964, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    B.C. Moore. Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Transactions on Automatic Control, 26:17–32, 1981.zbMATHCrossRefGoogle Scholar
  18. [18]
    R. Ober. Balanced realizations: canonical form, parameterization, model reduction. International Journal of Control, 46(2):643–670, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    R. Ober. Topology of the set of asymptotically stable systems. International Journal of Control, 46(l):263–280, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    R. Ober. Connectivity properties of classes of linear systems. International Journal of Control, 50:2049–2073, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    R. Ober. Balanced parametrization of classes of linear systems. SIAM Journal on Control and Optimization, 29:1251–1287, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    R. Ober and D. McFarlane. Balanced canonical forms for minimal systems: a normalized coprime factor approach. Linear Algebra and its Applications, Special Issue on Linear Systems and Control, 122–124:23–64, 1989.MathSciNetCrossRefGoogle Scholar
  23. [23]
    P. C. Opdenacker and E. A. Jonckheere. A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds. IEEE Transactions on Circuits and Systems, 35:184–189, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    J. C. Willems. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control, 16:621–634, 1971.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • R. Ober
    • 1
  • P. A. Fuhrmann
    • 2
  1. 1.Center for Engineering Mathematics, Programs in Mathematical SciencesThe University of Texas at DallasRichardsonUSA
  2. 2.Israeli Academy of Sciences, Department of MathematicsBen-Gurion University of the NegevBeer ShevaIsrael

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