Skip to main content
SpringerLink
Log in

Analysis and synthesis of modally balanced systems

  • Linear Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We study a special kind of linear dynamical systems, namely modally balanced systems, in which the transition function’s poles are proportional to Hankel singular values. We analyze existence conditions and properties of such systems and develop algorithms for their synthesis. Our results include algebraic modal balance criteria.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Moore, B.C., Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction, IEEE Trans. Automat. Control, 1981, vol. AC-26, pp. 17–32.

    Article  Google Scholar 

  2. Wilson, D.A., Convolution and Hankel Operator Norms for Linear Systems, IEEE Trans. Automat. Control, 1989, vol. AC-34, no. 1, pp. 94–97.

    Article  Google Scholar 

  3. Wilson, D.A., The Hankel Operator and Its Induced Norms, Int. J. Control, 1985, vol. 42, pp. 65–70.

    Article  MATH  Google Scholar 

  4. Ober, R. and McFarlane, D., Balanced Canonical Forms for Minimal Systems: A Normalized Coprime Factor Approach, Linear Algebra Appl., 1989, vol. 122–124, pp. 23–64.

    Article  MathSciNet  Google Scholar 

  5. Mironovskii, L.A., Hankel Operator and Hankel Functions of Linear Systems, Autom. Remote Control, 1993, vol. 53, no. 9, part 1, pp. 1368–1380.

    MathSciNet  Google Scholar 

  6. Glover, K., All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their L∞-error Bounds, Int. J. Control, 1984, vol. 39, no. 6, pp. 1145–1193.

    Article  MathSciNet  Google Scholar 

  7. Safonov, M.G., Chiang, R.Y., and Schur, A., Method for Balanced Model Reduction, IEEE Trans. Automat. Control, 1989, vol. AC-2, no. 7, pp. 729–733.

    Article  Google Scholar 

  8. Ober, R., Balanced Realizations: Canonical Form, Parametrization, Model Reduction, Int. J. Control, 1987, vol. 46, pp. 643–670.

    Article  MathSciNet  MATH  Google Scholar 

  9. Williams, T., Closed-form Gramians and Model Reduction for Flexible Space Structures, IEEE Trans. Automat. Control, 1990, vol. AC-35, pp. 379–382.

    Article  Google Scholar 

  10. Antoulas, A.C., Sorensen, D.C., and Zhou, Y., On the Decay Rate of Hankel Singular Values and Related Issues, Syst. Control Lett., 2002, no. 46, pp. 323–342.

    Google Scholar 

  11. Opdenacker, Ph.C. and Jonckheere, E.A., State-space Approach to Approximation by Phase Matching, Int. J. Control, 1987, vol. 45, pp. 671–679.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ober, R., Asymptotically Stable Allpass Transfer Functions: Canonical Form, Parametrization and Realization, Proc. IFAC World Congr., 1987.

    Google Scholar 

  13. Mironovskii, L.A., Linear Systems with Multiple Singular Values, Autom. Remote Control, 2009, vol. 70, no. 1, pp. 43–63.

    Article  MathSciNet  MATH  Google Scholar 

  14. Mironovskii, L.A. and Kurmaev, I.R., Synthesis of Trisingular Dynamical Systems, Informatsionno-Upravlyayushchie Sist., 2010, no. 6, pp. 77–85.

    Google Scholar 

  15. Liu, W.Q., Sreeram, V., and Teo, K.L., Model Reduction for State-space Symmetric Systems, Syst. Control Lett., 1998, vol. 34, pp. 209–215.

    Article  MathSciNet  MATH  Google Scholar 

  16. Srinivasan, B. and Myszkorowski, P., Model Reduction of Systems with Zeros Interlacing the Poles, Syst. Control Lett., 1997, vol. 30, no. 1, pp. 19–24.

    Article  MathSciNet  MATH  Google Scholar 

  17. De Abreu-Garcia, J.A. and Fairman, F.W., Balanced Realization of Orthogonally Symmetric Transfer Function Matrices, IEEE Trans. Circuits Syst., 1987, vol. CAS-34, no. 9, pp. 997–1010.

    Article  Google Scholar 

  18. De Abreu-Garcia, J.A. and Fairman, F.W., On Using Permutation Symmetric Jordan Realizations to Achieve SISO Balancing, Int. J. Syst. Sci., 1987, vol. 18, pp. 441–448.

    Article  MATH  Google Scholar 

  19. Sveinsson, J.R. and Fairman, F.W., Minimal Balanced Realization of Transfer Function Matrices Using Markov Parameters, IEEE Trans. Automat. Control, 1985, vol. AC-30, no. 10, pp. 1014–1016.

    Article  MathSciNet  Google Scholar 

  20. Horn, A., On the Eigenvalues of a Matrix with Prescribed Singular Values, Proc. Am. Math. Soc., 1954, vol. 5, pp. 4–7.

    Article  MathSciNet  MATH  Google Scholar 

  21. Ober, R.J., Balanced Parameterization of Classes of Linear Systems, SIAM J. Control Optim., 1991, vol. 29, no. 6, pp. 1251–1287.

    Article  MathSciNet  MATH  Google Scholar 

  22. Mironovskii, L.A., Relations between Parallel and Balanced Canonical Forms, Elektron. Modelir., 1989, no. 6, pp. 8–10.

    Google Scholar 

  23. Mironovskii, L.A. and Solov’eva, T.N., On Properties of Regular Dynamical Systems, Probl. Upravlen., 2011, no. 3, pp. 12–19.

    Google Scholar 

  24. Mironovskii, L.A., Modelirovanie lineinykh sistem (Modeling Linear Systems), St. Petersburg: GUAP, 2009.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State University of Aerospace Instrumentation, St. Petersburg, Russia

    L. A. Mironovskii & T. N. Solov’eva

Authors
  1. L. A. Mironovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. N. Solov’eva
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © L.A. Mironovskii, T.N. Solov’eva, 2013, published in Avtomatika i Telemekhanika, 2013, No. 4, pp. 59–79.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mironovskii, L.A., Solov’eva, T.N. Analysis and synthesis of modally balanced systems. Autom Remote Control 74, 588–603 (2013). https://doi.org/10.1134/S0005117913040036

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117913040036

Keywords

Search

Navigation