Skip to main content
SpringerLink
Log in

On the role of homogeneous forms in robustness analysis of control systems

  • Chapter
  • First Online:
Multidisciplinary Research in Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 289))

Abstract

This paper considers the problem of positivity (or nonnegativity) of one-parameter families of homogeneous polynomial forms, i.e. of forms whose coefficients depend on a scalar parameter which is allowed to vary over either a continuous or a discrete set. The main contribution of the paper is an efficiently computable sufficient condition for positivity (or nonnegativity), which requires the solution of a Linear Matrix Inequalities (LMI) optimization problem for each value of the scalar parameter. Moreover, necessity of such a condition is investigated and proven to hold for some families of homogeneous forms. The paper also shows that several important problems in the analysis of control systems can be recast as positivity (or nonnegativity) of suitable one-parameter families of homogeneous forms. In particular, two robustness problems involving linear systems and the problem of computing all the equilibria of polynomial nonlinear systems are discussed in detail.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. K. Bose (Ed.). Special issue on multidimensional systems. Proc. IEEE, 65(6), 1977.

    Google Scholar 

  2. N. K. Bose. Applied Multidimensional Systems Theory. Van Rostrand Reinhold, New York, 1982.

    MATH  Google Scholar 

  3. R. W. Brockett. Lie algebra and Lie groups in control theory. In D.Q. Mayne and R.W. Brockett, editors, Geometric Methods in Systems Theory, pages 43–82. Dordrecht, Reidel, 1973.

    Google Scholar 

  4. G. Hardy, J.E. Littlewood, and G. Pólya. Inequalities: Second edition. Cambridge University Press, Cambridge, 1988.

    MATH  Google Scholar 

  5. B. Reznik. Some concrete aspects of Hilbert’s 17th problem. In C. N. Delzell and J.J. Madden, editors, Real Algebraic Geometry and Ordered Structures, Cont. Math., pages 251–272, 2000.

    Google Scholar 

  6. G. Chesi, A. Garulli, A. Tesi, and A. Vicino. LMI-based techniques for solving quadratic distance problems. In Proc. of 40-th IEEE CDC, 2001. To appear in IEEE Trans. Aut. Control, 2003.

    Google Scholar 

  7. N.Z. Shor. Class of global minimum bounds of polynomial functions. Cybernetics, 23:731–734, 1987. (Russian orig.: Kibernetika, no. 6, pp. 9–11, 1987).

    Article  MATH  Google Scholar 

  8. G. Chesi, R. Genesio, and A. Tesi. Optimal ellipsoidal stability domain estimates for odd polynomial systems. In Proc. of 36-th IEEE CDC, pages 3528–3529, San Diego, December 1997.

    Google Scholar 

  9. G. Chesi, A. Tesi, A. Vicino, and R. Genesio. A convex approach to a class of minimum norm problems. In A. Garulli, A. Tesi, and A. Vicino, editors, Robustness in Identification and Control, pages 359–372. Springer, 1999.

    Google Scholar 

  10. G. Chesi, A. Garulli, A. Vicino, and R. Cipolla. Estimating the fundamental matrix via constrained least squares: a convex approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(3):397–401, 2002.

    Article  Google Scholar 

  11. G. Chesi, A. Tesi, A. Vicino, and R. Genesio. An LMI approach to constrained optimization with homogeneous forms. Systems and Control Letters, 42:11–19, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. Thesis, California Institute of Technology, 2000.

    Google Scholar 

  13. Y. Nesterov. Squared functional systems and optimization problems. In H. Frenk, K. Roos, T. Terlaky, and S. Shang, editors, High Performance Optimization. Kluwer, 2000.

    Google Scholar 

  14. Y. Genin, Y. Hachez, Y. Nesterov, and P. Van Dooren. Optimization problems over positive pseudo-polynomial matrices. Technical report, http://www.auto.ucl.ac.be/~vdooren/download/Welcome.html, 2001.

  15. A. Rantzer and P. A. Parrilo. On convexity in stabilization of nonlinear systems. In Proc. of 39th IEEE Conference on Decision and Control, Sydney, December 2000.

    Google Scholar 

  16. B. Tibken. Estimation of the domain of attraction for polynomial systems via LMI’s. In Proc. of 39th IEEE Conference on Decision and Control, Sydney, December 2000.

    Google Scholar 

  17. G. Chesi, A. Garulli, A. Tesi, and A. Vicino. An LMI-based approach for characterizing the solution set of polynomial systems. In Proc. of 39-th IEEE Conf. on Decision and Control, pages 1501–1506, Sydney, December 2000.

    Google Scholar 

  18. G. Chesi, A. Garulli, A. Tesi, and A. Vicino. LMI-based construction of homogeneous Lyapunov functions for systems with structured uncertainties. In Proc. of 41-th IEEE CDC, Las Vegas, December 2002.

    Google Scholar 

  19. Y. Nesterov and A. Nemirovsky. Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia, 1993.

    Google Scholar 

  20. S. P. Bhattacharyya, H. Chapellat, and L. H. Keel. Robust Control: The Parametric Approach. Prentice Hall, NJ, 1995.

    Google Scholar 

  21. A. Tesi and A. Vicino. Robust stability of state space models with structured uncertainties. IEEE Transactions on Automatic Control, 35:191–195, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  22. D.D. Šiljak. Nonlinear Systems: Parametric Analysis and Design. John Wiley & Sons, New York, 1969.

    Google Scholar 

  23. R.K. Brayton and C. H. Tong. Stability of dynamical systems: a constructive approach. IEEE Transactions on Circuits and Systems, 26:224–234, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  24. K. Zhou and P. P. Khargonekar. Stability robustness bounds for linear state-space models with structured uncertainties. IEEE Transactions on Automatic Control, 32:621–623, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  25. F. Blanchini. Nonquadratic Lyapunov functions for robust control. Automatica, 31:451–461, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  26. L. Xie, S. Shishkin, and M. Fu. Piecewise Lyapunov functions for robust stability of linear time-varying systems. Systems and Control Letters, 31:165–171, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  27. A. L. Zelentsovsky. Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems. IEEE Transactions on Automatic Control, 39:135–138, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  28. H. P. Horisberger and P. R. Belanger. Regulators for linear time invariant plants with uncertian parameters. IEEE Transactions on Automatic Control, 21:705–708, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  29. G. Chesi, A. Garulli, A. Tesi, and A. Vicino. Homogeneous Lyapunov functions for systems with structured uncertainties. Technical report, Dipartimento di Ingegneria dell’Informazione, Università di Siena, 2002. ftp://ftp.ing.unisi.it/pub/users/garulli/hplf.pdf.

  30. A. Tarski. A Decision Method for Elementary Algebra and Geometry. Univ. California Press, Berkeley, 1951.

    MATH  Google Scholar 

  31. N. Jacobson. Lectures in Abstract Algebra, Vol. III. Van Nostrand, Princeton, 1964.

    MATH  Google Scholar 

  32. D. Cox, J. Little, and D. O’Shea. Using Algebraic Geometry. Springer, 1998.

    Google Scholar 

  33. A. Morgan. Solving polynomial systems using continuation for engineering and scientific problems. Prentice-Hall, New Jersey, 1987.

    MATH  Google Scholar 

  34. R.W. Newcomb. Nonlinear differential systems: a canonic, multivariable theory. Proc. IEEE, 65(6):930–935, 1977.

    Article  MATH  Google Scholar 

  35. R.R. Mohler. Bilinear Control Processes. Academic Press, New York, 1973.

    MATH  Google Scholar 

  36. A. Benallou, D. A. Mellichamp, and D. E. Seborg. Characterization of equilibrium sets for bilinear systems with feedback control. Automatica, 19:183–189, 1983.

    Article  MATH  Google Scholar 

  37. G. Chesi, A. Garulli, A. Tesi, and A. Vicino. Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach. Technical report, Dipartimento di Ingegneria dell’Informazione, Università di Siena, 2002. ftp://ftp.ing.unisi.it/pub/users/garulli/poly.pdf.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Informazione, Università di Siena, Via Roma 56, 53100, Siena, Italia

    G. Chesi, A. Garulli & A. Vicino

  2. Dipartimento di Sistemi e Informatica, Università di Firenze, Via di S. Marta, 3, 50139, Firenze, Italia

    A. Tesi

Authors
  1. G. Chesi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Garulli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Tesi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Vicino
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dip. di Ingegneria dell’Automazione e dei Sistemi (DIAS), Università di Palermo, Viale delle Scienze, 90128, Palermo, Italy

    Laura Giarré

  2. Dept. of Mechanical and Environmental Engineering, University of California at Santa Barbara (UCSB), 93106, Santa Barbara, CA, USA

    Bassam Bamieh

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Chesi, G., Garulli, A., Tesi, A., Vicino, A. (2003). On the role of homogeneous forms in robustness analysis of control systems. In: Giarré, L., Bamieh, B. (eds) Multidisciplinary Research in Control. Lecture Notes in Control and Information Sciences, vol 289. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36589-3_13

Download citation

  • DOI: https://doi.org/10.1007/3-540-36589-3_13

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00917-7

  • Online ISBN: 978-3-540-36589-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation