Parametric Stability

Part of the Communications and Control Engineering book series (CCE)


Mathematical descriptions of large-scale systems often include a set of parameters whose values cannot be predicted with any precision. Ensuring stability in the presence of such uncertainties is a fundamental theoretical problem, which has lead to the development of numerous robust control strategies. Given the wealth of existing results in this field, it is fair to say that even a superficial overview would clearly exceed the scope of this book. In view of that, we will focus our attention on two specific issues that are related to parametric stability – the existence of moving equilibria and the design of parametricallydependent control laws. In the following sections we will consider these problems in some detail, and will show how they can be resolved in the framework of linear matrix inequalities. For the sake of clarity, we will examine only smaller systems, with the understanding that this approach can be easily extended to large-scale problems along the lines proposed in Chaps. 2 and 3.

We begin with several brief observations regarding the existence and stability of equilibria in nonlinear dynamic systems. In the analysis of such systems, it is common practice to treat these two properties separately. Establishing the existence of an equilibrium usually involves the solution of a system of nonlinear algebraic equations, using standard numerical techniques such as Newton’s method (e.g., Ortega and Rheinboldt, 1970). Once the equilibrium of interest is computed, it is translated to the origin by an appropriate change of variables. Stability properties and feedback design are normally considered only after such a transformation has been performed. This two-step methodology has been widely applied to systems that contain parametric uncertainties, and virtually all control schemes developed along these lines implicitly assume that the equilibrium remains fixed at the origin for the entire range of parameter values (e.g., Šiljak, 1969, 1989; Ackermann, 1993; Barmish, 1994; Bhattacharya et al., 1995).


Linear Matrix Inequality Parametric Stability Reference Input Gain Schedule Schedule Variable 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ackermann, J. (1993). Robust Control: Systems with Uncertain Physical Parameters. Springer, New York.MATHGoogle Scholar
  2. Barmish, B. R. (1994). New Tools for Robustness of Linear Systems. MacMillan, New York.MATHGoogle Scholar
  3. Bhattacharya, S. P., H. Chapellat and L. H. Keel (1995). Robust Control: The Parametric Approach. Prentice-Hall, Upper Saddle River, NJ.Google Scholar
  4. Fink, L. H. (Ed.) (1994). Bulk Power System Voltage Phenomena III – Voltage Stability, Security and Control. Proceedings of the ECC/NSF Workshop, Davos, Switzerland.Google Scholar
  5. Ikeda, M., Y. Ohta and D. D. Šiljak (1991). Parametric stability. In: New Trends in Systems Theory, G. Conte, A. M. Perdon and B. Wyman (Eds.), Birkhauser, Boston, MA, 1–20.Google Scholar
  6. Kwatny, H. G., A. K. Pasrija and L. Y. Bahar (1986). Static bifurcations in electric power networks: Loss of steady – state stability and voltage collapse. IEEE Transactions on Circuits and Systems, 33, 981–991.MATHCrossRefGoogle Scholar
  7. Lakshmikantham, V. and S. Sivasundaram (1998). Stability of moving invariant sets and uncertain dynamic systems on time scale. Computers and Mathematics with Applications, 36, 339–346.MATHCrossRefMathSciNetGoogle Scholar
  8. Lawrence, D. A. and W. J. Rugh (1995). Gain scheduling dynamic linear controllers for a nonlinear plant. Automatica, 31, 381–390.MATHCrossRefMathSciNetGoogle Scholar
  9. Leela, S. and N. Shahzad (1996). On stability of moving conditionally invariant sets. Nonlinear Analysis, Theory, Methods and Applications, 27, 797–800.MATHCrossRefMathSciNetGoogle Scholar
  10. Leonessa, A., W. M. Haddad and V. Chellaboina (2000). Hierarchical Nonlinear Switching Control Design with Application to Propulsion Systems. Springer, London.Google Scholar
  11. Martynyuk, A. A. and A. S. Khoroshun (2008). On parametric asymptotic stability of large-scale systems. International Applied Mechanics, 44, 565–574.CrossRefGoogle Scholar
  12. Ohta, Y. and D. D. Šiljak (1994). Parametric quadratic stabilizability of uncertain nonlinear systems. Systems and Control Letters, 22, 437–444.MATHCrossRefMathSciNetGoogle Scholar
  13. Ortega, J. and W. Rheinboldt (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York.MATHGoogle Scholar
  14. Packard, A. (1994). Gain scheduling via linear fractional transformation. Systems and Control Letters, 22, 79–92.MATHCrossRefMathSciNetGoogle Scholar
  15. Rugh, W. J. (1991). Analytical framework for gain scheduling. IEEE Control Systems Magazine, 11, 79–84.CrossRefGoogle Scholar
  16. Shamma, J. and M. Athans (1990). Analysis of gain scheduled control for nonlinear plants. IEEE Transactions on Automatic Control, 35, 898–907.MATHCrossRefMathSciNetGoogle Scholar
  17. Shamma, J. and M. Athans (1992). Gain scheduling: Potential hazards and possible remedies. IEEE Control Systems Magazine, 12, 101–107.CrossRefGoogle Scholar
  18. Šiljak, D. D. (1969). Nonlinear Systems: The Parameter Analysis and Design. Wiley, New York.MATHGoogle Scholar
  19. Šiljak, D. D. (1978). Large-Scale Dynamic Systems: Stability and Structure. North-Holland, New York.MATHGoogle Scholar
  20. Šiljak, D. D. (1989). Parameter space methods for robust control design: A guided tour. IEEE Transactions on Automatic Control, 34, 674–688.MATHCrossRefGoogle Scholar
  21. Silva, G. and F. A. Dzul (1998). Parametric absolute stability of a class of singularly perturbed systems. Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, 1422–1427.Google Scholar
  22. Stamova, I. M. (2008). Parametric stability of impulsive functional differential equations. Journal of Dynamical and Control Systems, 14, 235–250.CrossRefMathSciNetGoogle Scholar
  23. Stilwell, D. J. and W. J. Rugh (2000). Stability preserving interpolation methods for the synthesis of gain scheduled controllers. Automatica, 36, 665–671.MATHCrossRefMathSciNetGoogle Scholar
  24. Sundarapandian, V. (2002). A necessary condition for local asymptotic stability of discrete-time nonlinear systems with parameters. Applied Mathematics Letters, 15, 271–273.MATHCrossRefMathSciNetGoogle Scholar
  25. Sundarapandian, V. (2006). New results on the parametric stability of nonlinear systems. Mathematical and Computer Modeling, 43, 9–15.MATHCrossRefMathSciNetGoogle Scholar
  26. Wada, T., M. Ikeda, Y. Ohta and D. D. Šiljak (2000). Parametric absolute stability of multivariable Lur’e systems. Automatica, 36, 1365–1372.MATHCrossRefGoogle Scholar
  27. Zečević, A. I. (2004). Analytic gain scheduling using linear matrix inequalities. Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 11, 589–607.Google Scholar
  28. Zečević, A. I. and D. M. Miljković (2002). The effects of generation redispatch on Hopf bifurcations in electric power systems. IEEE Transactions on Circuits and Systems, 49, 1180–1186.CrossRefGoogle Scholar
  29. Zečević, A. I. and D. D. Šiljak (2003). Stabilization of nonlinear systems with moving equilibria. IEEE Transactions on Automatic Control, 48, 1036–1040.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Aleksandar I. Zečević
    • 1
  • Dragoslav D. Šiljak
    • 1
  1. 1.Dept. Electrical EngineeringSanta Clara UniversitySanta ClaraUSA

Personalised recommendations