Abstract
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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ackermann, J. (1993). Robust Control: Systems with Uncertain Physical Parameters. Springer, New York.
Barmish, B. R. (1994). New Tools for Robustness of Linear Systems. MacMillan, New York.
Bhattacharya, S. P., H. Chapellat and L. H. Keel (1995). Robust Control: The Parametric Approach. Prentice-Hall, Upper Saddle River, NJ.
Fink, L. H. (Ed.) (1994). Bulk Power System Voltage Phenomena III – Voltage Stability, Security and Control. Proceedings of the ECC/NSF Workshop, Davos, Switzerland.
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.
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.
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.
Lawrence, D. A. and W. J. Rugh (1995). Gain scheduling dynamic linear controllers for a nonlinear plant. Automatica, 31, 381–390.
Leela, S. and N. Shahzad (1996). On stability of moving conditionally invariant sets. Nonlinear Analysis, Theory, Methods and Applications, 27, 797–800.
Leonessa, A., W. M. Haddad and V. Chellaboina (2000). Hierarchical Nonlinear Switching Control Design with Application to Propulsion Systems. Springer, London.
Martynyuk, A. A. and A. S. Khoroshun (2008). On parametric asymptotic stability of large-scale systems. International Applied Mechanics, 44, 565–574.
Ohta, Y. and D. D. Šiljak (1994). Parametric quadratic stabilizability of uncertain nonlinear systems. Systems and Control Letters, 22, 437–444.
Ortega, J. and W. Rheinboldt (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York.
Packard, A. (1994). Gain scheduling via linear fractional transformation. Systems and Control Letters, 22, 79–92.
Rugh, W. J. (1991). Analytical framework for gain scheduling. IEEE Control Systems Magazine, 11, 79–84.
Shamma, J. and M. Athans (1990). Analysis of gain scheduled control for nonlinear plants. IEEE Transactions on Automatic Control, 35, 898–907.
Shamma, J. and M. Athans (1992). Gain scheduling: Potential hazards and possible remedies. IEEE Control Systems Magazine, 12, 101–107.
Šiljak, D. D. (1969). Nonlinear Systems: The Parameter Analysis and Design. Wiley, New York.
Šiljak, D. D. (1978). Large-Scale Dynamic Systems: Stability and Structure. North-Holland, New York.
Šiljak, D. D. (1989). Parameter space methods for robust control design: A guided tour. IEEE Transactions on Automatic Control, 34, 674–688.
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.
Stamova, I. M. (2008). Parametric stability of impulsive functional differential equations. Journal of Dynamical and Control Systems, 14, 235–250.
Stilwell, D. J. and W. J. Rugh (2000). Stability preserving interpolation methods for the synthesis of gain scheduled controllers. Automatica, 36, 665–671.
Sundarapandian, V. (2002). A necessary condition for local asymptotic stability of discrete-time nonlinear systems with parameters. Applied Mathematics Letters, 15, 271–273.
Sundarapandian, V. (2006). New results on the parametric stability of nonlinear systems. Mathematical and Computer Modeling, 43, 9–15.
Wada, T., M. Ikeda, Y. Ohta and D. D. Šiljak (2000). Parametric absolute stability of multivariable Lur’e systems. Automatica, 36, 1365–1372.
Zečević, A. I. (2004). Analytic gain scheduling using linear matrix inequalities. Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 11, 589–607.
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.
Zečević, A. I. and D. D. Šiljak (2003). Stabilization of nonlinear systems with moving equilibria. IEEE Transactions on Automatic Control, 48, 1036–1040.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Zečević, A.I., Šiljak, D.D. (2010). Parametric Stability. In: Control of Complex Systems. Communications and Control Engineering. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1216-9_5
Download citation
DOI: https://doi.org/10.1007/978-1-4419-1216-9_5
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-1215-2
Online ISBN: 978-1-4419-1216-9
eBook Packages: EngineeringEngineering (R0)