On the Robust Stability of Time-Varying Linear Systems

  • Mohamed Mansour
  • Brian D. O. Anderson
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 121)


This paper considers the robust stability of time-varying linear systems described by a linear differential equation whose coefficients vary inside given intervals and with restricted magnitudes of the rates of change of the coefficients. This problem can be considered as a generalization of the Kharitonov problem, which is in turn a generalization of the Hurwitz problem, and it was formulated as an open problem in [7]. To solve this problem Lyapunov theory is used where a Lyapunov function is obtained (using characteristics of positive real functions [2], [3]) which is multiaffine in the polynomial coefficients. With this Lyapunov function extreme point results are obtained. The structure of the Lyapunov matrix as well as the structure of the conditions for the solution of a robust positive real function problem are characterized. A second approach based on the critical stability conditions is also suggested but the Lyapunov matrix thus obtained is no longer in general multiaffine in the parameters. Examples of low order systems are given. The resulting stability conditions are only sufficient.


Lyapunov Function Robust Stability Lyapunov Equation Lyapunov Matrix Strict Positive Real 
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Copyright information

© Birkhäuser Verlag Basel 1996

Authors and Affiliations

  • Mohamed Mansour
    • 1
  • Brian D. O. Anderson
    • 2
  1. 1.Institut für AutomatikSwiss Federal Institute of TechnologySwitzerland
  2. 2.Research School of Information Sciences and Engineering and Cooperative Research Centre for Robust and Adaptive SystemsAustralian National UniversityAustralia

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