Dynamics and Control

, Volume 4, Issue 1, pp 5–20 | Cite as

Globally optimal quadratic Lyapunov functions for robust stability of systems with structured uncertainty

  • A. Olas
  • F. Ahmadkhanlou


The robust stability problem for a nominally linear system with nonlinear, time-varying structured perturbations is considered. The system is of the form
$$\dot x = A_N x + \sum\limits_{j = 1}^q { p_j A_j x} $$

The Lyapunov direct method has been often utilized to determine the bounds for nonlinear, time-dependent functions pj which can be tolerated by a stable nominal system. In most cases quadratic forms are used either as components of vector Lyapunov function or as a function itself. The resulting estimates are usually conservative. Optimizing the Lyapunov function reduces the conservatism of the bounds. The main result of this article is a theorem formulating the necessary and sufficient conditions for a quadratic Lyapunov function to be globally optimal. The theorem is constructive and makes it possible to propose a recursive procedure of a design of optimal Lyapunov function. Examples demonstrate application of the proposed method.


Linear System Quadratic Form Lyapunov Function Stability Problem Robust Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • A. Olas
    • 1
  • F. Ahmadkhanlou
    • 1
  1. 1.Department of Mechanical EngineeringOregon State UniversityCorvallis

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