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Stability and Feedback Stabilization

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it representsa first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or theattainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring thestabilization of some quantity (typically, some sort of “error” signal). A very special case, when there are no inputs, is that ofstability.

Introduction

This article discusses the problem of stabilization of an equilibrium, which we take withoutloss of generality to be the origin, for a finite‐dimensional system \( {\dot x=f(x,u) } \). The objective is to find a  feedbacklaw \( { u=k(x) }\) which renders the origin of the “closed‐loop” system \( { \dot x=f(x,k(x)) } \)globally asymptotically stable. Theproblem of stabilization of...

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Abbreviations

Stability:

A globally asymptotically stable equilibrium is a state with the property that all solutions converge to this state, with no large excursions.

Stabilization:

A system is stabilizable (with respect to a given state) if it is possible to find a feedback law that renders that state a globally asymptotically stable equilibrium.

Lyapunov and control-Lyapunov functions:

A control-Lyapunov functions is a scalar function which decreases along trajectories, if appropriate control actions are taken. For systems with no controls, one has a Lyapunov function.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, USA

    Eduardo D. Sontag

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  1. Eduardo D. Sontag
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Sontag, E.D. (2009). Stability and Feedback Stabilization. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_515

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