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# Control bifurcations

Krener, Arthur J. and Kang, Wei and Chang, Dong Eui (2004) Control bifurcations. IEEE Transactions on Automatic Control, 49 (8). pp. 1231-1246. ISSN 0018-9286. http://resolver.caltech.edu/CaltechAUTHORS:KREieeetac04

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## Abstract

A parametrized nonlinear differential equation can have multiple equilibria as the parameter is varied. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves. This typically happens because some eigenvalues of the parametrized linear approximating differential equation cross the imaginary axis and there is a change in stability of the equilibrium. The topological nature of the solutions is unchanged by smooth changes of state coordinates so these may be used to bring the differential equation into Poincare/spl acute/ normal form. From this normal form, the type of the bifurcation can be determined. For differential equations depending on a single parameter, the typical ways that the system can bifurcate are fully understood, e.g., the fold (or saddle node), the transcritical and the Hopf bifurcation. A nonlinear control system has multiple equilibria typically parametrized by the set value of the control. A control bifurcation of a nonlinear system typically occurs when its linear approximation loses stabilizability. The ways in which this can happen are understood through the appropriate normal forms. We present the quadratic and cubic normal forms of a scalar input nonlinear control system around an equilibrium point. These are the normal forms under quadratic and cubic change of state coordinates and invertible state feedback. The system need not be linearly controllable. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations.

Item Type: Article © Copyright 2006 IEEE. Reprinted with permission. Manuscript received March 29, 2001; revised May 15, 2002 and June 30, 2003. [Posted online: 2004-08-16] Recommended by Associate Editor Hua Wang. Control bifurcation, fold control bifurcation, Hopf control bifurcation, normal form, transcritical control bifurcation CaltechAUTHORS:KREieeetac04 http://resolver.caltech.edu/CaltechAUTHORS:KREieeetac04 http://dx.doi.org/10.1109/TAC.2004.832199 No commercial reproduction, distribution, display or performance rights in this work are provided. 5056 CaltechAUTHORS Archive Administrator 25 Sep 2006 15 Jan 2015 20:41

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