Available online at www.sciencedirect.com Systems & Control Letters 49 (2003) 267–278 www.elsevier.com/locate/sysconle Resonant terms and bifurcations of nonlinear control systems with one uncontrollable mode Boumediene Hamzi a ; ∗ , Wei Kang b a Department of Mathematics, University of California, Davis, CA 95616, USA b Mathematics Department, Naval Postgraduate School, Monterey, CA 93943, USA Received 10 May 2002; received in revised form 18 November 2002; accepted 21 December 2002 Abstract In this paper we provide a simple algorithm of feedback design for systems with uncontrollable linearization with only quadratic degeneracy, such as transcritical and saddle-node bifurcations. This approach avoids the computation of nonlinear normal forms. It is based only on quadratic invariants which can be determined directly from the quadratic terms in the uncontrollable dynamics. c  2003 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear systems; Bifurcations; Resonant terms; Feedback design 1. Introduction Nonlinear parameterized dynamical systems exhibit complicated performance around bifurcation points. As the parameter of a system is varied, changes may occur in the qualitative structure of its solutions around an equilibrium point [17]. For control systems, a change of control properties may happen around an equilibrium point when the linearization is uncontrollable at this point [16]. This change of control properties results also in a lack of robustness. Hence the analysis for the bifurcation of control systems can be viewed as a “robustness analysis” in the context of control theory. The use of feedbacks to stabilize a system with bifurcation has been studied by several authors, and some fundamental results can be found in [1–4,6,9], the Ph.D. thesis [10–14], and the references therein. In [12,13], two related bifurcation problems of control systems are introduced. The rst problem is the bifurcation of control systems. It concerns the change of the properties of a control system at a critical point, including the change of controllability and the change of stabilizability (see [12]). The second problem concerns the feedback control of systems around a critical point. It has been found that around a critical point the closed-loop control system has generic bifurcation phenomenon. However, feedback controller can change the nature of a bifurcation. For systems with a single uncontrollable mode, bifurcations with control feedbacks are classied for nonlinear systems with a single input (see [13]). * Corresponding author. Tel.: +1-530-7549385; fax: +1-530-7526635. E-mail addresses: hamzi@math.ucdavis.edu (B. Hamzi), wkang@nps.navy.mil (W. Kang). 0167-6911/03/$-see front matter c  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-6911(02)00345-6