BIFURCATION TAILORING OF NONLINEAR SYSTEMS Xiao Fan Wang 1,2,3 , M. Di Bernardo 2 , M. H. Lowenberg 1 , D. P. Stoten 3 , G. Charles 1,2,3 1 Department of Aerospace Engineering, University of Bristol, Bristol, BS8 1TR, UK 2 Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TR, UK 3 Department of Mechanical Engineering, University of Bristol, Bristol, BS8 1TR, UK Abstract: We discuss a novel approach to control bifurcations in nonlinear systems. The aim of bifurcation tailoring is to design an appropriate control law such that the controlled system has a desired bifurcation diagram. After describing two open-loop bifurcation tailoring techniques, this paper proposes two alternative modified bifurcation tailoring methods based on the use of the Newton-flow algorithm and the so-called Minimal Control Synthesis adaptive control strategy. The novel technique is applied to the Duffing system as an illustration example. Copyright © 2002 IFAC Keywords: Bifurcation; Open-loop control; Adaptive control; Nonlinear system. 1. INTRODUCTION In recent years, there has been rapidly growing interest in control of bifurcations in nonlinear dynamical systems (Chen, 1999; Chen et al., 2000). The goal of bifurcation control is typically achieved by delaying the onset of an inherent bifurcation and/or stabilizing an existing one. Some of the bifurcation control approaches to solve these problems presented in the literature include linear or nonlinear static feedback (Abed and Fu, 1986), washout filter-aided dynamic feedback (Wang & Abed, 1995), harmonic balance approximation (Berns et al., 1998) and normal forms-based feedback (Kang, 1998). In a broader sense, bifurcation control can be referred to as the task of designing a controller to modify the bifurcation properties of a given nonlinear system. Hence the goal becomes that of achieving a set of desirable asymptotic behaviours of the system as its parameters are varied. Recently, motivated by the control of bifurcations in complex flight dynamics, Lowenberg at al proposed the concept of bifurcation tailoring. This novel method is aimed at changing the bifurcation diagram of a given system to a desired one by appropriately varying extra system parameters in addition to the bifurcation parameter (Lowenberg, 1998a, 1998b; Lowenberg and Richardson, 1999). The original bifurcation tailoring technique involves an ‘inversion’ of the bifurcation continuation method as used in software such as AUTO (Doedel & Wang, 1995), therefore, is open-loop in nature from a control point of view. In other words, it cannot guarantee the stability of the desired behavior (equilibrium points or limit cycles) at any given value of the bifurcation parameter. Therefore, in addition to the feedforward control, an effective feedback mechanism should be added to the original bifurcation tailoring technique to address disturbances and modeling errors, so as to guarantee the stability and robustness of the controlled system. This paper is concerned with the development of such a feedback mechanism through the synthesis of novel bifurcation tailoring methods. These are based on the combined use of an on-line continuation technique (the Newton-flow algorithm) and a sophisticated adaptive control strategy, the Minimal Control Synthesis Algorithm or MCS (Stoten and Benchoubane, 1990a, 1990b). The rest of the paper is outlined as follows. Definition of bifurcation tailoring is presented in Section 2. In Section 3, open-loop bifurcation tailoring techniques and their limitations are discussed. Section 4 proposes two open-loop plus close-loop bifurcation tailoring methods. An illustration example is presented in Section 5. 2. BIFURCATION TAILORING: STATEMENT OF THE PROBLEM Consider a continuous-time dynamical system described by Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain