Reconstruction of pitchfork bifurcation with exogenous disturbances based on equivalent-input-disturbance approach
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ORIGINAL PAPER
Reconstruction of pitchfork bifurcation with exogenous disturbances based on equivalent-input-disturbance approach Jinhua She · Xiang Yin · Min Wu · Daiki Sato · Kouhei Ohnishi
Received: 5 August 2020 / Accepted: 30 October 2020 © Springer Nature B.V. 2020
Abstract Reconstructing a pitchfork bifurcation that is destroyed by an exogenous disturbance is important to control it. This paper extends the equivalent-inputdisturbance (EID) approach to solve the problem of reconstructing a pitchfork bifurcation when there is an exogenous disturbance for the first time. A nonlinear term and a bifurcation parameter are designed in the system to reconstruct the pitchfork bifurcation. The bifurcation characteristics of the equilibriums of the system are analyzed by exploring the features of the EID-based system. This provides us analytic solutions for a system with a degree higher than four for the first time. Unlike other pitchfork-bifurcation control systems, the EID-based control system only rejects the disturbance and retains the nonlinearity. NumeriJ. She · X. Yin (B) · M. Wu School of Automation, China University of Geosciences, Wuhan 430074, Hubei, China e-mail: [email protected] J. She · X. Yin · M. Wu Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan 430074, Hubei, China J. She School of Engineering, Tokyo University of Technology, Hachioji, Tokyo 192-0982, Japan D. Sato FIRST, Tokyo Institute of Technology, Yokohama 226-8503, Japan K. Ohnishi Haptics Research Center, Keio University, Kawasaki, Kanagawa 212-0032, Japan
cal examples demonstrate the effectiveness of the presented method and show its superiority over others. The robustness of the reconstruction with respect to measurement noise and control of a pitchfork bifurcation are also shown by the examples. Keywords Bifurcation control · Bifurcation analysis · Disturbance rejection · Equivalent input disturbance
1 Introduction Bifurcation is a phenomenon that the topological type of a system changes as system parameters change [1]. While bifurcation is considered to be baleful and a control law is designed in a conventional nonlinear control system to avoid or reject it, a controller is designed in a bifurcation control system to make use of it to achieve desirable dynamical behavior. Bifurcationcontrol problems contain delaying the onset of an inherent bifurcation [2,3], stabilizing a bifurcated solution or branch [4], and others [5]. Bifurcation control was first presented for the control of a pitchfork bifurcation by Abed and Fu [6]. Later, they studied the problem of controlling a Hopf bifurcation [7]. Since then, bifurcation control has been receiving a great number of attention. And the last two decades have witnessed a huge growth in the field of bifurcation control. For example, a state-feedback controller was designed to eliminate the Neimark–Sacker bifurcation in a hybrid predator–prey economic model
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[8]; delayed feedback was applied to suppress bifurcations and chaot
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