Hopf bifurcation control for the main drive delay system of rolling mill

In this work, the Hopf bifurcation of the main drive delay system of rolling mill is controlled and analyzed by designing a nonlinear controller. The time-delay is selected as a bifurcation parameter, and the following conclusions are obtained through analysis: (1) in the absence of state feedback control, the system will generate the Hopf bifurcation at the expense of its stability when the bifurcation parameter exceeds the threshold value; (2) in the state under feedback control, the occurrence of Hopf bifurcation is effectively delayed and the stable region of the system is also well extended. More importantly, we can change the properties of bifurcation periodic solutions by selecting the appropriate gain parameters. Some numerical simulations reveal that under the nonlinear feedback control, the vibration amplitude of the system can be effectively reduced.


Introduction
For decades, we have witnessed the development and improvement of nonlinear dynamics theory. The rolling mill vibration model [1][2][3] is a class of nonlinear dynamic systems that evolves from a simple linear model to a complex nonlinear dynamic model. According to the structure and load characteristics of the actual equipment, the nonlinear torsional vibration model is established with time delay [4], nonlinear stiffness [5,6], and damping as nonlinear characteristic parameters to reveal more vibration mechanism and phenomena.
Many studies have shown that frictional vibrations encountered in actual production often cause bifurcations, which can seriously threaten the stable operation of the system. The Hopf bifurcation [7][8][9][10][11] is a common and important bifurcation phenomenon. The so-called Hopf bifurcation refers to the phenomenon that a closed orbit will occur in the vicinity equilibrium point when the stability of the singularity of the system is reversed, which can be used to explain many of the vibration problems in engineering.
In view of the system instability caused by Hopf bifurcation, some literature works [12][13][14][15][16] have studied the various bifurcation control methods to modify the bifurcation characteristics, which can obtain some expected dynamical behaviors, for example, in order to postpone the onset of Hopf bifurcation [17], to change the properties of Hopf bifurcation [18], and to reduce the amplitude of vibration [19]. Xiao et al. [20] controlled unstable or steady states and periodic orbits for a novel congestion control model by using the state feedback method. Xu [21] et al. used the polynomial function as a state feedback controller to realize the control of Hopf bifurcation for an internet congestion system.
In 2014, Zhang et al. established the main drive delay system of rolling mill [4]: where ζ , ξ , η, τ , c , and p are the real parameters. The specific meaning of parameters can be seen in [4]. Zhang et al. only gave the conditions for Hopf bifurcation to exist and some properties of bifurcating periodic solutions were not discussed. Furthermore, works seldom pay attention to the problem of Hopf bifurcation control for the main drive delay system of rolling mill. Our work will adopt the state feedback to control Hopf bifurcations. The main contributions of this paper are as follows: (1) A nonlinear controller is established to control the Hopf bifurcation of the main drive delay system of rolling mill; (2) The conditions for the existence of Hopf bifurcation in the main drive delay system of the rolling mill without control and with control are given, respectively; (3) In the state under feedback control, the occurrence of Hopf bifurcation is effectively delayed and the nature of bifurcating periodic solutions can be changed by selecting proper gain parameters of the nonlinear part of the controller.

Hopf bifurcation under control
In this section, we design a nonlinear controller to control the Hopf bifurcation in the main drive delay system of rolling mill. The nonlinear state feedback controller is as follows: where b 1 , b 2 , and b 3 are positive feedback parameters. The rolling mill main drive system Eq.

Theorem 4 For system
. Furthermore, it generates a Hopf bifurcation at E 0 when τ = τ k2 .
In the following, we will explore the nature of the Hopf bifurcation for controlled system (8) by implementing the normal form (NF) and the center manifold reduction (CMR) [22].

Figure 12
Waveforms and phase diagrams of uncontrolled system (2) and controlled system (8) with τ = 3.1 state feedback control are shown in Fig. 9. From Fig. 9, we note that the threshold value of the bifurcation increases with the increase of the control parameter b 1 and reaches the maximum value around b 1 = 0.6, then the threshold value decreases with the increase of b 1 .
In Figs. 10, 11, 12, we show waveforms and phase diagrams of the uncontrolled system and controlled system (2) with the same time delay, respectively. The result indicates that the vibration amplitude of the system can be effectively reduced under nonlinear feedback control. Therefore, the proposed control strategy is feasible.

Conclusion
In this paper, the main drive delay system of the rolling mill is considered and a nonlinear controller is designed to control the Hopf bifurcation in the system. We give the conditions for the Hopf bifurcation to exist in the main drive delay system of rolling mill without control and under control, respectively. In the state under feedback control, the occurrence of Hopf bifurcation is effectively delayed, and the nature of bifurcating periodic solutions can be changed by selecting proper gain parameters of the nonlinear part of the controller. Besides, the relation graph between the stability of system (8) and control parameter is given. Some numerical results validate the above analysis.