Control of Neimark–Sacker bifurcation in a type of weak impulse excited centrifugal governor system
Introduction
As one of the research focuses of nonlinear dynamic system control, bifurcation control has attracted the attention of many scholars [1], [2], [3], [4], [5], [6], [7]. The engineering applications of bifurcation control in power systems for impending voltage collapse and catastrophe in designing an alert system [8], regulating human heart rhythms [9], controlling flight dynamics [10] and thermal runaway of reactors, and reducing vibration of hard disk drives [11] has been widely reported. Generally, the goal of bifurcation control is to alter the bifurcation features of a nonlinear system, pushing forward the formation of an intrinsic bifurcation [12] and changing the amplitudes [13] and stability [14] of bifurcated states. In addition, it has been reported that the inverse problem of conventional bifurcation analysis, namely the generation of bifurcation with the desired characteristics [15], [16], [17], [18].
Neimark–Sacker bifurcation is a universal phenomenon in the engineering areas of physical, electronic, biological, chemical, and so on. In general, Neimark–Sacker bifurcation outcomes an oscillatory behavior of torus solutions. The oscillating behavior can be helpful in many functional applications, namely mixing, fault diagnosis and monitoring in electromechanical systems, low-energy navigation and control. Recently, in different types of nonlinear dynamic systems, Neimark–Sacker bifurcation designed under control with specific oscillatory characters of torus states has been the focus of attention. Chatterjee and Mallik [19] investigated a class of self-excited oscillators with quasi-periodic vibro-impact effects. Luo and Xie [20] studied Hopf bifurcations in a two-degree-of-freedom undamped vibro-impact system with periodic motion of single-impact under strong resonant. Xie and Wen [21] analyzed Hopf bifurcation of a vibro-impact system with period two.
Controlling amplitude and frequency of a bifurcated oscillatory state has always been a concern. Tang et al. [22] controlled the oscillating magnitude of the limit cycle in a van der Pol system with weakly nonlinearity. Cui et al. [23] used the explicit control formulae in order to control the amplitude of bifurcated limit cycles in a Langford system. Although there are many researches on bifurcated limit cycle [24], the control of amplitude and frequency of a tori is lacking. Wen et al. [25] obtained the control equations of discrete-time nonlinear systems by using center manifold method, normal form technique as well as the Iooss’s Hopf bifurcation theory and effectively created Hopf bifurcations in discrete-time systems. Wen et al. [26] attained a controller design a torus solution of a dry friction system by using Hopf–Hopf interaction bifurcation criterion without evaluating eigenvalues.
In engineering, impulse-excited vibrations are often encountered [27], [28]. Due to impulse excitation, a system may exhibit complex dynamic behaviors, such as flutter in bearing or braking systems. Matteo et al. [29] considered the response estimation problem of nonlinear systems under multiplicative impulsive input and obtained the jump evaluation in closed form for two general classes of nonlinear multiplicative functions. Pilipchuk et al. [30], [31] attained solutions of a nonlinear Duffing oscillator under periodic impulsive excitation using a non-smooth temporal transformation. What is more, in rotational systems, square wave excitations are often utilized [32], [33], [34], [35], [36], [37], especially for motors [32], [36], [37], which is urgent to study dynamic behavior of rotational systems under square wave excitations.
The centrifugal governor plays a significant part in many rotating machines, e.g. internal combustion engines, steam engines, various fueled turbines, which is typically applied to maintain the speed of engine by controlling the amount of fuel. However, due to some practical demands, the requested speed fluctuates continuously over and below the mean speed, which recognizes the centrifugal governor to have a periodic oscillation, that is, a limit cycle or torus. In recent years, more and more researchers pay attention to centrifugal microfluidic devices, which can realize parallelization, miniaturization and autonomous analysis operations, greatly reducing sample and reagent consumption, processing time and total cost. [38], [39], [40], [41]. Cai et al. [42] designed a magnetically actuated valving system integrated a flyball governor to control the distance between the magnets during operation. In this system, the valve is controlled by manipulating the spinning speed of the flyball governor. Using this system, Cai et al. [43] developed chaotic mixing principles enhancing mixing efficiency of liquids, which is generated in a cyclic compressing–releasing process by controlling the spinning rate of the disk. Therefore, in the case of some applications, it is of practical importance to create a torus with desired stability, amplitude and frequency in centrifugal governor through control.
In this paper, a control method with feedback is addressed by generating a Neimark–Sacker bifurcation in a centrifugal governor system. Firstly, by utilizing an explicit criterion, without the use of eigenvalue calculations, we determine the linear gains for the Neimark–Sacker bifurcation containing transversality conditions and eigenvalue assignment that trigger Neimark–Sacker bifurcation at any a pre-determined parameter point. Secondly, normal form reduction and center manifold theory are used to control the stability of bifurcated torus after the nonlinear gains derived. Thirdly, the control equations for the mean radius of cross section of bifurcated torus are derived to choose linear gains. Finally, numerical experiments indicate that a bifurcated torus can be created in the centrifugal governor system at a predefined parameter position. With help of control criterion, some practical demands, e.g. mixture, could be satisfied in mechanical systems. We expect that the feedback control method will enable new capabilities in bifurcation control in mechanical systems, mixer or vibrating screen design, and drive new applications of nonlinear dynamics.
Section snippets
The mechanical model
The rotating machine represented by the flywheel with angular velocity is connected to the shaft having angular velocity through cone gear pair with transmission ratio of . The spring with stiffness and two balls assembling arms of length are connected to a sleeve. The horizontal distance from the vertical axis of the shaft to one end of the arm is . The deviation angle of the arm from the vertical direction is denoted by . Based on the Lagrangian method, the equation of motion of
The feedback control with polynomial functions
A polynomial functions state feedback control model is applied to establish the controller. By applying to the centrifugal governor system (5), we have the close-loop control system where the controller where is an equilibrium of the system (4), which satisfies . is the control gain vector. With the control we can change the equilibrium structure of the original system (4) during
Numerical experiments
For original system (5), it is assumed that the parameter is the bifurcation parameter, while other system parameters are . [24] The parameter is pre-determined as the desired bifurcation location for controlling Neimark–Sacker bifurcation, at which the motion state of the system is shown in Fig. 3. We obtain Poincaré map in Fig. 3(b) with constant phase surface as Poincaré section. Lyapunov characteristic exponents computations are implemented
Conclusions and remarks
In this paper, the feedback control problem of designing Neimark–Sacker bifurcation in a centrifugal governor system is addressed. A feedback control method is proposed to achieve three aspects of controlling problem including existence, stability, and adjusting mean radius of cross section of the limit torus to be designed. An explicit criterion including eigenvalue assignment and transversality conditions, without using eigenvalue computation, is utilized to derive the linear gains
CRediT authorship contribution statement
Zengyao Lv: Study conception and design, Initiated the project, System modeling, Controller design, System simulation, Data analysis, First draft of the manuscript was written, Commented on previous versions of the manuscript, Read and approved the final manuscript. Huidong Xu: Study conception and design, Initiated the project, System modeling, Controller design, Commented on previous versions of the manuscript, Read and approved the final manuscript. Zihao Bu: Study conception and design,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding
Huidong Xu acknowledge the financial supports by Applied Basic Research Program of Shanxi Province of China (No. 201801D121021), and Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi, China (No. RZ18100267).
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