Abstract
In this paper, a 4D fractional-order centrifugal flywheel governor system is proposed. Dynamics including the multistability of the system with the variation of system parameters and the derivative order are investigated by Lyapunov exponents (LEs), bifurcation diagram, phase portrait, entropy measure, and basins of attraction, numerically. It shows that the minimum order for chaos of the fractional-order centrifugal flywheel governor system is q = 0.97, and the system has rich dynamics and produces multiple coexisting attractors. Moreover, the system is controlled by introducing the adaptive controller which is proved by the Lyapunov stability theory. Numerical analysis results verify the effectiveness of the proposed method.
Highlights
Erefore, it is of great theoretical significance to introduce fractional calculus into the study on chaos and control of the centrifugal flywheel governor system
Ge et al [20] determined the existence of chaos in the fractional-order autonomous and nonautonomous nonlinear centrifugal flywheel governor system by using the bifurcation diagram and phase portrait and realized chaos control of the system by the linear feedback control method
The Fractional-Order Centrifugal Flywheel Governor System e mechanics model of the centrifugal flywheel governor with external disturbance is depicted in Figure 1, where l, m, r, and φ represent the length of the rod, the mass of the fly ball, the distance between the rotational axis and the suspension joint, and the angle between the rotational axis and the rod, respectively. e motor drives the flywheel to rotate with angular velocity ω. e flywheel is joined to the axis through a gear box, so the axis rotates with angular velocity nω. n is the proportional coefficient, k is the stubborn coefficient of the spring, and g is the gravitational acceleration
Summary
Flywheel compute LEs of the fractional-order system can be obtained according to reference [28]. e predictor-corrector algorithm is an effective method to solve the fractional-order partial differential equation. Fix s 5, other conditions remain unchanged, and the responding basins of attraction in the x-y plane with different values of q are shown in Figures 8(d)–8(f ). Where x, y, z, and w are state variables and u1, u2, u3, and u4 are external active control inputs. en, the adaptive control system can be defined by u1 u2
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