Stability and Hopf bifurcation of the maglev system with delayed position and speed feedback control

Abstract

In this paper, the dynamic behavior of suspension system of maglev train with time-delayed position and velocity feedback signal is considered with rigid guideway. The stability conditions of the system are obtained with characteristic root method. The Hopf bifurcation direction and stability of the system at the critical point are also investigated. Based on center manifold reduction and Poincaré normal form theory, the general formula for the direction, the estimation formula of period and stability of Hopf bifurcating periodic solution are also given. It is shown that time delays can change the current complicated dynamic behavior. And the condition that the bifurcation may occur is given to restrain the dynamic response and vibration between vehicle and guideway of the system with time-delayed position and velocity signal.

Nonlinear analysis of a maglev system with time-delayed feedback control

“…Note that k p influences the steady-state error and hence the stiffness, k d controls the suspension damping and k a the overall stability margin. Some previous work [11,12] assumed that the time delay occurs only in one or two of the feedback control variables, however, we make the more reasonable assumption, as in [13], that all the feedback control variables have a time delay. We use z aτ = z a (t − τ ),ż aτ andz aτ to denote, respectively, the position, velocity and acceleration feedback control signals with time delay.…”

“…We use realistic values [11,12] for the physical parameters as given in Table 1. We choose the control parameters k p and k a according to …”

“…τ value at the intersection point, m, if it exists, otherwise τ c is the τ value at the maximum, n. We have tried different pairs of k p and k a and the shapes of the curves defined by (12) and (13) Note that the points on the curves defined by equations (12) and (13) First we consider the transversality condition of the Hopf bifurcation theorem, i.e., whether a pair of complex eigenvalues crosses the imaginary axis with non-zero speed. Differentiating (8) with respect to c, and evaluating the real part at the bifurcation point gives:…”

“…Using various techniques (centre manifold reduction, pseudo-oscillator analysis), previous work [11,12,13] studied the stability and Hopf bifurcation of the single-degree-of-freedom suspension system with time delay, and found that the dynamic behavior can be changed by adjusting time delay. But the time delay of state feedback signals for the maglev system is not adjustable in practice.…”

“…Finally, in Section 5, we draw conclusions about our work and highlight the implications for the maglev system. The maglev model we study is similar to that presented in [11,12,13,20]. Following the approach of [11,12,13], we make the simplification that the deformation of the track is zero.…”