Abstract
Considering the nonlinear properties of spring and damping of suspension, a quarter-car model with time-delayed control is established. The Routh–Hurwitz stability criterion and stability switching method are used to analyze the stability of the system and obtain the stability region diagram. The multiobjective optimization function is established by considering the ride comfort, driving safety, and handling stability. The optimal control parameters are obtained by the optimization and simulation of the system under harmonic excitation and random excitation. In addition, the responses of the active suspension system with optimal time-delay control and the passive suspension system without control are compared. The results show that the active suspension system with time-delay displacement feedback control can reduce the vibration of the system, and there is an optimal feedback parameter combination to optimize the vehicle running state. The design of multiobjective function optimization proposed in this paper can improve ride comfort, driving safety, and handling stability and provide guidance for comprehensively improving vehicle performance.
Highlights
It is found that time delay hinders the operation of the system and often leads to the generation of limit cycles, loss of stability, and bifurcation chaos
Xu and Lu [20] studied the influence of time-delay feedback control on the dynamic behavior of nonlinear systems with periodic external excitation. rough numerical analysis, it was observed that time delay could lead to instability of the static and dynamic equilibrium states of the system, resulting in Hopf bifurcation, period-doubling bifurcation, almost periodic motion, chaos, and other motion behaviors
When the system was not in the time-delay-independent region, stability switch might occur with the change of the time delay. ese admissible switches corresponded to Hopf bifurcation when the time delay crossed the critical value, and the properties of periodic solution direction and stability of Hopf bifurcation were determined by using normal form theory and central manifold theorem
Summary
When τ > 0, according to the first-order approximation stability theory for nonlinear systems [25], when all the real parts of the first-order approximation equation are negative, the equilibrium solution of the original nonlinear equation is asymptotically stable. En, the critical condition for the instability of the system is that the characteristic equation has a pure imaginary root λ iω By substituting it into equation (15), substituting with Euler’s formula: e− λτ e− iωτ cos(ωτ) − i sin(ωτ), separating the real and imaginary parts of the equation, and setting them equal to zero, respectively, the following equation can be obtained: PR(ω) + QR(ω)cos(ωτ) + QI(ω)sin(ωτ) 0,. When g falls into other regions, as the time delay increases, the system undergoes several stabilization switches and eventually becomes unstable [6]. ese admissible switches correspond to the Hopf bifurcation that occurs when the delay spans the critical value. e shaded parts are the stable regions
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