Active disturbance rejection control(ADRC) is a practical control method with a two-degree-of-freedom structure. Due to its capability of handling multifarious disturbances in a straightforward and effective manner, ADRC has been successfully applied to many mechanical systems. However, the limit cycle vibration may be induced when employing the ADRC for mechanical systems with friction. At present, there is no precise analysis work about the friction induced vibration under the ADRC framework. Therefore, this paper investigates this problem by using the analysis tools of nonlinear dynamic systems. First, two representative friction models, static switch model and dynamic LuGre model, respectively, are considered, and active disturbance rejection controllers of different orders are designed for a class of second-order motion systems. Equivalent forms of the controllers are obtained and their relationships with the proportional-integral-derivative(PID) controller are revealed. Then, the limit cycle is calculated by using the shooting method combined with the pseudo arc-length continuation approach. Based on the Floquet theory, the stability, occurrence and type of bifurcation of the limit cycle can be determined. In addition, the local stability of the equilibrium points is analyzed based on the Jacobian matrix and approximate numerical method. Finally, the effects of the model and parameter of friction, the order and parameters of the ADRC on the limit cycle are investigated by numerical calculations. As shown by the calculation results, the parameter $\beta$, which determines the negative slope of the Stribeck effect, has a significant effect. When $\beta>1$, closed-loop systems with these two friction models have the same characteristics. Cyclic fold bifurcation(CFB) of the limit cycle occurs and the set of equilibrium points is locally stable. However, characteristics of these two closed-loop systems are totally different when $\beta<1$. As for the ADRC order, it is found that the order does not affect the conclusions in terms of the existence and stability of the limit cycle, and the stability of the set of equilibrium points. Moreover, low-order ADRC has superior performance in tackling the conflict between the friction compensation and stability robustness. These results can provide some guidelines on the understanding of practical phenomena, selection of the ADRC order, and parameter tuning.
2020, 52(5): 1485-1497.