Stability Analysis And Bifurcation Of A Kind Of Man-Machine System With Delay

Along with the development of science and technology, man-machine system is increasingly large and complex. High-precision and high-performance of the machine equipment cause the work responsibility which the people undertake be more significant. It is possible for fatal accident by human errors. So we not only in the practical work, but also solve the stable problem of the man-machine system theoretically. In this paper, a structurally unstable man-machine system is considered. It is representative of problems with severe non-linearity. This problem relates to explanation of construction of biped robots in robotics and controlling the vertical deviation of a space shuttle during take-off, and this man-machine system is closely related to the balancing of an inverted pendulum. As an example of the man-machine system, inverted pendulum is a typically nonlinear system of high-level, multi-variable and serious instability. Its behavior is great similar with the flight of rocket and construction of biped robots. So it has theoretical and practical significance for studying it. Because of inverted pendulum system itself with above characteristics precisely, it becomes the experimental system of in-depth studying, research and confirming the effectiveness of control theory.Firstly, the paper presents the man-machine system model that we will study. A mathematical analysis is performed to study the influence of the delay in the control force of man-machine system on Hopf bifurcation. It is found that the Hopf bifurcation occurs when the delay passes through a critical value. Then the explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived using the theory of normal form and center manifold, and some numerical simulations are carried out to illustrate the results found. Then we certify that there is Hopf-zero bifurcation if some conditions are satisfied. Finally, the mathematical model relatively of inverted pendulum system is presented. The condition to ensure the stability of the zero solution and its stable domain are decided by analyzing the characteristic equation of the linearization. The study of the Hopf bifurcation is given, and the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived. Then some numerical simulations are carried out to illustrate the results found.