Stabilization Of A Kind Of Mechanical Systems Of Multiple Degrees Of Freedom Via Delayed Feedback

Last two decades have witnessed a rapid development of active control of various mechanical systems, owing to the recent advances in computing, sensing and driving technology. However, with increasingly strict requirements for control speed and system performance, the unavoidable time delays in control loop have become a severe limitation to the development and application of active control. On the other hand, several studies have shown that the voluntary introduction of time delays can also benefit the control. Moreover, the nonlinear time-delay systems exhibit rich interesting dynamic behaviors, which can be employed to design controls to achieve some special kinds of motion. Therefore, dynamics and control of time-delay systems are of great scientific significance and practical value.This dissertation mainly focuses on the stabilization of mechanical systems of multiple degrees of freedom with multiple unstable equilibrium points via delayed feedback and the conrresponding nonlinear dynamics, so as to deepen understanding of dynamics and control of time-delay systems. The contents and contribution of the dissertation are as follows.1. According to the research objective, an experimental setup of a double pendulum on a cart is designed and established to study the dynamics and control of mechanical system with delayed feedback. Effects of digital filters on the stability and dynamic behaviors of the controlled double pendulum are studeied theoretically, numerically and experimentally. The experiment results show that an over demanding selection of the filter specifications will lead to the instability of the closed-loop system through a Hopf bifurcation, because the group delay of the filter exceeds a critical value. In the theoretical and numerical analysis, the filters are modeled into the dynamic equation of closed loop system as the components of pure time delays. This simplification yields a continuous time-delay system so that one can make not only the stability analysis, but also the Hopf bifurcation analysis. The experimental results show a good agreement with the theoretical and numerical results, and positively confirm the simplification of the digital filters as the components of pure time delays.2. A brief review on three existing state-transformation based delayed linear quadratic (LQ) control methods is presented. A discrete delayed LQ controller is successfully applied to the stabilization experiment of the double pendulum system to avoid the instablility induced by the group delay of a digital filter. The experimental results indicate the effectiveness of the control methods and support the adequacy of simplifying the digital filter as a time delay component.3. The relations between a continous transformation and a discrete transformation for the system with an input delay are revealed and inspired by which, a new continuous delayed LQ control is proposed. The new control method has a very interesting property that the feedback gain matrix keeps the same for different values of an input delay. Based on this property, the new delayed LQ control method, intended to deal with the dynamic system with a constant delay, can also be applied to the dynamic system with a slowly time-varying input delay.4. A systematic approach to stabilizing a kind of linear undamped systems of multiple degrees of freedom by using both position and delayed position (PDP) feedbacks is proposed. For the fully-actuated system, the approach enables one to complete the design of controller directly through the use of modal decoupling and a stability chart. For the under-actuated system, the approach includes two steps. The first step is to move all the eigenvalues of the system on the imaginary axis of the complex plane by using a position feedback, and the second step is to drag all the eigenvalues of the system to the open left half of the complex plane through the use of a delayed position feedback, which can be determined on the basis of sensitivity analysis of eigenvalues. Two illustrative examples well demonstrate the design procedure of PDP feedback controllers and their efficacy.