Design and Stability Analysis of an Integral Time-Delay Feedback Control Combined With an Open-Loop Control for an Infinitely Variable Transmission System

The kinematic model of an infinitely variable transmission (IVT) is introduced, and the nonlinear differential equation for the dynamic model of the IVT system with a permanent magnetic direct current (DC) motor and a magnetic brake is derived. To make the average of the input speed converge to a desired constant for any input power and output load, an integral time-delay feedback control combined with an open-loop control is used to adjust the speed ratio of the IVT. The speed ratio for the open-loop control is obtained by a modified incremental harmonic balance (IHB) method. Existence and convergence of a periodic solution are proved under a condition for parameters of the IVT system, and uniqueness of the periodic solution is proved by converting the nonlinear differential equation to a new differential equation that is Lipchitz in the dependent variable and piecewise continuous in the independent variable. A time-delay variable that is an approximation of the average of the input speed is used as the feedback to control the changing rate of the speed ratio. The IVT system with the time-delay control variable can be converted to a distributed-parameter system. Thus, the spectral Tau method is used to design the time-delay feedback control so that the IVT system is locally exponentially stable. The static error from the open-loop control is eliminated; the feedback control variable with time-delay is smoother than that without time-delay, which yields a lower control effort and more robust control design, since the time-delay variable that acts as a low-pass filter reduces the effect of the instantaneous change of the IVT system.