On Controlling Non-Autonomous Time-Delay Feedback Systems

Time-delay feedback oscillators of non-autonomous type are considered in the paper. These oscillators have been studied extensively for many decades in a broad set of fields such as sensor design, manufacturing, and machine dynamics. A time-delay model system having one time-delay constant and several nonlinear feedback terms in the governing differential equation is first studied. Many researches have demonstrated that a time-delay feedback even in the form of a small perturbation is able to perturb the oscillator to exhibit complex dynamical responses including bifurcation and route-to-chaos. These motions are harmful as they have a very negative impact on the stability, and thus output quality, of the system. For example, manufacturing processes that are characterized by time-delay feedback all have an operation limit on speed because the chaotic behaviors which are unpredictable and extremely unstable are difficult to control. With a viable control solution, the performance, quality, and capacity of manufacturing can be improved enormously. A novel concept capable of simultaneous control of vibration amplitude in the time-domain and spectral response in the frequency-domain has been demonstrated to be feasible for the control of dynamic instability including bifurcation and route-to-chaos in many nonlinear systems. The concept is followed to create a control configuration that is feasible for the mitigation of non-autonomous time-delay feedback oscillators. Featuring wavelet adaptive filters for simultaneous time-frequency resolution and filtered-x least mean square algorithm for online identification, the controller design is shown to successfully moderate the dynamic instability of the time-delay feedback oscillator and unconditionally warrant a limit cycle. The controller design that integrates all these features is able to mitigate dynamical deterioration in both the time and frequency domains and properly regulate the responses with the desired reference signal. Specifically the qualitative behavior of the controlled oscillator output follows a definitive fractal topology before settling into a stable manifold. The controlled response is categorically quasi-periodic and of the prescribed vibration amplitude and frequency spectrum. The control scheme is novel and requires no linearization. By applying wavelet domain analysis approach to the nonlinear control of instability, the true dynamics of the time-delay feedback system as delineated by both the time and frequency information are faithfully retained without being distorted or misinterpreted. Through employing adaptive technique, the high sensitivity of the time-delay feedback system to external disturbances is also properly addressed.