The Optimization of Time-Frequency Control: Using Non-Autonomous Time-Delay Feedback Systems as Example

A novel time-frequency nonlinear scheme demonstrated to be feasible for the control of dynamic instability including bifurcation, non-autonomous time-delay feedback oscillators, and route-to-chaos in many nonlinear systems is applied to the control of a time-delayed system. The control scheme features wavelet adaptive filters for simultaneous time-frequency resolution. Specifically Discrete Wavelet transform (DWT) is used to address the nonstationary nature of a chaotic system. The concept of active noise control is also adopted. The scheme applied the filter-x least mean square (FXLMS) algorithm which promotes convergence speed and increases performance. In the time-frequency control scheme, the FXLMS algorithm is modified by adding an adaptive filter to identify the system in real-time in order to construct a wavelet-based time-frequency controller capable of parallel on-line modeling. The scheme of such a construct, which possesses joint time-frequency resolution and embodies on-line FXLMS, is able to control non-autonomous, nonstationary system responses. Although the controller design is shown to successfully moderate the dynamic instability of the time-delay feedback oscillator and unconditionally warrant a limit cycle, parameters are required to be optimized. In this paper, the setting of the control parameters such as control time step, sampling rate, wavelet filter vector, and step size are studied and optimized to control a time-delay feedback oscillators of a nonautonomous type. The time-delayed oscillators have been applied in a broad set of fields including sensor design, manufacturing, and machine dynamics, but they can be easily perturbed to exhibit complex dynamical responses even with a small perturbation from the time-delay feedback. These responses for the system have a very negative impact on the stability, and thus output quality. Through employingfrequency-time control technique, the time responses of the time-delay feedback system to external disturbances are properly mitigated and the frequency responses are also suppressed, thus rendering the controlled responses quasi-periodic.