Wavelet Based Nonlinear Time-Frequency Control Theory With Local Adaptability

Abstract

Abstract A novel nonlinear control concept featuring simultaneous control of vibration amplitude in the time domain and spectral response in the frequency domain is developed and subsequently incorporated to maintain dynamic stability in these nonlinear dynamics by denying bifurcation and route-to-chaos from coming to pass. This study mathematically examines the time-frequency control theory and derives the ranges of the regression step sizes that ensure fast convergence and solution stability of the controller design. The local adaptability is novel, intelligent, self-adjusting, and universally applicable to addressing bifurcated and chaotic responses. Incorporating the control theory into designing dynamic systems governed by discontinuous, time-delayed nonlinear oscillators would realize unconditional stability, high-performance quality, and operation efficiency. In the end, a novel time-delayed vibro-impact oscillator was investigated. Its instability in manufacturing would result in premature tool breakage, increased wear rate, and poor workpiece quality. Improvements in the discontinuous system in performance and stability were consequential in implementing the wavelet-based time-frequency controller with parallel online modeling. The controller demonstrated the most unique property in the frequency domain. As was evident in the instantaneous frequency domain, the controlled motions of the oscillator were unconditionally stable, stationary, and periodic.