Model Decomposition for Multiple Time-Scale Nonlinear Dynamical Systems

University Of California-Irvine, Irvine CA

Investigators

Abstract

As the requirements for machines, structures, vehicles, and other mechanical systems become more ambitious and more demanding, the order and complexity of the mathematical models that must be confronted for analysis and design increases. Physical insight and analytical methods become less effective. Yet as the system dimension increases, so does the likelihood of multiple time-scales. The presence of two or more widely separated time-scales offers the opportunity for system decomposition and consequent simplified analysis and design. For a linear time-invariant system, both time and frequency domain methods are available to exploit this opportunity. For a nonlinear system, the analytical singular perturbation method is available. But this method is only applicable to a mathematical model in singularly perturbed form, and this form is not generally obtainable without a priori knowledge of the multiple time-scale structure. The research objective is thus to develop a methodology for time-scale identification and reduced-order model development for application to finite dimensional nonlinear dynamical systems. Under previous NSF funding, the foundations of the methodology, which is based on finite-time Lyapunov exponents and vectors, have been developed. In the current project the methodology is used to investigate the time-scale structure in nonlinear dynamical systems that model mechanical systems. Exemplary systems that have yielded to the analytical singular perturbation method, and thus are known to have two or more time-scales, are investigated first to further refine the methodology. Next systems either known or suspected to have multiple time-scale behavior are investigated. An approach is then developed for using the time-scale information to construct a state transformation that will bring the system model into a form amenable to reduced-order analysis and control design. The research will establish a significant new capability for nonlinear system analysis and design. The order reduction, better conditioning, and physical understanding -- previously obtainable only for low-order systems by clever analysts applying the analytical singular perturbation method -- will be obtainable for general nonlinear dynamical systems