Parametric identification of chaotic/nonlinear systems and reduced order models based on proper orthogonal decomposition

In the dissertation, the parametric identification and proper orthogonal modes (POMs) were investigated on chaotic/nonlinear systems such that a reliable and general-purpose process can be developed to reconstruct the mathematical models of nonlinear and/or chaotic systems.; First, a parametric identification method was examined for different chaotic systems, e.g. whirling, multi-degree-of-freedom, and strong nonlinearity, and by simulation and experiments. The original parametric identification method of chaotic systems is a hybrid time and frequency domain method based upon the harmonic balance method applied to unstable periodic orbits (UPOs), and solved by least mean squares. A chaotic base-excited single pendulum system was simulated. The identification method was modified for the whirling system with measured data of angular displacement. The nonlinearity was also approximated by two types of function series: linear interpolation functions and harmonic functions. Poincare sections showed that the identified system and the original simulation system had similar chaotic behavior. Then, the identification method was applied to an experimental chaotic double pendulum under vertical base excitation. Several noise reduction techniques were applied to reduce the identification errors due to the noise contamination in the experimental data and the strong nonlinearity. Meanwhile, an error optimization process, defined by the linear regression and statistics, was proposed to improve the identified parameters by selecting sub-harmonics of the unstable periodic orbits. An energy balance method, as a second-step identification after the harmonic balance method, was applied to give a more accurate estimation of the small damping coefficients.; It was also found that any chaotic orbit can be an approximated representation of some UPO, and the longer the orbit the better the approximation. Thus, the UPO extraction process can be neglected. The identification process can be simplified to a frequency domain method. Examples were examined to show the success of the simplification.; The study then goes to parametric identification and the POM for building reduced order models of unknown systems. The special interest here is systems with strong nonlinearity, where the reduced order models from experimental POM was limited in simulating the unknown systems other than the neighborhood set plane where the POM data is in the phase space. An added-constraint method was proposed and examined by two simple systems: a two-mass system with nonlinear spring and a mass-pendulum system. It showed that the added constraint can improve applicability of the POM reduced order model as well as increasing the accuracy of the simulation result. Nevertheless, the method is to be tested by high dimensional nonlinear systems; to which the proposed added constraint method can really make a big difference in applications.