PYI: Nonlinear Control and Order Reduction

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This PYI award will fund research into the development of a methodology for order reduction of nonlinear systems which (i) respects and preserves the nonlinearity of the original model, (ii) does not require exact knowledge of the relative speeds of various system variables, and (iii) incorporates the effect of an exogenous control input. The results will have implications for nonlinear controller design of systems with fast and slow modes. There are two major reasons for developing a nonlinear model order reduction technique. The first is common to all large dynamical engineering systems: the large number of governing equations. The second reason concerns the fact that nonlinear effects can not be neglected since they will often be the deciding influence in the performance of many advanced engineering systems, such as high-performance aircraft, rapidly maneuvering spacecraft, and chemical reactors. Only the first of these issues is considered in the standard technique of first linearizing the dynamical equations and then using a linear model order reduction technique. Clearly, any nonlinear effect (such as bifurcations, or even nonlinear transient response) will be lost using this approach. The main analytical framework for this study consists of multiparameter singular perturbation theory and center manifold theory. These two powerful tools will be merged and extended. Use of functional-analytic results such as the so-called hard implicit function theorem will be made. The research supported under this PYI award concerns the development of a methodology for model order reduction of nonlinear systems. The approach will attempt to preserve the nonlinear nature of the original system since the nonlinearities may be the deciding influence in many advanced engineering systems.