Simplification and Order Reduction of Parametrically Excited Nonlinear Dynamical Systems

Abstract

System simplification is, by far, the most common theme behind various techniques available for the analysis and controller design of nonlinear dynamic systems. One such technique that is widely used by engineers is the direct linearization about an equilibrium point. Most other techniques, however, involve carefully designed state transformations so that the system in the transformed domain preserves nonlinear characteristics of the original system and facilitates efficient analysis and controller design. This dissertation presents a few of such methodologies for analysis and control of nonlinear dynamic systems. However, the emphasis is on nonlinear systems with parametric excitations. The main focus is on the development of techniques for reduced order modeling of nonlinear dynamic systems that are influenced by certain external inputs. These reduced order models accurately approximate the dominant dynamics of original systems and are much simpler to analyze and control. Two types of inputs are considered; namely, external periodic excitations and a nonlinear, stabilizing state feedback control. The approach is based on the construction of an invariant subspace such that any motion initiated on this subspace, often called an invariant manifold, remains confined there for all time, t. As a result, the system dynamics can be modeled by a smaller number of differential equations which govern system evolution on the manifold, alone. Techniques to identify suitable state variables which span these manifolds are suggested and approximate solutions of the resulting partial differential equations, which govern the geometry of such manifolds, are presented. Reduced order models for systems with periodic excitation inputs can be integrated directly to analyze the dynamics of original systems. For situations where there is a state feedback input, nonlinear controllers can be designed in the reduced order domain. A technique to synthesize such controllers is also discussed in this work. Another important simplification technique proposed in this work is the construction of normal forms (simplest forms) of nonlinear systems by direct application of time-periodic near-identity transformations and Poincaré normal form theory. This is an independent analysis technique for systems subjected to periodic forces. It also provides a basis for construction of time-dependent invariant manifolds for reduced order modeling of such systems. Some typical dynamic systems of engineering interest are considered and the effectiveness of proposed methodologies is illustrated.