| A functional transformation method to solve nonlinear ordinary differential equations (ODEs) is presented and used for systems with a variety of dynamic behavior. The ultimate goal is to develop input-output relationships for dynamic systems that can be potentially used for control system design and analysis.; For nonlinear systems that are hyperbolic, with a single stable equilibrium point, the method provides closed-form solutions with explicit dependence on system parameters. With a relatively small number of expansion terms, system dynamics are captured very accurately. Convergence of the expansion solution is also studied and shown that for systems with a single stable equilibrium point in the absence of limit cycle, the global convergence of the expansion is guaranteed. For single unstable equilibrium point systems or systems that exhibit limit cycle behavior or for multiple equilibrium point systems, although global convergence of the expansion solution is not guaranteed, the local convergence can be proven. The boundaries of the local convergence depend on the degree of nonlinearity of the system behavior.; For nonlinear systems that exhibit limit cycles and unstable equilibrium points and multiple equilibrium points (both stable and unstable), the expansion solutions are not trivial to develop. For practical purposes, a numerical technique, referred to as “partial trajectories”, is used to maintain local convergence properties and thus provide accurate solutions. The solution trajectories are clearly sensitive to the degree of nonlinearity of the system, the character of the equilibrium point(s) and the initial conditions.; Open-loop and closed-loop transfer functions are also developed using fun expansions, and their advantages as well as shortcomings are discussed. |
