Numerical methods for stable inversion of nonlinear systems

Inversion is the process of computing reference trajectories for the plant input and state variables that are consistent with exact tracking of some given reference trajectory for the plant output. Stable inversion insists that the computed reference trajectories be bounded, whereas classical inversion forces the computed reference trajectories to be causal. Stable inversion reduces to classical inversion for the special case of minimum-phase systems. For the general case of nonminimum-phase systems, the existing numerical methods for stable inversion are based on a Picard process that requires decoupling coordinate transformations to separate the stable and unstable parts of the inverse system. The Picard iteration is implemented using a combination of forward-time and backward-time numerical integration or, if desired, by discrete Fourier transform techniques.; This thesis addresses several numerical methods applicable to stable inversion problems for nonlinear systems. Compared with the existing Picard method, the new methods introduced in this thesis (finite difference and relaxation methods) possess several advantages. The most noticeable advantage is their superior convergence properties, i.e. second order convergence speed and ability to converge even with aggressive trajectories and strong nonlinearities. The second advantage is their ability to handle systems with implicit inverse dynamics, or even to solve the stable inversion problem directly from the state equations or modified nodal equations (direct stable inversion). The third advantage is their ability to solve stable inversion problems for steady state periodic trajectories. These advantages give the new methods significantly better practical value than the existing Picard method. Both mathematical analysis and applications to physical systems are included.