Kinetic State Tracking for a Class of Singularly Perturbed Systems

The trajectory-following control problem for a general class of nonlinear multi-input/multi-output two time-scale system is revisited. While most earlier works used singular perturbation theory and assumed that an isolated real root exists for the nonlinear set of algebraic equations that constitute the slow subsystem, here, two time-scale systems are analyzed in the context of integral manifolds. It is shown that the singularly perturbed system has a center manifold and, for small values of the slow state, an approximate solution of the nonlinear set of transcendental equations can be computed. Geometric singular perturbation theory is used as the model-reduction technique, and modified composite control design is used to formulate the stabilizing control laws for slow state tracking. The control laws are independent of the scalar perturbation parameter and an upper bound for it, and the closed-loop error signals are determined such that uniform boundedness of the closed-loop system is guaranteed. Additionally, asymptotic stabilization is shown for the nonlinear regulation problem. The methodology is demonstrated through numerical simulation of a nonlinear generic two-degree-of-freedom kinetic model and a nonlinear, coupled, six-degree-of-freedom model of the F/A-18A Hornet. Results demonstrate that the methodology permits close tracking of a reference trajectory while maintaining all control signals within specified bounds.