On the control and stability of variable-order mechanical systems

Abstract

This work investigates the control and stability of nonlinear mechanics described by a system of variable-order (VO) differential equations. The VO behavior results from damping with order varying continuously on the bounded domain. A model-predictive method is presented for the development of a time-varying nominal control signal generating a desirable nominal state trajectory in the finite temporal horizon. A complimentary method is also presented for development of the time-varying control of deviations from the nominal trajectory. The latter method is extended into the time-invariant infinite temporal horizon. Simulation error dynamics of a reference configuration are compared over a range of damping coefficient values. Using a normal mode analysis, a fractional-order eigenvalue relation—valid in the infinite horizon—is derived for the dependence of the system stability on the damping coefficient. Simulations confirm the resulting analytical expression for perturbations of order much less than unity. It is shown that when deviations are larger, the fundamental stability characteristics of the controlled VO system carry dependence on the initial perturbation and that this feature is absent from a corresponding constant (integer or fractional) order system. It is then empirically demonstrated that the analytically obtained critical damping value accurately defines—for simulations over the entire temporal horizon—a boundary between rapidly stabilizing solutions and those which persistently oscillate for longtimes.