On the linear control of nonlinear mechanical systems

This article describes the design of a linear observer-linear controller-based robust output feedback scheme for output reference trajectory tracking tasks in a large class of fully actuated nonlinear mechanical systems whose generalized position coordinates are measurable. The unknown, possibly state-dependent, additive nonlinearity influencing the input-output description, in terms of the tracking error dynamics, is modeled as an absolutely bounded, additive, unknown “time-varying perturbation” input signal. This procedure simplifies the system tracking error description to that of independent chains of integrators with, known, position-dependent control gains, while additively being perturbed by an unknown, smooth, time-varying signal which is trivially observable. The total state-dependent uncertain input is assumed to be locally approximated by an arbitrary element of, a, fixed, sufficiently high degree family of Taylor polynomials for which a linear observer may be readily designed. Generalized Proportional Integral (GPI) observers, which are the dual counterpart of GPI controllers (see), are shown to naturally estimate, in an arbitrarily close manner, the unknown perturbation input of the simplified system and a certain number of its time derivatives, thanks to its embedded, internal time-polynomial model of the unknown, state-dependent, perturbation input. This information is used to advantage on the linear, observer-based, feedback controller design via a simple cancelation effort. The results are applied to the control of a laboratory prototype in a trajectory tracking problem.