  We obviate the use of observers for the purpose of output feedback tracking
control of Lagrangian systems and solve some long-standing yet well-documented
open problems. As often implemented in control practice, we replace unavailable
derivatives with approximate differentiation. Our contribution consists in
establishing uniform global asymptotic stability in closed-loop, for Lagrangian
systems without dissipative forces (friction) using only position feedback.
Firstly, for fully-actuated relative-degree-two systems, the controller is
reminiscent of passivity-based controllers for robot manipulators and consists
in a linear dynamic system together with a globally-Lipschitz control law.
Establishing a global uniform result, all the more with such a simple
controller, is particularly valuable relatively to the literature of
output-feedback control of systems with non-globally-Lipschitz nonlinearities
in the unmeasured variables. This first contribution solves a long-standing
open problem and, as a matter of fact, recasted in a general context this
result is at the edge of what is achievable -see [24]. Then, we show that our
control approach may be applied to a more general problem, that of tracking
control of Lagrangian systems augmented by a chain of integrators (with
relative degree greater than two). As a corollary, we solve the global-tracking
position-feedback control problem for flexible-joint robots but also for
systems coupled with output-feedback linearizable actuator dynamics. Finally,
we discuss remaining open problems of fairly general interest in the realm of
analysis and design of robust nonlinear systems.
