Observers are Unnecessary for Output-Feedback Control of Lagrangian Systems

Abstract

International audienceWe address and solve some long-standing yet well– documented open problems on output feedback tracking control of Euler-Lagrange systems with arbitrarily high relative degree; this includes underactuated systems. Our main contribution is to establish a theoretical foundation for the use of so-called dirty derivatives, a common " ad-hoc " replacement of unavailable state measurements such as generalized velocities, whence obviating the the use of observers for the purpose of position-feedback tracking control. Reminiscent of passivity-based control for robot manipulators, our control law is globally Lipschitz and the controller dynamics is linear. For relative-degree-two fully-actuated Lagrangian systems without dissipative forces (friction) and with bounded inertia matrix we establish uniform global asymptotic stability in closed loop. Furthermore, we show that our control approach applies to Lagrangian systems augmented by a chain of integrators (relative degree m + 2 systems). The design method, which is based on a recursive procedure in the spirit of backstep-ping control, is intuitive as it exploits structural properties such as passivity and inherent input-output stability. As a corollary, we solve an output feedback global-tracking control problem for flexible-joint robots but also for systems coupled with output-feedback linearizable actuator dynamics. In addition, we discuss remaining open problems of fairly general interest in the realm of analysis and design of robust nonlinear systems

A Separation Principle for Underactuated Lossless Lagrangian Systems

“…In this paper we extend the results of [18], [19] by assuming that the integrators' variables are not measured. We establish a separation principle for the same class of systems and we prove uniform global asymptotic stability of the origin.…”

“…In [18] is presented a dynamic output-feedback controller for (1) that guarantees uniform global asymptotic stability of the origin. The controller in the latter reference, however, is not of the form (5) since it depends also on the output y := [q ξ ]; for the case of flexible-joint robots, this is tantamount to assuming that the actuator's velocities are measurable.…”

“…In [18] is presented an output-feedback control method that applies to Lagrangian systems augmented by a chain of integrators (see Eq. (1) below) hence, underactuated systems.…”

Distributed Global Output-Feedback Control for a Class of Euler–Lagrange Systems

“…Future research directions may include distributed global output-feedback control for a class of underactuated Euler-Lagrange systems, such as nonholonomic wheeled mobile robots studied in [23]. We are also interested in distributed observerless global outputfeedback control for Euler-Lagrange systems with parameter uncertainties reported recently in [34].…”

Finite‐time sliding mode stabilization using dirty differentiation and disturbance compensation

“…Since the resulting estimation error might be of large amplitude, the stability of the closed‐loop system is hardly possible when the output derivative estimate is straightforwardly applied to the controlled system input. Recently, the so‐called dirty‐derivatives filters (DDF) were shown to be tolerant while being used in standard proportional‐derivative controllers, asymptotically stabilizing Lagrangian systems. More specifically, a Lagrangian system was shown to asymptotically be stabilized by a standard stabilizing proportional‐derivative controller fed by the DDF output rather than the real output derivative provided that the filter parameters met a special tuning rule.…”