On the Linear Control of Underactuated Nonlinear Systems Via Tangent Flatness and Active Disturbance Rejection Control: The Case of the Ball and Beam System

Abstract

In this paper, a systematic procedure for controller design is proposed for a class of nonlinear underactuated systems (UAS), which are non-feedback linearizable but exhibit a controllable (flat) tangent linearization around an equilibrium point. Linear extended state observer (LESO)-based active disturbance rejection control (ADRC) is shown to allow for trajectory tracking tasks involving significantly far excursions from the equilibrium point. This is due to local approximate estimation and compensation of the nonlinearities neglected by the linearization process. The approach is typically robust with respect to other endogenous and exogenous uncertainties and disturbances. The flatness of the tangent model provides a unique structural property that results in an advantageous low-order cascade decomposition of the LESO design, vastly improving the attenuation of noisy and peaking components found in the traditional full order, high gain, observer design. The popular ball and beam system (BBS) is taken as an application example. Experimental results show the effectiveness of the proposed approach in stabilization, as well as in perturbed trajectory tracking tasks.

Flatness-Based Active Disturbance Rejection Control for a PVTOL Aircraft System with an Inverted Pendular Load

“…The term ξ α represents the total uniformly absolutely bounded disturbance due to external disturbances and internal perturbations, i.e., ξ α < δ 1 ξ. A necessary and sufficient condition for having the observation error O α ultimately, uniformly, convergent toward a sufficiently small neighborhood of the acceleration estimation error phase space [3,33], consists in choosing the observer control parameters, λ jα j = 1, 2, 3, 4, such that the characteristic polynomial associated with the linear dominant dynamics is Hurwitz, making the linear injection error dynamics stable. The controller is thus given by…”

“…According to [33,37], a set of coupled, high-gain, extended linear Luenberger observers, for the simultaneous estimation of the phase variables associated with the flat output and the time-polynomial approximation variable, can be proposed as follows:…”

On the tracking of fast trajectories of a 3DOF torsional plant: A flatness based ADRC approach

“…In order to overcome the inconvenience of computing the high order time derivatives, the differential flatness approach is a control alternative for simple input simple output (SISO) systems. This methodology has been applied to solve the trajectory tracking problem of four order nonlinear underactuated systems (see [35,36]), where the nonlinear system is linearized around an equilibrium point. Therefore, from the linearized system, the flat output is often easily found by inspection.…”

“…A single observer (with poor noise rejection features) is replaced by its suitable decomposition into three observers of reduced order. These are not only easier‐to‐tune, with better transient response, but also with better noise filtering features [35]). The first block is controlled by the torque input, , with block output represented by .…”

“…In particular, it is shown that the flatness property of the plant enables the calculation of the even order time derivatives of at the output [35]. This means that only the odd order time derivatives of the output need to be estimated through a reduced order observer.…”