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This paper proposes an application of linear flatness control along with active disturbance rejection control (ADRC) for the local stabilization and trajectory tracking problems in the underactuated ball and rigid triangle system. To this end, an observer-based linear controller of the ADRC type is designed based on the flat tangent linearization of the system around its corresponding unstable equilibrium rest position. It was accomplished through two decoupled linear extended observers and a single linear output feedback controller, with disturbance cancelation features. The controller guarantees locally exponentially asymptotic stability for the stabilization problem and practical local stability in the solution of the tracking error. An advantage of combining the flatness and the ADRC methods is that it possible to perform online estimates and cancels the undesirable effects of the higher-order nonlinearities discarded by the linearization approximation. Simulation indicates that the proposed controller behaves remarkably well, having an acceptable domain of attraction.

In the last 3 decades, there has been increasing interest in the control of underactuated mechanical systems. These systems are characterized by having fewer actuators than degrees of freedom [

The ball and rigid triangle system.

Fortunately, the linearized tangent model of this system is locally controllable around the unstable equilibrium point, meaning that it is locally flat. Hence, the stabilization and tracking problems both can be solved locally, from a combined perspective of flatness approach and active disturbance rejection control (ADRC). To accomplish this, we can propose two decoupled extended linear observers, assuming that only the position variables of this system are available. These observers allow us to simultaneously estimate the time derivatives of the nonavailable flat output and recover the uncertain underlying nonlinear dynamics. Subsequently, these estimations, together with the ADRC approach, allow us to propose a control scheme to solve the aforementioned control problems. The main difference between flatness controllers [

In this article, an ADRC scheme with a flatness-based approach is proposed to practically solve the output feedback stabilization and the output feedback trajectory tracking problem for the BRT system. Our control approach assumes a lack of knowledge of the system parameters, the nonlinearities, and exogenous disturbance signals. The scheme not only estimates the unknown dynamics and the unknown state variables, but also reduces the tracking control problem to that defined on a chain of integrators after online active disturbance cancelations. Finally, the control algorithm is tested in several numerical simulations, showing excellent results for the output stabilization problem and output tracking problem. It is worth mentioning that the basin idea of this methodology is sustained by the use of high-gain observers, which are used as identifiers of the uncertain dynamics, assuming that in some operational region this uncertainty can be considered as being smooth and bounded. This kind of methodology has been used in [

The rest of this study is organized as follows. In Section

Evidently, the system above admits the following Euler-Lagrange representation:

Consider the tangent linearization of system (

The state-dependent input-coordinate transformation,

Recall that the time derivatives

In this section, we solve once again the previously mentioned control problem, assuming that only the position is available for measurements and that some dynamics of the original system are unknown. Before developing our control strategy, we give a summary on the ADRC subject.

Let us consider the uncertain plant of

Consider system (

then

From the linearized control model (

Matrix

From the discussion above, the following can be concluded.

Selecting, for instance,

From Facts

In order to avoid the large initial peaking phenomena found in the response of observer variables, we suggest using a clutch function to smooth these transient peaking responses in all the observer variables used in the controller. The “clutch” is defined as a time function smoothly increasing from 0 to 1, during a small time interval

As already mentioned, an advantage of combining the flatness and the ADRC methods is that it allows online estimates and cancels the undesirable effects of the higher-order nonlinearities discarded by the linearization approximation. Because our result is based on this advantage, it is important to provide some arguments to validate it. The ADRC is based on restriction to (1) flat systems and their corresponding input-to-flat outputs dynamics (multivariable or monovariable) and (2) a brute force exact linearization of the underlying input output dynamics (monovariable) or the set of statically or dynamically decoupled nonlinear dynamics, written in Isidori’s canonical form. This is achieved by online estimating and online feedback cancelling of absolutely everything that perturbs the forced linear dynamics from its desired nominal behaviour (trajectory tracking or stabilization). This uncertainty cancelling includes poorly known additive expressions containing state-dependent nonlinearities and the unpredictable effects of unmodeled dynamics and of external unknown disturbances. Very many examples and applications regarding the effectiveness of this technique as well as the theoretical results backing the methodology actually constitute sufficient proof of the assertion made in Abstract of the paper. References [

Closed loop response of the ADRC for two different initial conditions:

Figure

A window time, from 5 to 30 seconds, of the tracking trajectory error when the reference is a sinusoidal signal.

Finally, Figure

Phase space of the position and angular variables.

The differential flatness approach, in conjunction with ADRC, allows systematic solutions to a number of interesting nonlinear control problems. In this instance, we have exploited the local flatness property for the efficient stabilization and tracking of the underactuated BRT. It is important to emphasize that this system is not feedback-linearizable and that its relative degree is not well defined. Moreover, as far as we know, the stabilization of the BRT remained unsolved by using either the shaping energy approach or the IDA-PBC method. Nevertheless, this problem can be partially solved using linear control theory. The fact that the tangent linearization of this system, around the unstable equilibrium point, is locally controllable implies that the system is also locally flat. This allows us to use the robust ADRC in the efficient online estimation of the locally neglected nonlinearities and their active feedback cancelation. The solution, which is quite robust with respect to unmodeled disturbances and neglected nonlinearities, is, in fact, a linear controller with an online compensator. It is based on a set of decoupled linear extended observers and a single linear output feedback controller, with disturbance cancelation features. The proposed controller guarantees locally exponentially asymptotic stability for the stabilization problem and practical and local stability in the solution of the tracking error. To assess the effectiveness of the proposed methodology, numerical simulations were carried out. From the simulation results, we can demonstrate that the proposed controller behaves remarkably well, having an acceptable domain of attraction.

The following proof is based on the previous works [

Under Assumptions (A2) and (A3), the matrix

Besides, for

Resuming the stability proof, because

(

(

The authors declare that they have no competing interests.

This research was supported by the Centro de Investigación en Computación of the Instituto Politécnico Nacional (CIC-IPN) and by the Secretaría de Investigación y Posgrado of the Instituto Politécnico Nacional (SIP-IPN), under Research Grants 20160268 and 20161637. This research was done while Dr. Carlos Aguilar-Ibanez was on sabbatical leave from the Departamento de Mecatrónica del CINVESTAV.