Output feedback trajectory stabilization of the uncertainty DC servomechanism system

Abstract

This work proposes a solution for the output feedback trajectory-tracking problem in the case of an uncertain DC servomechanism system. The system consists of a pendulum actuated by a DC motor and subject to a time-varying bounded disturbance. The control law consists of a Proportional Derivative controller and an uncertain estimator that allows compensating the effects of the unknown bounded perturbation. Because the motor velocity state is not available from measurements, a second-order sliding-mode observer permits the estimation of this variable in finite time. This last feature allows applying the Separation Principle. The convergence analysis is carried out by means of the Lyapunov method. Results obtained from numerical simulations and experiments in a laboratory prototype show the performance of the closed loop system.

Introduction

The direct current motor-pendulum system (DCMP) is widely used as a test bed for assessing the effectiveness of several control techniques. This choice is due to the fact that its model captures some of the features found in more complex systems, as in the case of industrial robot manipulators [1], [2]. Related to this topic we mention some interesting works; for instance, Ref. [3] applies the well known Generalized Proportional–Integral controller to the tracking control problem for a linear DCMP. In [4] the authors solve the regulation problem using the sliding-mode super-twisting based observer (STBO) method, in conjunction with a twisting controller. An interesting work dealing with the control of the DCMP system using the STBO observer, combined with an identification scheme can be found in [5, Chapter 2]. A close related work is developed by Davila et al. [6]. A closed-loop input error approach for on-line estimation of a continuous-time model of the DCMP was developed in [7]; while in [8] a parameter identification methodology based on the discrete-time Least Squares algorithm and a parameterization using the Operational Calculus is proposed. In [9] an adaptive neural output feedback controller design is used to solve the tracking problem of the system studied here, having the advantage of including the model of the actuator. Ref. [10] employs $H_{\infty }$ techniques to deal with uncertainties; performance of the proposed approach is evaluated through numerical simulations. Another interesting work [11] reports the application of second order sliding mode control applied to an uncertain DC motor; the motor under control receives disturbance torques produced by another motor directly coupled to its shaft of the first motor. A smooth hyperbolic switching function eliminates chattering phenomena. The Hoekens mechanical system is the subject of research in [12]; here the authors apply a sliding mode control technique with uncertainty estimation combined with a learning technique. A key feature of this approach is the fact that it applies the switching to the plant indirectly through the learning process and the disturbance estimator thus reducing the occurrence of chattering; experiments validate the findings.

In this context, perhaps one of the most challenging control problems consists in designing a smooth output-feedback stabilization algorithm for an uncertain and perturbed DCMP [13], [14]. Generally speaking, this problem is by no means easy because it is not possible to fully compensate the effects of the system uncertainty without having information about the time derivative of this uncertainty [15]. This control problem has been solved using a combination of neuronal networks and adaptive control theory, as in [9], [16]. On the other hand, the problem is easily solved using a sliding mode controller (SMC), a methodology ensuring robustness against disturbances and parameter variations. However, its main drawback is that it may generate high frequency violent control signals, a behavior known as chattering. Moreover, the presence of chattering may excite unmodeled high-frequency dynamics, resulting in unforeseen instability and damage to the actuators. [17], [18], [19]. There exists essentially three main approaches to eliminate the chattering effects. The first approach uses continuous approximations, as the saturation function, of the sign function appearing in the sliding mode controllers [1], [20], [21]. This approach trades the robustness on the sliding surface and the system convergences to a small domain [22]. Observed-based approaches are another way of overcoming chattering; it consists in bypassing the plant dynamics by a chattering loop, then reducing the robust control problem to an exact robust estimation problem. However, this action can deteriorate the robustness with respect to the plant uncertainties and disturbances [19], [23], [24]. The last approach, based on the high-order sliding-mode method guaranties convergence to the origin of the sliding variable and its corresponding derivatives; the high-order sliding-mode algorithms translate the discontinuity produced by the sign function to the higher order derivatives, producing continuous control signals; however, these algorithms require a great computing effort [6], [25], [26], [27], [28].

In this work we introduce a smooth controller for output feedback trajectory tracking in an uncertain DCMP. The solution consists of a PD controller and a robust uncertain estimator. A super-twisting second-order sliding-mode observer estimates motor velocity. The observer finite time of convergence ensures that the estimation error will vanish after a finite time transient, the allowing the use of the Separation Principle. The corresponding convergence analysis is carried out using the Lyapunov method. This work continues with Section 2, where the model of the DCMP system and the problem statement are presented. In Section 3 the control strategy and the corresponding convergence analysis are developed. 4 Numerical and experimental results, 5 Conclusions are devoted respectively to the numerical and experimental results and the conclusions.