Adaptive sliding mode control for uncertain Euler–Lagrange systems with input saturation

https://doi.org/10.1016/j.jfranklin.2021.08.027Get rights and content

Abstract

In this paper, the tracking control problem of uncertain Euler–Lagrange systems under control input saturation is studied. To handle system uncertainties, a leakage-type (LT) adaptive law is introduced to update the control gains to approach the disturbance variations without knowing the uncertainty upper bound a priori. In addition, an auxiliary dynamics is designed to deal with the saturation nonlinearity by introducing the auxiliary variables in the controller design. Lyapunov analysis verifies that based on the proposed method, the tracking error will be asymptotically bounded by a neighborhood around the origin. To demonstrate the proposed method, simulations are finally carried out on a two-link robot manipulator. Simulation results show that in the presence of actuator saturation, the proposed method induces less chattering signal in the control input compared to conventional sliding mode controllers.

Introduction

In many mechanical systems, the actuator often cannot generate a required force level due to its physical limitations [1], [2].The actuator saturation may deteriorate the control performance or even make the control system be unstable. In recent years, many attentions have been paid on the research about the control input saturation for uncertain systems in the literatures [3], [4], [5], [6], [7], [8]. In [2], robust adaptive variable structure output feedback control (RAVSOFC) is designed for a saturated linear uncertain system. However, this method is only valid for linear time-invariant systems. In addition, the adaptive gain cannot decrease, which may cause overly-large control consumptions. In [9], sliding mode control (SMC) for saturated linear systems is proposed by introducing an adaptive piece-wise scheme. The weakness of this method lies in that the size of the dead zone must be larger than the amplitude of the sliding variable or it will cause persistent increasing of the control gain. Simultaneously, a too large dead zone size leads to a poor control precision. All the physical dynamic terms are treated as uncertainty which is handled all together by an adaptive algorithm in Chen et al. [7]. An adaptive integral sliding mode control scheme is developed in Guo et al. [8] but the inherent constant gain in the switching control component may cause unnecessary chattering. An auxiliary system is employed in the backstepping control design for a saturated unmanned surface vessel. Nevertheless, conservative assumption on the system’s initial tracking error is required because of the inherent barrier Lyapunov function [10]. In recent years, dynamic surface control, neural control, fuzzy control, backstepping control, etc. have also been investigated for the control of saturated systems [11], [12], [13], [14]. Euler–Lagrange system is a second-order differential equation that presents a large class of mechanical systems [15], [16]. A mechanical system, such as gear system, industrial manipulator, wheeled mobile robot, etc., is often required to be modeled as an Euler–Lagrange dynamic model before control design [17]. However, most of the above approaches are only proposed for a dedicated real system and are not valid for a general-form Euler–Lagrange system especially that under saturation [18], [19], [20].

SMC has been investigated in various different mechanical systems in recent years [21], [22], [23], [24], [25], [26]. Due to the sliding mode invariance, SMC is well known for its strong robustness against system parametric variations and external disturbances [27], [28]. In conventional SMC, the disturbance upper bound should be known before the controller design. To ensure the system stability under disturbances, a conservative control gain of the discontinuous control input is often selected, which as a result causes severe system chattering [29]. Adaptive sliding mode (ASM) control removes the need for the disturbance upper bound but one main problem is that in most of the current methods, the adaptive gain cannot decrease when the disturbance becomes small [30], [31], [32]. That means overestimation still exists when the disturbance amplitude drops. To alleviate this phenomenon, in Roy et al. [33], a first-order differential equation-type adaptive law is proposed for ASM which can make the control gain fall down when the disturbance becomes small. By combining with a dual layer nested adaptive law provided in Edwards and Shtessel [34], a similar structure is installed in a disturbance observer to measure the disturbance derivative upper bound in Shao et al. [35]. In [36], an adaptive law with a leakage term is introduced for the adaptive robust control of constrained mechanical systems. In fact, the aforementioned adaptive laws are of a special form of leakage-type (LT) adaptive law [37]. The LT adaptive law is effective for cancelling overestimation and is of a simple first-order form which is convenient for real implementation [38].

In this paper, we propose a general form of adaptive control solution for uncertain Euler–Lagrange systems with input saturation. To thoroughly remove overestimation, a leakage-type adaptive law is introduced for the adaptive sliding mode control of the system. The main contribution of this paper lies in that due to the leakage-type adaptation, not only the requirement for disturbance upper bound is removed, but also the adaptive parameter can approximate the disturbance variations such that overestimation is globally removed. Therefore, the chattering signal in the control input can be effectively suppressed. In addition, to cancel the saturation nonlinearity, an auxiliary dynamics is constructed, based on which the impacts of actuator saturation are compensated by a certain control input component in the proposed SMC. Compared to common researches, this paper investigates the control of the general-form saturated Euler–Lagrange systems rather than any dedicated real systems as those have been discussed in many existing literatures [19], [20]. To achieve this goal, the unique properties of Euler–Lagrange systems [39], such as the symmetric positive definiteness of inertia matrix, etc., are fully considered in the control design and subsequently in the stability analysis. Lyapunov analysis demonstrates that under the proposed control, the closed-loop system can asymptotically converge to a small neighborhood around zero in the presence of actuator saturation and system uncertainties. For illustration of the proposed leakage-type adaptive sliding mode (LTASM) control, numerical simulation is finally carried out on a two-link robot manipulator benchmark model and compared to the conventional sliding mode controllers developed in Guo et al. [8] and Wang et al. [9]. Simulation results show that compared to the one-way increasing adaptive gain in ASM control which subsequently generates rising chattering, and the constant control gain in integral sliding function-based adaptive sliding mode (IASM) control, the proposed method effectively suppresses the chattering owing to the LT algorithm.

This paper is organized as follows: Section 2 gives the system model and some essential mathematical fundamentals. The controller design and its stability analysis will be discussed in detail in Section 3. A simulation example is given in Section 4 for illustration of the proposed method and Section 5 concludes this paper.

Section snippets

Problem formulation

Generally, a mechanical system can be formulated as a second-order Euler–Lagrange equation via Newton–Euler or Lagrange dynamic modeling approach. An Euler–Lagrange system with actuator saturation can be generally described by Sun et al. [15], Roy et al. [39]:M(q(t))q¨(t)+C(q(t),q˙(t))q˙+G(q(t))=sat(u(t))+F(q(t),q˙(t))+d(q(t),q˙(t),q¨(t),t)where q, q˙, and q¨n denote the displacement, velocity, and acceleration vectors, respectively; M(q)n×n is the inertia matrix; C(q,q˙)n×n denotes the

Controller design

To construct the controller, auxiliary dynamics is first constructed to consider the impacts of control input saturation, which is subsequently cancelled by the proposed sliding mode control scheme. A leakage-type (LT) adaptive law is also introduced to deal with the unknown system uncertainties. Lyapunov analysis is finally used to demonstrate the control stability of the system under actuator saturation and system uncertainties.

Simulation example

To illustrate the proposed LTASM controller, a two-link robot manipulator benchmark model is used for simulation [46]. As shown in Fig. 1, based on Lagrangian mechanism, the dynamic model of the manipulator is obtained given by[a11(q2)a12(q2)a12(q2)a22]Mq¨+[β12(q2)q˙2β12(q2)(q˙1+q˙2)β12(q2)q˙10]Cq˙+[γ1(q1,q2)gγ2(q1,q2)g]G=sat(u)+dwhere q=[q1q2], sat(u)=[sat(u1)sat(u2)] is the control input, and d=[d1d2] is the external force disturbance.a11(q2)=(m1+m2)r12+m2r22+2m2r1r2cos(q2)+J1a12(q2)=m

Conclusions

This paper proposed an adaptive control scheme for general uncertain Euler–Lagrange systems under control input saturation. An adaptive sliding mode controller is proposed by introducing a novel leakage-type adaptive law by which the disturbance upper bound is not required a priori. To deal with the effects of actuator saturation, auxiliary dynamics is constructed to provide auxiliary variables for the controller design. The stability analysis verifies that under the proposed control, the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. U1813216 and 62003186), the Natural Science Foundation of Guangdong Province (No. 2020A1515010334).

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