Adaptive neural dynamic surface control of mechanical systems using integral terminal sliding mode
Introduction
Motivation. Being rooted in the integrator backstepping [1], dynamic surface control (DSC) is a systematic design method for controlling uncertain nonlinear systems [2], [3]. Similar to backstepping, DSC tackles a tracking control problem by introducing a set of coordinates that combine the tracking errors with auxiliary control variables. However, unlike backstepping, DSC uses low-pass-filters to avoid direct differentiation of these variables. A number of papers have shown that augmenting the DSC error surfaces with suitable integral terms improves steady-state tracking and robustness [4], [5], [6], [7]. The motivation for the present paper stems from [7] where the robustness of the DSC is enhanced significantly by adding nonlinear proportional-integral (PI)-type terms. The generalization with respect to [7] is twofold: (i) we consider a general n-degrees-of-freedom (DOF) mechanical system conforming to Euler–Lagrange equations rather than a 3-DOF gyroscope; (ii) instead of a quasi-sliding mode term defined by hyperbolic tangent functions, we consider a more general structure based on terminal sliding mode (TSM) theory.
Related literature. Based on the variational principles of mechanics, the Euler–Lagrange equations provide an effective mathematical framework for the analysis of many-body, many-DOF systems [8], [9]. In this regard, the control problems of mechanical systems have been extensively studied in the literature, in the context of either Euler–Lagrange systems [9] or robotics [10], [11]. Hence, numerous control methods are proposed for the tracking problem of fully-actuated mechanical systems. The majority of the reported controllers rely upon feedback linearization and/or proportional-derivative (PD)-type control inspired by the celebrated Slotine and Li controller [9], [10], [11], [12]. Using differential geometric methods, Bullo and Murray [13] studied the problem in a coordinate-free manner. In [3, chapter 6], a convex optimization-based dynamic surface controller was proposed and its disturbance attenuation properties were investigated. By applying the tools of convex analysis, Miranda-Villatoro et al. [14] proposed a class of set-valued controllers for the robust tracking control of Euler–Lagrange systems. For constant desired positions, Cruz-Zavala et al. [15] presented a finite-time controller that modifies the intrinsic energy functions of an Euler–Lagrange system.
Conventional sliding mode (SM) theory is based on the stabilization of a linear switching surface that reduces the order of the system dynamics and induces asymptotic stability of the system trajectories [16]. Sliding mode control (SMC) have been used to achieve strong robustness against a wide variety of uncertainties such as deterministic/stochastic disturbances, unknown nonlinear dynamics, and time-varying additive/multiplicative faults in the actuators and/or sensors [16], [17], [18]. TSM improves upon the conventional SM by incorporating nonlinear switching surfaces that result in the finite-time stability of the sliding motion [19], [20], [21]. Different variations of TSM have been used for robust, finite-time, high-precision control of dynamic systems in various applications [19], [22]. In the area of robotics and mechanical systems, the merits of the TSM have attracted the attention of researchers. Nonsingular and continuous forms of TSM have been used for robust and finite-time control of rigid manipulators [23], [24]. For fully-actuated mechanical systems, some papers have combined TSM with fractional derivatives/integrals to obtain a better convergence speed [25], [26].
Inspired by the biological neurons, artificial neural networks (ANNs) – neural networks (NNs) in short – are powerful tools for data clustering, pattern classification, optimization, and universal function approximation [27]. The latter is of particular interest for control community as there exists a vast literature on the adaptive NN approximation-based control of nonlinear systems, for example, see [28], [29], [30]. Particularly, the approximation capabilities of NNs provides an effective framework for adaptive control of uncertain mechanical and robotic systems [28]. Sun et al. [31] used Gaussian radial basis functions (RBFs) to design an adaptive SM controller for robot manipulators; the controller is able to estimate the unmeasured joint velocities. Tran and Kang [32] combined radial basis function neural networks (RBFNNs) approximation with TSM for adaptive finite-time control of robot manipulators. Zhou et al. [33] presented a backstepping-based, adaptive RBFNN control for robot manipulators subject to dead-zone nonlinearity in their control inputs. Liu et al. [34] designed an adaptive NN controller, with the optimal dimension of the hidden NN layer, for robot control in task space.
Contribution. The main contribution of this paper is to present a new tracking controller for mechanical systems by combining the merits of DSC, TSM, and RBFNNs. The novelties and distinctive features of the proposed control method, in comparison with the existing ones, are as follows:
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The recursive design structure of DSC, equipped with integral action, enables us to induce a quasi-TSM at the kinematic level of the tracking control. Moreover, the nonlinear integral TSM surface adjusts the convergence rate of the tracking error according to its distance from the origin.
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Instead of signum functions, we use hyperbolic tangent functions to produce the quasi-TSM. Thereby, the TSM error surface is sufficiently differentiable for DSC design purposes, and the chattering issue in the obtained control law is considerably alleviated.
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We tackle the issue of robustness at two levels. First, at the kinematic level, the proposed TSM grants a certain level of robustness to the position tracking error. Second, at the dynamic level, we employ nonlinear damping [1] in the DSC design. At this level, we also use an RBFNN to adaptively estimate the uncertainty upper-bound and to prevent conservative, high-gain control inputs. We note that, from the function approximation perspective, estimation of the upper-bound of the norm of a vector function demands less computational burden than estimating the function itself.
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In the RBFNN, we use raised-cosine basis functions in lieu of the commonly used Gaussian functions. The raised-cosine functions, unlike Gaussians, enjoy the property of having compact supports, resulting in the computational efficiency [35], [36]. Besides, the recent paper [37] has shown that RBFs with compact supports can improve adaptive NNs’ learning performance by enhancing the level of excitation of their regressors.
Paper organization. The organization of the remainder of this paper is as follows. Section 2 summarizes notation and required mathematical preliminaries. Section 3 states the control problem. Section 4 presents the main results on the integral DSC design. Section 5 contains further results about the TSM-based convergence of the tracking error. Section 6 elaborates a simulation example of controlling a two-link robot arm. Lastly, Section 7 concludes the paper.
Section snippets
Notation
Throughout this paper, denotes the set of nonnegative real numbers. We consider the Euclidean n-space, equipped with its standard topology. ‖.‖ stands for the vector 2-norm. returns a vector stacked by the given arguments. tr(.) gives the trace of a matrix (.). and return the minimum and the maximum eigenvalues of a real symmetric matrix (.), respectively.
Mathematical preliminaries
Here, we review some important inequalities that we will use for the convergence analysis of TSM. Lemma 1 Let
System dynamics and formulation of the control problem
Consider an n-DOF mechanical system whose configuration space is parameterized by the generalized coordinates, . Through Euler–Lagrange equations approach, the dynamics of the mechanical system is governed by the following differential equation [10]:where is the vector of generalized coordinates, is the vector of generalized velocities, is the vector of generalized accelerations,
Integral dynamic surface control
To obtain a suitable tracking error dynamics, using the DSC formulation, we introduce the following tracking errors:
The vector is a direct measure of the tracking performance, is an auxiliary control vector, is the low-pass-filtered version of φ, is the filtering error, and is a positive definite matrix. The intermediary tracking error is defined to enable a recursive control design.
To improve the robustness of
Tracking error convergence
In Section 4, we designed a dynamic surface controller and established its stability. Now, we further investigate the convergence of the tracking error using the continuous TSM. According to (25a), the dynamics of the tracking error is governed bywhereConsider the geometric setting of Theorem 1 and define the compact setas the projection of Ωe into the e1-space. Theorem 2 Consider the mathematical setting of
System description
To illustrate the efficacy of the proposed control method, we consider the two-link robot arm depicted in Fig. 1 [14]. The configuration space of the robot is parameterized by the joint angles q1 and q2. Through Euler–Lagrange approach, the robot dynamics is obtained as where
Conclusion
We combined the merits of DSC, TSM, and NN function approximation to present a novel control method for mechanical systems. An integral TSM term is introduced into the conventional DSC design structure. We used RBFNNs with compact supports to adaptively adjust the control gain according to the current magnitude of the disturbances. Comparative simulations showed that the integral DSC outperforms its conventional counterpart in terms of error convergence and robustness. Extending the integral
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.
Acknowledgment
This research was supported by the University of Tabriz’s plan to increasing research effectiveness under the contract no. s/858.
Jafar Keighobadi received the Ph.D. degrees in Mechanical Engineering and Control Systems from Department of Mechanical Engineering, Amirkabir University of Technology, Iran, in 2008. He is currently an Associate Professor of Mechanical Engineering Department at the University of Tabriz. His research interests include artificial intelligence, estimation and identification, nonlinear robust control, and GNC.
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Jafar Keighobadi received the Ph.D. degrees in Mechanical Engineering and Control Systems from Department of Mechanical Engineering, Amirkabir University of Technology, Iran, in 2008. He is currently an Associate Professor of Mechanical Engineering Department at the University of Tabriz. His research interests include artificial intelligence, estimation and identification, nonlinear robust control, and GNC.
Mehran Hosseini-Pishrobat received his B.Sc. and M.Sc. degrees, both in Mechanical Engineering, from the University of Tabriz, Iran, in 2013 and 2016, respectively. Since 2016, he has been a research assistant at the Faculty of Mechanical Engineering, the University of Tabriz. His research interests include nonlinear control, disturbance estimation-based control, and nonlinear observers.
Javad Faraji received his M.Sc. degree in Mechatronic Engineering from the University of Tabriz, Iran. Since 2014, he has been working toward the Ph.D. degree with the Department of Mechanical Engineering, the University of Tabriz. His research interests include integrated navigation systems, estimation and optimization theory, inertial and non-inertial sensors calibration.