Second-Order Sliding Mode Research Articles

An Adaptive Backstepping Sliding Mode Controller to Improve Vehicle Maneuverability and Stability via Torque Vectoring Control

To improve the maneuverability and stability of a vehicle and fully leverage the advantages of torque vectoring technology in vehicle dynamics control, a finite-time yaw rate and sideslip angle tracking controller is proposed by combining a second-order sliding mode (SOSM) controller with the backstepping method in this paper. However, existing research indicates that first-order sliding mode (FOSM) control suffers from the chattering problem, while the traditional SOSM controller requires knowing the bound of the uncertain term in advance to obtain the switching gain, which is difficult in practice. To address these problems, this paper proposes an adaptive second-order sliding mode (ASOSM) controller based on the backstepping method by adding the high-frequency switching term to the first derivative of the sliding mode variable, which implies that the actual control can be acquired after an integration process. The switching gain in the ASOSM controller is obtained by an adaptive algorithm without knowing any information of the uncertainty. The proposed algorithm is compared with FOSM and SOSM in different scenarios to demonstrate its applicability and robustness. Simulation results show that the bandwidth of the vehicle transient response can be improved by 21%. In addition, ASOSM and SOSM controllers are insensitive to vehicle mass and tire type, implying their robustness to such disturbances. Furthermore, ASOSM requires less control action because of the adaptive law when it performs similarly with SOSM and FOSM.

Path Following Control of Autonomous Four-Wheel-Independent-Drive Electric Vehicles via Second-Order Sliding Mode and Nonlinear Disturbance Observer Techniques

In this article, we aim to investigate the path following control problem for four-wheel-independent-drive electric vehicles with consideration of modeling errors and complex driving scenarios. In order to guarantee operational safety and robustness, a supertwisting second-order sliding mode (SOSM) control strategy is employed to suppress the heavy chattering issue existing in the traditional sliding mode control (SMC). To degrade the lumped disturbance, a nonlinear disturbance observer (NDOB) technique is proposed to estimate the disturbance and compensate the modeling error. Then, a composite control strategy, which combines the SMC algorithm and NDOB technique is proposed based on stability analysis of the corresponding closed-loop system. Finally, hardware-in-the-loop experimental results show that the supertwisting SOSM control algorithm with the NDOB technique can achieve better tracking performance and suppress the chattering phenomenon.

Torque Vectoring and Rear-Wheel-Steering Control for Vehicle's Uncertain Slips on Soft and Slope Terrain Using Sliding Mode Algorithm

Off-road vehicles always experience serious uncertain longitudinal and lateral slips when running on soft and slope terrains, and some parameters of the vehicles, such as the cornering stiffness and the slip of wheels, are always not constants. In this paper, control strategies for the torque of each wheel and the rear-wheel-steering angle are proposed to maintain a stable velocity and approach an ideal reference model for the off-road vehicle by using second-order sliding mode (SOSM) techniques. An observer is constructed to estimate the actual sideslip angle of the vehicle with the consideration of uncertainties and disturbances of the system. Then, with conditions of bounded uncertainties and disturbances, composite super-twisting (ST) controllers combined with a velocity controller are designed to generate the total torque, the differential torque, and the rear-wheel-steering angle. On this basis, the proposed controllers have been verified to lead good robustness for maintaining the stable velocity and approaching the ideal reference model by using an optimal torque allocation controller at a lower layer. In comparison with conventional yaw moment controllers without the rear-wheel-steering control, the proposed controllers are shown to be more effective.

Implementation of second order sliding mode disturbance observer for a one-link flexible manipulator using Dspace Ds1104

In this paper, a second-order sliding mode (SOSM) disturbance observer will be designed and implemented for a single link manipulator using Ds1104. The target system model is derived via the expression of the Lagrange equations of movement based on an energy approach. The flexible link is a clamped-free Euler–Bernoulli beam, and the assumed modes method is used to approximate the elastic movement. The proposed observer is meant to estimate the single-link flexible manipulator state, and the equivalent output injection method is applied to reconstruct the determinist unknown time varying disturbance. To appraise the performance of the proposed strategy, the observer is implemented using dSPACE DS1104 Controller board Real-Time Interface (RTI) running under MATLAB/Simulink software and ControlDesk 4.2 graphical interface. The illustrated results, either of the simulation or the experimental setting up, show that the proposed strategy provides finite time and robust estimation of single link nonlinear flexible manipulator state. Furthermore, the observer gives with accuracy the reconstruction of external perturbation.

Second-Order Sliding Mode-Based Direct Torque Control of Variable-Flux Memory Machine

The variable-flux memory machine (VFMM) exhibits high efficiency over a wide speed range because its airgap magnetic flux can easily be regulated by varying the magnetization state (MS) of the employed low coercivity force permanent magnets. This paper proposes a second-order sliding mode (SM)-based direct torque control (DTC) strategy for the VFMM, which features a relatively low computational complexity and less dependence on machine parameters. The operating principle and mathematical model of the VFMM are first described and established. On this basis, a novel control scheme combining the DTC strategy with MS manipulation is proposed, in which the machine is controlled by utilizing the DTC strategy incorporating the $i_{\mathrm {d}} = 0$ condition, while the MS manipulation is implemented by energizing reference stator flux linkage pulses. A super-twisting second-order SM controller is subsequently developed to achieve strong robustness. The developed control structure avoids a complicated decoupling algorithm and is simpler than that of the conventional field-oriented control (FOC) method. Finally, the effectiveness of the proposed control scheme is validated by simulation and experimental results.

Higher-Order Super-Twisting Control for Trajectory Tracking Control of Skid-Steered Mobile Robot

In this paper, Higher-Order Super-Twisting control (HOST) is designed and implemented for trajectory tracking control of four wheels Skid-Steered Mobile Robot (SSMR). The conventional Super-Twisting (ST) Second-Order Sliding Mode Control (SOSMC) is robust, yet it has two main drawbacks for such a system, 1) the chattering phenomena persists on the system inputs and outputs which leads to vibration in the robot motors and 2) the SSMR system has a relative degree equal to two. Therefore, the ST control design requires a sliding variable containing error derivatives, which only permit asymptotic convergence. For that, a higher-order controller is required for this control structure, which is designed for second order sliding variable, in order to obtain a robust controller with finite-time convergence. The HOST control can reduce chattering in steady-state compared with conventional ST-SOSMC and converge the state variables in finite time. The performance of this controller has been validated experimentally on Pioneer P3AT SSMR robot. The experimental results show good performance under parametric uncertainty variations and external disturbance with neglected chattering.

Time-Varying Second-Order Sliding Mode Control for Systems Subject to External Disturbance

In this article, the design problem of a class of novel second-order sliding mode controller for systems subject to external disturbance is investigated, where the sliding variable is affected by a time-varying regulation. On the one hand, for the unmatched disturbance bounded by positive functions, a second-order sliding mode controller is proposed to prove the finite-time stability of aforementioned system through the backstepping-like method and virtual control strategy. On the other hand, for the non-vanishing bounded disturbance, a non-singular terminal sliding mode surface containing disturbance observations is designed via a finite-time disturbance observation method and the finite-time stability of the system controlled output is realized. The claimed performances of the proposed methods are validated by two simulation examples.

Adaptive Second Order Sliding Mode Control for the Regulation of Active Magnetic Bearing

The active magnetic bearing (AMB) has been widely used for the corrective actions in the fast rotating machines, which often faces the problem of deviation in the shaft position. The AMB system, with the use of an electromagnetic control force, regulates the displaced rotor and maintains it to the nominal location. This paper investigates the application of a sliding mode based robust control technique for the control design of five degrees of freedom (DOF) AMB system, which is subjected to the model uncertainties and bounded disturbances with unknown bounds. The proposed controller is formulated by combining the adaptive, integral second-order sliding mode control (ISOSMC) techniques. The adaptive laws estimate the gains of the controller without the upper bound information of the lumped disturbance. The use of ISOSMC ensures the robustness of the controller, along with the finite-time convergence of system states. Moreover, the ISOS also significantly reduces the chattering in the control input. The theoretical analysis under the proposed scheme for the AMB system guarantees the finite-time convergence using the Lyapunov and homogeneity theories. At last, numerical analysis is carried out to validate the offered methodology.

Adaptive Second-Order Sliding Mode Control for a Tilting Quadcopter with Input Saturations

Abstract In this paper, we address the problem of tracking control for a novel tilting quadcopter in the presence of uncertainties and disturbance. To handle these effects, an adaptive second-order sliding mode control (ASOSMC) is proposed to obtain fast convergence rate and high precision with alleviating chattering effect. Considering the input saturations, a new scheme is designed toward the time derivative of the control inputs where an auxiliary system is adopted to the controller. Moreover, the overall uncertainties are expressed in a linearly parametric form without any prior knowledge. It can be proved that the sliding mode manifolds can fast converge to a small neighborhood around zero within finite-time, and then tracking errors exponentially. Finally, simulation results are carried out to demonstrate the effectiveness and robustness of the proposed controller.

High-Performance Second-Order Sliding Mode Control for NPC Converters

In this article, a linear extended state observer (LESO) based second-order sliding mode (SOSM) control strategy with the direct power control is proposed for a three-phase neutral-point-clamped (NPC) power converter connected to a dc microgrid. Comparing with the PI control method, the proposed approach implements the advanced SOSM controller into the voltage regulation loop and instantaneous power tracking loop to enhance the dynamic and steady state performance. Furthermore, saturation function is applied in the SOSM method to weaken the chattering phenomenon. On the other hand, since the dc load is regarded as an external disturbance, an efficient LESO is designed in the voltage regulation loop to reject this disturbance. The design process of the proposed control strategy is shown based on the continuous averaged model of the NPC converter. Finally, comparison experiments among PI, LESO-based PI, and proposed LESO-based SOSM control strategies are implemented, which validate the superiority of the proposed approach.

Second-Order Sliding-Mode-Based Synchronization Control of Cable-Driven Parallel Robots

Cable-driven parallel robots (CDPRs) can provide a high speed and a heavy payload in a large workspace. The main challenge of the CDPRs stems from the dynamic control, in which all the cables must coordinate with each other to remain in tension, and the cable-driven form may lead to model uncertainties. To solve it, the second-order sliding mode (SOSM) is combined with the multicable synchronization idea to propose a new SOSM-based synchronization control (SOSM-SC) strategy for the CDPRs. The goal of the cross-coupled control approach is to improve the synchronization motion relationship between all the cables and guarantee better trajectory tracking control, whereas the SOSM can restrain the model uncertainties and external disturbances. The Lyapunov theory is utilized to prove the asymptotic stability of the closed-loop control system of CDPRs. Trajectory tracking experiments indicate that compared with the existing synchronization control strategy and the augmented proportional-derivative strategy, the proposed SOSM-SC strategy significantly reduces the cable tracking error and the cable synchronization error, and ultimately improves the control accuracy for the mobile platform. The tracking experiments of the antidisturbance indicate that the SOSM-SC strategy can retain good tracking accuracy under external disturbances.

Second-Order Sliding Mode Controller Design with a Larger Domain of Attraction

In this paper, by combining a second-order sliding mode (SOSM) algorithm with the saturation technique, a novel SOSM control scheme has been presented. The feature of the proposed SOSM controller lies that there is a saturation function imposed on the sliding variable, which could significantly enlarge the domain of attraction for the closed-loop system. The geometric method has been utilized to prove that all the sliding variables will be steered to the origin in a finite time. Meanwhile, the relation between the control parameters and the shape of the phase trajectory has also been discussed. Finally, the proposed method has been applied to the tracking control problem for a robotic manipulator.

Design a Second Order Sliding Mode Controller for Electrical Servo Drive Systems

The aim of this paper is to design and study a powerful second-order sliding mode controller for electrical servo drive systems. The suggested controller can successfully overcome the chattering problem that was usually facing such systems during operation. The first (1-SMC) and second (2-SMC) sliding mode controllers are nonlinear controllers’ techniques capable of stabilizing the output of a plant, even though a disturbance and parameter uncertainty is present. The asymptotically stable is the significant property of 1-SMC as well as 2-SMC. Despite the robustness of the 1- SMC, in real-time but it suffers from a large settling time and a chattering (undesirable rapid oscillations) of system trajectory close to the sliding surface. The chattering must be reduced because of its negative impact on system stability. The chattering can be reduced by replacing the sign function, used in classical sliding mode, by a saturation function. In the current study, the Second Order Sliding Mode Controller (2-SMC) is used to overcome the drawbacks of 1- SMC by reducing both the chattering and the settling time of the control action. The Electrical Servo drive system was adopted in this paper for testing; both, the 1-SMC as well as the 2-SMC. The comparison of results between the two controllers indicated smaller chattering and settling time in the 2-SMC than that in the 1-SMC. The simulation results of this work were obtained by using the Matlab programming.

Modified adaptive second order sliding mode control: Perturbed system response robustness

Sensitivity to parameters, which is negative for any system; for an aircraft is destructive. In this respect, an Adaptive Second-order Sliding Mode Control (AS-SMC) is proposed to manage the unknown time-varying aircraft parameter uncertainty and un-modeled coupling perturbations. The employed adaptivity relaxes the required knowledge of the perturbation bound, the suggested sliding surface improves the system stability and bypassing the SMC reaching phase enhances the performance robustness. The effectiveness of the scheme in providing low tolerance responses with respect to the basic controller is illustrated through extensive simulations. The response is more appreciated when the system operates in a high angle of attack/sideslip conditions.

Adaptive and Robust Second Order Sliding Mode Controller Dedicated to Nonlinear Uncertain Pendulum Application System

The research work presented within this paper deals with an innovative second-order sliding mode control (SOSMC) allocated to adaptive gain and associated with nonlinear systems subject to unknown but bounded uncertainties. The derived controller guarantees the control gain dynamical adaptation for the sake of counteracting the system’s uncertainties and to mitigate the chattering phenomenon. The Lyapunov method is also used to analyses the stability of any closed loop system (CLS) within a finite-time under bounded uncertainties assumptions. To assess how effective is the approach considered within this paper, the adaptive controller has been carefully studied on a benchmark of nonlinear systems on a damped overturned pendulum.

Adaptive and Robust Second Order Sliding Mode Controller Dedicated to Nonlinear Uncertain Pendulum Application System

The research work presented within this paper deals with an innovative second-order sliding mode control (SOSMC) allocated to adaptive gain and associated with nonlinear systems subject to unknown but bounded uncertainties. The derived controller guarantees the control gain dynamical adaptation for the sake of counteracting the system’s uncertainties and to mitigate the chattering phenomenon. The Lyapunov method is also used to analyses the stability of any closed loop system (CLS) within a finite-time under bounded uncertainties assumptions. To assess how effective is the approach considered within this paper, the adaptive controller has been carefully studied on a benchmark of nonlinear systems on a damped overturned pendulum.

Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine

This paper addresses a second-order sliding mode control method for the formation problem of multirobot systems. The formation patterns are usually symmetrical. This sliding mode control is based on the super-twisting law. In many real-world applications, the robots suffer from a great diversity of uncertainties and disturbances that greatly challenge super-twisting sliding mode formation maneuvers. In particular, such a challenge has adverse effects on the formation performance when the uncertainties and disturbances have an unknown bound. This paper focuses on this issue and utilizes the technique of an extreme learning machine to meet this challenge. Within the leader–follower framework, this paper investigates the integration of the super-twisting sliding mode control method and the extreme learning machine. The output weights of this extreme learning machine are adaptively adjusted so that this integrated formation design has guaranteed closed-loop stability in the sense of Lyaponov. In the end, some simulations are implemented via a multirobot platform, illustrating the superiority and effectiveness of the integrated formation design in spite of uncertainties and disturbances.

Adaptive-Gain Second-Order Sliding Mode Direct Power Control for Wind-Turbine-Driven DFIG under Balanced and Unbalanced Grid Voltage

In a wind turbine system, a doubly-fed induction generator (DFIG), with nonlinear and high-dimensional dynamics, is generally subjected to unbalanced grid voltage and unknown uncertainty. This paper proposes a novel adaptive-gain second-order sliding mode direct power control (AGSOSM-DPC) strategy for a wind-turbine-driven DFIG, valid for both balanced and unbalanced grid voltage. The AGSOSM-DPC control scheme is presented in detail to restrain rotor voltage chattering and deal with the scenario of unknown uncertainty upper bound. Stator current harmonics and electromagnetic torque ripples can be simultaneously restrained without phase-locked loop (PLL) and phase sequence decomposition using new active power expression. Adaptive control gains are deduced based on the Lyapunov stability method. Comparative simulations under three DPC schemes are executed on a 2-MW DFIG under both balanced and unbalanced grid voltage. The proposed strategy achieved active and reactive power regulation under a two-phase stationary reference frame for both balanced and unbalanced grid voltage. An uncertainty upper bound is not needed in advance, and the sliding mode control chattering is greatly restrained. The simulation results verify the effectiveness, robustness, and superiority of the AGSOSM-DPC strategy.

Variable-Gain Second-Order Sliding Mode Controller With Globally Fixed-Time Stability Guarantees

In this brief, the globally fixed-time control problem is investigated for a general class of uncertain nonlinear systems. The novelties of this brief are two folds. One novelty lies in the propose of a novel variable-gain second-order sliding mode (SOSM) controller. It overcomes the principal shortcomings of first-order sliding mode control and makes it possible to achieve fixed-time stability. The conventional SOSM algorithms can only achieve finite-time stability whose settling time will grow unboundedly when the initial conditions tend to infinity. Another novelty is the extension of constant upper bounds to time-varying upper functions. This enables us to propose a global result, while only local results can be obtained in most of the existing literatures. It is indicated that the proposed novel variable-gain SOSM controller can achieve the establishment of SOSMs in a fixed-time independent of initial conditions. In addition, Lyapunov analysis is given to show that the closed-loop system is globally fixed-time stable. An application to variable-length pendulum is given.

HYBRID FUZZY SECOND-ORDER SLIDING MODE CONTROL SPEED FOR DIRECT TORQUE CONTROL OF DUAL STAR INDUCTION MOTOR

This paper presents a Speed Fuzzy Second-Order Sliding Mode Control (FSOSMC) for Direct Torque Controlled Dual Star Induction Motor (DTC-DSIM). A DTC uses the instantaneous values of voltage vector where each reference voltage vector is computed with a DTC algorithm. The sliding mode control is characterized by its robustness against modeling disturbances and uncertainties; however, this technique has a major disadvantage which is the phenomenon of chattering that produces torque fluctuations. In order to overcome this drawback, a new control scheme that uses the FSOSMC for speed control is proposed. . It is shown that the proposed control strategy conserves the main advantages of the DTC in addition to reducing considerably the chattering effect. It is also robust against modeling disturbances and uncertainties. Several simulation tests are performed illustrating the high accuracy of the proposed control scheme.