Backstepping Control Law Research Articles

Feedback deep brain stimulation for rehabilitation in Parkinson’s disease via unknown input observer

In this article, a feedback mechanism is used for Deep Brain Stimulation of Basal Ganglia (BG) in order to control excessive tremor caused by Parkinson’s disease. Due to the nonlinearity in the Basal Ganglia model, there are some limitations in performance of previously used feedback control schemes. We enhance the linear approximation of the model by introducing an unknown input (disturbance) to the model which represents the approximation error. Then we apply an unknown input observer to estimate the system states. The estimated state is used to implement a double loop feedback with back stepping and proportional control laws simultaneously stimulating the Globus pallidus internal and Sub thalamic nucleus sections. Simulations show improvements in estimation and control.

A Neural Network-Based Adaptive Backstepping Control Law With Covariance Resetting for Asymptotic Output Tracking of a CSTR Plant

A robust nonlinear adaptive controller merging a backstepping approach with neural networks is proposed for a nonlinear non-affine model. The work presented here is evaluated on a complex uncertain model of a continuous stirred tank reactor plant including an unknown varying parameter that enters the complexity model. By exploiting NN and adaptive backstepping approximation methods, an equivalent adaptive NN controller is constructed to achieve robust asymptotic output tracking control. The robustness to uncertainties as well as the lack of informative process data is the main enhancement of this work. This is attained through the implementation of the covariance resetting algorithm in the least square estimation of the NN weight tuning algorithm. The proposed novel control algorithm has been analyzed using Lyapunov analysis. In addition to excellent output trajectory tracking performance, the proposed approach has a profound benefit in terms of substantially lower control effort in comparison to the established work in the literature. In terms of applications in the petrochemical industry, lower control effort can translate to a more energy-efficient actuator, leading to lower costs over a long-run operation. The proposed method's feasibility for chemical process control was shown via numerical simulation.

Adaptive Backstepping Attitude Control Law with L2-Gain Performance for Flexible Spacecraft

In this paper, an observer-based adaptive backstepping attitude maneuver controller (briefly, OBABC) for flexible spacecraft is presented. First, an observer is constructed to estimate the flexible modal variables. Based on the proposed observer, a backstepping control law is presented for the case where the inertia matrix is known. Further, an adaptive law is developed to estimate the unknown parameters of the inertia matrix of the flexible spacecraft. By utilizing Lyapunov theory, the proposed OBABC law can guarantee the asymptotical convergence of the closed-loop system in the presence of the external disturbance, incorporating with the L2-gain performance criterion constraint. Simulation results show that the attitude maneuver can be achieved by the proposed observer-based adaptive backstepping attitude control law.

Nonlinear control by backstepping applied to the web winding system

The key system in the reversible cold rolling mill is the web winding system (WWS). The WWS is a complex process characterized by multiple variables with varying parameters in time, a strong coupling and a non-linearity, various operating conditions. For many products, such as thin film tapes, highly precise control of web tension and web velocity to prevent bending or breaking of the web is of paramount importance throughout the manufacturing process. Because of the radius variation and broad range of roller inertia, the dynamic system changes strongly during the winding process. In this work, we present the design of a new version of the nonlinear backstepping control law for the WWS, in order to guarantee that the final product does not undergo deformations or breaks. It is an adaptive control approach of backstepping with introduction of integral actions. Using Lyapunov stability theory, we verify that trajectory dynamics are asymptotically stable. Simulation studies with the MATLAB/SIMULINK environment are presented to exemplify the effectiveness and performances of the proposed controllers.

Fuel-Efficient En Route Speed Planning and Tracking Control of Truck Platoons

This paper investigates the problem of speed planning and tracking control of a platoon of trucks on highways. We present a two-layered hierarchical framework for truck platoon coordination: a speed planning layer for en route speed profile calculation and a control layer for vehicle speed tracking. The speed planning algorithm is derived with regard to an average vehicle based on a combined fuel-time cost and receding dynamic programming. The idea of using an average vehicle instead of the platoon leader for speed planning makes the resulting speed profile more fuel-efficient for platoons comprised of vehicles differing in weight and size. The vehicle controller, a discrete time back-stepping control law, is designed on the basis of a nonlinear vehicle model considering road slope and heterogeneity of the vehicles. What makes the control algorithm more interesting is that it is strengthened by a novel string stability criterion. The effectiveness of the method is demonstrated by both numerical simulations and experiments with laboratory-scale Arduino cars.

DSP in the Loop Implementation of a Backstepping Controller for Wind Energy Conversion System Based on a Doubly Fed Induction Generator Connected to Grid

This paper deals with the design and implementation of the control strategy of a grid connected Doubly Fed Induction Generator wind turbine on a TMS320F28335 Digital Signal Processor (DSP). The synthesized controller is a nonlinear Backstepping control law whose stability is guaranteed in the context of Lyapunov stability theory. The control aim is to optimize the power extracted by the generator for large wind speed variations through Maximum Power Point Tracking (MPPT) and Unity Power Factor (UPF) control. These command strategies are implemented in order to achieve the following specifications: good static precision for optimal energy conversion; electrical dynamics as high as possible regardless of slow mechanical dynamics, smooth transient without overshoots that could damage the lifecycle of the machine and converters, rapid disturbance isolation and robustness to any parametric variations or uncertainties. The results of the Hardware in the Loop (HIL) implementation on DSP show satisfactory performance in terms of response time, setpoint tracking, stability, and robustness under stochastic wind speed profile and parameter variations.

Implementation and validation of backstepping control for PMSG wind turbine using dSPACE controller board

This paper describes the design and implementation in real time of a nonlinear control for a wind energy conversion system (WECS). The Backstepping control has been implemented to improve the performance of the conversion system based on a permanent magnet synchronous generator (PMSG) connected to the grid. Two static back-to-back converters ensure grid connection and are controlled by Pulse Width Modulation (PWM). The proposed control algorithm ensures proper speed control to extract maximum power. First of all, a WECS full review has been discussed. Thereafter, a Backstepping control laws detailed description based on the Lyapunov stability technique has been reported. Consequently, these control thus helping it possible to operate the complete system in the best performances in the static and dynamic regimes. The second part of this article has been devoted to the Backstepping control experimental validation using the dSPACE DS1104 control board and the Matlab–Simulink environment in order to check and validate the system efficiency. The results achieved have been clearly responded to the requirements of robustness and follow-up of references even under fluctuating wind conditions, and confirmed the control effectiveness in both static and dynamic operating modes.

Backstepping-based control methodology for aircraft roll dynamics

This article investigates a backstepping-based control method for aircraft roll dynamics. The research starts with a formulation of backstepping control law for a general class of a strict-feedback form of nonlinear dynamic systems. The backstepping control law is formulated by introducing a normal tracking error. Then, control and virtual control inputs are selected by addressing each layer of the design process with a chosen corresponding control Lyapunov function. The parameter assignment in each design layer is selected to ensure the stability of the entire system. Next, a backstepping-based control algorithm with online-gain schedule or variable gains is provided for the standard strict-feedback system. In order to validate the proposed method, application of roll dynamics of aircraft is implemented. Dynamic equations of free-to-roll aircraft model is restructured in a standard strict-feedback model for formulating the backstepping control. Then, a backstepping control–based control strategy is provided for aircraft free-to-roll dynamics. Indoor experimental and simulation studies of roll angle control for the L-59 free-to-roll aircraft model at NASA Langley Research Center are implemented to verify and validate the proposed approach.

Dynamic infinity‐norm constrained control allocation for attitude tracking control of overactuated combined spacecraft

In this study, a dynamic L ∞ (infinity-norm) constrained control allocation scheme is designed for attitude tracking control of combined spacecraft with inertia uncertainties and external disturbances. A disturbance-observer-based constrained backstepping control law is developed to generate the control command signals considering non-symmetric constraints on control input, where the lumped disturbance containing inertia uncertainties and external disturbances is compensated by the output of stable non-linear disturbance observer. The control scheme can guarantee that the attitude and angular velocity tracking error converge to small neighbourhood of zero by appropriately tuning the control parameters. With the consideration of physical amplitude and rate constraints on actuators, the dynamic L ∞ constrained control allocation problem is solved by linear programming technique. Numerical examples demonstrate the effectiveness of the proposed disturbance-observer-based constrained backstepping control method and the dynamic L ∞ constrained control allocation algorithm.

Robust Adaptive Neural-Network Backstepping Control Design for High-Speed Permanent-Magnet Synchronous Motor Drives: Theory and Experiments

This paper presents a robust adaptive backstepping control (RABC) for high-speed permanent-magnet synchronous motor (HSPMSM) drive system. The proposed RABC achieves high performance operation by incorporating an ideal backstepping controller (IBC), a recurrent radial basis function neural network (RRBFNN) uncertainty observer, and a robust controller. The Lyapunov stability theorem is utilized to design the IBC as a position controller of the HSPMSM servo drive system. To enhance the disturbance rejection capability during parameter changes, certain information is needed within the backstepping control law so that the system performance would not sorely be affected. To mitigate the need for the lumped parameter uncertainties within the backstepping controller, an online adaptive observer based on RRBFNN is designed to estimate the nonlinear parameter uncertainties. Moreover, the robust controller is intended to retrieve the remaining of the RRBFNN approximation errors. To assure the stability of the proposed RABC, the Lyapunov stability analysis is used to derive the online adaptive control laws. The performance of the proposed RABC is verified by simulation and experimental analysis under different operating conditions and parameter uncertainties. The test results validate the effectiveness of the proposed RABC scheme to achieve preferable tracking performance regardless of external disturbances and parameter uncertainties.

Dynamic surface method–based adaptive backstepping control for the permanent magnet synchronous motor on parameter identification

Aiming at the problem of rotor inertia uncertainty and load torque disturbance of the permanent magnet synchronous motor control system, this article proposes an adaptive backstepping speed control strategy for the permanent magnet synchronous motor based on parameter identification. First, an online identification method of rotor inertia based on disturbance observer is designed. Furthermore, to solve the problem of the external load torque disturbance, combined with the identified rotor inertia, the load torque adaptive law and backstepping control law are designed according to Lyapunov’s stability theorem, in which the derivation of the virtual control law is calculated by dynamic surface control. Finally, simulation and experiment on MATLAB and dSPACE platform are accomplished to show the effectiveness of the proposed method.

Adaptive Backstepping Tracking Control for an over–Actuated DP Marine Vessel with Inertia Uncertainties

Abstract Designing a tracking control system for an over-actuated dynamic positioning marine vessel in the case of insufficient information on environmental disturbances, hydrodynamic damping, Coriolis forces and vessel inertia characteristics is considered. The designed adaptive MIMO backstepping control law with control allocation is based on Lyapunov control theory for cascaded systems to guarantee stabilization of the marine vessel position and heading. Forces and torque computed from the adaptive control law are allocated to individual thrusters by employing the quadratic programming method in combination with the cascaded generalized inverse algorithm, the weighted least squares algorithm and the minimal least squares algorithm. The effectiveness of the proposed control scheme is demonstrated by simulations involving a redundant set of actuators. The evaluation criteria include energy consumption, robustness, as well accuracy of tracking during typical vessel operation.

A fuzzy adaptive backstepping control based on mass observer for trajectory tracking of a quadrotor UAV

SUMMARYIn this paper, a fuzzy adaptive backstepping control law that combines mass observer and fuzzy control strategy with backstepping control is developed for the trajectory tracking control of a quadrotor unmanned aerial vehicle. In the control law, the mass change disturbance to the flight control could be restrained by the mass observer to estimate the real‐time mass and correct the mass parameter of the quadrotor, and the time‐varying external disturbance to the flight control could be restrained by fuzzy compensation of the fuzzy control law. Lyapunov stability theorem is used to analyze the stability of the closed‐loop control system. Two cases of trajectory tracking numerical simulations are presented in MATLAB/SIMULINK simulation environment. In addition, the quadrotor hovering experiment was completed on the actual flight experimental system with the time‐varying disturbance from wind outdoors. The simulation and actual flight results illustrate that the proposed control strategy is effective and stable under the simulating environment and the actual environment.

Robust three-dimensional path-following control for an under-actuated stratospheric airship

This paper proposes a robust three-dimensional (3-D) path-following controller for an under-actuated stratospheric airship in the presence of uncertainties. The resultant control system exhibits an inner-outer loop control structure. In the outer control loop, the path-following error dynamics is constructed in a moving Serret-Frenet frame and a new guidance law with the sideslip and attack angle compensation is designed, which decreases the path-following steady-state error. In the inner control loop, a disturbance observer based backstepping control law is proposed to achieve the desired dynamic behavior on the path. Furthermore, a new velocity tracking control strategy is developed which aligns the resultant velocity tangent to the path. Finally, numerical simulation results are shown to verify the effectiveness of the proposed controller.

Comparative study of attitude control methods based on Euler angles, quaternions, angle–axis pairs and orientation matrices

This paper presents a comparative study about the attitude control methods based on four commonly used error indicators, namely the triad of 3-2-1 deviational Euler angles, the error quaternion, the deviational angle–axis pair and the orientation error matrix. These error indicators are used here with the same backstepping control law to have a common basis of comparison. This control law makes the controller track a restoring angular velocity generated here specifically for each error indicator. This comparative study shows that all these error indicators can be used satisfactorily even in the critical orientations associated with them by taking special measures. For the deviational Euler angle triad, a critical orientation is a singularity, at which the angles become indefinite. Unless the vehicle is stationary, this indefiniteness is resolved here by applying L’Hopital’s Rule on the angular velocity information. The Euler angle triad has also a multiplicity problem. It is solved here by using the novel criterion of minimal deviation angles. For the other error indicators, a critical orientation is an antipodal orientation, which is opposite to the desired one. In an antipodal orientation, the error quaternion and the deviational angle–axis pair cannot be determined through the customary formulas. They are determined here by using the novel specially introduced formulas. Besides, they may suffer from the unwinding phenomenon in an ordinary orientation. This phenomenon is prevented here by keeping the scalar part of the error quaternion non-negative and the deviation angle between 0° and 180°. For the orientation error matrix, a stationary and undisturbed antipodal orientation is an unstable equilibrium, in which the ordinary backstepping control law becomes ineffective for driving the system into action. This problem is solved here by adding an extra term to the ordinary backstepping control law.

Adaptive Control Based Harvesting Strategy for a Predator-Prey Dynamical System.

This paper deals with designing a harvesting control strategy for a predator-prey dynamical system, with parametric uncertainties and exogenous disturbances. A feedback control law for the harvesting rate of the predator is formulated such that the population dynamics is asymptotically stabilized at a positive operating point, while maintaining a positive, steady state harvesting rate. The hierarchical block strict feedback structure of the dynamics is exploited in designing a backstepping control law, based on Lyapunov theory. In order to account for unknown parameters, an adaptive control strategy has been proposed in which the control law depends on an adaptive variable which tracks the unknown parameter. Further, a switching component has been incorporated to robustify the control performance against bounded disturbances. Proofs have been provided to show that the proposed adaptive control strategy ensures asymptotic stability of the dynamics at a desired operating point, as well as exact parameter learning in the disturbance-free case and learning with bounded error in the disturbance prone case. The dynamics, with uncertainty in the death rate of the predator, subjected to a bounded disturbance has been simulated with the proposed control strategy.

Command-Filtered Integral Backstepping Control of Longitudinal Flapping-Wing Flight

The current work is concerned with longitudinal trajectory tracking control and simulation of a flapping-wing micro air vehicle. For the purpose of accurate simulation studies, a nonlinear and time-periodic dynamic model that accounts for the wings inertial effects is developed using Lagrange’s equations for quasi coordinates. Moreover, the aerodynamic loads are formulated, based on quasi-steady assumptions, taking into account the rotational circulation as well as the leading-edge vortex. Using the averaging method, the dynamic model is transformed to a time-invariant one that is the basis for controller design. Employing two control inputs, the flapping frequency and bias, an integral backstepping control law is synthesized. To ensure smooth oscillation-free control signals, a low-pass command filter is introduced in the control law. The choice of the control signals is based on Lyapunov stability criterion, while stability of the error dynamics is proved by the singular perturbation method. Simulation studies indicate that the proposed backstepping controller successfully achieves trajectory tracking. In addition, simulation results highlight the role of integral backstepping in achieving robust performance against model uncertainties and external disturbances and the importance of the command filter in maintaining smooth feasible control signals.

Safe control of trailing UAV in close formation flight against actuator fault and wake vortex effect

In the close formation flight, the wake vortex induced by the leading aircraft will have adverse effects on the safe flight of the trailing unmanned aerial vehicle (UAV). Hence, this paper investigates a difficult problem of safe control for the trailing UAV against actuator faults, input saturation, and wake vortex effect. By using disturbance observers, external wake vortex, disturbances, and internal actuator faults are estimated. Then, with the help of estimated knowledge of disturbance, backstepping control laws are developed for the longitudinal dynamics and the lateral-directional dynamics, respectively. One of the key features of the proposed strategy is that, the inherent problem, i.e., “explosion of complexity” in conventional backstepping control, is solved by the presented dynamic surface control scheme. Another key feature is that external wake vortex, disturbances, and internal actuator faults, input saturation are simultaneously considered. It is shown by Lyapunov stability analysis that the closed-loop system is uniformly ultimately bounded with safety requirements guaranteed even in the presence of wake vortex and actuator faults. The effectiveness of the proposed approach is further validated by simulation results.

2-step integral backstepping control of the two-rotor aero-dynamical system (TRAS)

This work proposes a simplified nonlinear backstepping control method for fast trajectory tracking of a 2-DOF laboratory helicopter called the Two Rotor Aero-dynamical System (TRAS). The relative degree 3 system is decomposed into the yaw subsystem (YS) and the pitch subsystem (PS) and an integral backstepping controller (IBC) is designed for each subsystem with the coupling effects considered as uncertainties. The control design considers the system as having relative degree 2, resulting in a much less complex 2-step backstepping control law requiring partial state feedback. Real time experiments show good tracking ability of the proposed method to different input waveforms within a wide operating range.

Spacecraft reorientation control in presence of attitude constraint considering input saturation and stochastic disturbance

This paper proposes a novel feedback control law for spacecraft to deal with attitude constraint, input saturation, and stochastic disturbance during the attitude reorientation maneuver process. Applying the parameter selection method to improving the existence conditions for the repulsive potential function, the universality of the potential-function-based algorithm is enhanced. Moreover, utilizing the auxiliary system driven by the difference between saturated torque and command torque, a backstepping control law, which satisfies the input saturation constraint and guarantees the spacecraft stability, is presented. Unlike some methods that passively rely on the inherent characteristic of the existing controller to stabilize the adverse effects of external stochastic disturbance, this paper puts forward a nonlinear disturbance observer to compensate the disturbance in real-time, which achieves a better performance of robustness. The simulation results validate the effectiveness, reliability, and universality of the proposed control law.