To overcome the above problems, in , a combination of backstepping control and ANN has been proposed. According to this method, in the process of backstepping controller design, two ANN are used to estimate two nonlinear functions. Therefore there is no need to find the regression matrix for on-line estimation of unknown parameters. In , using the ANN, the theory of robustbackstepping control has been presented for strictly feedback nonlinear systems. This method has been applied to a single arm robot in  and to a rotor flux Field Oriented Control (FOC) IM drives in .
rejection capability in different weather conditions. Unmanned autonomous helicopter control system should make these performances achieved by improving the tracking performance and disturbance rejection capability in different weather conditions. Therefore, robustness is one of the critical issues which must be considered in the control system design for small unmanned autonomous helicopter, especially those covering large flight envelope. One major problem rarely addressed by researchers to date is that of a wind disturbance. To cope with such a problem, researchers have considered different approaches such as robust feedback linearization, robust nonlinear H∞ controller [1, 2], neural network [3, 4], adaptive backstepping  etc. This paper presents a robustbackstepping controller design to stabilize the inner loop and the outer loop of a small scale helicopter in the presence of external wind gusts. The control of a hover flight in a gusty environment is always a challenging control problem due to the inherent nonlinearity, dynamical instability and unsteady operating conditions. The position control of a small scale unmanned autonomous helicopter under the vertical and horizontal wind gusts using the feedback method is addressed in , wherein the wind disturbance is assumed to be the sum of a fixed number of sinusoids with unknown amplitudes, frequencies and phases. In , the authors present a robustbackstepping technique of autonomous scale helicopter subject to parameters uncertainties and uniform time varying three-dimensional wind gusts. With the assistance of an unknown input observer technique (UIO), the controller is reported to be able to handle the effect of these uncertainties on an autonomous helicopter. A robust H ∞ controller for a hover
This paper has presented a new robustbackstepping sliding mode controller (RBSMC) for reference input wheel slip tracking based on the single-corner model with the actuator dynamics. The proposed RBSMC combines the merit of backstepping method that has simplified and flexible design procedure with the mer- it of sliding mode control that is insensitive to system uncertainty and external disturbance, and the Lya- punov-based method is used to derive that the closed- loop wheel dynamic system is L 2 -gain stable. Then,
This paper presents a robustbackstepping control (BC) methodbased on nonlinear disturbance observer (NDOB) for trajectory tracking of the nonholonomic wheeled mobile robot (WMR) in the presence of external disturbances and parameters uncertainties. At first, a bounded Fuzzy logic based backstepping controller (BFLBC) is designed to control the WMR without considering the effects of the external disturbances and the parameters uncertainties. Typically, the conventional BC controller depends upon the state tracking errors analysis, where unbounded velocity signal is produced for the applications that have huge tracking errors. Therefore, a fuzzy logic controller (FLC) is introduced in this research in order to normalize the state tracking errors, so that the input errors to the BC are bounded to a finite interval. Finally, the designed BFLBC is integrated with the nonlinear disturbance observer in order to attenuate the external disturbances and model uncertainties. The simulation results show the effectiveness of the proposed controller to generate a bounded velocity signal as well as to stabilize the tracking errors to zero. In addition, the results prove that the proposed controller provide an excellent disturbance attenuation as well as robustness against the parameters uncertainties.
Abstract—A novel Lyapunov function, which contains a concept of transfer function, was constructed to prove the rightness of a new kind of robustbackstepping filter derivative control method. Meanwhile, the relationship between Lyapunov function and transfer function was established, which is an important concept that can be applied in a large family of control systems. Also, the backstepping design technology is perfectly integrated with the PID control method, which was testified by the simulation result. And comparing with pure derivative method, better performance was achieved by the adopting of filter derivative method.
It should be emphasized that very few research works in the literature, are taking into account the ground effect and external wind gusts in the altitude control of a small helicopter near ground surface. In the previous version of this paper , the authors designed a robustbackstepping controller for altitude control of a small helicopter by considering both the effects of ground effect and external wind gusts but in [15, 16, 17] the author did not consider the ground effect. Throughout this paper, for an unmanned autonomous helicopter altitude control near the ground surface, both ground effect compensation and external wind gusts are taken into account within the system model. However, in our case, we concentrate on horizontal wind gusts with a typical level of 10 m/s. Vertical wind gusts can be neglected compared with its horizontal counterparts since the main factor influencing thrust in hover comes from the horizontal gusts, particularly close to the ground where the vertical gust component is near zero .
An approach for tracking and synchronization problems of hyperchaotic systems using active backstepping method has been proposed in this paper. The mentioned method has been applied to the hyperchaotic Rössler and Wang systems successfully. Three numerical simulations are used to demonstrate the effectiveness of the designed active backstepping controllers.
The autonomous underwater vehicle (AUV) mostly has fewer control inputs than the degree of freedoms (DOFs) in motion and be classified into underactuated system. It is a difficult tasks to stabilize that system because of the highly nonlinear dynamic and model uncertainties. It is usually required nonlinear control method and this paper presents the stabilization of an underactuated X4-AUV using backstepping based PID nonlinear control techniques. The X4-AUV system is executed by separating system into two subsystems which is translational and rotational subsystems. Integral backstepping control is applied for translational subsystem and PID backstepping control for the rotational subsystem. As a results the x-position and all angles is stabilized into desired point. The effectiveness of the proposed control technique for an underactuated X4-AUV demonstrates through simulation.
proposed a new adaptive sliding mode dynamic controller with an integrator for non-complete wheeled mobile robots. The simulation results showed that the control scheme has zero steady-state error, fast error convergence and robustness in the presence of disturbances and uncertainties. Yokoyama et al.  designed an adaptive controller for the trajectory tracking problem of mobile robots, which combined the advantages of the backstepping method and immersion and invariant control (I&I) method. Wu et al.  proposed a hybrid robust control algorithm based on backstepping kinematics control and fuzzy sliding mode robust dynamics control, which solved the trajectory tracking control problem of the mobile robot model under uncertainties and external interferences.
In this paper, a contraction-based backstepping nonlinear control technique was proposed. The proposed controller syn- thesis technique utilizes both the recursive nature of backstepping control and of contraction analysis. This approach results in a contracting closed-loop dynamics, with exponential stability. The use of the hierarchical contraction form in the control problem formulation also results in the exponential convergence of controlled variables and can be easily applied to non-autonomous systems. A flight path angle controller was synthesized and simulated using the proposed technique to demonstrate the exponential convergence achieved by the backstepping controller design.
The studies in quadrotor control designing have been increasing rapidly in recent years. Linear controllers design for quadrotors have been achieved in sev- eral work, like the Linear Quadratic Regulator(LQR) and a Proportional Integral Derivative (PID) (Khatoon et al., 2014) (Reyes-Valeria et al., 2013). Nonlinear control design has also been achieved with different techniques, like Backstepping (Das et al., 2009), Slid- ing Mode (Runcharoon and Srichatrapimuk, 2013) and Feedback Linearisation (Saif, 2009). (Castillo et al., 2005) compared the performance of a nonlin- ear control algorithm with a LQR control law. Results show the unstable response of a linear controller ap- plied to a nonlinear system, while the nonlinear con- troller shows stable response. (Gomes et al., 2016) used an AR.Drone quadrotor and a Vicon motion cap- ture system to track a moving target with a Propor- tional Derivative (PD) controller for linear position- ing. (Mashood et al., 2016) showed experimental re- sults of two AR.Drone following a squared path us- ing VICON system and MATLAB/SIMULINK for feedback and control implementation. This was possible with AR Drone Simulink Development Kit (ARDSDK). (Campbell et al., 2012) showed the de- sign and implementation of a quadrotor aircraft au- topilot, allowing the UAV to take off, transit from one location to another and land at a desired loca-
Characteristics of the network of the system can aect the control law design. So, at rst step of the con- troller design, general characteristics of the networks considered in the present work should be dened. After that, decentralized control laws are designed to track the velocity bounded trajectories by formation of integrator and double-integrator agents. Robust controller design is the nal step of the controller design.
Background: A nonlinear control method is considered for stabilizing all attitudes and positions of an underactuated X4-AUV with four thrusters and six degrees-of-freedom (DOF). Objective: The objective of this paper is to stabilize all the attitudes and positions for an X4-AUV system using backstepping control method. A control system composed of translational and rotational subsystems. A controller for the translational subsystem stabilizesthe positions, whereas controllers for the rotational subsystems generate the desired roll, pitch and yaw angles. Results:Thus, the rotational controllers stabilize all the attitudes of the X4-AUV at a desired position of the vehicle. Some numerical simulations are conducted to demonstrate the effectiveness of the proposed controllers. Conclusion: From the simulation, write the main conclusion for your paper. From the simulation and exprimental results, when using the FLC controller and the MRAC in coordination in which the control law is used to cope with parameter variation and load disturbances, the controlled system can be robustly stabilized all the time.
Chaos synchronization of two systems is presented in this paper by using Backstepping control method. However, due to the effectiveness of the cost and density with one controller backstepping is effective. The advantage of this procedure is that there is only one controller no matter how much dimension are there in the system to be synchronized.
For a long time, age-structured epidemic models have been studied by many authors (see, for instance, Iannelli , Inaba , Inaba and Nishiura  and Tudor ). Mathematically, these models can be regarded as the boundary-value problems of partial differential equations. On the other hand, in engineering, the backstepping method has recently been developed by Smyshlyaev and Krstic  to obtain the boundary feedback control for stabilizing the systems of partial differential equations and has widely been studied by many authors (see, for instance, Susto and Krstic  and Baccoli et al. ). The aim of this study is to make use of the backstepping method for epidemiological con- siderations. Specifically, we will develop a new method for the prediction of increase or decrease of infected population.
Nowadays, autonomous helicopters are used in different military and civilian applications such as border patrolling, geological surveying, traffic monitoring, area mapping, search and rescue and reconnaissance missions. Quadrotors are a special class among different types of helicopters. Quadrotor helicopters have gained a lot of research interest in the past few years, due to the clear advantages posed by their vertical take-off and landing (VTOL), hovering capability, and slow precise movements. Belonging to the helicopter rotorcraft class, quadrotors are highly nonlinear systems that are difficult to stabilize. The main challenges of the quadrotor helicopters are mainly due to its unstable nature and complexity of the dynamic model. Furthermore, this type of helicopter is a highly nonlinear system and very susceptible to external disturbances. Many approaches and techniques have been proposed to control a quadrotor helicopter, such as linear quadratic regulator (LQR) control , proportional-integral- derivative (PID) control , fuzzy logic (FL) control , sliding mode control , and backstepping control [5-8]. The backstepping control scheme is a nonlinear control method based on the Lyapunov theorem. The backstepping control design techniques have received great attention because of its systematic and recursive design methodology for nonlinear feedback control. The advantage of backstepping compared with other control methods lies in its design
This paper has numerically studied the dynamical behaviors of a fractional-order single-machine infinite-bus (FOSMIB) power system. Periodic motions, period- doubling bifurcations and chaotic attractors are observed in the FOSMIB power sys- tem. The existence of chaotic behavior is affirmed by the positive largest Lyapunov exponent (LLE). Based on the fractional-order backstepping method, an adaptive controller is proposed to suppress chaos in the FOSMIB power system. Numerical simulation results demonstrate the validity of the proposed controller.
In this paper, using the integrator backstepping method develops the problem of nonlinear position control of a 5 DoF robot manipulator. It consists of elaborating a control method  that guarantees the asymptotic stability and the tracking of desired position and velocity trajectories. A major advantage of this method is its flexibility to build the control law by avoiding the cancellation of useful non-linearities . Simulation results presented in this paper show that the system has global stability.
This paper is organized as follows. In section 2, adaptive backstepping control consisting of ideal backstepping control and GRBABC is designed. The evolutionary algorithms are described in section 3. In section 4, evolutionary algorithms to compute the optimal parameters of GRBABC are designed. Finally, to show the effectiveness of these methods, they are applied to two chaotic systems, Duffing system and Lü system in section 5. The paper is concluded in section 6.