The designed BSMC provides an effective robust control approach for quadrotor system Eq. (18). How- ever, the control laws Eq. (25), Eq. (26), Eq. (27), Eq. (28), Eq. (29), Eq. (30) and Eq. (31) can cause the chattering phenomenon resulting from the use of the sign function. In this context, high switching gain k i will lead to an increase in oscillations of the con- trol input signal, and therefore an excitation of high- frequency dynamics will take place, as results, a chat- tering phenomenon will be created. However, increas- ing the gain causes an increase of the oscillations in input control around the sliding surface. Moreover, a decrease in switching gain k i can reduce the chatter- ing phenomenon and improve the tracking performance despite large external disturbances. To achieve more appropriate performance, this gain must be adjusted. This adjustment is based on the distance between the system states and the sliding surface. i.e., when the trajectory of the system state deviates from the sliding surface, the switching gain should be increased in order to reduce chattering and vice versa. This idea can be realised by combining Fuzzy logic with Backstepping- SlidingModeControl to construct FBSMC to facilitate the adaptive switch-gain (see Fig. 2) according to some appropriate fuzzy rules. The Architecture of the FLS is shown in Fig. 3.
nonlinear adaptive controlbased on backstepping techniques [4,5]. The increasing numbers of works involve linear control, nonlinear control and intelligent control methods. It is found that nonlinear control strategy has been an efﬁcient approach . Veysi  designed an optimal fuzzyslidingmode controller for controlling the end-effector position in the task space. Feedback linearization method and slidingmode con- trol (SMC) were employed to overcome the uncertain- ties. Soltanpour  designed a slidingmode controller to control the position tracking of robot manipulator. The controller had global asymptotical stability in the presence of uncertainties and unmodelled dynamics. Yao  synthesized a robust integral of the sign of the error controller and an adaptive controller via back- stepping method for motion control of a hydraulic rotary actuator. The controller guaranteed asymptotic tracking performance in the presence of various uncer- tainties. Fuzzycontrol using linguistic information possesses several advantages such as robustness, model-free and universal approximation theorem. Wang and Mendel [10,11] proposed that fuzzy logic systems were capable of uniformly approximating any nonlinear function. Fuzzy logic system is useful for plants having difﬁculties in deriving mathematical models. SMC has been given much attention by indus- trial and academic communities. It is usually utilized to control the systems facing uncertainty. It has advan- tages in having a simple structure and being robust against the parameter uncertainties . The disad- vantage of SMC is the chattering problem of control action due to its discontinuous switching part in
especially in traffic jams or long-distance driving cir- cumstances. However, the wheel slip control is the basis of active safety control systems and intelligent driver assistance systems. For instance, the anti-lock brake system (ABS) regulates the slip of each wheel at its optimum value to prevent it from locking during braking, such that the shortest stopping distance is achieved and the capability of directional stability and steer-handling is maintained . The electronic stability program (ESP) may produce additional yaw moment by commanding the target slip of one or two wheels to prevent vehicle from spinning and drifting out of lane . Finally, the adaptive cruise control system (ACC) can follow target speed or forward ve- hicle at the desired safety headway distance by com- manding the target slip of the wheels and the target torque of the power system . As a consequence, de- signing the wheel slip controller has important theo- retical and practical significance for active safety con- trol systems and intelligent driver assistance systems. In recent years, many control approaches which are robust against system uncertainty and external dis- turbance have been proposed for the wheel slip con- trol due to the modeling errors, the measurement or estimation errors, and the changing of external en- vironment conditions of the wheel dynamic system, such as slidingmodecontrol , hybrid control  and fuzzycontrol , etc. Johansen et al.  established the speed-dependent nominal linearized slip model with a perturbation term as a basis for the wheel slip control, and utilized gain-scheduled LQR approach to design the gain matrices of the control- ler. Pasillas-Lépine  adopted wheel deceleration logic-based switching and wheel dynamic model to design the five-phase anti-lock brake algorithm, and proved the existence and stability of limit cycles by the Poincaré map. Hsu  proposed an intelligent exponential sliding-modecontrol strategy for ABS, and a functional recurrent fuzzy neural network uncertainty estimator was designed to reduce the chattering of the exponential sliding-modecontrol strategy by approximating and compensating the unknown nolinear term of ABS dynamics on-line. Jing et al.  presented a switched control strategy for the anti-lock brake system and then analyzed the stability condition of the closed-loop system by Lyapunov-based method in the Filippov frame- work. The proposed control strategy in [7-9, 19] may
In this paper, a robust chattering free backsteppingslidingmode controller is successfully developed for attitude stabilization and trajectory tracking of a quadrotor helicopter perturbed by external disturbances. First, the mathematical model of the quadrotor is introduced. Then, the proposed robust control system comprises a backstepping and a switching function is developed. The backsteppingcontrol design is derived based on Lyapunov function, so that the stability of the system can be guaranteed, while switching function is used to attenuate the effects caused by external disturbances. In order to eliminate the chattering phenomena, the sign function is replaced by the fuzzy system. Finally, the developed control scheme is applied to autonomous quadrotor helicopter. Simulation results show that a satisfactory control performance can be achieved by using the developed control system. Furthermore, it can be seen that a fuzzy system can be utilized to eliminate a chattering phenomena in the control inputs of the BSMC.
The application of adaptive neural networks to robot manipulator is presented in Perez et al. (2012) which explain recurrent neural networks and Lyapunov function methodology. An adaptive type-2 FLC for flexible-joint manipulators with structured and unstructured dynamical uncertainties have introduced in (Chaoui et al., 2012). In Abdel et al. (2011), the author has proposed fuzzy partition to the state variables based on the Lyapunov synthesis. Authors in Zeinali and Notash (2010) and Ho et al. (2009) have presented a methodology that enables the designer to systematically derive the rule base of the control. In Kohrt et al. (2013), authors have discussed on-line robust control for robot manipulator. This paper presents a new adaptive SMC for 2DOF robot manipulator; an adaptive tracking controller with a PID sliding surface. The adaptive SMC algorithm can estimate the value of switching gain constant (K w ) and boundary
There are all kinds of abundant marine resources in the ocean, such as mineral, marine organisms and ocean energy. Most of these resources are not fully developed to date. Therefore, many new devices and equipments are developed for ocean applications such as ocean surveillance and measurements. Unmanned underwater vehicles (UUVs) are the most widely used equipment nowadays. The features of UUVs are unoccupied, reliable and highly maneuverable. An underwater vehicle will complete missions as instructed, and exchange information and data with the ship or station. Generally speaking, UUVs can be applied for scientific research, acquiring information of oceans. They can also be used for inspections and operations of the facilities under the water [1,2,3] .
In recent years, advanced control has been applied in many engineering practices [20-22]. Advanced control in- cludes predictive control, fuzzycontrol, neural control, nonlinear control and robust control et al, and it has better control effect than the traditional PI controller. From the section II, we know that the mathematical model we estab- lished is a second-order model. Thus, we construct the virtual controller by using backsteppingcontrol. Backsteppingcontrol is a kind of nonlinear control, and it is widely used in the control of higher-order models . In order to reduce the differential process of the controller, we introduce the command-filter in the controller . The work in  applies command-filtered backsteppingcontrol to nonlinear multi-agent tracking problems. The controller is designed and the stability of the system is verified based on Lyapunov stability theory. It can be seen from simulation that the designed controller has satisfactory control performance. In the model established for the PV system, the pa- rameters of the resistance, inductance, and capacitance in the circuit are included. However, the measurement of these
factors, but the acquisition of fuzzy rules depends on the knowledge of the designers. Ref.  provides a neural network direct adaptive control method, and this method is applied to the control of AUV with bounded external disturbances and bounded neural network approxima- tion errors, and verified by dynamic positioning and sin- gle degree of freedom (DOF) trajectory tracking. In Ref. , the ocean current disturbances and AUV dynamic modeling uncertainty are considered as uncertainties, the above two uncertainties are combined; then the radial basis function (RBF) neural network is used to approxi- mate the uncertainties online, and the adaptive slidingmodecontrol is adopted to control the AUV. However, in Refs. [6–9], the forward neural network (such as RBF or BP neural network) is difficult to reflect the time series influence of system input and output variables. There- fore, when the target trajectory of AUV has an abrupt change, the neural network weights may take a long time to convergence, leading to poor transition characteris- tics in trajectory mutation process, and may result in the overshoot of trajectory tracking.
Abstract. In this paper, an adaptive fuzzybacksteppingcontrol using dynamic slidingmodecontrol (AFBCDSMC) is presented for a micro-electromechanical system (MEMS) vibratory z-axis gyroscope. Based on an adaptive fuzzybacksteppingcontrol method, a dynamic slidingmodecontrol is proposed to compensate and adjust the external disturbances and model uncertainties. The fuzzycontrol method with adaptive backstepping controller can approximate the system nonlinearities well without accurate system model and. It can make the controller have the ability to learn and adjust the fuzzy parameters in real time. In addition, dynamic slidingmodecontrol can transfer discontinuous terms to the first-order derivative of the control input in order to effectively reduce the chattering. Simulation studies are investigated to demonstrate the satisfactory performance of the proposed method.
PMs have strong non-linearity and time-varying properties , which may cause difficulties in implementing precise control . In order to solve these problems, a variety of control approaches have been developed. Zhao et al.  used neural network to adjust the parameters of PID controller. However, the method has the problems of long response time, poor tracking on desired trajectory and low tracking accuracy in the step response experiment. Zhang et al.  proposed a hybrid fuzzy controller to control the elbow exoskeleton robot actuated by PMs. However, this method cannot estimate the external disturbance when chattering happens, resulting in a large overshoot of step response. For the safety of human-robot interaction, Choi et al.  proposed a new approach to control the compliance and associated position independently. However, when an external disturbance occurs suddenly, the control method cannot detect the external disturbance quickly and it takes a long time to re-track the desired trajectory. Meng et al.  proposed an iterative feedback tuning control method for the repetitive training. However, the actual trajectory changed in a ladder shape because the external disturbance cannot be estimated. Jiang et al. also  proposed an adaptive fuzzycontrol algorithm based on neural network optimization to control the humanoid lower limb device driven by pneumatic muscles. However, this method cannot achieve high-accuracy tracking control and the error would significantly increase when the external load changes.
In this paper, a composite nonlinear controller has been proposed for the IM flux and speed control that is robust to all electromechanical parameter variations and uncertainties. First a backstepping controller is designed for a three-phase IM. This controller provides speed and flux tracking for desired trajectories, but it needs electrical and mechanical parameters to be known and if some uncertainties exist in the control system, the controller can not do its rule properly. So, to
The rollover stability has become the focus of the world automobile research in recent years.Differential braking can reduce the inhibition of lateral acceleration, yaw velocity and does not change the purpose of the driver, but the cost is low.Therefore,using differential braking to improve the vehicle roll stability is a better choice.The paper  has proposed a nonlinear asymptotic tracking control strategy to ensure the reference yaw angle control strategy to prevent rollover. The paper  has improved the stability of the car through direct yaw moment control, using optimal LQR control strategy. The paper  has established a nonlinear vehicle model developing a differential braking stability control and a fuzzy logic controller. The paper has built a vehicle stability control by using model predictive control theory framework tolimit the maximum roll angle of the car. The linear matrix inequality (LMI) used in this paper is a powerful tool for the design of control field, to construct the effective computer algorithm for solving problems. Before the linear matrix inequality (LMI) are widely used in control ,most problems are solved by method of Riccati equation or inequality method [6-8].In the processing of the solution of the Riccati equation or inequality, there are a large number of parameters and the positive definite symmetric matrices to be adjusted. The linear matrix inequality method can make up the shortcomings of Riccati equation method.
The slidingmodecontrol is robust to plant uncertainties and insensitive to external disturbances. It is widely used to obtain good dynamic performance of controlled systems. However, the chattering phenomena due to the finite speed of the switching devices can affect the system behavior significantly. Additionally, the slidingcontrol requires the knowledge of mathematical model of the system with bounded uncertainties. Reduced chattering may be achieved without sacrificing robust performance by combining the attractive features of fuzzycontrol with SMC. Fuzzy logic is a potent tool for controlling ill-defined or parameter-variant plants. By generalizing fuzzy rules, a fuzzy logic controller can cope well with severe uncertainties. Fuzzy schemes with explicit expressions for tuning can avoid the heavy computational burden [11-14].
A lot of efforts have been devoted to PV micro-grid in the past. In reference , a fractional order slidingmode controller was applied to an islanded distributed energy resource system, and the output voltage tracking control was realized. However, in the simulation, the authors used the DC source instead of the distributed energy . When the PV array was connected to the DC side of the system, the control effect of the designed controller was unknown. The work in  proposed a voltage and frequency control strategy for the distributed energy resource system, different types of loads were used to be simulated the control effect , and the parallel operation of multiple distributed-resource units was simulated. However, there was no use of the distributed energy resource in the author’s simulation, and the energy storage unit was not considered in the system. Also in the work of , the maximum power point tracking (MPPT) was realized through the use of slidingmodecontrol. Then the Lyapunov function-basedcontrol method was utilized in VSC, the simulation showed that the controller had better control effect. However, the control algorithm presented in this work possessed some limitations, and the control effect also had space for improvement , in addition, the energy storage unit was also not considered in the system.
Voice coil motor (VCM) has excellent characteristic of its simple structure, low friction between motor and motor device and ability of high-frequency repeatability movement. The VCM is used in the applications of linear positioning control systems with small control range, such as an optical read/write head of a hard disk . However, the dynamic equation of a VCM is difficult to obtain due to the nonlinear and time- varying motor behaviors. From the control design viewpoint, the conventional control technologies are always based on a good understanding of the control system dynamics, so it is very difficult to control a VCM by using conventional control theory.
In order to improve the performance of single-input single-output (SISO) nonlinear systems with uncertainties, an adaptive fuzzyslidingmode controller (AFSMC) that combines linearization feedback is presented in this paper. The fuzzy logic system is used to approximate the unknown system function and the AFSMC algorithm is designed by used of slidingmodecontrol techniques. Based on the Lyapunov theory, the continuous function is designed to eliminate the chatting action of the control signal. The simplicity of the proposed scheme facilitates its implementation and the overall control scheme guarantees the global asymptotic stability if all the signals involved are uniformly bounded. Simulation results have shown that the proposed controller shows superior tracking performance. AFSMC can effectively achieve desired performance and have much more advantages over conventional SMC.
The most common form of electromechanical drive for industrial, commercial and domestic applications is an Induction machine due to its cost, reliability and performance. Induction machines have simpler and more rugged structure, higher maintainability and economy than DC motors . Basically, there are two types of instantaneous electromagnetic torque-controlled AC drives used for high performance applications. They are Vector Control(Field oriented control) and Direct Torque control. Based on operating principles these two methods are different, but their aims are same. Vector Control is based on stator current control in the field rotating reference using PWM inverter control. And Direct Torque Control (DTC) is based on stator flux control in the stator fixed reference frame using direct control of the inverter switching.
In this paper, an adaptive fuzzysliding-mode controller (AF-SMC) based on fractional calculus for uncertain chaotic nonlinear systems were proposed. Three fuzzy logic systems were designed to approximate the unknown system functions and the switching term of SMC, respectively. Stability of the closed-loop system was proved and adaptive laws for the tuned parameters were obtained by using Lyapunov arguments. Numerical simulations are done on the chaotic nonlinear gyroscope system and results show that the proposed fractional
linear matrix inequalities techniques to control the velocity of a dc motor driven by a dc/dc Buck converter, was presented. The results showed, via numerical simulations, that the proposed control scheme guarantees fast angular velocity tracking with minimal duty cycles. More recently, in 2013, Sira-Ram´ ırez and Oliver-Salazar  described a robust control law based on active disturbance rejection control and flatness-basedcontrol, taking into account an unknown time-varying load, for two combinations of dc/dc Buck converters and dc motors. Numerical simulations showed the robustness of this technique for the angular velocity control of the motor shaft. Finally, in 2013, Silva- Ortigoza et al.  introduced a two-stage controlbased on differential flatness for the angular velocity control without taking into account velocity measurements of a dc/dc Buck converter–dc motor system. They showed, through numerical simulations that included an Σ − Δ modulator, that the proposed control scheme effectively provides robustness to the tracking performance when parametric uncertainties related to the system appear. Additionally, important contributions related to the connection of other dc/dc power converters and dc motors have been reported in , –.
In , an active safety system was developed based on the by-wire technologies to improve yaw moment control system that depends on the braking action on the left and right wheels. Other types of active steering systems have been reported such as Active Front Steering (AFS)   , Active Rear Steering (ARS)  and Active Four Wheel Steering (W4S) . Active steering is an effective way that can improve drivers comfort and handling. The vehicle handling and lateral stability can be controlled at the same time, if both the external yaw moment and active steering angle are adopted .