Abstract: In this note, we consider the same class of systems as in a previous paper, i.e., a class of uncertain dynamic nonlinear systems preceded by unknown backlash-like hysteresis nonlinearities, where the hysteresis is modeled by a differential equation, in the presence of bounded external disturbances. By using backstepping technique, robust adaptive backstepping control algorithms are developed. Unlike some existing control schemes for systems with hysteresis, the developed backstepping controllers do not require the uncertain parameters within known intervals. Also, no knowledge is assumed on the bound of the "disturbance-like" term, a combination of the external disturbances and a term separated from the hysteresis model. It is shown that the proposed controllers not only can guarantee global stability, but also transient performance.

Abstract: In this paper, two different backstepping neural network (NN) control approaches are presented for a class of affine nonlinear systems in the strict-feedback form with unknown nonlinearities. By a special design scheme, the controller singularity problem is avoided perfectly in both approaches. Furthermore, the closed loop signals are guaranteed to be semiglobally uniformly ultimately bounded and the outputs of the system are proved to converge to a small neighborhood of the desired trajectory. The control performances of the closed-loop systems can be shaped as desired by suitably choosing the design parameters. Simulation results obtained demonstrate the effectiveness of the approaches proposed. The differences observed between the inputs of the two controllers are analyzed briefly.

Abstract: In this paper, a robust adaptive tracking control problem is discussed for a general class of strict-feedback uncertain nonlinear systems. The systems may possess a wide class of uncertainties referred to as unstructured uncertainties, which are not linearly parameterized and do not have any prior knowledge of the bounding functions. The Takagi-Sugeno type fuzzy logic systems are used to approximate the uncertainties. A unified and systematic procedure is employed to derive two kinds of novel robust adaptive tracking controllers by use of the input-to-state stability (ISS) and by combining the backstepping technique and generalized small gain approach. One is the robust adaptive fuzzy tracking controller (RAFTC) for the system without input gain uncertainty. The other is the robust adaptive fuzzy sliding tracking controller (RAFSTC) for the system with input gain uncertainty. Both algorithms have two advantages, those are, semi-global uniform ultimate boundedness of adaptive control system in the presence of unstructured uncertainties and the adaptive mechanism with minimal learning parameterizations. Four application examples, including a pendulum system with motor, a one-link robot, a ship roll stabilization with actuator and a single-link manipulator with flexible joint, are used to demonstrate the effectiveness and performance of proposed schemes.

Abstract: A mathematical model of a 4-wheel skid-steering mobile robot is presented in a systematic way. The robot is considered as a subsystem consisting of kinematic, dynamic and drive levels. Next, a designing process of a kinematic controller based on the algorithm introduced by (Dixon et al., 2001) is shown. An extension of the kinematic control law at the dynamic and motor levels using the Lyapunov analysis and the backstepping technique is developed. To validate the designed algorithm, extensive simulation results for trajectory tracking and set-point cases are discussed. Some deliberations concerning the tuning of the controller are presented, too.

Abstract: In this paper, adaptive neural network (NN) control is investigated for a class of multiinput and multioutput (MIMO) nonlinear systems with unknown bounded disturbances in discrete-time domain. The MIMO system under study consists of several subsystems with each subsystem in strict feedback form. The inputs of the MIMO system are in triangular form. First, through a coordinate transformation, the MIMO system is transformed into a sequential decrease cascade form (SDCF). Then, by using high-order neural networks (HONN) as emulators of the desired controls, an effective neural network control scheme with adaptation laws is developed. Through embedded backstepping, stability of the closed-loop system is proved based on Lyapunov synthesis. The output tracking errors are guaranteed to converge to a residue whose size is adjustable. Simulation results show the effectiveness of the proposed control scheme.

Abstract: In this paper, an adaptive backstepping design is proposed to synchronize two uncertain chaos systems. This method can be applied to a variety of chaos systems which can be transformed into the so-called general strict-feedback form no matter whether it contains external excitation or not. Rossler system and Duffing oscillator are used as examples for detailed description. Numerical simulations show the effectiveness and feasibility of the method.

Abstract: This note presents a time-varying global adaptive controller at the torque level that simultaneously solves both tracking and stabilization for mobile robots with unknown kinematic and dynamic parameters. The controller synthesis is based on Lyapunov's direct method and backstepping technique. Simulations illustrate the effectiveness of the proposed controller.

Abstract: We identify a class of feedforward nonlinear systems that are linearizable by a coordinate change. Then we develop explicit expressions for the Lyapunov-based integrator forwarding recursive procedure of Sepulchre, Jankovic, and Kokotovic, which has its roots in a coordinate transformation proposed by Mazenc and Praly. The explicit expressions that we develop allow us to also find closed-form control laws for several classes of systems that are not feedback linearizable, including some that are in the feedforward form and others that are in what we refer to as the "block-feedforward" form. Performance advantages of Lyapunov-based forwarding controllers over nested saturation controllers have been well illustrated in the literature on examples. The analytical expressions for the Lyapunov functions and the control laws allow us to give quantitative performance bounds.

Abstract: In this paper, adaptive synchronization of two uncertain chaotic systems is presented using adaptive backstepping approach. The master system is any smooth nonlinear chaotic system, while the slave system is a nonlinear chaotic system in the feedback form. Global stability and exponential synchronization between the master and slave systems can be achieved. The proposed approach offers a systematic design procedure for adaptive synchronization of a large class of continuous-time chaotic systems in the chaos research literature. Computer simulations are provided to verify the operation of the designed synchronization scheme.

Abstract: In recent years, backstepping method has been developed in the field of nonlinear control, such as controller, observer and output regulation. In this paper, an effective backstepping design is applied to chaos synchronization. There are some advantages in this method for synchronizing chaotic systems, such as (a) the synchronization error is exponential convergent; (b) only one variable information of the master system is needed; (c) it presents a systematic procedure for selecting a proper controller. Numerical simulations for the Chua's circuit and the Rossler system demonstrate that this method is very effective.

Abstract: This paper describes a control strategy to stabilize the position of a micro air vehicle in wind gusts despite unknown aerodynamic efforts. The proposed approach allows us to overcome the problem of gyroscopic coupling by taking advantage from both the structure of the thrust mechanism, which is made of two counter rotating propellers, and the control strategy which involves a decoupling of the yaw rate dynamics from the rest of the system dynamics. The controller is designed by means of backstepping techniques allowing the stabilization of the vehicle's position while on-line estimating the unknown aerodynamic efforts.

Abstract: Three different adaptive controllers for a permanent magnet linear synchronous motor (PMLSM) position-control system are proposed. The proposed controllers include: a backstepping adaptive controller, a self-tuning adaptive controller, and a model reference adaptive controller. The detailed systematic controller design procedures are discussed. A PC-based position control system is implemented. Several experimental results including transient responses, load disturbance responses, and tracking responses of square-wave, sinusoidal-wave, and triangular-wave commands are discussed and compared. The proposed system has a good robustness performance even though the inertia of the system is increased to 10 times. The experimental results validate the theoretical analysis.

Abstract: Including integral action in a nonlinear backstepping design is the topic of this paper. Two methods for adding integral feedback are proposed and analyzed. These are compared to the more traditional methods: 1) adaptive backstepping, and 2) plant augmentation that adds an extra relative degree and thus gives one extra step of backstepping. A test plant is used to compare the different control laws. Based on the theoretical analysis and the simulations, some interesting conclusions are made for each integral control strategy.

Abstract: This article presents a new method for on-line approximation based backstepping control in the presence of known magnitude, rate, or bandwidth constraints on the intermediate states or actuators The presentation is based on developing the design and analysis for a second-order system - these results can be recursively extended to higher order systems The new results allow on-line learning to continue even when known magnitude, rate, or bandwidth constraints are in effect, even though those constraints do not allow the control objectives to be met for the duration of those constraints

Abstract: Chaos synchronization and parameters identification of single time scale brushless dc motors are studied in this paper. In order to analyze a variety of periodic and chaotic phenomena, we employ several numerical techniques such as phase portrait, bifurcation diagram, and Lyapunov exponents. By the adaptive control, the improved backstepping design method, the Gerschgorin theorem, and by addition of a monitor, chaos synchronization of two identical BLDCM systems are presented. Then, by the adaptive control, and the random optimization method, parameters identification is approached.

Abstract: This paper presents an adaptive, integrated guidance and control approach for ballisticmissile interceptors using backstepping control techniques. The primary benefits of backstepping are that control loops are derived in a rigorous and systematic manner, and unmatched uncertainties in the plant dynamics are handled readily. An on-line neural network is used to provide robustness to parametric uncertainties in the missile aerodynamics. The adaptive integrated approach enables shorter design times, and more ecient designs, by removing the need for an iterative integration process required by traditional methods. The algorithm development is restricted to the pitch-plane, but can be readily extended to all three axes. Numerical simulation results, including Monte-Carlo results, are presented from a high-fidelity, nonlinear simulation to demonstrate the ecacy of the approach.

Abstract: A fresh nonlinear backstepping design scheme, which is developed for the control of half-car active suspension systems to improve the inherent tradeoff between ride quality and suspension travel, is proposed in this paper. Since ride quality is dependent on a combination of vertical and angular displacements of a vehicle body, the design of active suspensions must have the potential to minimize heave and pitch movements in order to guarantee the ride comfort of passengers. The other important factor to be emphasized in the design of active suspensions is the suspension travel which means the space variation between the car body and the tires. In order to avoid damaging vehicle components and generating more passenger discomfort, the active suspension controllers must be capable of preventing the suspension from hitting its travel limits. Our design strategy, with two intentionally additional nonlinear filters, shows the potential to achieve these conflicting control objectives. The novelty of our active s...

Abstract: In this paper, an adaptive parallel control architecture to stabilize a class of nonlinear systems which are nonminimum phase is proposed. For obtaining an on-line performance and self-tuning controller, the proposed control scheme contains recurrent fuzzy neural network (RFNN) identifier, nonfuzzy controller, and RFNN compensator. The nonfuzzy controller is designed for nominal system using the techniques of backstepping and feedback linearization, is the main part for stabilization. The RFNN compensator is used to compensate adaptively for the nonfuzzy controller, i.e., it acts like a fine tuner; and the RFNN identifier provides the system's sensitivity for tuning the controller parameters. Based on the Lyapunov approach, rigorous proofs are also presented to show the closed-loop stability of the proposed control architecture. With the aid of the RFNN compensators, the parallel controller can indeed improve system performance, reject disturbance, and enlarge the domain of attraction. Furthermore, computer simulations of several examples are given to illustrate the applicability and effectiveness of this proposed controller.

Abstract: We present a design of a tracking controller for an unstable, nonminimum phase nonlinear plant using trajectory linearization method. The controller consists of two parts: a dynamic inverse and a tracking error stabilizing control law. The nominal control computed by nonlinear pseudo-inverse using nonlinear coordinate transformation and backstepping stabilization, and trajectory stabilization is achieved using linear time-varying (LTV) PD-eigenstructure assignment. Simulation case studies show that significant improvement in tracking performance, robustness and disturbance rejection over the classical and modern gain scheduled controllers can be achieved using rational combinations of nonlinear and LTV control techniques.

Abstract: A robust controller design methodology for high speed reentry vehicles (HSRV) is presented in this paper. The design methodology is based on a nonlinear backstepping technique. High speed reentry vehicles often contain signiflcant nonlinearities in the dynamics equation of motion which are partially caused by the high velocities and ∞ight path trajectories. These two factors tend to produce signiflcant aerodynamic forces and moments on the vehicle. The nonlinear backstepping control technique provides an alternative to methods such as piecewise linearization-based gain scheduling and feedback linearization. Backstepping approaches utilize Lyapunov theory to guarantee stability of the closed system and the methods are inherently recursive. Nonlinear optimization is also used to guarantee the accuracy of the mapping between the actuators and applied moments and an adaptive term is included to compensate for the moment bias. The proposed design procedure turns out to be simpler than designs based on alternative methods and the design appears relatively easy to implement.

Abstract: The dynamics of hydraulic systems are highly nonlinear and the system may be subjected to non-smooth and discontinuous nonlinearities due to directional change of valve opening, friction, etc. Aside from the nonlinear nature of hydraulic dynamics, hydraulic servo systems also have large extent of model uncertainties. To address these challenging issues, a robust state-feedback controller is designed by employing backstepping design technique such that the system output tracks a given signal arbitrarily well, and all signals in the closed-loop system remain bounded. Moreover, a relevant disturbance attenuation inequality is satisfied by the closed-loop signals. Compared with previously proposed robust controllers, this paper’s robust controller based on backstepping recursive design method is easier to design, and is more suitable for implementation.

Abstract: An adaptive displacement tracking control using only displacement feedback is proposed for a piezo-positioning mechanism. In order to develop a dynamic model to represent the overall system dynamics of the controlled piezo-positioning mechanism, a specific function is proposed. This function that describes the hysteresis of the controlled mechanism contains information on the mechanical motion dynamics, hysteresis friction, disturbance load and parameter variations. Based on the developed model, an adaptive backstepping displacement tracking control is proposed, in which the backstepping adaptation of the specific function and the estimation of the reconstructed state with an unknown specific function are presented. The proposed control design leads to an improved tracking performance, and robustness to the external load-disturbance, and variations in system parameters. The validity of the proposed control design is demonstrated by experimental results.

Abstract: Previous study showed that a third-order phase-locked loop (PLL) with sinusoidal phase detector characteristics experienced a Hopf bifurcation point as well as chaotic behavior. As a result, this behavior drives the PLL to the out-of-lock (unstable) state. The analysis was based on a modern nonlinear theory such as bifurcation and chaos. The main goal of this paper is to control this chaotic behavior. A nonlinear controller based on the theory of backstepping is designed. The study showed the effectiveness of the designed nonlinear controller in controlling the undesirable unstable behavior and pulling the PLL back to the in-lock state.