Backstepping Procedure Research Articles

Observers-based event-triggered adaptive fuzzy backstepping synchronization of uncertain fractional order chaotic systems.

In this paper, for a class of uncertain fractional order chaotic systems with disturbances and partially unmeasurable states, an observer-based event-triggered adaptive fuzzy backstepping synchronization control method is proposed. Fuzzy logic systems are employed to estimate unknown functions in the backstepping procedure. To avoid the explosion of the complexity problem, a fractional order command filter is designed. Simultaneously, in order to reduce the filter error and improve the synchronization accuracy, an effective error compensation mechanism is devised. In particular, a disturbance observer is devised in the case of unmeasurable states, and a state observer is established to estimate the synchronization error of the master-slave system. The designed controller can ensure that the synchronization error converges to a small neighborhood around the origin finally and all signals are semiglobal uniformly ultimately bounded, and meanwhile, it is conducive to avoiding Zeno behavior. Finally, two numerical simulations are given to verify the effectiveness and accuracy of the proposed scheme.

A composite adaptive robust control for nonlinear systems with model uncertainties

AbstractThis article is concerned with a composite adaptive robust control (CARC) for non‐strict feedback nonlinear systems with model uncertainties including parametric uncertainties and uncertain nonlinearities. It is assumed that the model uncertainties of the nonlinear system are bounded but unknown. Firstly, prescribed tracking performance is achieved by the CARC control law even the unknown parameters are time‐varying. This is accomplished firstly by designing a discontinuous projection that confines the unknown parameters in an estimated region. Then the robustness of the closed‐loop system is guaranteed by leveraging the deterministic robust control philosophy. Also, a command filter has been designed to avoid computing complex derivatives of virtual control law in the backstepping procedure. Secondly, a composite adaptation law that intelligently integrates information of both tracking error and prediction error is constructed for improving the performance of controllers. The high adaptation gain of CARC leads to higher parameter convergence speed and smaller tracking errors. Moreover, if the actual parameters are within the design range, asymptotic stability can be achieved in presence of parametric uncertainties only. In addition, exponential convergence of tracking errors and parameters can be shown under certain conditions. Finally, numerical examples are made to validate the superiority and effectiveness of the proposed control scheme.

Stabilization of linear systems cascaded with a chain of infinite-delayed integrators

This paper considers stabilization of linear systems cascaded with a chain of infinite-delayed integrators. Distributed infinite input delay is also taken into consideration. The systems under consideration can be seen as a special class of linear strict-feedback systems with infinite delays. A novel predictor feedback controller is proposed via an n-step backstepping procedure. The Lyapunov theorem developed for infinite-delayed systems in our previous work is applied. It is shown that the proposed controller can globally stabilize the concerned systems. To our best knowledge, it is a first work on control of systems with simultaneous distributed infinite state and input delays. A simulation example is provided to illustrate the effectiveness of our controller.

Adaptive Fault-Tolerant Tracking Control for Uncertain Nonlinear Systems With Unknown Control Directions and Limited Resolution

This article addresses the adaptive fault-tolerant tracking control (FTTC) problem for a family of strict-feedback uncertain nonlinear systems subject to limited sensor resolution and unknown control directions. Both partial loss-of-effectiveness (LOE) and lock-in-place (LIP) faults of actuators are investigated. An adaptive control strategy based on a Nussbaum-type function and neural networks is presented by introducing a backstepping approach to make the system output track a desired reference output signal with bounded tracking error in the case of faulty actuators. The effect of the limited resolution is decoupled from the nonlinear system and is approximated using a neural network. The impact of disturbances is effectively compensated by utilizing adaptive parameter estimation terms in the backstepping procedure. It is proven that the proposed FTTC strategy can ensure the boundedness of all signals and guarantee that the output tracking error can converge into a small neighborhood of the origin. Two simulation examples are given to illustrate the effectiveness of the proposed FTTC strategy.

Adaptive Neural Network Control Using Nonlinear Information Gain for Permanent Magnet Synchronous Motors.

In this study, an adaptive neural network (NN) control using nonlinear information (NI) gain for permanent magnet synchronous motors (PMSMs) is proposed to improve control and estimation performance. The proposed method consists of a nonlinear controller, a three-layer NN approximator, and NI gain. The nonlinear controller is designed via a backstepping procedure for position tracking. The commutation scheme is designed to implement the PMSM control without the direct-quadrature (DQ) transform. The three-layer NN approximator is designed to estimate the unknown complex function generated by the recursive backstepping process. The NI gains are designed to enhance the control and estimation performance according to the increased tracking errors owing to the load torque and the desired position variations. All of signals in the closed-loop system guarantee the semiglobal uniformly ultimately boundness (UUB) using the Lyapunov stability theorem and the input-to-state stability (ISS) property. The performance of the proposed method was validated by experiments performed using a PMSM testbed.

Adaptive fault‐tolerant control of switched nonlinear systems based on tuning function method

AbstractThis paper solves the problem of adaptive fault‐tolerant control of uncertain switched nonlinear systems with unknown parameters and actuator failures. Based on the multi‐Lyapunov function method, a novel adaptive switched controller with tuning function is designed using the backstepping procedure. Meanwhile, a switched fault‐tolerant adaptation mechanism is also given to compensate the fault efficiency. Furthermore, it is certificated that all the signals of the closed‐loop system are uniformly bounded and the tracking error restrain to a small neighbor. Finally, the simulation results verify the feasibility of the proposed method.

Transients Improvement in the Output-Feedback Adaptive Backstepping Control of Nonlinear Systems with Input Constraints

The paper addresses the problem of transient performance improvement in the output-feedback adaptive backstepping control of parametrically uncertain nonlinear system represented in the cascaded output-feedback form. The system state is inaccessible for measurement, the system input is constrained, and the control gain is assumed unknown (its sign is known). The problem is solved by applying specially modified K-filters with adjustable parameter, using the backstepping procedure taking into account the input constraints, and involving the Kreisselmeier-like adaptation algorithm with a modified swapping scheme. The algorithm enables us to improve the closed-loop transient performance without increasing initial “swings” in the control error transients. The latter property is achieved by including the high-order time derivatives of the adjustable parameters, calculated by the adaptation algorithm, in the control law. The properties of the closed-loop system are illustrated via simulation.

Gradient-Descent-Based Learning Gain for Backstepping Controller and Disturbance Observer of Nonlinear Systems

This paper proposes a gradient-descent-based learning (GL) gain for backstepping controller and disturbance observer (DOB) of nonlinear system. The proposed method consists of the GL gain update law, controller, and DOB. The GL gain update law is proposed to adapt the control gain and DOB gain according to the direction that minimizes the cost function. The mathematical analysis reveals that the GL gain always has a positive sign and upper bound. The controller is designed via a backstepping procedure to track the desired output with GL control gain. The DOB is designed to estimate the unknown external disturbance with the GL DOB gain. Because the control and DOB gains are simultaneously tuned to achieve improved performance, the time consumption for tuning can be reduced. In addition, the peaking phenomenon can be avoided initially by a small initial value of GL gains. The stability of the closed-loop system is guaranteed using the input-to-state stability property. The performance of the proposed method was validated via simulations and experiments using a DC motor.

Finite-Time Adaptive NN Backstepping Dynamic Surface Control for Input-Delay Fractional-Order Nonlinear Systems

This article focuses on a finite-time adaptive dynamic surface control (DSC) approach for a kind of nonstrict fractional-order nonlinear systems (FONSs) with input delay. An auxiliary compensation function is presented by using the integral function of the input signal to handle input delay problem. To overcome the problem of inherent computational complexity, a fractional-order filter is applied to approach the virtual controller and its fractional-order derivative in each step of the backstepping procedure. By using backstepping technology and neural network (NN), the finite-time adaptive DSC approach is developed, and finite-time stability criteria based on fractional-order Lyapunov method are introduced to prove the finite-time convergence of the tracking error into a small region around the origin. The effectiveness of the presented control scheme is demonstrated by two examples.

Secure practical tracking of an uncertain robotic system under coarse conditions on reference trajectory and obstacles

AbstractThis article investigates practical tracking joint with obstacle‐avoidance for a class of uncertain robotic systems. The remarkable features are revealed from two aspects. First, more serious system uncertainties are allowed than the related literature due to the presence of unknown system matrices and mismatched disturbance in the investigated system. Second, only coarse conditions are required on reference trajectory and obstacles since their time derivatives are not necessarily to be available for feedback but those of the related literature muse be, while the reference trajectory is only required to be first rather than second order continuously differentiable. For this, a secure adaptive control framework is established by incorporating adaptive dynamic compensation mechanism into the backstepping procedure with the smart choice of adaptive law and the skillful injection of a barrier function. Consequently, a state‐feedback controller is explicitly designed which guarantees the obstacle‐avoidance of the robotic system while all the signals of the resulting closed‐loop system are bounded with system output practically tracking the reference trajectory. Finally, simulation results are provided for two specified robot systems to validate the effectiveness of the theoretical results.

Stabilization of chains of linear parabolic PDE–ODE cascades

Over the past decade, stabilization problems have been solved for various cascade and “sandwich” configurations involving linear ODEs and PDEs of both hyperbolic and parabolic types. In this paper, we consider systems in which the output of the (i+1)th ODE subsystem is the control input of the ith PDE subsystem, and in which the state of the ith PDE subsystem enters as control into the ith ODE subsystems. We extend the existing results, among which a representative one is for the case where the ODEs in the chain are scalar and the PDEs are pure delays, in two major directions. First, we allow for the virtual inputs to be affected by PDE dynamics different from pure delays: we allow the PDEs to include diffusion, i.e., to be parabolic, and to even have counter-convection, and, in addition, for the PDE dynamics to enter the ODEs not only with the PDE’s boundary value but also in a spatially-distributed (integrated) fashion. Second, we allow the ODEs in the chain to be not just scalar ODEs in a strict-feedback configuration but general LTI systems. We develop an n-step backstepping procedure and prove that the resulting closed-loop system is exponentially stable. A simulation example is provided to illustrate the effectiveness of our controllers.

Prescribed Performance Control for Multiagent Systems via Fuzzy Adaptive Event-Triggered Strategy

This article discusses the prescribed performance control problem for multiagent systems involving the state triggering and the controller output triggering simultaneously. To successfully apply the backstepping technique in the event-triggered control design, the virtual control signal is constructed by the original system state. Compared with the existing results on the prescribed performance, a new barrier Lyapunov function is developed with considering the characteristics of multiagent systems, and a fuzzy adaptive event-triggered control protocol is proposed by the backstepping procedure. It guarantees that the consensus tracking error converges to a predefined region of the origin in a preset finite time. At the same time, all other closed-loop signals remain bounded without the Zeno behavior. Finally, a simulation example confirms the availability of the developed control scheme with a comparison.

Saturation-Tolerant Prescribed Control for a Class of MIMO Nonlinear Systems.

This article proposes a saturation-tolerant prescribed control (SPC) for a class of multiinput and multioutput (MIMO) nonlinear systems simultaneously considering user-specified performance, unmeasurable system states, and actuator faults. To simplify the control design and decrease the conservatism, tunnel prescribed performance (TPP) is proposed not only with concise form but also smaller overshoot performance. By introducing non-negative modified signals into TPP as saturation-tolerant prescribed performance (SPP), we propose SPC to guarantee tracking errors not to violate SPP constraints despite the existence of saturation and actuator faults. Namely, SPP possesses the ability of enlarging or recovering the performance boundaries flexibly when saturations occur or disappear with the help of these non-negative signals. A novel auxiliary system is then constructed for these signals, which bridges the associations between input saturation errors and performance constraints. Considering nonlinearities and uncertainties in systems, a fuzzy state observer is utilized to approximate the unmeasurable system states under saturations and unknown actuator faults. Dynamic surface control is employed to avoid tedious computations incurred by the backstepping procedures. Furthermore, the closed-loop state errors are guaranteed to a small neighborhood around the equilibrium in finite time and evolved within SPP constraints although input saturations and actuator faults occur. Finally, comparative simulations are presented to demonstrate the feasibility and effectiveness of the proposed control scheme.

Fault detection and isolation for a class of uncertain nonlinear systems: A dual-monitoring model-based approach

This paper presents a fault detection and isolation (FDI) scheme for a class of uncertain nonlinear systems with unknown faults, in which fault isolation process and control reconfiguration are integrated via a supervisory switching strategy. A group of adaptive estimators is performed together with the switching process. The proposed FDI approach, namely dual-monitoring model-based approach, can exploit information both from backstepping procedure and estimators, by which a set of monitoring functions is constructed to detect the occurrence of fault and also identify the specific fault type. In particular, based on some principles, the fault isolation process can be accelerated under a novel interacted isolation scheme. It can be seen that the tracking error can be always guaranteed within a prescribed performance in the existence of system faults.

Adaptive fuzzy asymptotic control design for MIMO nonlinear systems with state constraints

This paper addresses the asymptotic control problem of uncertain multi-input and multi-output (MIMO) nonlinear systems. The considered MIMO systems contain unknown virtual control coefficients (UVCCs) and state constraints. A creative Lyapunov function by associating with the lower bounds of UVCCs is presented to counteract the adverse effect deriving from UVCCs. The state constraints are ensured by utilising the barrier Lyapunov function. Moreover, the asymptotic tracking controller is recursively constructed by combining the backstepping technique with fuzzy logic systems. The remarkable character of the designed controller is that the asymptotic tracking performance can be achieved by introducing some smooth functions into adaptive backstepping procedure. In contrast to the existing results, the conditions on the UVCCs are relaxed. Finally, the new control design is illustrated by a practical example.

Saturated-command-deviation based finite-time adaptive control for dynamic positioning of USV with prescribed performance

This paper investigates the finite-time adaptive dynamic positioning (DP) control problem of an unmanned surface vehicle (USV) with prescribed performance subject to time-varying disturbances, unknown model parameters and input saturation. First, the error transformation is performed to provide a framework for the design of model-free prescribed performance DP controller, which eliminates the need for accurate prior information of the model and external disturbances. Then, a saturated-command-deviation based compensated law is proposed by incorporating the fuzzy supervisory approach to account for the excess control efforts resulted by input saturation. Subsequently, the fuzzy logic systems (FLS) integrated with minimum learning parameters technique are employed to approximate the model-based and high-order terms, such that the explosion of computational complexity is removed and requirements for accurate modeling are eliminated. Meanwhile, to suppress the negative effects induced by time-varying disturbances and unknown approximation errors, the adaptive technique is adopted to estimate the upper boundary of compounded disturbances online. Finally, by incorporating the softening sign signal and piecewise function into recursive backstepping procedure, a DP controller is developed to avoid potential singularity and achieve finite time convergence. Theoretical analysis and simulation results successfully illustrate the effectiveness of the proposed scheme.

Decentralized Adaptive Funnel Control of Uncertain Large-Scale Interconnected Nonlinear System

In this paper, a decentralized adaptive funnel control for a class of uncertain large-scale nonlinear systems is studied. By using a new funnel control method with a barrier Lyapunov function, a new adaptive decentralized funnel controller is recursively constructed to achieve output tracking of given transient behavior. To solve the interconnection terms issues in controller design, an auxiliary nonlinear function is introduced. Meanwhile, dynamic surface control is applied to the backstepping design in order to avoid the “explosion of complexity” problems in backstepping procedure. It is demonstrated that the proposed control scheme proves that all signals of the closed-loop system are bounded and that the tracking error always evolves in the funnel. Finally, the effectiveness of the scheme is verified by a numerical example and a third-order power network system model.

Secure Adaptive Trajectory Tracking Control for Nonlinear Robot Systems Under Multiple Dynamic Obstacles: Safety Barrier Certificates

This article studies the problem of obstacle-avoidance tracking control for a class of uncertain nonlinear robot systems in multiple-dynamic-obstacles environment. The main challenge focuses on how to simultaneously ensure tracking performance and obstacle avoidance. The existing collision-avoidance tracking control schemes cannot guarantee the tracking performance inside the obstacle detection region, since the use of additive Lyapunov-barrier function (LBF) generates the dynamic mismatching. To overcome this difficulty, a novel integral-multiplicative LBF is constructed. The adaptive mechanism is designed to compensate for the mismatching uncertainties. By incorporating the barrier function into backstepping procedure, a secure adaptive tracking control scheme is proposed. Compared with the existing results, the proposed control scheme can simultaneously achieve the obstacle avoidance and tracking performance regardless of being inside the obstacle sensing region and unknown nonlinear uncertainties.

Adaptive Backstepping Control of Uncertain Sandwich-Like Nonlinear Systems With Deadzone Nonlinearity

A systematic differentiator-based adaptive backstepping control methodology is proposed for a class of sandwich-like nonlinear system with unknown state-dependent deadzone nonlinearity and parametric uncertainties. The novelty of our approach is that a high-order sliding mode differentiator is utilized to estimate the nonstrict feedback coupling term resulting from the sandwiched deadzone, and all the outputs of the differentiator are integrated into the backstepping procedure based on Lyapunov functions with flat zone recursively. By this approach, all the unknown parameters are estimated online, the discontinuity of the virtual input caused by bound estimations is avoided. It is shown that the ultimate boundedness of all the closed-loop signals is achieved and the output tracking error converges to a preset set. Simulation is performed to verify the theoretical findings.

Adaptive Tracking for Uncertain Switched Nonlinear Systems With Prescribed Performance Under Slow Switching

This article considers the problem of adaptive prescribed performance tracking control via slow switching for a general class of uncertain switched nonlinear systems (SNSs) with unmodeled dynamics (UDs) and nonstrict-feedback structure. The UDs are not in their form of the input-to-state practical stability and their state information is unmeasurable. By reassigning the function variables, the coupling effects between UDs and the prescribed performance function are eliminated through the iterative process. In virtue of the neural networks (NNs) approximation capability, a novel adaptive backstepping procedure is proposed without adding extra first-order filters. By choosing an appropriate slow switching law, all the signals of the closed-loop system are bounded, and the system output tracks the reference signal with a prescribed performance level (PPL).