Globally Stable Adaptive Backstepping Neural Network Control for Uncertain Strict-Feedback Systems With Tracking Accuracy Known a Priori

Predefined Accuracy Adaptive Tracking Control for Nonlinear Multiagent Systems With Unmodeled Dynamics

This article focuses on an adaptive dynamic surface tracking control issue of nonlinear multiagent systems (MASs) with unmodeled dynamics and input quantization under predefined accuracy. Radial basis function neural networks (RBFNNs) are employed to estimate unknown nonlinear items. A dynamic signal is established to handle the trouble introduced by the unmodeled dynamics. Moreover, the predefined precision control is realized with the aid of two key functions. Unlike the existing works on nonlinear MASs with unmodeled dynamics, to avoid the issue of "explosion of complexity," the dynamic surface control (DSC) method is applied with the nonlinear filter. By using the designed controller, the consensus errors can gather to a precision assigned a priori. Finally, the simulation results are given to demonstrate the effectiveness of the proposed strategy.

Adaptive neural identification and non-singular control of pure-feedback nonlinear systems

This paper proposes a new constructive identification and adaptive control method for nonlinear pure-feedback systems, which remedies the 'explosion of complexity' and potential control singularity encountered in the traditional adaptive backstepping controllers. First, to avoid using the backstepping recursive design, alternative state variables and the corresponding coordinate transformation are introduced to reformulate the pure-feedback system into an equivalent canonical model. Then, a high-order sliding mode (HOSM) observer is used to reconstruct the unknown states for this canonical model. To remedy the potential singularity in the control, the unknown system dynamics are lumped to derive an alternative identification structure and one-step control synthesis, where two radial basis function neural networks (RBFNN) are adopted to online estimate these lumped dynamics. In this framework, the online estimation of control gain is not in the denominator of controller, and thus the division by zero in the controllers is avoided. Finally, a new online learning algorithm is constructed to obtain the RBFNNs' weights, ensuring the convergence to the neighborhood of true values and allowing accurate identification of unknown dynamics. Theoretical analysis elaborates that the convergence of both the tracking error and the estimation error is obtained simultaneously. Simulations and practical experiments on a hydraulic servo test-rig verify the effectiveness and utility of the suggested methods.

Prescribed Performance Tracking Control Under Uncertain Initial Conditions: A Neuroadaptive Output Feedback Approach

This work is concerned with the prescribed performance tracking control for a family of nonlinear nontriangular structure systems under uncertain initial conditions and partial measurable states. By combining neural network and variable separation technique, a state observer with a simple structure is constructed for output-based finite-time tracking control, wherein the issue of algebraic loop arising from a nontriangular structure is circumvented. Meanwhile, by using an error transformation, the developed control scheme is able to ensure tracking with a prescribed accuracy within a pregiven time at a preassigned convergence rate under any bounded initial condition, eliminating the long-standing initial condition dependence issue inherited with conventional prescribed performance control methods, and guaranteeing the predeterminability of convergence time simultaneously. Two simulation examples also demonstrate the effectiveness of the presented control strategy.

Globally Adaptive Neural Network Output-Feedback Control for Uncertain Nonlinear Systems

In this article, a globally neural-network-based adaptive control strategy with flat-zone modification is proposed for a class of uncertain output feedback systems with time-varying bounded disturbances. A high-order continuously differentiable switching function is introduced into the filter dynamics to achieve global compensation for uncertain functions, thus further to ensure that all the closed-loop signals are globally uniformity ultimately bounded (GUUB). It is proven that the output tracking error converges to the prespecified neighborhood of the origin. The effectiveness of the proposed control method is verified by two simulation examples.

Dynamic Gain Reduced-Order Observer-Based Global Adaptive Neural-Network Tracking Control for Nonlinear Time-Delay Systems

In this article, a globally adaptive neural-network tracking control strategy based on the dynamic gain observer is proposed for a class of uncertain output-feedback systems with unknown time-varying delays. A reduced-order observer with novel dynamic gain is proposed. An n th-order continuously differentiable switching function is constructed to achieve the continuous switching control of the system, thus further ensuring that all the closed-loop signals are globally uniformly ultimately bounded (GUUB). It is proved that by adjusting the designed parameters, the tracking error converges to a region which can be adjusted to be small enough. The effectiveness of the control scheme is demonstrated by two simulation examples.

Guaranteeing Global Stability for Neuro-Adaptive Control of Unknown Pure-Feedback Nonaffine Systems via Barrier Functions

Most existing approximation-based adaptive control (AAC) approaches for unknown pure-feedback nonaffine systems retain a dilemma that all closed-loop signals are semiglobally uniformly bounded (SGUB) rather than globally uniformly bounded (GUB). To achieve the GUB stability result, this article presents a neuro-adaptive backstepping control approach by blending the mean value theorem (MVT), the barrier Lyapunov functions (BLFs), and the technique of neural approximation. Specifically, we first resort the MVT to acquire the intermediate and actual control inputs from the nonaffine structures directly. Then, neural networks (NNs) are adopted to approximate the unknown nonlinear functions, in which the compact sets for maintaining the approximation capabilities of NNs are predetermined actively through the BLFs. It is shown that, with the developed neuro-adaptive control scheme, global stability of the resulting closed-loop system is ensured. Simulations are conducted to verify and clarify the developed approach.

NN event-triggered finite-time consensus control for uncertain nonlinear Multi-Agent Systems with dead-zone input and actuator failures

This paper develops a Neural Network (NN) event-triggered finite-time consensus control method for uncertain nonlinear Multi-Agent Systems (MASs) with dead-zone input and actuator failures. In practical applications, actuator failures would inevitably arise in MASs. And the time, pattern, and value of the failures are unknown. Besides, the actuators of MASs also suffer from dead-zone nonlinearity. No matter actuator failures or dead-zone input would dramatically affect the performance and stability of MASs. To address these issues, finite-time adaptive controllers capable of simultaneously compensating for actuator failures and dead-zone input are constructed by adopting the backstepping technology. Meanwhile, the NN control scheme is adopted to handle the unknown nonlinear dynamics of each agent. Furthermore, an event-triggered control mechanism is established that no longer requires continuous communication on the control network. Under the proposed control method, all followers achieve finite-time synchronization, irrespective of the presence of limited bandwidth, unknown failures, and dead-zone input. These results are demonstrated by simulations.

A novel prescribed performance controller for strict-feedback nonlinear systems with input constraints

This paper is devoted to the prescribed performance control (PPC) for a class of strict-feedback nonlinear systems with input saturation constraints. With the help of an improved tuning function, the system can achieve the desired steady-state and transient performance in the pre-designed time. A new error transformation function is introduced, which has inherent robustness, so it does not need to use any approximation technique or calculate the analytical derivative. Compared with the relevant results, the proposed scheme has the same lower complexity, but better transient and steady-state performance, although there exists uncertain nonlinearity and uncertain disturbances in the system. Finally, the correctness of the above algorithm is verified by simulation experiments.

Finite-Time Robust Intelligent Control of Strict-Feedback Nonlinear Systems With Flight Dynamics Application

The tracking control is investigated for a class of uncertain strict-feedback systems with robust design and learning systems. Using the switching mechanism, the states will be driven back by the robust design when they run out of the region of adaptive control. The adaptive design is working to achieve precise adaptation and higher tracking precision in the neural working domain, while the finite-time robust design is developed to make the system stable outside. To achieve good tracking performance, the novel prediction error-based adaptive law is constructed by considering the estimation performance. Furthermore, the output constraint is achieved by imbedding the barrier Lyapunov function-based design. The finite-time convergence and the uniformly ultimate boundedness of the system signal can be guaranteed. Simulation studies show that the proposed approach presents robustness and adaptation to system uncertainty.

Adaptive fuzzy command filtered backstepping control for uncertain pure-feedback systems

This paper considers tracking control of non-affine pure-feedback systems with uncertain disturbances; a new adaptive fuzzy command filtered backstepping controller is presented. First, the non-affine difficulty of pure-feedback systems is solved with the help of the mean-value theorem, and the fuzzy logic systems (FLSs) are adopted to estimate the system uncertainties which contain external disturbances. Next, a novel command-filtered technology is proposed to avoid the so-called "explosion of complexity" in the backstepping design process. The compensation mechanism is employed to eliminate the shortcoming of the traditional dynamic surface control. The presented control scheme ensures that all closed-loop signals are bounded, and the system output can track the given reference signal. The main contribution of this paper is that the developed approach is not only generalized to uncertain non-affine pure-feedback systems, which extends the practical applications of classical command filtered schemes, but also has an error compensation system, which improves tracking performance in comparison with existing dynamic surface control proposals. Finally, several simulation experiments validate the effectiveness and superiority of the proposed control scheme.

Event-Triggered Adaptive Neural Control for Fractional-Order Nonlinear Systems Based on Finite-Time Scheme

This article addresses the finite-time event-triggered adaptive neural control for fractional-order nonlinear systems. Based on the backstepping technique, a novel adaptive event-triggered control scheme is proposed, and finite-time stability criteria are introduced with the aim to ensure that the tracking error enters into a small region around the origin in finite time. Finally, the stability of the closed-loop system is ensured via a fractional Lyapunov function theory and two simulation examples were used to prove the validity of the designed control scheme.

Motion Planning and Adaptive Neural Tracking Control of an Uncertain Two-Link Rigid-Flexible Manipulator With Vibration Amplitude Constraint

This article deals with an uncertain two-link rigid-flexible manipulator with vibration amplitude constraint, intending to achieve its position control via motion planning and adaptive tracking approach. In motion planning, the motion trajectories for the two links of the manipulator are planned based on virtual damping and online trajectories correction techniques. The planned trajectories can not only guarantee that the two links can reach their desired angles, but also have the ability to suppress vibration, which can be adjusted to meet the vibration amplitude constraint by limiting the parameters of the planned trajectories. Then, the adaptive tracking controller is designed using the radial basis function neural network and the sliding mode control technique. The developed controller makes the two links of the manipulator track the planned trajectories under the uncertainties including unmodeled dynamics, parameter perturbations, and persistent external disturbances acting on the joint motors. The simulation results verify the effectiveness of the proposed control strategy and also demonstrate the superior performance of the motion planning and the tracking controller.

Observer-Based Neuro-Adaptive Optimized Control of Strict-Feedback Nonlinear Systems With State Constraints

This article proposes an adaptive neural network (NN) output feedback optimized control design for a class of strict-feedback nonlinear systems that contain unknown internal dynamics and the states that are immeasurable and constrained within some predefined compact sets. NNs are used to approximate the unknown internal dynamics, and an adaptive NN state observer is developed to estimate the immeasurable states. By constructing a barrier type of optimal cost functions for subsystems and employing an observer and the actor-critic architecture, the virtual and actual optimal controllers are developed under the framework of backstepping technique. In addition to ensuring the boundedness of all closed-loop signals, the proposed strategy can also guarantee that system states are confined within some preselected compact sets all the time. This is achieved by means of barrier Lyapunov functions which have been successfully applied to various kinds of nonlinear systems such as strict-feedback and pure-feedback dynamics. Besides, our developed optimal controller requires less conditions on system dynamics than some existing approaches concerning optimal control. The effectiveness of the proposed optimal control approach is eventually validated by numerical as well as practical examples.

Human-Gait-Based Tracking Control for Lower Limb Exoskeleton Robot

Research shows that it is practical for the normal human movement mechanism to assist the patients with stroke in robot-assisted gait rehabilitation. In passive training, the effect of rehabilitation training for patients can be improved by imitating normal human walking. To make the lower limb exoskeleton robot (LLER) move like a normal human, a tracking control scheme based on human gait data is proposed in this paper. The real human gait data is obtained from healthy subjects using a three-dimensional motion capture platform (3DMCP). Furthermore, the normal human motion characteristics are adopted to enhance the scientificity and effectiveness of assistant rehabilitation training using LLER. An adaptive radial basis function network (ARBFN) controller based on feed-forward control is presented to improve the trajectory tracking accuracy and tracking performance of the control system, where the ARBFN controller is deployed to predict the uncertain model parameters. The feed-forward controller based on the tracking errors is used to compensate for the input torque of LLER. The effectiveness of the presented control scheme is confirmed by simulation results based on experimental data.

Adaptive Approximation-Based Tracking Control for a Class of Unknown High-Order Nonlinear Systems With Unknown Powers

In this article, the problem of adaptive tracking control is tackled for a class of high-order nonlinear systems. In contrast to existing results, the considered system contains not only unknown nonlinear functions but also unknown rational powers. By utilizing the fuzzy approximation approach together with the barrier Lyapunov functions (BLFs), we present a new adaptive tracking control strategy. Remarkably, the BLFs are employed to determine a priori the compact set for maintaining the validity of fuzzy approximation. The primary advantage of this article is that the developed controller is independent of the powers and can be capable of ensuring global stability. Finally, two illustrative examples are given to verify the effectiveness of the theoretical findings.

Adaptive Neural Dynamic Surface Control With Prespecified Tracking Accuracy of Uncertain Stochastic Nonstrict-Feedback Systems

This article addresses the adaptive neural tracking control problem for a class of uncertain stochastic nonlinear systems with nonstrict-feedback form and prespecified tracking accuracy. Some radial basis function neural networks (RBF NNs) are used to approximate the unknown continuous functions online, and the desired controller is designed via the adaptive dynamic surface control (DSC) method and the gain suppressing inequality technique. Different from the reported works on uncertain stochastic systems, by combining some non-negative switching functions and dynamic surface method with the nonlinear filter, the design difficulty is overcome, and the control performance is analyzed by employing stochastic Barbalat's lemma. Under the constructed controller, the tracking error converges to the accuracy defined a priori in probability. The simulation results are shown to verify the availability of the presented control scheme.

Adaptive Fuzzy Tracking Control for a Class of Uncertain Switched Nonlinear Systems With Full-State Constraints and Input Saturations

In this article, an adaptive fuzzy tracking control scheme is developed for a class of uncertain switched nonlinear systems with input saturations and full-state constraints. First to surmount the design difficulty with respect to a saturation nonlinearity controller, a nonlinear smooth function approximating the nondifferential saturation function is introduced to establish a standard switched adaptive tracking control strategy based on the mean-value theorem and the input compensation technique. Then, invoking fuzzy-logic systems (FLSs), a novel analysis method of average dwell time for switched nonlinear systems with full-state constraints is proposed by using an artful logarithmic inequality. Furthermore, the designed adaptive controller can ensure that all the states of uncertain switched nonlinear systems are not to violate the set constraint bounds by employing barrier Lyapunov functions (BLFs), and that the system output tracking error can converge to a desired neighborhood of the origin within a suitable compact set. Finally, the simulation results are given to demonstrate the validity of the presented approach.

Adaptive Neural Network Finite-Time Dynamic Surface Control for Nonlinear Systems

This article addresses the problem of finite-time neural network (NN) adaptive dynamic surface control (DSC) design for a class of single-input single-output (SISO) nonlinear systems. Such designs adopt NNs to approximate unknown continuous system functions. To avoid the "explosion of complexity" problem, a novel nonlinear filter is developed in control design. Under the framework of adaptive backstepping control, an NN adaptive finite-time DSC design algorithm is proposed by adopting a smooth projection operator and finite-time Lyapunov stable theory. The developed control algorithm means that the tracking error converges to a small neighborhood of origin within finite time, which further verifies that all the signals of the controlled system possess globally finite-time stability (GFTS). Finally, both numerical and practical simulation examples and comparing results are provided to elucidate the superiority and effectiveness of the proposed control algorithm.

Neuroadaptive Fault-Tolerant Control Under Multiple Objective Constraints With Applications to Tire Production Systems

Many manufacturing systems not only involve nonlinearities and nonvanishing disturbances but also are subject to actuation failures and multiple yet possibly conflicting objectives, making the underlying control problem interesting and challenging. In this article, we present a neuroadaptive fault-tolerant control solution capable of addressing those factors concurrently. To cope with the multiple objective constraints, we propose a method to accommodate these multiple objectives in such a way that they are all confined in certain range, distinguishing itself from the traditional method that seeks for a common optimum (which might not even exist due to the complicated and conflicting objective requirement) for all the objective functions. By introducing a novel barrier function, we convert the system under multiple constraints into one without constraints, allowing for the nonconstrained control algorithms to be derived accordingly. The system uncertainties and the unknown actuation failures are dealt with by using the deep-rooted information-based method. Furthermore, by utilizing a transformed signal as the initial filter input, we integrate dynamic surface control (DSC) into backstepping design to eliminate the feasibility conditions completely and avoid off-line parameter optimization. It is shown that, with the proposed neuroadaptive control scheme, not only stable system operation is maintained but also each objective function is confined within the prespecified region, which could be asymmetric and time-varying. The effectiveness of the algorithm is validated via simulation on speed regulation of extruding machine in tire production lines.

Novel adaptive impedance control for exoskeleton robot for rehabilitation using a nonlinear time-delay disturbance observer

A new adaptive impedance, augmented with backstepping control, time-delay estimation, and a disturbance observer, was designed to perform passive-assistive rehabilitation motion. This was done using a rehabilitation robot whereby humans' musculoskeletal conditions were considered. This control scheme aimed to mimic the movement behavior of the user and to provide an accurate compensation for uncertainties and torque disturbances. Such disturbances were excited by constraints of input saturation of the robot's actuators, friction forces and backlash, several payloads of the attached upper-limb of each patient, and time delay errors. The designed impedance control algorithm would transfer the stiffness of the human upper limb to the developed impedance model via the measured user force. In the proposed control scheme, active rejection of disturbances would be achieved through the direct connection between such disturbances from the observer's output and the control input via the feedforward loop of the system. Furthermore, the computed control input does not require any precise knowledge of the robot's dynamic model or any knowledge of built-in torque-sensing units to provide the desirable physiotherapy treatment. Experimental investigations performed by two subjects were exhibited to support the benefits of the designed approach.

Event-triggered adaptive prescribed performance control of uncertain nonlinear systems with unknown control directions

The problem of event-triggered prescribed performance control for a class of uncertain nonlinear systems with unknown control directions and faults is investigated. Compared with the existing methods, a new set of error transformation functions is defined for the first time. Although no approximate structure is adopted, prescribed performance control (PPC) and event triggered control (ETC) are realized simultaneously for the nonlinear system considered in this paper for the first time. The proposed control scheme can guarantee that all closed-loop signals are bounded, and the tracking error, as well as all state errors, converges within the adjustable constraint functions. Finally, two simulation experiments verify the effectiveness of the proposed algorithm.

Sensorless speed control of SPIM using BS_PCH novel control structure and NNSM_SC MRAS speed observer

This paper propose a novel Port Controlled Hamiltonian_Backstepping (PCH_BS) control structure with online tuned parameters, in combination with the modified Stator Current Model Reference Adaptive Syatem (SC_MRAS) based on speed and flux estimator using Neural Networks(NN) and sliding mode (SM) for sensorless vector control of the six phase induction motor (SPIM). The control design is based on combination PCH and BS techniques to improve its performance and robustness. The combination of BS_PCH controller with speed estimator can compensate for the uncertainties caused by the machine parameter variations, measurement errors, and external load disturbances, enables very good static and dynamic performance of the sensorless drive system (perfect tuning of the speed reference values, fast response of the motor current and torque, high accuracy of speed regulation) in a wide speed range, and robust for the disturbances of the load, the speed variation and low speed. The proposed sensorless speed control scheme is validated through Matlab-Simulink. The simulation results verify the effectiveness of the proposed control and observer.

Barrier Function-Based Adaptive Control for Uncertain Strict-Feedback Systems Within Predefined Neural Network Approximation Sets

In this article, a globally stable adaptive control strategy for uncertain strict-feedback systems is proposed within predefined neural network (NN) approximation sets, despite the presence of unknown system nonlinearities. In contrast to the conventional adaptive NN control results in the literature, a primary benefit of the developed approach is that the barrier Lyapunov function is employed to predefine the compact set for maintaining the validity of NN approximation at each step, thus accomplishing the global boundedness of all the closed-loop signals. Simulation results are performed to clarify the effectiveness of the proposed methodology.

Echo State Network-Based Backstepping Adaptive Iterative Learning Control for Strict-Feedback Systems: An Error-Tracking Approach

In this article, an echo state network (ESN)-based backstepping adaptive iterative learning control scheme is proposed for nonlinear strict-feedback systems performing the same operation repeatedly over a finite-time interval. Different from most of the output tracking approaches, an error-tracking approach is presented using the backstepping technique, such that the tracking error can follow a prespecified error trajectory without any requirement on the initial value of system states. Then, a novel Lyapunov function is constructed to deal with the unknown state-dependent gain function of the controller design. The uncertain nonlinearities are approximated by employing ESNs with simple feedback structures, and the weight update laws are developed by combining the parameter adaptation in the time domain and iteration domain. Moreover, the proposed control scheme is further extended to handle the strict-feedback systems with input saturations. Through the Lyapunov-like synthesis, the closed-loop stability and error convergence of the proposed error-tracking control scheme are analyzed in the presence of the approximation errors. Numerical simulations are provided to verify the effectiveness of the proposed scheme.

Neural-Network-Based Adaptive Funnel Control for Servo Mechanisms With Unknown Dead-Zone

This paper proposes an adaptive funnel control (FC) scheme for servo mechanisms with an unknown dead-zone. To improve the transient and steady-state performance, a modified funnel variable, which relaxes the limitation of the original FC (e.g., systems with relative degree 1 or 2), is developed using the tracking error to replace the scaling factor. Then, by applying the error transformation method, the original error is transformed into a new error variable which is used in the controller design. By using an improved funnel function in a dynamic surface control procedure, an adaptive funnel controller is proposed to guarantee that the output error remains within a predefined funnel boundary. A novel command filter technique is introduced by using the Levant differentiator to eliminate the "explosion of complexity" problem in the conventional backstepping procedure. Neural networks are used to approximate the unknown dead-zone and unknown nonlinear functions. Comparative experiments on a turntable servo mechanism confirm the effectiveness of the devised control method.

Adaptive Global Sliding-Mode Control for Dynamic Systems Using Double Hidden Layer Recurrent Neural Network Structure

In this paper, a full-regulated neural network (NN) with a double hidden layer recurrent neural network (DHLRNN) structure is designed, and an adaptive global sliding-mode controller based on the DHLRNN is proposed for a class of dynamic systems. Theoretical guidance and adaptive adjustment mechanism are established to set up the base width and central vector of the Gaussian function in the DHLRNN structure, where six sets of parameters can be adaptively stabilized to their best values according to different inputs. The new DHLRNN can improve the accuracy and generalization ability of the network, reduce the number of network weights, and accelerate the network training speed due to the strong fitting and presentation ability of two-layer activation functions compared with a general NN with a single hidden layer. Since the neurons of input layer can receive signals which come back from the neurons of output layer in the output feedback neural structure, it can possess associative memory and rapid system convergence, achieving better approximation and superior dynamic capability. Simulation and experiment on an active power filter are carried out to indicate the excellent static and dynamic performances of the proposed DHLRNN-based adaptive global sliding-mode controller, verifying its best approximation performance and the most stable internal state compared with other schemes.

Adaptive neural network terminal sliding mode control for uncertain spatial robot

The tracking control problem for uncertain spatial robot is investigated by means of adaptive terminal sliding mode control in this article. To approximate unknown nonlinear functions of these systems, a neural network model is employed. By using Lyapunov stability theory, adaptive terminal sliding mode controller is given, which guarantees that the tracking error converges to an arbitrary small region of zero and all the signals remain bounded. Finally, numerical simulation is given to confirm the effectiveness of the proposed method.

Adaptive Reinforcement Learning Control Based on Neural Approximation for Nonlinear Discrete-Time Systems With Unknown Nonaffine Dead-Zone Input

In this paper, an optimal control algorithm is designed for uncertain nonlinear systems in discrete-time, which are in nonaffine form and with unknown dead-zone. The main contributions of this paper are that an optimal control algorithm is for the first time framed in this paper for nonlinear systems with nonaffine dead-zone, and the adaptive parameter law for dead-zone is calculated by using the gradient rules. The mean value theory is employed to deal with the nonaffine dead-zone input and the implicit function theory based on reinforcement learning is appropriately introduced to find an unknown ideal controller which is approximated by using the action network. Other neural networks are taken as the critic networks to approximate the strategic utility functions. Based on the Lyapunov stability analysis theory, we can prove the stability of systems, i.e., the optimal control laws can guarantee that all the signals in the closed-loop system are bounded and the tracking errors are converged to a small compact set. Finally, two simulation examples demonstrate the effectiveness of the design algorithm.

Adaptive Neural Dynamic Surface Control for Nonstrict-Feedback Systems With Output Dead Zone

This paper focuses on the problem of adaptive output-constrained neural tracking control for uncertain nonstrict-feedback systems in the presence of unknown symmetric output dead zone and input saturation. A Nussbaum-type function-based dead-zone model is introduced such that the dynamic surface control approach can be used for controller design. The variable separation technique is employed to decompose the unknown function of entire states in each subsystem into a series of smooth functions. Radial basis function neural networks are utilized to approximate the unknown black-box functions derived from Young's inequality. With the help of auxiliary first-order filters, the dimensions of neural network input are reduced in each recursive design. A main advantage of the proposed method is that for an -order nonlinear system, only one adaptation parameter needs to be tuned online. It is rigorously shown that the proposed output-constrained controller guarantees that all the closed-loop signals are semiglobal uniformly ultimately bounded and the tracking error never violates the output constraint.

Robust Adaptive Fuzzy Control of a Class of Uncertain Nonlinear Systems With Unstable Dynamics and Mismatched Disturbances

This paper studies the robust stabilization problem for a class of uncertain nonlinear systems with unstable zero dynamics. The considered zero dynamic is not assumed to be input-to-state practically stable and contains nonlinear uncertainties and mismatched external disturbances. A new robust adaptive fuzzy control method is developed by combining theory with backstepping technique. First, an ideal virtual control function is designed, which can guarantee the zero dynamic asymptotically stable with a suboptimal performance. Then, based on some non-negative functions and backstepping design, the actual controller is constructed for the overall system, which ensures that the tracking error for the ideal virtual control signal converges to a priori accuracy regardless of external disturbances. In this design, an auxiliary signal is introduced to overcome the difficulties from the unavailable virtual reference signal. By exploiting the implicit function theorem, the proposed design technique is directly applied to a special case, where the zero dynamic is partially linear. A two inverted pendulums is used to illustrate the application and effectiveness of the proposed design method.

Neuroadaptive Control With Given Performance Specifications for MIMO Strict-Feedback Systems Under Nonsmooth Actuation and Output Constraints

This paper studies the prescribed performance tracking control problem for a class of multi-input multi-output strict-feedback systems with asymmetric nonsmooth actuator characteristics and output constraints as well as unexpected external disturbances. By combining a novel speed transformation with barrier Lyapunov function, a neural adaptive control scheme is developed that is able to achieve given tracking precision within preassigned finite time at prespecified converging mode. At each of the first $n-1$ steps of backstepping design, we make use of the radial basis function neural networks to cope with the uncertainties arising from unknown and time-varying virtual control gains, and in the last step, we introduce a matrix factorization technique to remove the restrictive requirement on the unknown control gain matrix and its NN-approximation, simplifying control design. Furthermore, to reduce the number of parameters to be online updated, we introduce a virtual parameter to handle the lumped uncertainties, resulting in a control scheme with low complexity and inexpensive computations. The effectiveness of the proposed control strategy is validated by systematic stability analysis and numerical simulation.

Event-Based Adaptive NN Tracking Control of Nonlinear Discrete-Time Systems

This paper is concerned with the simultaneous design of a neural network (NN)-based adaptive control law and an event-triggering condition for a class of strict feedback nonlinear discrete-time systems. The stability and tracking performance of the closed-loop network control system under the event-triggering strategy is formally proven based on the Lyapunov theory in a hybrid framework. The proposed Lyapunov formulation yields an event-triggered algorithm to update the control input and NN weights based on conditions involving the closed-loop state. Different from the existing traditional NN control schemes where the feedback signals are transmitted and executed periodically, the feedback signals are transmitted and executed only when the event-trigger error exceeds the specified threshold, which can largely reduce the communication load. The effectiveness of the approach is evaluated through a simulation example.

Passive and active rehabilitation control of human upper-limb exoskeleton robot with dynamic uncertainties

This paper investigates the passive and active control strategies to provide a physical assistance and rehabilitation by a 7-DOF exoskeleton robot with nonlinear uncertain dynamics and unknown bounded external disturbances due to the robot user's physiological characteristics. An Integral backstepping controller incorporated with Time Delay Estimation (BITDE) is used, which permits the exoskeleton robot to achieve the desired performance of working under the mentioned uncertainties constraints. Time Delay Estimation (TDE) is employed to estimate the nonlinear uncertain dynamics of the robot and the unknown disturbances. To overcome the limitation of the time delay error inherent of the TDE approach, a recursive algorithm is used to further reduce its effect. The integral action is employed to decrease the impact of the unmodeled dynamics. Besides, the Damped Least Square method is introduced to estimate the desired movement intention of the subject with the objective to provide active rehabilitation. The controller scheme is to ensure that the robot system performs passive and active rehabilitation exercises with a high level of tracking accuracy and robustness, despite the unknown dynamics of the exoskeleton robot and the presence of unknown bounded disturbances. The design, stability, and convergence analysis are formulated and proven based on the Lyapunov–Krasovskii functional theory. Experimental results with healthy subjects, using a virtual environment, show the feasibility, and ease of implementation of the control scheme. Its robustness and flexibility to deal with parameter variations due to the unknown external disturbances are also shown.

Adaptive Neural Output-Feedback Control for a Class of Nonlower Triangular Nonlinear Systems With Unmodeled Dynamics

This paper presents the development of an adaptive neural controller for a class of nonlinear systems with unmodeled dynamics and immeasurable states. An observer is designed to estimate system states. The structure consistency of virtual control signals and the variable partition technique are combined to overcome the difficulties appearing in a nonlower triangular form. An adaptive neural output-feedback controller is developed based on the backstepping technique and the universal approximation property of the radial basis function (RBF) neural networks. By using the Lyapunov stability analysis, the semiglobally and uniformly ultimate boundedness of all signals within the closed-loop system is guaranteed. The simulation results show that the controlled system converges quickly, and all the signals are bounded. This paper is novel at least in the two aspects: 1) an output-feedback control strategy is developed for a class of nonlower triangular nonlinear systems with unmodeled dynamics and 2) the nonlinear disturbances and their bounds are the functions of all states, which is in a more general form than existing results.

Observer-Based Adaptive Fault-Tolerant Tracking Control of Nonlinear Nonstrict-Feedback Systems

This paper studies an output-based adaptive fault-tolerant control problem for nonlinear systems with nonstrict-feedback form. Neural networks are utilized to identify the unknown nonlinear characteristics in the system. An observer and a general fault model are constructed to estimate the unavailable states and describe the fault, respectively. Adaptive parameters are constructed to overcome the difficulties in the design process for nonstrict-feedback systems. Meanwhile, dynamic surface control technique is introduced to avoid the problem of "explosion of complexity". Furthermore, based on adaptive backstepping control method, an output-based adaptive neural tracking control strategy is developed for the considered system against actuator fault, which can ensure that all the signals in the resulting closed-loop system are bounded, and the system output signal can be regulated to follow the response of the given reference signal with a small error. Finally, the simulation results are provided to validate the effectiveness of the control strategy proposed in this paper.

Differentiation-Free Multiswitching Neuroadaptive Control of Strict-Feedback Systems

Issues of differentiation-free multiswitching neuroadaptive tracking control of strict-feedback systems are presented. It mainly consists of a set of nominal adaptive neural network compensators plus an auxiliary switched linear controller that ensures the semiglobally/globally ultimately uniformly bounded stability when the unknown nonlinearities are locally/globally linearly bounded, respectively. In particular, the so-called explosion of complexity is annihilated in two steps. First, a set of first-order low-pass filters are constructed for solving such a problem in the nominal neural compensators. In contrast to most existing dynamic surface control-based schemes, bounded stability of the filter dynamics is ensured by virtue of the localness and hence boundedness of the neural compensators. Separation of controller-filter pairs is thus achieved in this paper. Next, an auxiliary switched linear state feedback control is synthesized to further solve such a problem in the nonneural regions. Besides being differentiation-free, such an approach provides more flexibility for meeting various control objectives at a time. An earlier proposed smooth switching algorithm is also incorporated to tackle the control singularity problem. Finally, simulation works are presented to demonstrate the validity of the proposed scheme.

Pattern-Based NN Control of a Class of Uncertain Nonlinear Systems

This paper presents a pattern-based neural network (NN) control approach for a class of uncertain nonlinear systems. The approach consists of two phases of identification and another two phases of recognition and control. First, in the phase (i) of identification, adaptive NN controllers are designed to achieve closed-loop stability and tracking performance of nonlinear systems for different control situations, and the corresponding closed-loop control system dynamics are identified via deterministic learning. The identified control system dynamics are stored in constant radial basis function (RBF) NNs, and a set of constant NN controllers are constructed by using the obtained constant RBF networks. Second, in the phase (ii) of identification, when the plant is operated under different or abnormal conditions, the system dynamics under normal control are identified via deterministic learning. A bank of dynamical estimators is constructed for all the abnormal conditions and the learned knowledge is embedded in the estimators. Third, in the phase of recognition, when one identified control situation recurs, by using the constructed estimators, the recurred control situation will be rapidly recognized. Finally, in the phase of pattern-based control, based on the rapid recognition, the constant NN controller corresponding to the current control situation is selected, and both closed-loop stability and improved control performance can be achieved. The results presented show that the pattern-based control realizes a humanlike control process, and will provide a new framework for fast decision and control in dynamic environments. A simulation example is included to demonstrate the effectiveness of the approach.

Model-Based Adaptive Event-Triggered Control of Strict-Feedback Nonlinear Systems

This paper is concerned with the adaptive event-triggered control problem of nonlinear continuous-time systems in strict-feedback form. By using the event-sampled neural network (NN) to approximate the unknown nonlinear function, an adaptive model and an associated event-triggered controller are designed by exploiting the backstepping method. In the proposed method, the feedback signals and the NN weights are aperiodically updated only when the event-triggered condition is violated. A positive lower bound on the minimum intersample time is guaranteed to avoid accumulation point. The closed-loop stability of the resulting nonlinear impulsive dynamical system is rigorously proved via Lyapunov analysis under an adaptive event sampling condition. In comparing with the traditional adaptive backstepping design with a fixed sample period, the event-triggered method samples the state and updates the NN weights only when it is necessary. Therefore, the number of transmissions can be significantly reduced. Finally, two simulation examples are presented to show the effectiveness of the proposed control method.

Composite Intelligent Learning Control of Strict-Feedback Systems With Disturbance

This paper addresses the dynamic surface control of uncertain nonlinear systems on the basis of composite intelligent learning and disturbance observer in presence of unknown system nonlinearity and time-varying disturbance. The serial-parallel estimation model with intelligent approximation and disturbance estimation is built to obtain the prediction error and in this way the composite law for weights updating is constructed. The nonlinear disturbance observer is developed using intelligent approximation information while the disturbance estimation is guaranteed to converge to a bounded compact set. The highlight is that different from previous work directly toward asymptotic stability, the transparency of the intelligent approximation and disturbance estimation is included in the control scheme. The uniformly ultimate boundedness stability is analyzed via Lyapunov method. Through simulation verification, the composite intelligent learning with disturbance observer can efficiently estimate the effect caused by system nonlinearity and disturbance while the proposed approach obtains better performance with higher accuracy.