Pure-feedback Nonlinear Systems Research Articles

Prescribed Performance Tracking Control of a Class of Uncertain Pure-Feedback Nonlinear Systems With Input Saturation

In this paper, we address the issue of the prescribed performance control of a class of pure-feedback nonlinear systems with uncertainties and input saturation. The active disturbance rejection control is adopted at each step of the backstepping technology. In detail, the extended state observer is employed to estimate unknown functions to compensate for uncertain items. The tracking differentiator is used to take place of the derivative of the virtual control signals, which overcomes the repeated differentiations of nonlinear functions. The control input limitation is handled with the auxiliary system, and predefined tracking performance functions are utilized to improve the tracking accuracy and speed. By means of the input-to-state stability and Lyapunov stability theory, the tracking error is proven to converge to arbitrarily small neighborhood of the origin. Two simulation examples are provided to demonstrate the effectiveness of the presented results.

Adaptive Fault-Tolerant Compensation Control and Its Application to Nonlinear Suspension Systems

In this paper, an adaptive fault-tolerant method is proposed to synthesize a compensation controller for the uncertain nonlinear pure-feedback systems possessing dead-zone actuators and stochastic failures. Each actuator’s failure mode is described by a scalar Markovian type function, which is not only much more practical in control engineering but also challenging in control theory. By exploring the adaptive backstepping methodology, a compensation scheme for actuator failure is presented to guarantee that the solution of the closed-loop system is a unique and bounded in probability. Importantly, the proposed control method can achieve arbitrarily small tracking error in the presence of nonlinear actuators with random failures. The case study of active suspension system using the adaptive fault-tolerant compensation controller shows the effectiveness of the presented method.

Adaptive finite-time output-feedback control design for switched pure-feedback nonlinear systems with average dwell time

In this paper, the issue of adaptive fuzzy tracking control is addressed for a class of completely non-affine uncertain switched pure-feedback nonlinear systems with unmeasurable states. In such a system, the average dwell time switching rule is adopted, and its stability is discussed for the first time in the sense of finite time. A fuzzy logic system-based switched observer is constructed to approximate unmeasurable states, which reduces the conservativeness stemming from utilizing a common observer for all subsystems. In addition, filter signals are applied to avoid algebraic loop problem existing in adoption of the controller. Then, a novel adaptive finite-time control strategy is developed on the basis of multiple Lyapunov functions approach, dynamic surface control and backstepping technique. It is proved that the designed controllers of subsystems can ensure that all the closed-loop signals are bounded under a class of switching signals with average dwell time, while the tracking error can converge to a small area of the origin in a finite time. Finally, two simulation examples are given to demonstrate the validity of the proposed control scheme.

Approximation-Based Fixed-Time Adaptive Tracking Control for a Class of Uncertain Nonlinear Pure-Feedback Systems

This paper examines approximation-based fixed-time adaptive tracking control for a class of uncertain nonlinear pure-feedback systems. Novel virtual and actual controllers are designed that resolve the meaninglessness of virtual and actual controllers at the origin and in the negative domain, and the sufficient condition for the system to have semiglobal fixed-time stability is also provided. Radial basis function neural networks are introduced to approximate unknown functions for solving the fixed-time control problem of unknown nonlinear pure-feedback systems, and the mean value theorem is used to solve the problem of nonaffine structure in nonlinear pure-feedback systems. The controllers designed in this paper ensure that all signals in the closed-loop system are semiglobally uniform and ultimately bounded in a fixed time. Two simulation results show that appropriate design parameters can limit the tracking error within a region of the origin in a fixed time.

Adaptive neural prescribed performance output feedback control of pure feedback nonlinear systems using disturbance observer

SummaryIn this study, an adaptive output feedback control with prescribed performance is proposed for unknown pure feedback nonlinear systems with external disturbances and unmeasured states. A novel prescribed performance function is developed and incorporated into an output error transformation to achieve tracking control with prescribed performance. To handle the unknown non‐affine nonlinearities and avoid the algebraic loop problem, the radial basis function neural network (RBFNN) is adopted to approximate the unknown non‐affine nonlinearities with the help of Butterworth low‐pass filter. Based on the output of the RBFNN, the coupled design between sate observer and disturbance observer is presented to estimate the unmeasured states and compounded disturbances. Then, the adaptive output feedback control scheme is proposed for unknown pure feedback nonlinear systems, where a first‐order filter is introduced to tackle with the issue of “explosion of complexity” in the traditional back‐stepping approach. The boundedness and convergence of the closed‐loop system are proved rigorously by utilizing the Lyapunov stability theorem. Finally, simulation studies are worked out to demonstrate the effectiveness of the proposed scheme.

Adaptive Dynamic Surface Control for a Class of Nonlinear Pure-Feedback Systems with Parameter Drift

In order to solve the problem of unknown parameter drift in the nonlinear pure-feedback system, a novel nonlinear pure-feedback system is proposed in which an unconventional coordinate transformation is introduced and a novel unconventional dynamic surface algorithm is designed to eliminate the problem of “calculation expansion” caused by the use of backstepping in the pure-feedback system. Meanwhile, a sufficiently smooth projection algorithm is introduced to suppress the parameter drift in the nonlinear pure-feedback system. Simulation experiments demonstrate that the designed controller ensures the global and ultimate boundedness of all signals in the closed-loop system and the appropriate designed parameters can make the tracking error arbitrarily small.

BLF-based adaptive DSC for a class of stochastic nonlinear systems of full state constraints with time delay and hysteresis input

This paper is concerned about barrier Lyapunov function(BLF)-based adaptive dynamic surface control(DSC) for a class of stochastic nonlinear systems of full state constraints with hysteresis input and time delay. All states of the system are guaranteed to be constrained in bounded compact set. The pure feedback nonlinear system is transformed into a strictly feedback nonlinear system with nonaffine nonlinear terms by using the mean value theorem. By using backstepping design of BLF combine with DSC technique, the explosion of complexity is avoided. Based on the neural network with the finite covering lemma, the time-delay functions are solved. Simultaneously, the backlash-like hysteresis input control in this paper is considered. The two cases of symmetric and asymmetric BLFs are discussed separately. An adaptive controller is designed to ensure that the output tracking error converges on a small region of the origin. Finally, the control scheme ensures that all signals in the closed-loop systems are semi-global uniformly ultimately bounded. Results of two simulation cases are presented to prove the effectivity of the theoretical analysis.

Fixed-Time Adaptive Neural Tracking Control for a Class of Uncertain Nonlinear Pure-Feedback Systems

This paper presents fixed-time adaptive neural tracking control for a class of uncertain nonlinear pure-feedback systems. To overcome the design difficulty arising from the nonaffine structure of nonlinear pure-feedback systems, the mean value theorem is introduced to separate the nonaffine appearance of nonlinear pure-feedback systems. Radial basis function (RBF) neural networks are employed to approximate designed unknown functions f̂ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ). By combining RBFs and Lyapunov functions, a novel fixed-time controller is designed, and semiglobal uniform ultimate boundedness of all signals in the closed-loop control system is guaranteed in a fixed time. Sufficient conditions are given to ensure that the system has semiglobal fixed-time stability. The main purpose of this paper is to design a controller for an unknown nonlinear pure-feedback system so that the system output y can track the reference signal yd. The simulation experiments indicate that the selection of sufficient design parameters makes the tracking error converge on a domain of the origin. Compared with the existing finite-time control and fixed-time control, the proposed fixed-time control scheme reduces the size of the tracking error.

High Order Sliding Mode Backstepping Control for a Class of Unknown Pure Feedback Nonlinear Systems

This paper aims at addressing the problem of finite time exact tracking control for a class of unknown nonlinear systems in pure feedback form while guaranteeing the closed-loop stability. The uncertain nonlinear system considered in this paper is not only in completely non-affine form but also explicitly dependent on the time and therefore, can cover a general class of nonlinear systems. By incorporating high order sliding mode controller into backstepping design procedure, a new robust control scheme with novel integrators is proposed, which is able to steer the tracking error to zero. Moreover, the closed loop stability can be proved with the help of the constructed integrators in every design step. Two simulation examples are presented to illustrate the correctness and effectiveness of the proposed control scheme.

Adaptive Neural Tracking Control for a Class of Pure-Feedback Systems With Output Constraints Based on Event-Triggered Strategy

In this paper, an adaptive event-triggered tracking control problem is considered for a class of pure-feedback nonlinear systems with output constraints. The mean value theorem is used to transform the pure-feedback system in non-affine form into a system in affine form. In addition, the radial basis function neural network (RBF NN) control is used to approximate the unknown nonlinear function in the system and the tracking error of the controller is limited to a small constant boundary by using the positive obstacle Lyapunov function. An adaptive controller for a class of pure-feedback systems is established, which based on the backstepping control theory and event-triggered control theory, it can ensure all the closed-loop signals are bounded and avoid the Zeno-behavior. The simulation results prove the effectiveness of the controller design.

Estimator-Based Control for Pure-Feedback Systems With Incomplete Measurements

This paper investigates estimator-based control problems for pure-feedback nonlinear systems with incomplete measurements due to the transmission packet losing or sensor saturation. The incomplete measurements can cause the state variables unavailable or distorted, which can degrade the performance of the system. To solve these problems, a state estimator is designed for the data-losing case, based on which two backstepping control methods are developed. The output of the system is subject to a prescribed constraint by using an obstacle Lyapunov function. By solving a linear matrix inequality, the stability conditions of the state estimator and closed-loop system are derived. It is proved that the control scheme can guarantee that all the signals of the closed-loop system are uniformly ultimately bounded in mean square. The effectiveness of the proposed methods is confirmed by simulations.

Intelligent Adaptive Tracking Controller Design for Stochastic Switched Pure-Feedback Nonlinear Systems With Input Saturation and Non-Lower Triangular Structure

This paper concentrates on the design of intelligent adaptive tracking controller for stochastic switched nonlinear pure-feedback systems with input saturation and non-lower triangular structure. It needs to be emphasized that both the issues of pure-feedback structure and non-differential saturation nonlinearity are involved in the studied system. With the help of the mean-value theorem, a novel intelligent adaptive tracking controller is developed in this work to overcome the difficulty resulted from pure-feedback structure, and the inherent property of Gaussian functions is utilized to handle functions that are unknown and include all state variables. Moreover, through the universal intelligent approximation technology, a novel control strategy is constructed under the framework of backstepping, which guarantees that the tracking error can converge to a small neighborhood near the origin in the sense of mean quartic value and all signals of the nonlinear closed-loop system can be bounded in probability. Eventually, the effectiveness of the presented scheme is further illustrated by the simulation of two practical examples.

Direct Adaptive Neural Network Control for Ship Manoeuvring Modelling Group Model-Based Uncertain Nonlinear Systems in Non-Affine Pure-Feedback Form

In this article, a direct neural network based adaptive backstepping control approach is proposed for a class of uncertain non-affine ship manoeuvring pure-feedback nonlinear systems. To carry out the backstepping design, the high fidelity 3 degrees of freedom Manoeuvring Modelling Group (MMG) model with external disturbances is transformed into the ship manoeuvring systems in non-affine pure-feedback form. Then, by combing the Implicit Function Theorem, Mean Value Theorem and dynamic surface control technique, the proposed approach is able to avoid completely the circularity problem and complexity growing problem exist in the adaptive neural network controller. During the controller design, the uncertain nonlinear functions are approximated by neural networks. Following this control approach, it is worth noting that the direct adaptive backstepping control for the high fidelity MMG model based ship manoeuvring nonlinear systems is achieved firstly, and the controller structure is simpler. Furthermore, it is shown via stability analysis that all signals in the closed-loop system are uniformly ultimately bounded. At last, two reference signals consist of a constant and a realistic performance requirement of ship are applied to simulation studies to illustrate the utility and merits of the proposed control scheme.

Robust adaptive prescribed performance dynamic surface control for uncertain nonlinear pure-feedback systems

A robust adaptive prescribed performance dynamic surface control design procedure for a class of nonlinear pure-feedback systems with both unknown vector parameters and unmodeled dynamics is presented. The unmodeled dynamics lie within some bounded functions, which are assumed to be partially known. A state transformation and an auxiliary system are proposed to avoid using the cumbersome formula to handle the non-affine structure. A radical-type Lyapunov function and L function are designed to ensure the prescribed performance of the pure-feedback system. Bounded value of finding the partial derivative of the non-affine function is avoided by designing special virtual controllers in using dynamic surface control. As illustrated by examples, the proposed adaptive prescribed performance dynamic surface control scheme is shown to guarantee the semi-global uniformly ultimately bounded, and the dynamic performance and steady-state performance of the tracking error depend on the prescribed performance functions.

Adaptive Control for Pure-Feedback Nonlinear Systems Preceded by Asymmetric Hysteresis

An adaptive control scheme is proposed for a class of uncertain pure-feedback nonlinear systems preceded by asymmetric hysteresis nonlinearity. The asymmetric property is described by the modified Bouc-Wen model based on the proposed asymmetric factor. State variables in the controller design are directly replaced with nonaffine functions to address the control problem caused by nonaffine appearance. Moreover, the control method can handle systems with external disturbances and guarantee the global stability of all the signals in the closed-loop system. The feasibility of the control scheme is verified by a simulation example and experimental results.

Event-triggering based adaptive neural tracking control for a class of pure-feedback systems with finite-time prescribed performance

In this paper, a finite-time tracking control problem is considered for a class of pure-feedback nonlinear systems with event-triggered strategy. The implicit function theorem and the mean value theorem are used to transform the pure-feedback nonlinear systems into strict feedback nonlinear systems. The neural network is adopted to approximate the unknown function and the tracking error is limited to a pre-given boundary by prescribed performance at a finite time. In addition, an improved event-triggered control strategy is proposed to obtain a larger threshold, and also the proposed controller can avoid the Zeno-behavior. Based on Lyapunov stability theory, the adaptive neural network controller can ensure that all the signals in the closed-loop system are uniformly ultimately bounded. Finally, the feasibility of this control scheme is proved by simulation.

Neuroadaptive Control Design for Pure-Feedback Nonlinear Systems: A One-Step Design Approach.

In this article, we propose a one-step control design approach for pure-feedback nonlinear systems in the presence of unmatched and nonvanishing external disturbances. Different from the commonly utilized backstepping design, the proposed method, integrated with the dynamic surface control (DSC) technique, only involves one-step design with one single Lyapunov function in the whole control synthesis, which derives the actual control and the intermediate controls simultaneously in a collective way, avoiding the repetitive design procedures and multiple Lyapunov functions, yet circumventing the issue of "explosion of complexity." Furthermore, with this method, the increase in system order does not increase the design and analysis complexity. Numerical simulation examples confirm and validate the effectiveness of the proposed method.

Observer-Based Fuzzy Adaptive Event-Triggered Control for Pure-Feedback Nonlinear Systems With Prescribed Performance

This paper studies the problem of fuzzy adaptive event-triggered control for a class of pure-feedback nonlinear systems, which contain unknown smooth functions and unmeasured states. Fuzzy logic systems are adopted to approximate unknown smooth functions and a fuzzy state observer is designed to estimate unmeasured states. Via the event-triggered control technique, the control signal of the fixed threshold strategy is obtained. By converting the tracking error into a new virtual error variable, an observer-based fuzzy adaptive event-triggered prescribed performance control strategy is designed. The key advantage is that the proposed method does not require a priori knowledge of partial derivatives of system functions, i.e., it relaxes the restrictive condition that the partial derivatives of system functions need to be known for pure-feedback nonlinear systems. Simulation results confirm the efficiency of the proposed method.

A novel alleviating fuzzy control algorithm for a class of nonlinear stochastic systems in pure-feedback form

In this paper, a novel alleviating computation adaptive fuzzy control scheme is presented for a class of uncertain stochastic nonlinear pure-feedback systems. Different from the existing results that are based on traditional back-stepping schemes as well as approximation technique of fuzzy logic systems (FLSs) for stochastic nonlinear systems, this new approach assumes that the norm of optimal approximation parameter vector of FLSs is bounded by unknown parameter. At each design step of this new approach, fewer bounded adaptive parameters need to be adjusted. Thus, this new approach can alleviate the online computation burden and improve the robust control performance. Meanwhile, combining Lyapunov theorem analysis, it is proven that all the signals in the closed-loop system are uniformly ultimately bounded (UUB) and the tracking error can converge to a small neighborhood of zero. The good performance of this approach is well demonstrated in a simulation example.

Dynamic backstepping control for pure-feedback non-linear systems

Abstract A dynamic backstepping control method is proposed for non-linear systems in the pure-feedback form, for which the traditional backstepping method suffers from solving the implicit non-linear algebraic equation. This method treats the implicit algebraic equation directly via a dynamic way, by augmenting the (virtual) controls as states during each recursive step. Compared with the traditional backstepping method, one more Lyapunov design is executed in each step. As new dynamics are included in the design, the resulting control law is in the dynamic feedback form. Under appropriate assumptions, the proposed control scheme achieves the uniformly asymptotic stability and the closed-loop system is local input-to-state stable for various disturbance. Moreover, the control law may be simplified to the inverse-free form by setting large gains, which will alleviate the problem of `explosion of terms’. The effectiveness of this method is illustrated by the stabilization and tracking numerical examples.