Pure-feedback Nonlinear Systems Research Articles

Adaptive dynamic surface-based differential games for a class of pure-feedback nonlinear systems with output constraints

ABSTRACTThis paper investigates the zero-sum differential game problem for a class of uncertain nonlinear pure-feedback systems with output constraints and unknown external disturbances. A barrier Lyapunov function is introduced to tackle the output constraints. By constructing an affine variable at each dynamic surface control design step rather than utilising the mean-value theorem, the tracking control problem for pure-feedback systems can be transformed into an equivalent zero-sum differential game problem for affine systems. Then, the solution of associated Hamilton–Jacobi–Isaacs equation can be obtained online by using the adaptive dynamic programming technique. Finally, the whole control scheme that is composed of a feedforward dynamic surface controller and a feedback differential game control strategy guarantees the stability of the closed-loop system, and the tracking error is remained in a bounded compact set. The simulation results demonstrate the effectiveness of the proposed control scheme.

Fuzzy observer-based sampled-data control for a class of pure-feedback nonlinear systems

This paper aims at the sampled-data control problem for a class of pure-feedback nonlinear systems. A fuzzy state observer is constructed to evaluate the unavailable states. In this process, fuzzy logic systems are applied to approximate the uncertain nonlinear functions. Based on the new designed state observer, a sampled-data control scheme for the pure-feedback nonlinear systems is proposed. The designed sampled-data controller ensures the boundedness of the nonlinear systems. Finally, two numerical examples are used to demonstrate that the proposed method is efficient.

Finite-Time Adaptive Fuzzy Tracking Control Design for Nonlinear Systems

This paper addresses the finite-time tracking problem of nonlinear pure-feedback systems. Unlike the literature on traditional finite-time stabilization, in this paper the nonlinear system functions, including the bounding functions, are all totally unknown. Fuzzy logic systems are used to model those unknown functions. To present a finite-time control strategy, a criterion of semiglobal practical stability in finite time is first developed. Based on this criterion, a novel adaptive fuzzy control scheme is proposed by a backstepping technique. It is shown that the presented controller can guarantee that the tracking error converges to a small neighborhood of the origin in a finite time, and the other closed-loop signals remain bounded. Finally, two examples are used to test the effectiveness of proposed control strategy.

Adaptive Neural Control of Pure-Feedback Nonlinear Systems With Event-Triggered Communications.

This paper is concerned with the adaptive event-triggered control problem for a class of pure-feedback nonlinear systems. Unlike the existing results where the control execution is periodic, the new proposed scheme updates the controller and the neural network weights only when desired control specifications cannot be guaranteed. Clearly, this can largely reduce the amount of transmission data. Besides, since the event-trigger error is discontinuous because of the event-triggering mechanism, the stability analysis in the classical sense may not be guaranteed. To solve this problem, we formulate the event-triggered network control systems as a nonlinear impulsive dynamical system, and a novel Lyapunov theorem is used to show the stability properties of the closed-loop systems. Finally, two simulation examples are given to illustrate the effectiveness of the theoretical results.

Adaptive Neural Tracking Control of Switched Stochastic Pure-Feedback Nonlinear Systems With Unknown Bouc-Wen Hysteresis Input.

This paper aims to analyze the problem of adaptive neural network (NN) tracking control for a class of switched stochastic nonlinear pure-feedback systems with unknown direction hysteresis. In the light of recent studies on the hysteresis phenomenon in the field of nonlinear switched systems, this paper focuses on Bouc-Wen hysteresis model with unknown parameters and direction conditions. To simplify the control design, the following procedure is applied. Prior to tackling the unknown direction hysteresis problem based on the Nussbaum function and the backstepping techniques, the pure-feedback structure difficulty is governed by the mean value theorem. Furthermore, an optimized adaptation method is utilized to cope with computational burden. Universal approximation capability of radial basis function NNs and Lyapunov function method is synthesized to develop an adaptive NN tracking control scheme. It is demonstrated that under arbitrary deterministic switching, the presented controller can guarantee that all signals in the closed-loop system are semiglobally uniformly ultimately bounded in probability and the tracking error converges to a neighborhood of the origin. Finally, two simulation examples are given to illustrate the advantages of the proposed control design approach.

Fuzzy-approximation adaptive fault-tolerant control for nonlinear pure-feedback systems with unknown control directions and sensor failures

In this paper, the problem of adaptive fuzzy fault-tolerant control (FTC) is investigated for a class of nonlinear pure-feedback systems with unknown control directions and sensor failures. By utilizing the mean value theorem, the considered nonlinear pure-feedback system is transformed into a strict-feedback structure. The unavailability of state variables caused by sensor faults is dealt with by using parameter separation and regrouping technique. Based on cancelation and decoupled backstepping technique, an adaptive fuzzy fault-tolerant control method is developed. It is proved that the proposed adaptive FTC scheme can guarantee the exponential stabilization of the resulting closed-loop system. Furthermore, all the signals of the system remain semi-global uniformly ultimately bound (SGUUB). The simulation results are provided to illustrate the effectiveness of the proposed control scheme.

Adaptive Fuzzy Control for Pure-Feedback Nonlinear Systems With Nonaffine Functions Being Semibounded and Indifferentiable

An adaptive fuzzy control approach is proposed for nonaffine pure-feedback systems with nonaffine functions being semibounded and possibly indifferentiable. A semibounded and continuous condition for nonaffine function is presented to guarantee the controllability of system. To overcome the difficulty from this mild condition, a compact set is introduced, and the nonaffine nonlinear functions are modeled in a new way based on this compact set, which is proved to be invariant set eventually. By using the dynamic surface control (DSC) technique, the problem of explosion of complexity inherent in backstepping method is avoided in the proposed adaptive fuzzy control scheme. Robust compensators are used to minify the influence of uncertainties and disturbances. Furthermore, it is proved that all the closed-loop signals are bounded and the tracking error converges to a small residual set asymptotically. Finally, simulation examples are provided to demonstrate the effectiveness of the designed method.

Robust event-driven tracking control with preassigned performance for uncertain input-quantized nonlinear pure-feedback systems

This paper presents a simplified design methodology for robust event-driven tracking control of uncertain nonlinear pure-feedback systems with input quantization. All nonlinearities and quantization parameters are assumed to be completely unknown. Different from the existing event-driven control approaches for systems with completely unknown nonlinearities, the main contribution of this paper is to design a simple event-based tracking scheme with preassigned performance, without the use of adaptive function approximators and adaptive mirror models. It is shown in the Lyapunov sense that the proposed event-driven low-complexity tracker consisting of nonlinearly transformed error surfaces and a triggering condition can achieve the preselected transient and steady-state performance of control errors in the presence of the input quantization.

Nussbaum gain adaptive neural control for stochastic pure-feedback nonlinear time-delay systems with full-state constraints

Abstract In this paper, the problem concerned with adaptive approximation-based control is discussed for a class of stochastic pure-feedback nonlinear time-delay systems with unknown direction control gains and full-state constraints. In the controller design process, the approximation capability of neural networks is utilized to identify the unknown nonlinearities, the appropriate Lyapunov–Krasovskii functionals are constructed to compensate the unknown time-delay terms, barrier Lyapunov functions (BLFs) are designed to ensure that the state variables are constrained, and the Nussbaum-type gain function is used to solve the difficulties caused by the unknown virtual control gains. Then, based on adaptive backstepping technique and Lyapunov stability theory, a robust control scheme is presented, and the developed controller decreases the number of learning parameters and thus reduces the computational burden. It is shown that the proposed controller can guarantee that all the signals in the closed-loop system are 4-Moment (or 2 Moment) semi-globally uniformly ultimately bounded (SGUUB) and the tracking error converges to a compact set of the origin. Finally, two simulation examples are included to validate the effectiveness of the proposed approach.

Filter-Driven-Approximation-Based Control for a Class of Pure-Feedback Systems With Unknown Nonlinearities by State and Output Feedback

This paper presents a new approximation-based control approach for uncertain nonlinear pure-feedback systems. The main idea of this paper is to estimate unknown continuous nonlinear functions through a linear combination of first-order filtered signals of state variables and a control input in the nonadaptive control framework, instead of using conventional adaptive neural or fuzzy function approximators. Based on the proposed filter-driven approximation technique, we first present a state-feedback control scheme for pure-feedback systems with unknown nonaffine nonlinearities and a dead-zone input. Then, a filter-driven-approximation-based output-feedback control scheme is proposed via a system transformation and an observer to estimate unmeasurable state variables. Based on the Lyapunov stability theorem, the control errors and the filter-driven approximation errors are considered to prove that the controlled closed-loop system is semi-globally uniformly ultimately bounded. Finally, simulation results are provided to show that the proposed filter-driven-approximation-based controller and the existing function-approximation-based adaptive controllers have similar control performance for nonlinear pure-feedback systems.

Prescribed-Performance Control Design for Pure-Feedback Nonlinear Systems with Most Severe Uncertainties

This paper is devoted to the control design for uncertain pure-feedback nonlinear systems. Remarkably, the systems under investigation allow the completely nonaffine appearance of the virtual/actual controls and can be subjected to serious nonparametric uncertainties and unknown control directions, which are arguably the most severe uncertainties. Even so, a prescribed-performance control design scheme is proposed for the systems, where the system states are ensured to evolve within prescribed funnels. By suitably choosing the prescribed funnels, the controller designed can not only accomplish the basic steady performance on stabilization/tracking but also achieve other prescribed performance to guarantee the efficiency and reliability of the systems. Particularly, based on the scheme with suitably choosing the prescribed funnels, global stabilization with prescribed convergence rate and global/semiglobal practical tracking with prescribed transient performance (arrival time/maximum overshoot) are achieved for the uncertain pure-feedback systems.

Adaptive fuzzy dynamic surface control for uncertain discrete-time non-linear pure-feedback MIMO systems with network-induced time-delay based on state observer

ABSTRACTIn this paper, a state observer-based adaptive fuzzy dynamic surface control is developed for uncertain discrete-time non-linear pure-feedback multiple-input-multiple-output (MIMO) systems with network-induced time-delay. The uncertainties are approximated by a set of adaptive fuzzy logic systems, with the adjusted parameters updated by a simplified recursive least squares estimation algorithm, combined with a state observer. For a constant known network-induced time-delay, the proposed modified dynamic surface control utilising the predicted system states, expands the acceptable network-induced time-delay and stable operating range for a discrete-time non-linear pure-feedback MIMO system in the network. The simulation results indicate that the presented method is effective.

Adaptive fault-tolerant control with prescribed performance for switched nonlinear pure-feedback systems

In this paper, the problem of adaptive fuzzy fault-tolerant control is investigated for a class of switched uncertain pure-feedback nonlinear systems under arbitrary switching. The considered actuator failures are modeled as both lock-in-place and loss of effectiveness. By utilizing mean value theorem, the considered pure-feedback systems are transformed into a class of switched nonlinear strict-feedback systems. Under the framework of backstepping design technique and common Lyapunov function (CLF), an adaptive fuzzy fault-tolerant control (FTC) method with predefined performance bounds is developed. It is proved that under the proposed controller, all the signals of the close-loop systems are bounded and the state tracking error for each step remains within the prescribed performance bound (PPB) regardless of actuator faults and the system switchings. In addition, the tracking errors and magnitudes of control inputs can be reduced by adjusting the PPB parameters of errors in the first and last steps. The simulation results are provided to show the effectiveness of the proposed control scheme.

Adaptive neural control for a class of non-affine pure-feedback nonlinear systems

ABSTRACTA novel adaptive neural control scheme is designed for a class of pure-feedback nonlinear systems with non-affine functions possibly being discontinuous. The non-affine function is not necessary to be continuous with respect to control variables or input, and the bounds of non-affine function are unknown functions. Some compact sets are constructively introduced to investigate the bounds of non-affine function so as to cope with the difficulty from these unknown bounds. Moreover, the dynamic surface control technique has been utilised for handling with the problem of ‘explosion of complexity’, and the minimal learning parameter technique is also employed to overcome the problem of excessive parameters. Furthermore, it is highly proved that all the variables will always stay in the introduced compact sets, and all the signals in the closed-loop control system are semi-globally uniformly ultimately bounded by choosing the appropriate design parameters. Finally, simulation examples are provided to demonstrate the effectiveness of the designed approach.

Decentralised disturbance-observer-based adaptive tracking in the presence of unmatched nonlinear time-delayed interactions and disturbances

ABSTRACTThis paper proposes an approximation-based nonlinear disturbance observer (NDO) approach for decentralised adaptive tracking of uncertain interconnected pure-feedback nonlinear systems with unmatched time-delayed nonlinear interactions and external disturbances. Compared with the existing approximation-based NDO approach for uncertain interconnected nonlinear systems where the centralised design framework was proposed, the main contribution of this paper is to develop a decentralised and memoryless NDO-based adaptive control scheme in the presence of unknown time-varying delayed interactions and disturbances unmatched in the control inputs. The recursive design methodology is derived to construct the decentralised NDO and controller where the function approximators used in the decentralised NDO are employed to design the decentralised adaptive controller. From the Lyapunov stability theorem using Lyapunov--Krasovskii functionals, it is shown that all signals of the closed-loop system are semi-globally uniformly ultimately bounded and the tracking errors converge to an adjustable neighbourhood of the origin.

Barrier Lyapunov functions-based adaptive control for nonlinear pure-feedback systems with time-varying full state constraints

This paper studies the problem of controller design for pure-feedback nonlinear systems with asymmetric time-varying full state constraints. The mean value theorem is employed to transform a pure-feedback system into a strict-feedback structure with non-affine terms. For the transformed system, a time-varying asymmetric Barrier Lyapunov Function (ABLF) with the error variables is employed to ensure the time-varying constraints satisfaction. By allowing the barriers to vary with the desired trajectory in time, the initial condition requirements are relaxed efficiently. The presented control scheme can guarantee that all signals in the closed-loop system are ultimately bounded. It is also proved that the tracking error converges to an adjustable neighborhood of the origin even in the presence of disturbance. The performance of the ABLF-based control are illustrated through two examples.

Adaptive neural control for MIMO stochastic nonlinear pure-feedback systems with input saturation and full-state constraints

Abstract This paper investigates an adaptive neural tracking control problem for a class of multi-input multi-output (MIMO) stochastic nonlinear pure-feedback systems with input saturation and full-state constraints. First, the implicit function and mean value theorems are used to overcome the difficulty in the control of pure-feedback systems. Second, the Gaussian error function is employed to represent a continuous differentiable asymmetric saturation model, the neural network is employed to approximate the unknown nonlinearity, and a barrier Lyapunov function is designed to ensure that the full-state variables are restricted. At last, based on Lyapunov stability theory, a robust adaptive neural control method is obtained, which decreases the number of learning parameters and thus reduces the computational burden. It is shown that the designed controller can guarantee that all the signals in the closed-loop system are 4-Moment (or 2 Moment) semi-globally uniformly ultimately bounded (SGUUB) and the tracking error converges to a small neighborhood of the origin. The simulation example is given to further illustrate the effectiveness of the proposed approach.

Adaptive neural control for stochastic pure‐feedback non‐linear time‐delay systems with output constraint and asymmetric input saturation

In this study, the adaptive tracking control is investigated for a class of stochastic pure-feedback non-linear time-delay systems with output constraint and asymmetric input saturation non-linearity. First, the Gaussian error function is employed to represent a continuous differentiable asymmetric saturation model, and the barrier Lyapunov function is designed to cope with the output constraints. Then, the appropriate Lyapunov–Krasovskii functional and the property of hyperbolic tangent functions are used to address the effects of the unknown time-delay terms, and the neural network is employed to approximate the unknown non-linearities. At last, based on Lyapunov stability theory, a robust adaptive neural controller is proposed, which decreases the number of learning parameters and thus avoids the over-estimation problem. Under the designed neural controller, all the closed-loop signals are guaranteed to be 4-moment (or 2 moment) semi-globally uniformly ultimately bounded and the tracking error converges to a small neighbourhood of the origin for bounded initial conditions. Two simulation examples are presented to further illustrate the effectiveness of the designed method.

Backstepping active disturbance rejection control: a delayed activation approach

This study provides a solution to the problem of applying the active disturbance rejection control (ADRC) approach to the tracking control of pure feedback non-linear systems with large uncertainties. The design is based on the backstepping technique, and in each step two extended state observers (ESOs) are designed, one to approximate the derivative of the virtual control generated from the last step, while the other to estimate the system uncertainty. A key feature of the developed backstepping ADRC is that the activations of the ESOs are delayed successively. It is proved that under the new backstepping scheme, the closed-loop signals are guaranteed to be semi-globally uniformly ultimately bounded and the output tracking error can be made arbitrarily small.

Adaptive Neural Dynamic Surface Control of Pure-Feedback Nonlinear Systems With Full State Constraints and Dynamic Uncertainties

In this paper, adaptive neural dynamic surface control (DSC) is developed using radial basis function neural networks (NNs) for a class of pure-feedback nonlinear systems with full state constraints and dynamic uncertainties. Based on a one-to-one nonlinear mapping, the pure-feedback system with full state constraints is transformed into a novel pure-feedback system without state constraints. The dynamic uncertainties are dealt with using a dynamic signal. Using modified DSC and mean value theorem as well as Nussbaum function, two adaptive NN control schemes are proposed based on the transformed system. The designed control strategy removes the conditions that the upper bound of the control gain is known, and the lower bounds and upper bounds of the virtual control coefficients are known. It is shown that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded, and the full state constraints are not violated. Two numerical examples are provided to illustrate the effectiveness of the proposed approach.