Uncertain Pure-feedback Nonlinear Systems Research Articles

Predefined-Time Output-Feedback Leader-Following Consensus of Pure-Feedback Multiagent Systems.

This article delves into the predefined-time output-feedback leader-following consensus problem of uncertain pure-feedback nonlinear multiagent systems for the first time. To streamline subsequent design, the original systems in pure-feedback form are first transformed into canonical systems. Following this, a distributed predefined-time extended state observer (ESO) and a local predefined-time ESO are developed to reconstruct the unknown states/lumped disturbance of the transformed leader system and follower systems, respectively. Based on the estimated states and utilizing a bounded regulation function, two nonsingular and nonconservative predefined-time control laws are formulated to achieve consensus tracking. The proposed method showcases the following advantages: 1) the actual convergence time rather than the upper bound of the convergence time (UBCT) of the tracking errors can be explicitly specified a priori regardless of the initial conditions in a bounded region, optimizing control energy usage and 2) the system overshoot could be effectively reduced by selecting appropriate parameters for the regulation function. Finally, numerical examples are conducted to verify the obtained results.

Fixed-time safe tracking control of uncertain high-order nonlinear pure-feedback systems via unified transformation functions

Fixed-time safe tracking control of uncertain high-order nonlinear pure-feedback systems via unified transformation functions

Prescribed performance event-triggered fault-tolerant control of uncertain pure-feedback nonlinear systems

The prescribed performance event-triggered fault-tolerant control (FTC) problem is considered for a class of uncertain pure-feedback nonlinear systems with actuator failures. Contrary to the existing results, the novel decoupling condition and the universal construction mode of error transformation are first given. Based on this, a new radical constrained function is used to low-complexity controller design without involving the Nussbaum gain scheme or adaptation parameter updated structure, in spite of nonlinear coupling and event triggering input. It is proved that the desired prescribed accuracy of the closed-loop output tracking can be achieved and the number of event-trigger showed a significant reduction. The effectiveness of theory results is verified by simulation on a numerical example.

Dynamic Event-Triggered Communication-Based Adaptive Consensus Tracking of Uncertain Nonlinear Multiagent Systems in Pure-Feedback Form

This study is aimed at investigating a dynamic event-triggered communication-based adaptive distributed consensus control problem for a class of uncertain pure-feedback nonlinear multiagent systems with limited communication resources. The nonaffine nonlinear functions of multiagent systems are assumed to be unknown and heterogeneous, and intermittent inter-agent communication is considered to occur within a directed network. A novel adaptive distributed dynamic surface control strategy using neural networks is developed to manage non-differentiable virtual control laws associated with the intermittent communication between agents with uncertain nonaffine nonlinear parts. The key contribution of this research is the derivation of dynamic event-triggering conditions using distributed tracking errors to efficiently adjust inter-event times, while ensuring the consensus tracking performance and robustness against unexpected external disturbances. Compared with the existing static event-triggered communication approach, the proposed controller can alleviate the communication burden among agents. Using technical lemmas, it is shown that all signals of the considered system are semiglobally uniformly ultimately bounded, and Zeno behavior is strictly excluded. Comparative simulation results illustrate the effectiveness of the proposed dynamic event-triggered control approach.

Fixed-time prescribed performance fault compensation control of uncertain pure-feedback nonlinear systems

This study is primarily focused on the issue of low-complexity global fixed-time prescribed performance fault compensation control for uncertain nonlinear systems with actuator failures, and extra disturbance. First of all, a new fixed-time function in view of inverse trigonometric function is proposed and used to define the state errors. Then, proper root-type error conversion functions and smooth orientation functions are linked to design the continuous control signals in the state feedback controller. In addition, based on the idea of proof by contradiction, it is shown that the state errors converge to a predictable compact aggregate at a definite rate at a fixed time. Meanwhile, the boundedness of any closed-loop signals might be guaranteed. The simulation results and algorithm comparisons are delivered to demonstrate the effectiveness and superiority of the developed control strategy at last.

Quantized-State-Feedback-Based Neural Control for a Class of Switched Nonlinear Systems With Unknown Control Directions

This paper investigates the problem of unknown virtual control directions in a state-quantized adaptive recursive control design for a class of arbitrarily switched uncertain pure-feedback nonlinear systems in a band-limited network. State quantization is considered for state feedback control in a band-limited network. The primary contribution of this study is to provide a quantized state feedback adaptive control strategy to address the unknown control direction and arbitrarily switched nonaffine nonlinearities. Herein, a coupling problem between Nussbaum functions and quantization errors caused by quantized state feedback control laws is considered in the Lyapunov-based design and stability analysis. A state-quantized adaptive recursive control scheme using the function approximation is constructed without a priori knowledge of the signs of the control gain functions, where the estimated parameters and Nussbaum-type functions are adaptively updated via quantized states. Theoretical lemmas are derived to show that the adaptive parameters and quantization errors of the closed-loop signals are bounded using the proposed control scheme. The boundedness of the closed-loop signals and the convergence of tracking error to a neighborhood of the origin are proved using the common Lyapunov function approach. Two simulation examples are shown to illustrate the effectiveness of the proposed theoretical result.

Adaptive event-triggered tracking using nonlinear disturbance observer of arbitrarily switched uncertain nonlinear systems in pure-feedback form

An adaptive nonlinear disturbance observer (NDO) strategy is proposed for event-triggered tracking of switched uncertain pure-feedback nonlinear systems with mismatched external disturbances and arbitrary switching. Contrary to existing control results using NDOs for switched nonlinear systems, the main contribution of this study is the development of a common NDO-based event-triggered tracker with a triggering law to handle arbitrarily switched unknown nonaffine nonlinearities. Using the common Lyapunov function approach, a recursive NDO-based tracker design methodology is investigated to guarantee that all the closed-loop signals are uniformly ultimately bounded, and Zeno behavior is excluded in the resulting event-triggering mechanism. In each recursive step, the same adaptive estimate terms in the control laws and the NDO are derived via the function approximation technique to compensate for mismatched and switched unknown nonaffine nonlinearities. Finally, simulation examples are considered to verify the effectiveness of the proposed control technique.

Adaptive robust anti-disturbance control for pure feedback nonlinear systems with multiple constraints

To solve the control problem for a class of uncertain pure feedback nonlinear systems subjected to external disturbances and multiple constraints, an adaptive robust control methodology is proposed based on a disturbance observer. To handle unknown nonlinearity and external unknown disturbances, a nonlinear disturbance observer is constructed based on a radial basis function neural network, which uses a Butterworth low-pass filter to remove the algebraic loop problem. Then, to guarantee that the system can stably track the desired trajectory under the state constraints, input saturation, and prescribed tracking performance constraints, we developed a novel barrier Lyapunov function and a backstepping controller that combines an auxiliary bounded function, a Nussbaum function, and a first-order sliding-mode differentiator. Subsequently, the stability of the closed-loop system is rigorously proved by Lyapunov analysis. Finally, simulations are conducted to demonstrate the effectiveness of the proposed approach.

Neuroadaptive fault-tolerant control of state constrained pure-feedback systems: A collective backstepping design

In this work, we present a neuroadaptive and fault-tolerant tracking control scheme for uncertain nonlinear pure-feedback systems in the presence of time-varying and asymmetric full state constraints and unanticipated actuation failures. Instead of using multi-step recursive backstepping design, we employ a one-step approach for control development. By introducing a nonlinear coordinate transformation, we convert the original nonlinear system with asymmetrical state constraints into a new augmented one free from state constraints, which allows for the complete obviation of the feasibility conditions in the strategy. Furthermore, by making use of the feature from skew symmetric matrix in the augmented system, we develop the neural adaptive control algorithms collectively without the need for repetitive design procedure, in which only one Lyapunov function and one step derivation are involved, leading to a design approach whose synthesis complexity does not increase with the order of the system.

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.

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.

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.

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.

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.

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.

Approximation-Free Output-Feedback Non-Backstepping Controller for Uncertain SISO Nonautonomous Nonlinear Pure-Feedback Systems

A novel differentiator-based approximation-free output-feedback controller for uncertain nonautonomous nonlinear pure-feedback systems is proposed. Using high-order sliding mode observer, which is a finite-time exact differentiator, the time-derivatives of the signal generated using tracking error and filtered input are directly estimated. As a result, the proposed non-backstepping control law and stability analysis are drastically simple. The tracking error vector is guaranteed to be exponentially stable in finite time regardless of the nonautonomous property in the considered system. It does not require neural networks or fuzzy logic systems, which are typically adopted to capture unstructured uncertainties intrinsic in the controlled system. As far as the authors know, there are no research results on the output-feedback controller for the uncertain nonautonomous pure-feedback nonlinear systems. The results of the simulation show clearly the performance and compactness of the control scheme proposed.

Fault-Estimation-Based Output-Feedback Adaptive FTC for Uncertain Nonlinear Systems With Actuator Faults

This paper investigates the adaptive output-feedback fault-tolerant control (FTC) problem for a class of uncertain nonlinear pure-feedback systems subject to actuator faults. To address the challenges incurred by the actuator faults and unmeasured state, a novel fault compensation observer-based adaptive controller is designed by estimating fault parameters online. By combining the backstepping design framework with prescribed performance and command filters techniques, it is proved that the proposed FTC scheme guarantees that all the signals in the closed-loop system are bounded and that the tracking error remains within the predefined bounds even if actuator faults occur. Compared with the existing output-feedback FTC design methods, the compensation state observer removes the restriction that the fault parameters are known or the uncontrollability caused by input saturation. The simulation and experimental results verify the effectiveness of the proposed method.

Pattern-based NN control for uncertain pure-feedback nonlinear systems

Implementing human-like learning and control for nonlinear dynamical systems operating in different control situations is an important and challenging issue. This paper presents a pattern-based neural network (NN) control strategy for nonlinear pure-feedback systems via deterministic learning (DL). Firstly, an appropriately designed adaptive neural dynamic surface controller is proposed to achieve the finite time tracking control. By analyzing the recurrent property of NN input signals, a partial persistent excitation (PE) condition for radial basis function (RBF) network is established, the implicit desired control dynamics under different control situations are accurately identified via DL in the case that the dimension of NN input is reduced. And a set of pattern-based experienced actual and virtual controllers is constructed using the learned knowledge. Secondly, to classify different control situations, when the system is operating in different control situations but controlled by current normal experienced controller, the dynamics of each subsystem are accurately identified via DL, n sets of dynamical estimators are constructed using the learned knowledge. Thirdly, in the recognition phase, n sets of residuals are achieved by comparing each set of estimators with the monitored system, sudden change in the control situation is rapidly recognized based on the principle of the earliest occurrence of the minimum residual. Finally, in the control phase, according to the recognition result, the correct experienced actual and virtual controllers will be selected to control the plant, guaranteed stability and superior control performance are achieved without any further re-adaptation online. Simulation studies are given to verify the proposed scheme can not only acquire and memorize knowledge like humans, but also reuse the learned knowledge to achieve rapid recognition and control of current control situation.

Adaptive fuzzy dynamic surface control for uncertain nonlinear systems in pure-feedback form with input and state constraints using noisy measurements

ABSTRACTThis paper is concerned with the design of an adaptive fuzzy dynamic surface control for uncertain nonlinear pure-feedback systems with input and state constraints using a set of noisy measurements. The design approach is described as follows. The nonlinear uncertainties are approximated by using the fuzzy logic systems at the first stage, secondly the adaptive fuzzy dynamic surface control is introduced to remove the problem of the explosion of complexity for the derivation of the adaptive fuzzy backstepping control, thirdly a new saturation function for state constraints is proposed to design the controllers based on the Lyapunov function, fourthly the number of the adjustable parameters is reduced by using the simplified extended single input rule modules, and finally the weighted least squares estimator to take the estimates for the un-measurable states and the adjustable parameters is in a simplified structure designed. The proposed approach provides effective system performance in the simulation experiment.