Pure-feedback Nonlinear Systems Research Articles

Finite-time event-triggered output-feedback adaptive decentralized echo-state network fault-tolerant control for interconnected pure-feedback nonlinear systems with input saturation and external disturbances: A fuzzy control-error approach

In this paper, we introduce a decentralized Event-Triggered fault-tolerant echo-state network (ESN) direct adaptive controller for uncertain interconnected nonlinear systems in pure-feedback form. The proposed controller addresses input saturation, actuator faults, external disturbances, and unavailable states for measurement. Unlike the existing works in the literature that adapts the ESN weights using the tracking error, our method employs the control error that is estimated using a fuzzy inference system, to derive the adaptation laws. The complexity explosion typically seen with recursive back-stepping and Dynamic Surface Control (DSC) designs is completely avoided. Our control scheme addresses four types of state-dependent actuator faults: bias, drift, loss of accuracy, as well as loss of effectiveness. The ESNs approximate unknown ideal control laws, while robust terms are added to enhance the stability of the closed-loop system. Stability analysis ensures that tracking errors converge, in finite time, to a small compact set near the origin due to the strictly positive real (SPR) property. Simulation results of a quadrotor UAV and a mathematical system are presented. A comparative analysis is performed with another approach to emphasize the effectiveness of the proposed method.

Adaptive Barrier-Lyapunov-Functions Based Control Scheme of Nonlinear Pure-Feedback Systems with Full State Constraints and Asymptotic Tracking Performance

Adaptive Barrier-Lyapunov-Functions Based Control Scheme of Nonlinear Pure-Feedback Systems with Full State Constraints and Asymptotic Tracking Performance

Adaptive Projection and Fuzzy Tracking Design for Unknown Control Coefficients and References.

This article addresses the tracking control problem of nonlinear pure-feedback systems, where the control coefficients and the dynamics of the references are unknown. Fuzzy-logic systems (FLSs) are used to approximate the unknown control coefficients and at the same time the adaptive projection law is designed to allow each fuzzy approximation to cross zero, which yields that the proposed method avoids the assumption of using Nussbaum function, that is, the unknown control coefficients never cross zeros. Another adaptive law is designed to estimate the unknown reference and then it is intergraded into the saturated tracking control law to achieve the uniformly ultimately bounded (UUB) performance of the resulting closed-loop system. Simulations show the feasibility and effectiveness of the proposed scheme.

Backstepping-based adaptive fuzzy tracking control for pure-feedback nonlinear multi-agent systems

This work developed an adaptive fuzzy tracking control strategy for pure-feedback nonlinear multi-agent systems via backstepping. The relevant unknown functions are addressed by fuzzy logic systems (FLSs), which take into account entirely unknown nonlinearities. The non-affine pure-feedback form is dealt with a Butterworth low-pass filter (BLF), and the converted systems can be used to construct a controller. The implementation of the first-order sliding-mode differentiator overcomes the difficulty of calculation explosion. The backstepping method is applied to design the virtual controller step by step, and the actual controller is designed successfully. Furthermore, through Lyapunov stability theory, the stability of systems is ensured, and that the tracking error is kept to a minimum. Finally, the simulation result shows the validity of the designed control strategy.

Non-overshooting control of nonlinear pure-feedback systems

In this paper, the non-overshooting tracking control problem of nonlinear pure-feedback systems is studied. From the model, nonlinear pure-feedback systems are more general than nonlinear strict-feedback systems. In this way, the nonlinear pure-feedback system can express the physical system that the strict-feedback system cannot. Therefore, it is more meaningful to study the nonlinear pure-feedback system. At present, there are few reports on non-overshooting control of nonlinear systems. These documents are mainly aimed at nonlinear strict-feedback systems, and the obtained non-overshooting conditions are also conservative. Then, this paper aims at nonlinear pure-feedback systems, and obtains its non-overshooting conditions from another aspect. The specific narrative is as follows: the closed-loop system is structured by designing the auxiliary controller, and then the expressions of tracking error are derived. By analyzing these expressions, the conditions of non-overshooting tracking control of pure-feedback systems are obtained. Finally, the simulation results of three examples verify the effectiveness of this algorithm.

Finite-time adaptive prescribed performance DSC for pure feedback nonlinear systems with input quantization and unmodeled dynamics

<abstract><p>This paper presents a new prescribed performance-based finite-time adaptive tracking control scheme for a class of pure-feedback nonlinear systems with input quantization and dynamical uncertainties. To process the input signal, a new quantizer combining the advantages of a hysteresis quantizer and uniform quantizer has been used. Radial basis function neural networks have been utilized to approximate unknown nonlinear smooth functions. An auxiliary system has been employed to estimate unmodeled dynamics by producing a dynamic signal. By introducing a hyperbolic tangent function and performance function, the tracking error was made to fall within the prescribed time-varying constraints. Using modified dynamic surface control (DSC) technology and a finite-time control method, a novel finite-time controller has been designed, and the singularity problem of differentiating each virtual control scheme in the existing finite-time control scheme has been removed. Theoretical analysis shows that all signals in the closed-loop system are semi-globally practically finite-time stable, and that the tracking error converges to a prescribed time-varying region. Simulation results for two numerical examples have been provided to illustrate the validity of the proposed control method.</p></abstract>

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.

Event-Triggered Adaptive Bipartite Asymptotic Tracking Control Using Intelligent Technique for Stochastic Nonlinear Multiagent Systems

The problem of event-triggered based adaptive bipartite asymptotic tracking control for multi-agent stochastic pure-feedback nonlinear systems over sign digraph is considered. The agents classification optimization strategy (ACOS) is presented, which transforms a structurally unbalanced multi-agent topology graph into a structurally balanced multi-agent topology graph. Furthermore, how to deal with the nonaffine structure and unknown control gains for multi-agent stochastic pure-feedback nonlinear systems is a challenging issue for designing controllers. As a result, the mean value theorem is used to transfer the nonaffine systems into the affine systems, and the Nussbaum functions are used to eliminate the influence caused by the unknown control gains. In addition, the intelligent control technique is used to approximate the unknown nonlinearities, the event-triggered control strategy following the switching thresholds is introduced to save unnecessary communication resources, and the computation burden is eliminated by means of the dynamic surface technique. It is proved that all signals of the closed-loop systems are bounded in probability and the tracking errors asymptotically converge to zero in probability. Finally, the simulation results illustrate the validity of the proposed scheme. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Impact Statement</i> —In nature, there are a lot of interesting movements of multi-agents. In this paper, the multi-agent stochastic pure-feedback nonlinear systems over sign digraph is considered. How to handle the nonaffine structure and unknown control gains for the studied systems under interest is a difficult issue. Therefore, the mean value theorem and the Nussbaum functions are used to overcome the difficulty of designing controllers. The bipartition problem with the structurally unbalanced sign digraph is another design difficulty, then we propose the ACOS in the controller design process. In addition, the controllers designed in this paper, which are constructed by applying the backstepping method, event-triggered control strategy, and dynamic surface technique, can save the unnecessary communication resources and alleviate computation burden. Finally, the simulation results illustrate the validity of the proposed scheme.

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.

Global Predefined-Time Adaptive Neural Network Control for Disturbed Pure-Feedback Nonlinear Systems With Zero Tracking Error.

This article presents a global adaptive neural-network-based control algorithm for disturbed pure-feedback nonlinear systems to achieve zero tracking error in a predefined time. Different from the traditional works that only solve the semiglobal bounded tracking problem for pure-feedback systems, this work not only achieves that the tracking error globally converges to zero but also guarantees that the convergence time can be predefined according to the user specification. In order to get the desired predefined-time controller, first, a mild semibound assumption for nonaffine functions is skillfully proposed so that the design difficulty caused by the structure of pure feedback can be easily solved. Then, we apply the property of radial basis function (RBF) neural networks (NNs) and Young's inequality to derive the upper bound of the term that contains the unknown nonlinear function and external disturbances, and the designed adaptive parameters decide the derived upper and robust control gain. Finally, the predefined-time virtual control inputs are presented whose derivatives are further estimated by utilizing finite-time differentiators. It is strictly proved that the proposed novel predefined-time controller can guarantee that the tracking error globally converges to zero within predefined time and a practical example is shown to verify the effectiveness and practicability of the proposed predefined-time control method.

Neural Networks-Based Adaptive Dynamic Surface Control for Nonlinear Pure-Feedback Systems under Prescribed Performance and Zero Tracking Error

Neural Networks-Based Adaptive Dynamic Surface Control for Nonlinear Pure-Feedback Systems under Prescribed Performance and Zero Tracking Error

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

Adaptive Tracking Control for Output-Constrained Switched MIMO Pure-Feedback Nonlinear Systems with Input Saturation

Adaptive Tracking Control for Output-Constrained Switched MIMO Pure-Feedback Nonlinear Systems with Input Saturation

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.

Event-Triggered Adaptive Fuzzy Asymptotic Tracking Control of Nonlinear Pure-Feedback Systems With Prescribed Performance.

This article considers the problem of fixed-time prescribed event-triggered adaptive asymptotic tracking control for nonlinear pure-feedback systems with uncertain disturbances. The fuzzy-logic system (FLS) is introduced to deal with the unknown nonlinear functions in the system. By constructing a new type of Lyapunov function, the restrictive requirement that the upper bounds of the partial derivative of the unknown system functions need to be known is relaxed during the controller design process. At the same time, by developing a novel fixed-time performance function (FPF), the fixed-time prescribed performance (FPP) can be achieved, that is, the tracking error can converge to the neighborhood of the origin in a fixed time and finally converges to zero asymptotically. In addition, the event-triggered strategy is developed to reduce the waste of communication resources. The proposed control law can ensure that all the signals of the system are bounded. Meanwhile, the Zeno behavior can be effectively avoided. Finally, an example is provided to prove the effectiveness of the proposed scheme.

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.

Pattern-based learning and control of nonlinear pure-feedback systems with prescribed performance

Pattern-based learning and control of nonlinear pure-feedback systems with prescribed performance

Adaptive neural prescribed performance control for switched pure-feedback non-linear systems with input quantization

Purpose This paper aims to investigate an adaptive prescribed performance control problem for switched pure-feedback non-linear systems with input quantization. Design/methodology/approach By using the semi-bounded continuous condition of non-affine functions, the controllability of the system can be guaranteed. Then, a constraint variable method is introduced to ensure that the tracking error satisfies the prescribed performance requirements. Meanwhile, to avoid the design difficulties caused by the input quantization, a non-linear decomposition method is adopted. Finally, the feasibility of the proposed control scheme is verified by a numerical simulation example. Findings Based on neural networks and prescribed performance control method, an adaptive neural control strategy for switched pure-feedback non-linear systems is proposed. Originality/value The complex deduction and non-differentiable problems of traditional prescribed performance control methods can be solved by using the proposed error transformation approach. Besides, to obtain more general results, the restrictive differentiability assumption on non-affine functions is removed.

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.

A universal error transformation strategy for distributed event-triggered formation tracking of pure-feedback nonlinear multiagent systems with communication and avoidance ranges

A universal error transformation method is proposed for designing a distributed event-triggered formation tracker with preserved network connectivity and collision avoidance for multiple pure-feedback nonlinear multiagent systems connected in a directed network. It is assumed that the communication and avoidance ranges of agents are heterogeneous and that all nonaffine nonlinear functions are unknown. The primary contribution of this study is to establish a universal nonlinear transformation of relative output errors among agents for dealing with the initial network interaction, collision prevention, and predefined-time tracking performance problems in a fully distributed manner. To this end, nonlinear relative output errors among agents are designed using performance functions dependent on communication and avoidance ranges. Local event-triggered adaptive control laws are recursively designed using nonlinear relative output errors and neural networks. It is proved that the boundedness of the presented nonlinear errors ensures the achievement of network interaction preservation, collision prevention, and formation tracking with predefined-time convergence. Furthermore, the existence of a minimum inter-event time is established for local event-triggered control laws. Finally, the effectiveness of the proposed theoretical strategy is validated by simulation examples.