Strict-feedback Nonlinear Systems Research Articles

Optimized Backstepping Combined With Dynamic Surface Technique for Single-Input–Single-Output Nonlinear Strict-Feedback System

Optimized Backstepping Combined With Dynamic Surface Technique for Single-Input–Single-Output Nonlinear Strict-Feedback System

Asymptotic tracking control for a class of uncertain nonlinear systems with output constraint

In this paper, we investigate the asymptotic tracking control for a class of strict-feedback nonlinear systems with unknown nonlinear functions and output constraint. The nonlinear transformation function-based system transformation approach is employed to convert the initial constrained system into an unconstrained one. To achieve the objective of asymptotic tracking, the tracking error is scaled by an exponential proportional function. Furthermore, to address the challenge of repetitive derivation of virtual control laws that occurs in the traditional design process of high-order backstepping methods, a first-order filter is applied. The designed control scheme guarantees that the system output will asymptotically track a predefined reference tracking trajectory while ensuring that the output constraint is not violated. Moreover, it is shown that the filter error will converge asymptotically to the origin. The effectiveness of the developed control strategy is verified through simulation results.

Adaptive command filtered control for a class of nonlinear time delay systems with extreme learning machine

AbstractIn this paper, an extreme learning machine (ELM)‐based adaptive command filtered control method is investigated for a class of strict feedback nonlinear systems with unknown external disturbances and time delay. To begin with, the original system state equation is transformed into a new form on account of coordinate transformation. Subsequently, an ELM is adopted to approximate unknown functions which exist in the whole system states, without any prior knowledges of the ideal weight vectors and approximation errors. Secondly, a command filter is developed for the system under consideration, which can avoid the probem of “explosion of complexity” caused by repeated derivation of virtual control signals in traditional backstepping control. Meanwhile, error compensation signals are designed to conquer the shortcoming of a dynamic surface control (DSC) method. The combination of ELM and command filter technique has been used to construct corresponding controllers and adaptive laws. Effective handling of the impact of time delay terms is ensured through designing a novel Lyapunov–Krasovskii functional. The proposed strategy guarantees boundedness of all signals in the closed‐loop system and the tracking error asymptotically converges to a compact set around the origin. In the end, two continuous stirred tank reactors (CSTRs) are taken as example to further verify the efficiency of the put forward control method.

Non-singular control strategies for a class of strict-feedback nonlinear time-delay systems

This paper focuses on the non-singular control for strict-feedback nonlinear time-delay systems. The boundedness of all the signals of the closed-loop system is rigorously proved by introducing a class of Lyapunov-Krasovskii functionals, including exponential functions. Besides, novel auxiliary functions with continuous derivatives are proposed to construct the control law to avoid the discontinuity of the virtual controllers. Besides, the amount of computation is reduced by applying the new auxiliary functions when the other control methods require loads of calculations for handling singular problems. In the end, numerical examples are given to validate the effectiveness of the proposed control scheme.

Observer-Based Output Consensus Control Scheme for Strict-Feedback Nonlinear Multi-Agent Systems With Disturbances

Observer-Based Output Consensus Control Scheme for Strict-Feedback Nonlinear Multi-Agent Systems With Disturbances

Neural Network Boundary Approximation for Uncertain Nonlinear Spatiotemporal Systems and Its Application of Tracking Control.

This brief addresses the neural network (NN) approximation problem for uncertain nonlinear systems with time-varying parameters (that is, unknown nonlinear spatiotemporal systems). Due to the fact that the unknown spatiotemporal functions cannot be directly approximated by NNs, a so-called time-varying parameter extraction is given to separate time-varying parameters from uncertain nonlinear spatiotemporal functions. By using the supremum of Euler norm of the extracted time-varying parameters, the nonlinear spatiotemporal function is mapped to an unknown state-based boundary function, which can be approximated by NNs. Based on the time-varying parameter extraction, an adaptive neural tracking control law is designed for uncertain strict-feedback nonlinear spatiotemporal systems, which guarantees the convergence of the tracking error with a trajectory performance. The effectiveness of the designed method is verified by simulations.

Output feedback command filtered fuzzy controller design for uncertain strict-feedback nonlinear systems with unmodeled dynamics and event-triggered strategy

An adaptive control problem is considered for a class of nth-order uncertain nonlinear systems with unmodeled dynamics. Compared with the traditional backstepping method, this article improves the systems performance by introducing command filtered and event-triggered. The former primarily enhances the flexibility of controller design, while the latter employs a variable threshold triggering mechanism to achieve savings in communication resources. Simultaneously, the estimation error is compensated by the introduced hyperbolic tangent function. In addition, fuzzy logic system (FLS) is primarily used for observer design and approximation of unknown functions. The conclusion that the closed-loop signals are bounded is obtained through stability analysis. Finally, the numerical simulation results provide a comparative validation of the effectiveness of the proposed control scheme.

Adaptive tracking control of nonlinear systems with unknown time‐varying parameters via a monotonically increasing function

AbstractThis article considers the adaptive tracking control problem of a class of strict feedback nonlinear systems with unknown time‐varying parameters. Note that the recent adaptive method, that is, congelation of variables, is built upon the assumption that the known basis functions are zeros as zero states. To remove this assumption, a novel monotonically increasing function is designed to transform the output tracking error and the backstepping tracking errors into increasing forms, which is integrated into the congelation of variables to design the ‐modification adaptive tracking control law. It is noted that the proposed form of monotonically increasing function is general, which includes positive proportional functions, exponential functions, and logarithmic functions and so on. To yield the boundedness of increasing tracking errors, a barrier Lyapunov function is designed. The asymptotic convergence of the resulting adaptive tracking system is analyzed in detail. Simulations demonstrate the effectiveness of the proposed method.

Predefined Time and Accuracy Adaptive Fault-Tolerant Control for Nonlinear Systems with Multiple Faults

This work mainly studies the issue of predefined time and accuracy adaptive fault-tolerant control for strict-feedback nonlinear systems with multiple faults. The faults in the controlled system include actuator faults and external system faults. The unknown functions for nonlinear systems are approximated by fuzzy logic systems (FLSs). And then, according to the backstepping technique and the predefined time stability theory, an adaptive fuzzy control algorithm is presented, which can make sure that all closed-loop system signals remain predefined time bound and the tracking error converges to a predefined accuracy within the predefined time. Ultimately, the effectiveness of the presented control algorithm is proved through two simulation examples.

Funnel-Based Adaptive Neural Fault-Tolerant Control for Nonlinear Systems with Dead-Zone and Actuator Faults: Application to Rigid Robot Manipulator and Inverted Pendulum Systems

This study addresses an adaptive neural funnel fault-tolerant control problem for a class of strict-feedback nonlinear systems with actuator faults and input dead zone. To guarantee the boundedness of the tracking error, a modified transformation for funnel error is devised and incorporated into the control design process. To manage unknown nonlinear functions, radial basis function neural networks (RBFNN) are employed in designing an adaptive neural funnel fault-tolerant controller through the backstepping technique. The proposed controller guarantees the output tracking error stays within a predefined funnel, and all signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, simulations of a rigid robot manipulator system and an inverted pendulum system are conducted to validate the practicality and effectiveness of the proposed control method.

Event-triggered prescribed-time tracking for uncertain nonlinear systems with unknown control gain and output constraints

This paper uses a prescribed-time event-triggered approach to explore the control of uncertain strict-feedback nonlinear systems with output constraints and an unknown time-varying control gain. The proposed method utilizes a universal barrier Lyapunov function (BLF) to ensure compliance with the output constraints. Furthermore, a technique known as congelation of variables is employed to handle the unknown time-varying control gain. Approximating the unknown dynamics using radial basis function neural network (RBFNN), a neural prescribed-time controller is designed based on the backstepping approach. Additionally, the event-triggered mechanism is employed to conserve computational and communication resources. With this scheme, all signals are guaranteed to converge to a compact set within the prescribed time frame. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed algorithm.

Estimator-based neural adaptive event-triggered control for strict-feedback nonlinear systems with incomplete measurements

This article addresses a neural adaptive event-triggered tracking control for a class of strict-feedback nonlinear dynamics with incomplete measurements, which is novel on the research of Cyber-physical Systems. The incomplete measurement problem caused by packet loss, saturation, and other issues during data transmission can lead to the unavailability of system state variables, which can degrade system performance and even lead to instability. To solve these problems, a state estimator for data-losing case and two controllers for normal and data-losing cases are designed utilising event-triggered strategies which can reduce the burden of calculation and data transmission. Radial basis function neural networks are adopted to approximate the unknown nonlinear system functions. A strict stability analysis in probability shows that the control laws for the considered strict-feedback nonlinear system can guarantee all the closed-loop to be uniformly ultimately bounded in mean square. Two examples are performed to demonstrate the effectiveness of the provided control method.

Event-Triggered Adaptive Fuzzy Control for Strict-Feedback Nonlinear FOSs Subjected to Finite-Time Full-State Constraints

In this article, an event-triggered adaptive fuzzy finite-time dynamic surface control (DSC) is presented for a class of strict-feedback nonlinear fractional-order systems (FOSs) with full-state constraints. The fuzzy logic systems (FLSs) are employed to approximate uncertain nonlinear functions in the backstepping process, the dynamic surface method is applied to overcome the inherent computational complexity from the virtual controller and its fractional-order derivative, and the barrier Lyapunov function (BLF) is used to handle the full-state constraints. By introducing the finite-time stability criteria from fractional-order Lyapunov method, it is verified that the tracking error converges to a small neighborhood near the zero and the full-state constraints are satisfied within a predetermined finite time. Moreover, reducing the communication burden can be guaranteed without the occurrence of Zeno behavior, and the example is given to demonstrate the effectiveness of the proposed controller.

Neural Network-Based Fixed-Time Tracking Control for Input-Quantized Nonlinear Systems With Actuator Faults.

This study reports a fixed-time tracking control problem for strict-feedback nonlinear systems with quantized inputs and actuator faults where the total number of faults is allowed to be infinite. By taking advantage of radial basis function neural networks (RBFNNs), unknown nonlinear function terms in the system dynamic model can be effectively approached. In addition, based on the sector property of quantization nonlinearities and the structure of the actuator fault model, novel adaptive estimations and innovative auxiliary design signals are constructed to compensate for the influence caused by actuator faults and quantized inputs properly in the fixed-time convergence settings. Then, rigorous theoretical analysis manifests that the proposed control scheme can make the output tracking error converge to a small neighborhood of the origin within a fixed time, and the upper bound of the setting time not only does not depend on initial states of the system but also can be preassigned by selecting parameters appropriately. Meanwhile, all the signals in the closed-loop system remain bounded. Finally, a numerical example and a practical example of a single-link manipulator are presented to demonstrate the effectiveness of the proposed control algorithm.

Adaptive stabilisation of second-order strict-feedback systems based on FAS with tuning functions

This paper thoroughly investigates adaptive stabilisation for a specific class of second-order strict-feedback nonlinear systems. Drawing inspiration from the fully actuated system (FAS) approach, an adaptive control scheme incorporating FAS with tuning functions is proposed. Notably, this approach directly addresses second-order systems without the need for reducing them to first-order systems, streamlining the intricate procedures involved. To handle uncertain parameters within the system, a combined scheme of adaptive estimation and tuning functions is employed. Rigorous analysis using Lyapunov stability theory demonstrates that the designed controller ensures the system's asymptotic convergence to the equilibrium point, with all other signals in the closed-loop system bounded. Finally, two simulation experiments verify the effectiveness of the proposed strategy: numerical simulations compared with sliding mode control and backstepping control, and practical application on the single-link robotic arm with a comparative analysis against backstepping control.

Continual online learning-based optimal tracking control of nonlinear strict-feedback systems: application to unmanned aerial vehicles

A novel optimal trajectory tracking scheme is introduced for nonlinear continuous-time systems in strict feedback form with uncertain dynamics by using neural networks (NNs). The method employs an actor-critic-based NN backstepping technique for minimizing a discounted value function along with an identifier to approximate unknown system dynamics that are expressed in augmented form. Novel online weight update laws for the actor and critic NNs are derived by using both the NN identifier and Hamilton-Jacobi-Bellman residual error. A new continual lifelong learning technique utilizing the Fisher Information Matrix via Hamilton-Jacobi-Bellman residual error is introduced to obtain the significance of weights in an online mode to overcome the issue of catastrophic forgetting for NNs, and closed-loop stability is analyzed and demonstrated. The effectiveness of the proposed method is shown in simulation by contrasting the proposed with a recent method from the literature on an underactuated unmanned aerial vehicle, covering both its translational and attitude dynamics.

Neural-network-based adaptive control of strict-feedback nonlinear systems with actuator faults: Event-triggered communications strategy

This article considers the neural network-based event-triggered adaptive control problem for a class of strict-feedback nonlinear impulsive systems with actuator faults and external disturbances. It is well known that reducing computational resources and carrier loads is a long-standing task in the field of adaptive control. However, the event-triggered scheme used in traditional research still consumes relatively more resources. To solve the problem, the control scheme in this paper only updates the controller and weight parameters at the event-triggered moments. Dedicated efforts are made to certificate the stability of the system through a new Lyapunov theory, in such a way that the system maintains its stability well even under uncertainties. Finally, two emulation examples are exploited to demonstrate the efficacy and applicability of the proposed approach.

Finite-time adaptive optimal control of uncertain strict-feedback nonlinear systems based on fuzzy observer and reinforcement learning

This paper proposes an adaptive optimal control strategy of finite-time control for high-order uncertain strict-feedback nonlinear systems. Firstly, a reinforcement learning (RL) based an optimal control scheme is employed to design a optimal controller, to achieve global optimisation. Additionally, considering the unmeasurable states, we construct a fuzzy observer and utilise fuzzy logic systems to approximate the unknown functions. Meanwhile, the inclusion of command filtering and time-based control simplifies the controller design and enhances the system's response rapidity. Finally, the effectiveness and feasibility of the proposed approach are validated through a numerical simulation and a single link-robot system simulation.

Prescribed-time tracking with prescribed performance for a class of strict-feedback nonlinear systems

In this paper, we will investigate the prescribed-time tracking with prescribed performance for a class of strict-feedback nonlinear systems. A control strategy will be proposed to achieve the tracking error constrain by prescribed performance bound (PPB) and converge to zero in prescribed-time. The PPB constrains not only the convergence rate, but also the maximum overshoot of the tracking error. The tracking error will converge to zero in prescribed-time. Such a prescribed-time is irrespective of both initial conditions and design parameters. First, by incorporating the PPB into the original system via a transformation, a equivalently transformed system is obtained, which is unconstrained and convenient for control design. Then, with the help of backstepping, a controller is recursively designed to guarantee that the tracking error satisfies the PPB all the time and converges to zero in prescribed-time. The key to designing the controller is that a fractional term is introduced when designing the virtual stabilizing function during each step of backstepping, which in turn decreases the Lyapunov function to origin within prescribed-time. Finally, two examples are presented to demonstrate the validity of our proposed control strategy.

Existence of stationary distribution for stochastic coupled nonlinear strict-feedback systems with Markovian switching

In this paper, a class of stochastic coupled nonlinear strict-feedback systems with Markovian switching (SCNSSMs) are introduced, and the existence of stationary distribution for SCNSSMs under pinning control is studied for the first time. In particular, an appropriate technique for handling the impact of Markovian switching is proposed in the process of designing virtual controllers through the back-stepping method. Regarding the global Lyapunov function for an SCNSSM, we first construct a quartic Lyapunov function for each stochastic nonlinear strict-feedback system with Markovian switching of the SCNSSM, and then combine graph theory to construct the global Lyapunov function. After that, the theoretical results are applied to stochastic coupled oscillator systems with Markovian switching. Finally, the effectiveness of proposed results is verified by some numerical simulations.