Strict-feedback Nonlinear Systems Research Articles

Prescribed performance event-triggered fuzzy optimal tracking control for strict-feedback nonlinear systems

The prescribed performance event-triggered optimal tracking control problem is considered for a class of uncertain strict-feedback nonlinear systems with external disturbances. First, the disturbance observers are constructed to estimate external disturbances. Then, the controller contains an adaptive fuzzy controller and an optimal compensation term, in which the unknown nonlinearities and cost function are approximated by the fuzzy logic systems. An adaptive fuzzy controller is established by the dynamic surface control (DSC) technique, which is addressed to handle “computation complexity” issue occurred in conventional backstepping approach. Based on adaptive dynamic programming (ADP) method, an optimal compensation term is developed by minimizing the cost function. Furthermore, communication load is reduced via embedding event-triggered mechanism. In addition, the tracking error can be restricted in the prescribed region with the aid of the prescribed performance control. Thus, the whole control scheme can not only ensure that the tracking error converges to a boundary but also achieve optimization, reduce the communication burden and avoid “computation complexity” issue. The boundedness of all signals in the closed-loop is proved. Finally, the simulation examples illustrate the validity of the designed control scheme.

Observer-based dynamic event-triggered adaptive control for uncertain nonlinear strict-feedback systems

This paper proposes an observer-based dynamic event-triggered adaptive control approach for uncertain nonlinear strict-feedback systems. Initially, an observer for unmeasurable states is constructed. Subsequently, employing backstepping technique, the output-feedback adaptive control law and parameter adaptive law are developed. The tuning function method is used to avoid the over-parameterization problem in parameter adaptive law design. To lower the data exchange rate within the network, a dynamic event-triggered scheme is formulated to allow real-time update control signal. Through Lyapunov theory analysis, the resulting closed-loop system is demonstrated to be stabilized, and the occurrence of Zeno behavior is effectively prevented. Finally, the simulation example validates the obtained result of the presented design approach.

Tuning function-based fault-tolerant finite-time control for nonlinear systems with sensor/actuator faults

This paper discusses the adaptive fault-tolerant finite-time control tracking problem for a class of strict-feedback nonlinear systems with multiple faults and disturbances. Our essay focuses on the problem of over-parameterisation in traditional backstepping methods. A new fault-tolerant control scheme is proposed by combining tuning function with fuzzy control technology and finite-time control. This scheme greatly simplifies the process of derivation and computation and avoids high computational cost and high memory consumption. Thus the purpose of improving the reliability and robustness of the system is achieved. Finally, according to the Lyapunov finite-time stability theory, the proposed method can ensure that the closed-loop system remains semi-globally practically finite-time stable and the tracking errors converge to a small neighbourhood near the origin in finite time under the condition of unmatched disturbances and sensor/actuator faults simultaneously. The effectiveness of the proposed control strategy is verified by a simulation example.

Practically Predefined-Time Adaptive Fuzzy Tracking Control for Nonlinear Stochastic Systems.

This article addresses the practically predefined-time adaptive fuzzy tracking control problem of strict-feedback nonlinear stochastic systems, where the system under consideration includes stochastic disturbances and uncertain parameters. First, in this study, practically predefined-time stochastic stabilization (PPSS) in the p th moment sense is introduced, and a Lyapunov-type criterion for PPSS is proposed to assure the stabilization of the system considered. With these ideas, based on the backstepping design method, a semiglobally practically predefined-time adaptive fuzzy tracking control algorithm is proposed with a fuzzy system used to approximate the unknown part of the system. Moreover, the settling time of the system response can be arbitrarily adjusted in a mean-value sense, and such freedom can be used to improve the stochastic finite-/fixed-time control results. Finally, a practical example and a numerical example of a comparison are provided to validate the effectiveness of the proposed control strategy.

Event-Triggered Tracking Control for a Class of Nonlinear Systems: A Dynamic Gain Approach

This note presents a dynamic gain approach to the event-triggered tracking control for a class of strict-feedback nonlinear systems. A novel shifting function is designed to make the initial tracking error arbitrarily bounded, which significantly relaxes the requirement on the initial tracking error. To achieve the objective that the tracking error does not exceed the constraint after the pre-specified finite time, a barrier Lyapunov function and a new reduced-order observer are constructed by using the shifting function and the desired tracking signal. Based on the proposed dynamic gain approach, an event-triggered output feedback tracking controller is designed. It is proved that the developed control scheme guarantees that all the closed-loop signals are bounded and the tracking error constraint is strictly obeyed after a specified finite time. Compared with the celebrated backstepping method, the dynamic gain approach does not involve any virtual control variables, which avoids the complicated iterative procedure. The effectiveness of the proposed approach is verified by an example.

Prescribed Performance Fault-Tolerant Control of Strict-Feedback Systems via Error Shifting.

This article investigates the prescribed performance control (PPC) problem for a class of nonlinear strict-feedback systems with sensor/actuator faults. A shifting function is introduced to modify the output tracking error generated by the practically measured system state, based on which an improved PPC method is proposed to achieve the convergence of output tracking error to the prescribed region, and this convergence is shown to be independent of the initial tracking condition and insusceptible to sensor/actuator faults. The faults-induced uncertainties together with the nonlinear dynamics are compensated by involving a radial basis function neural network (RBFNN) to make the controller robust adaptive fault-tolerant without prior knowledge of fault coefficients. Via Lyapunov stability analysis, it is proven that all signals in the closed-loop system are semiglobally uniformly ultimately bounded. The effectiveness and superiority of the method are demonstrated by two simulation examples.

Asymptotic Tracking Control for Uncertain Nonlinear Strict-Feedback Systems With Unknown Time-Varying Delays.

It is nontrivial to achieve asymptotic tracking control for uncertain nonlinear strict-feedback systems with unknown time-varying delays. This problem becomes even more challenging if the control direction is unknown. To address such problem, the Lyapunov-Krasovskii functional (LKF) is used to deal with the time delays, and the neural network (NN) is applied to compensate for the time-delay-free yet unknown terms arising from the derivative of LKF, and then an NN-based adaptive control scheme is constructed on the basis of backstepping technique, which enables the output tracking error to converge to zero asymptotically. Besides, with a milder condition on time delay functions, the notorious singularity issue commonly encountered in coping with time delay problems is subtly settled, which makes the proposed scheme simple in structure and inexpensive in computation. Moreover, all the signals in the closed-loop system are ensured to be semiglobally uniformly ultimately bounded, and the transient performance can be improved with proper choice of design parameters. Both the theoretical analysis and numerical simulation are carried out to validate the relevance of the proposed method.

Adaptive Multi-Dimensional Taylor Network Tracking Control for a Class of Nonlinear Strict Feedback Systems

Nonlinear systems are very common in real life, but because they are not superposed and homogeneous, there are many difficulties in controlling nonlinear systems. Therefore, an adaptive control method based on a multi-dimensional Taylor network (MTN) is proposed for a class of nonlinear systems with strict feedback so that the output of the system can track the given signal. In order to achieve the control effect, we define a new state variable and transform the strict feedback system. After transformation, the original feedback system has a standard form, and two parameters to be identified are obtained. Then, the state observer is designed, and the two parameters are identified via the approximation of the MTN. On this basis, the controller design and a system stability analysis are completed. The lemma is introduced, and the stability condition is established by using this low-pass filter to ensure that all closed-loop signals are semi-globally uniform and finally bounded and the output tracking error converges to the residual set near zero. Finally, a numerical simulation of a hydraulic system is carried out to verify the effectiveness of the proposed method. Under the three indexes, the proposed method has obvious advantages.

Dynamic learning from adaptive neural control for full-state constrained strict-feedback nonlinear systems

This study focuses on the learning and control issues of strict-feedback systems with full-state constraints. To achieve learning capability under constraints, transformation mapping is utilized to convert the original system with full-state constraints into a quasi-pure-feedback unconstrained system. Utilizing the system transformation technique, only a single neural network (NN) is required to identify the unknown dynamics within the transformed system. Combining the dynamic surface control design, a novel adaptive neural control scheme is developed to ensure that all closed-loop signals are uniformly bounded, and every system state remains within the predefined constraint range. In addition, the precise convergence of NN weights is further transformed into an exponential stability problem for a category of linear time-varying systems under persistent excitation conditions. Subsequently, the converged NN weights are efficiently stored and utilized to create a learning controller to achieve better control performance while abiding by the full-state constraints. The viability of this control strategy is demonstrated via simulations.

Fixed-time command-filtered composite adaptive neural fault-tolerant control for strict-feedback nonlinear systems

The research investigates the fixed-time command-filtered composite adaptive neural fault-tolerant (FCCANF) control issue of strict-feedback nonlinear systems (SFNSs). There exist unknown functions and bounded disturbances in the considered systems. Radial basis function neural networks (RBFNNs) will be used in the estimate of the unknown functions. By the serial–parallel estimation models (SPEMs), the forecast biases and the track biases can change the weights of RBFNNs and the approximate characteristics of RBFNNs will be improved. Then, utilizing the novel fixed-time command filter and adaptive disturbance observers, the issue of complex explosion will be effectively solved and the external disturbance is effectively compensated. Subsequently, by utilizing the adaptive control technique, a novel FCCANF controller is developed. Additionally, we have that the system internal variables are bounded and the output variable inclines to a little interval around zero in fixed time which is not determined by the system initial variables. Eventually, numerical and practical examples are shown to prove the availability of the obtained control technique.

Practical finite-/fixed-time improved command-filtered backstepping control for nonlinear systems via immersion and invariance

This paper presents a practical finite-/fixed-time adaptive tracking control scheme based on improved command filtered backstepping (ICFBS) and immersion and invariance (I&I) for a class of single-input single-output (SISO) strict-feedback nonlinear systems with uncertainty parameterization. The conventional command-filtered backstepping method (CFBS) is improved by introducing any bounded function such that ICFBS relaxes the boundedness assumption of the control direction function and elevates the tunable space of the design scheme, with almost no increase in computational complexity. The higher-power composite term of a parameter estimation is designed in the parameter update law based on I&I and the inequality constructed by us proves that this form of the adaptive update law can be augmented to suit for not only uniform ultimate bounded (UUB) and practical finite-time stability but also practical fixed-time stability of the whole closed-loop system. The error compensation system is also extended to realize the practical finite-/fixed-time convergence of the error compensation signals. The proposed controller guarantees practical finite-/fixed-time stability of the closed-loop system. Therefore, our work solves two unsolved problems: both practical fixed-time and practical finite-/fixed-time ICFBS and I&I adaptive tracking control. Finally, the proposed control scheme is validated by a practical electromechanical system and a nonlinear system with unbounded control direction functions.

Observer-based event-triggered recursive optimal tracking control for a class of strict-feedback nonlinear systems

In this paper, an innovative event-triggered optimal tracking control algorithm is proposed for input saturated strict-feedback nonlinear systems with unknown dynamics. In order to reduce the requirement of configuring a complete suit of sensors and enhance the reliability of the controlled system, a neural networks (NNs) based adaptive state observer is developed firstly to reconstruct the system states. Subsequently, based on the state estimation information, a hybrid-triggered feedforward controller is designed to transform the original tracking control problem into an equivalent regulation issue, which is then solved by developing an event-triggered optimal controller. Therefore, the final controller consists of a hybrid-triggered feedforward controller and an event-triggered optimal controller. In order to make the actual input signals of the two controllers be updated simultaneously, a synchronization-oriented triggering rule is established by using multiple triggering errors. By virtue of this unique framework, the proposed control scheme can not only minimize the predefined cost function, but also greatly reduce the data transmission. What is more, the convergence properties of the proposed control strategy are achieved by using Lyapunov theory. It is important to note that unlike the widely adopted observer-controller framework, where the separation principle holds for the design of the state observer, there is a considerable coupling relationship between the error dynamics of the state observer and the event-triggered optimal controller in this paper. The distinguishing feature of the proposed method is its ability to ensure a satisfactory level of precision in both state estimation and tracking control, even in the presence of control saturation issues. At last, the proposed control strategy is applied to the tracking control problem of a high-order robot system and marine surface vehicle to demonstrate its effectiveness.

Adaptive practical prescribed-time formation tracking of networked nonlinear multiagent systems with quantized inter-agent communication

This paper presents an adaptive practical prescribed-time (PPT) control design for distributed time-varying formation tracking of networked uncertain strict-feedback nonlinear systems with quantized inter-agent communication under a directed network. The primary contribution of this study is to develop a time-varying formation tracking strategy to resolve the quantized inter-agent communication problem in the prescribed-time control field. A practical finite-time function is introduced to design an adaptive PPT formation tracker using quantized output information of neighbors, which can be used continuously after a prescribed time. A neural-network-based design methodology that utilizes the quantization-based distributed errors is presented to deal with the unknown virtual and actual control coefficient functions in the command-filtered backstepping design. Distributed adaptive compensating signals are constructed using the practical finite-time function to compensate for command-filter errors, unknown nonlinearities, and quantization errors in the PPT tracking framework. We prove that despite the presence of quantized inter-agent communication, the time-varying formation tracking errors converge to a compact set including the origin within the pre-assigned convergence time, where the compact set can be adjusted by choosing a design parameter of the practical finite-time function, and the prescribed settling time is independent of the initial system conditions and design parameters. The effectiveness of the proposed theoretical approach is confirmed through two simulation examples.

Predefined-time event-triggered adaptive tracking control for strict-feedback nonlinear systems with full-state constraints

This paper investigates the problem of predefined-time event-triggered adaptive tracking control for strict-feedback nonlinear systems (SFNSs) with full-state constraints. To handle asymmetric full-state constraints, a nonlinear state-dependent function (NSDF) that purely depends on the constraint state is introduced. Then, a switching threshold event-triggered mechanism (ETM) is designed to enhance usage efficiency of communication resources. Moreover, a predefined-time event-triggered adaptive tracking controller is constructed by incorporating a smooth tuning function into each step of the backstepping design based on dynamic surface technology. Under such a control scheme, the output tracking error can be steered to the small neighborhood of the origin within the user setting time. Compared to various previous control schemes developed for full-state constrained systems, our proposed method can circumvent the demanding feasibility conditions in controller designs. Finally, a simulation result is given to illustrate the effectiveness of the control scheme.

Fuzzy Adaptive Finite-Time Tracking Control for a Class of Nonlinear Systems: An Event-Triggered Quantized Control Scheme

This brief proposes a fuzzy adaptive event-triggered quantized finite-time control (ETQFTC) scheme for uncertain strict-feedback nonlinear systems with full-state constraints. By means of state transformation, filtered backstepping technique and fuzzy adaptive control with the idea of minimum learning parameter (MLP), a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C^{1}$</tex-math></inline-formula> ETQFTC scheme with low differential calculation and few parameter updates is developed. In the proposed scheme, the feasibility condition generally required in the existing state-constrained control, and the control chattering and singularity problem which are common in the existing FTC schemes are removed and avoided respectively. Finally, complete stability analysis and a set of comparative simulations are presented.

A novel adaptive coordinated tracking control scheme for a morphing aircraft with telescopic wings

Inspired by flight biology, morphing flight technology has great potential to improve the adaptability and maneuverability of aircraft. This paper is devoted to the flight control problem of morphing aircraft, and aimed at safe and fuel-saving flight through morphing actively. Specifically, the longitudinal dynamics of a morphing aircraft with telescopic wings is modelled as a strict-feedback nonlinear system. Through fitting the expression of aerodynamic parameters by the morphing ratio, the model uncertainties induced by morphing errors are embedded in the dynamics. To meet the safety and fuel-saving requirements, an Adaptive Coordinated Tracking Control Scheme (ACTCS) is then proposed, which consists of a morphing control module and a tracking control module. For the morphing control module, an on-line morphing decision model is given in an optimization process with respect to the morphing ratio, and a second-order tracking filter is introduced to smooth the decision output and ensure the physical realizability. For the tracking control module, the novel adaptive controllers for the velocity and altitude subsystems are proposed based on the dynamic surface control method, in which adaptive mechanisms are designed to compensate for the model uncertainties. Finally, the proposed ACTCS is simulated in nine different cases of the test flight mission, to verify its effectiveness, robustness and fuel-saving effect.

Multiple adaptive fuzzy Nussbaum-type functions design for stochastic nonlinear systems with fixed-time performance

Nussbaum design has been successfully exploited to deal with uncertain systems with a single unknown control coefficient. However, it is challenging for the traditional Nussbaum method to efficiently handle the feedback control design with multiple unknown time-varying control coefficients because the multiple Nussbaum-type gains might mutually cancel out, which makes the Lyapunov-based stability analysis infeasible. To tackle this issue, a novel type of Nussbaum function is designed for the tracking problem of a class of stochastic strict-feedback nonlinear systems. In the presented Nussbaum design, the multiple unknown control coefficients are separately compensated, where the effect of mutual cancellation no longer exists and the Lyapunov stability analysis can be directly applied. Moreover, the proposed Nussbaum design is combined with the fixed-time stability theory, which ensures the settling time is bounded and independent of the initial condition. Within the adaptive fuzzy backstepping framework, it is shown that all signals in the closed-loop system remain bounded through the theoretical discussions. The simulation experiment is conducted to verify the effectiveness of the presented design for the continuous-time stochastic nonlinear dynamical system.

Adaptive robust output tracking of uncertain strict-feedback nonlinear systems with full state constraints via control schemes with simple structure

In this paper, the problem of output tracking control is considered for a class of uncertain strict-feedback nonlinear systems with virtual control coefficients and full state constraints. It is assumed only that the nonlinear uncertainties of the systems are some smooth functions enough to guarantee the existence of solutions to the differential equations which describe uncertain systems, and that the virtual control coefficients are unknown. Based on backstepping algorithm, by combining the barrier Lyapunov function with an integral inequality reported in the current control literature, a design method is presented whereby a simple adaptive robust output tracking control scheme can be synthesised. The presented design method can evade the repeated differentiation problem appearing in using backstepping algorithm, and need not know all the nonlinear upper bounds of uncertainties, which are repeatedly employed at each step of backstepping procedure. It is also shown that the output tracking error of the uncertain nonlinear systems with both unknown virtual control coefficients and full state constraints can converge uniformly exponentially towards a ball which can be as small as desired. Finally, a numerical example is given, and the corresponding simulations are also implemented to demonstrate the simplicity of the proposed method and the validity of the theoretical results.

Control-error-based output-feedback adaptive decentralized neural network controller for interconnected uncertain strict-feedback nonlinear systems with input saturation

In this paper, a control-error-based decentralized neural network (NN) direct adaptive controller is presented for uncertain interconnected nonlinear systems, in strict-feedback form, subject to input saturation and external disturbances with unavailable states for measurement. Different from the existing results in the literature, the proposed approach is based on the control error instead of the tracking error resulting in a separation-like principle. Furthermore, the explosion of complexity due to back-stepping recursive design is completely avoided along with discarding all restrictive assumptions imposed on the unmatched interconnections. Actually, NNs are used to approximate the unknown ideal control laws, and auxiliary control terms are appended to deal with approximation errors and enhance the stability of the closed-loop system. Besides, fuzzy inference systems are introduced to estimate the unknown control errors, leading to simplified derivation of adaptive laws. Thanks to the strictly positive real (SPR) property, the tracking errors are proved to converge asymptotically to zero using Lyapunov theory, which is superior to bounded stability results usually found in the literature. Simulation results show the effectiveness of the proposed approach.

Lifelong learning‐based multilayer neural network control of nonlinear continuous‐time strict‐feedback systems

AbstractIn this paper, we investigate lifelong learning (LL)‐based tracking control for partially uncertain strict feedback nonlinear systems with state constraints, employing a singular value decomposition (SVD) of the multilayer neural networks (MNNs) activation function based weight tuning scheme. The novel SVD‐based approach extends the MNN weight tuning to layers. A unique online LL method, based on tracking error, is integrated into the MNN weight update laws to counteract catastrophic forgetting. To adeptly address constraints for safety assurances, taking into account the effects caused by disturbances, we utilize a time‐varying barrier Lyapunov function (TBLF) that ensures a uniformly ultimately bounded closed‐loop system. The effectiveness of the proposed safe LL MNN approach is demonstrated through a leader‐follower formation scenario involving unknown kinematics and dynamics. Supporting simulation results of mobile robot formation control are provided, confirming the theoretical findings.