This study investigates the fixed-time prescribed tracking control problem for the ship heading control system subject to marine stochastic noise and actuator faults. First, the fuzzy logic system (FLS) is used to approximate the unknown nonlinear terms. Then, a fault compensation mechanism based on an adaptive technique is developed to handle the actuator faults. Next, a fixed-time performance function is introduced to address the output tracking error constraint issue and make the output follow the reference signal within a fixed time. Furthermore, a fixed-time prescribed tracking controller and adaptive laws are designed. Within the proposed control framework, the tracking error is demonstrated to conform to the fixed-time prescribed performance. Furthermore, all closed-loop signals are bounded in probability, as substantiated through the application of stochastic Lyapunov stability theory. Finally, a series of simulation experiments is conducted to validate the efficacy of the proposed algorithm.

This paper investigates an adaptive fault-tolerant control problem for a class of strict-feedback incommensurate fractional-order multi-input multi-output (MIMO) nonlinear systems with intermittent actuator faults. To avoid the complexity explosion inherent in conventional backstepping methods, a control strategy without backstepping is developed. Firstly, via state transformation, the original system is reformulated into a structure suitable for the proposed framework, and an error observer is designed to estimate the unmeasurable states caused by the aforementioned transformation. To effectively compensate for intermittent actuator faults, a fault-tolerant controller incorporating the hyperbolic tangent function is constructed. Furthermore, fuzzy logic systems (FLSs) are employed to approximate the unknown nonlinear functions arising in the reconstructed system, thereby supporting the controller design. To facilitate stability analysis without resorting to frequency-distributed models, a sum-separable Lyapunov function is established. Due to the multi-order dynamic coupling in incommensurate fractional-order systems, a convex optimization problem arises. Therefore, a novel feasibility search algorithm for Riccati-type LMI constraints is put forward to systematically obtain feasible solutions. Subsequently, stability analysis demonstrates that all signals in the closed-loop system are uniformly bounded. Meanwhile, the prespecified control objectives are successfully achieved. Finally, a simulation example of the fractional-order Chua-Hartley system is provided to verify the effectiveness of the proposed fault-tolerant control scheme.

This work investigates fuzzy adaptive output-feedback control issue for incommensurate fractional-order nonlinear systems with input and output quantization. The input and output are quantized via a sector bounded quantizer. Due to the case that system states are partially measurable, a fuzzy state observer with quantized signals is constructed. Fuzzy logic systems are employed to model the nonlinear dynamics. Further, the fractional-order dynamic surface control strategy is developed to overcome the complexity brought by adaptive backstepping technique. Then, for the purpose of ensuring the boundability of a series of errors caused by continuous original state in stability analysis and discontinuous quantization state in control, a novel fractional-order projection operator with smooth property is proposed. Finally, by means of the frequency distribution model and indirect Lyapunov method, it is proved that all closed-loop signals are bounded. The advantages of the proposed control method are illustrated by a numerical example.

This paper addresses adaptive fuzzy tracking control for fractional-order nonlinear systems (FONSs) subject to asymmetric state constraints without imposing separate feasibility conditions on intermediate virtual controllers. Fuzzy logic systems are used to approximate the unknown nonlinear functions. By exploiting the boundedness of the hyperbolic tangent function, a coordinate transformation and an asymmetric fractional barrier Lyapunov function (AFBLF) are developed to construct bounded virtual control signals. Within a backstepping framework, an adaptive fuzzy controller is designed. Fractional-order Lyapunov analysis establishes semi-global uniform ultimate boundedness of all closed-loop signals and preservation of the prescribed asymmetric constraints. Comparative and benchmark simulations demonstrate constraint satisfaction, moderate control effort, and suppression of high-frequency chattering-like oscillations.

This paper addresses the adaptive resilient bipartite formation control problem for second-order nonlinear multi-unmanned ground vehicle systems under sensor data attacks, where system states become unmeasurable due to attacks and nonlinear dynamics are unknown. To solve this issue, the distributed resilient control method based on an adaptive fuzzy observer is proposed. Specifically, a distributed observer is designed to estimate the attacked states online, and adaptive technology is utilized to compensate for the unknown dynamics and the impact of attacks. This method can still achieve the preset bipartite time-varying formation under continuous attacks and ensure that the observation error is uniformly bounded. The Lyapunov function is constructed via linear matrix inequalities, and the stability of the closed-loop system is strictly proven. Simulation results demonstrate that the proposed method can maintain the formation configuration with stable observation errors under attacks of varying intensities.

An adaptive fuzzy command filtering backstepping technique is given for strict-feedback fractional-order uncertain nonlinear systems. The controlled system includes unknown nonlinear functions, as well as state and input quantization. Considering fractional-order nonlinear systems that do not satisfy matching conditions, unknown nonlinear functions are approximated by fuzzy logic systems, and a sector bounded quantizer is used to quantify all input and state variables. During the plan process, command filtering backstepping scheme is used to avoid the use of nonsmooth states. Subsequently, in order to ensure the boundedness of a series of errors caused by continuous original states for stability analysis and discontinuous quantization states for control, a sufficiently smooth fractional-order projection operator is proposed. In addition, the fractional-order uniformly bounded criterion has been established and strictly proven, which solves the problem of uniformly bounded error signals in the fractional-order sense under the premise of known parameter boundedness. Thus, the boundedness of all closed-loop signals is ensured by the fractional-order uniformly bounded criterion. Finally, the simulation results have confirmed the efficacy of the method.

This paper addresses the active fault-tolerant control problem for fractional-order nonlinear systems with output constraint. Firstly, fuzzy logic systems are employed to approximate the unknown nonlinear functions. Secondly, a state observer is designed for the unmeasurable states, while a fault detection mechanism is established to detect the occurrence of faults, and a fault compensation term is constructed to counteract their effects. Then, an asymmetric tangent-type Barrier Lyapunov Function (ATBLF) is introduced to handle the output constraints. Concurrently, an auxiliary function is incorporated into the virtual control signals to compensate for the coupling terms arising from the BLF. Finally, by combining backstepping techniques with a fuzzy adaptive update mechanism, a novel control strategy is designed. The Dynamic Surface Control (DSC) technique is utilized to avoid the “explosion of complexity” problem. Based on the fractional-order Lyapunov stability theory, it is proven that all signals in the closed-loop system are semiglobally uniformly ultimately bounded, and the output of the system can be constrained within the given bounds.

In this paper, an adaptive control tactic is designed for a class of nonlinear systems with nonconvex parameterization by using the backstepping method. During the backstepping control process, an improved nonlinear filter with a compensation term is introduced to deal with the repetitive differentiation problem of the designed virtual controllers, which removes the boundary layer error simultaneously. The nonconvex pa-rameterized terms are discussed in light of the extensions of explicit realizability assumptions. Under the constructed control algorithms, the bound ness of all signals in the closed-loop system is proved, and the tracking error can reach 0 as time approaches to infinity. Finally, the effectiveness of the proposed method is demonstrated through a simulation example.

In this paper, the fault-tolerant control (FTC) problem of a single-link robotic arm (SLRA) with sensor faults is investigated. Fuzzy logic systems (FLSs) are exploited for approximating unknown nonlinearities. According to the introduce the specific auxiliary functions, the unknown coupling terms with uncertain parameters generated by sensor faults are solved effectually. Under the backstepping control framework, a fault-tolerant controller is constructed. Then, the stability of the closed loop system is proved via Lyapunov stability theory. Finally, a numerical simulation example is provided to further show the feasibility of the proposed method.

This paper presents a nonlinear filters-based adaptive fuzzy control design for strict-feedback nonlinear systems with unknown asymmetric dead-zone output and virtual control coefficients. First, a novel smooth approximation of the non-smooth asymmetric output dead-zone nonlinearity is constructed to improve the approximation performance and speed. Then, to dilute the effects of unknown dead-zone and virtual control coefficients, an adaptive compensation mechanism is presented by utilizing projection operators techniques and the properties of fuzzy basis functions and hyperbolic tangent functions. Further, a novel adaptive backstepping control design method based on the nonlinear filters is proposed, which not only avoids the issue of explosion of complexity inherent in the backstepping procedure, but also completely compensates the effects of the boundary errors generated by the introduced filters in spite of unknown virtual control coefficients. From the properties of the smooth projection operator, and based on Lyapunov synthesis, it is shown that all closed-loop signals are ensured uniformly bounded. Finally, the applicability of the proposed control method is rigorously verified by a example of one-link manipulator with a brushed dc motor. Note to Practitioners —Strict-feedback nonlinear systems can be utilized to model many practical engineering systems, such as one-link manipulator, permanent magnet synchronous motor, robots with flexible joints, spacecrafts, etc. Due to the physical limitations and other factors, non-smooth dead-zone nonlinearity often presents in system components, which will influence the control performance and threat the operation security. Therefore, this paper considers the control design problem for strict-feedback nonlinear systems with unknown asymmetric dead-zone output nonlinearity, and the proposed control algorithm is successfully applied to one-link manipulator with a brushed dc motor. In addition, the virtual control coefficients are often assumed to be known constants or known nonlinear functions with known bounds in the existing results, which is unrealistic on account of the economic costs and environmental factors. To erase such limitation, this paper presents a novel adaptive compensation mechanism based on smooth auxiliary function dependent on the property of tanh-function and the projection operator technique.

For uncertain fractional-order nonlinear systems (UFONS) with unknown control coefficients and intermittent actuator faults, the asymptotic tracking control problem is investigated in this paper. Firstly, to weaken the influence of virtual control coefficients and intermittent actuator faults, a smooth fractional-order projection operator-based adaptive compensation mechanism is presented. Additionally, a fractional-order nonlinear filter is constructed to replace the fractional-order derivative of virtual control functions approximately, which not only avoids the issue of complexity explosion existed in backstepping control frame, but fully compensates the effects of boundary errors caused by the employed filter in spite of the unknown virtual control coefficient. By constructing a fractional Lyapunov function from the property of projection operator, it is proved that all signals in the closed-loop system are bounded, and the asymptotic tracking control object is achieved. Definitively, a simulation study is presented to verify the availability of the presented method.

In this work, an adaptive fuzzy backstepping fault-tolerant control (FTC) issue is tackled for uncertain fractional-order (FO) nonlinear systems with sensor and actuator faults. A fuzzy logic system is exploited to manage unknown nonlinearity. In addition, a novel FO nonlinear filter-based dynamic surface control (DSC) method is constructed, effectively avoiding the inherent complexity explosion problem in the backstepping recursive process, and in the light of the construction of auxiliary functions, compensating the coupling term introduced by faults. On account of certain assumptions, the stability criterion of the FO Lyapunov function is applied to guarantee the stability of the closed-loop system. Finally, the simulation example verifies the validity of the presented control strategy.

This paper investigates the problem of fixed-time distributed time-varying optimization of a nonlinear fractional-order multiagent system (FOMAS) over a weight-unbalanced directed graph (digraph), where the heterogeneous unknown nonlinear functions and disturbances are involved. The aim is to cooperatively minimize a convex time-varying global cost function produced by a sum of time-varying local cost functions within a fixed time, where each time-varying local cost function does not have to be convex. Using a three-step design procedure, a fully distributed fixed-time optimization algorithm is constructed to achieve the objective. The first step is to design a fully distributed fixed-time estimator to estimate some centralized optimization terms within a fixed time T0. The second step is to develop a novel discontinuous fixed-time sliding mode algorithm with nominal controller to derive all the agents to the sliding-mode surface within a fixed time T1, and meanwhile the dynamics of each agent is described by a single-integrator MAS with nominal controller. In the third step, a novel estimator-based fully distributed fixed-time nominal controller for the single-integrator MAS is presented to guarantee all agents reach consensus within a fixed time T2, and afterwards minimize the convex time-varying global cost function within a fixed time T3. The upper bound of each fixed time Tm(m=0,1,2,3) is given explicitly, which is independent of the initial states. Finally, a numerical example is provided to validate the results.

We present a method to precisely determine the $^{2}D_{3/2}$ state lifetime of a single trapped $^{174}\mathrm{Yb}^{+}$ ion. This method is based on the detection of photons emitted from excited states, where a highly synchronized measurement sequence for laser control and an intensity-alternating sequence for atomic excitation and photon counting are used to minimize systematic errors. This method is easy to implement and is immune to fluctuations of magnetic field, laser intensity, and frequency detuning. Combined with the real-time approach of background photon correction, the radiative lifetime of the $^{2}D_{3/2}$ state is determined to be $54.83\ifmmode\pm\else\textpm\fi{}0.18$ ms, which represents an order of magnitude improvement in measurement precision. The accurately determined lifetime sets a benchmark for many-body atomic theories and is particularly useful to determine the coherence time limit of the optical clock.

This study is devoted to a nonlinear filter-based adaptive fuzzy output-feedback control scheme for uncertain fractional-order (FO) nonlinear systems with unknown external disturbance. Fuzzy logic systems (FLSs) are applied to estimate unknown nonlinear dynamics, and a new FO fuzzy state observer based on a nonlinear disturbance observer is established for simultaneously estimating the unmeasurable states and mixed disturbance. Then, with the aid of auxiliary functions, a novel FO nonlinear filter is given to approximately replace the virtual control functions, together with the corresponding fractional derivative, which not only erases the inherent complexity explosion problem under the framework of backstepping, but also completely compensates for the effects of the boundary errors induced by the constructed filters compared to the previous FO linear filter method. Under certain assumptions, and in line with the FO stability criterion, the stability of the controlled system is ensured. An FO Chua–Hartley simulation study is presented to verify the validity of the proposed method.

This paper studies the issue of adaptive fuzzy output-feedback event-triggered control (ETC) for a fractional-order nonlinear system (FONS). The considered fractional-order system is subject to unmeasurable states. Fuzzy-logic systems (FLSs) are used to approximate unknown nonlinear functions, and a fuzzy state observer is founded to estimate the unmeasurable states. By constructing appropriate Lyapunov functions and utilizing the backstepping dynamic surface control (DSC) design technique, an adaptive fuzzy output-feedback ETC scheme is developed to reduce the usage of communication resources. It is proved that the controlled fractional-order system is stable, the tracking and observer errors are able to converge to a neighborhood of zero, and the Zeno phenomenon is excluded. A simulation example is given to verify the availability of the proposed ETC algorithm.

This paper studies a fuzzy adaptive fixed-time tracking control issue for nonlinear high-order largescale systems. Fuzzy logic systems (FLSs) are utilized to identify unknown nonlinearities. Through using adaptive backstepping and adding a power integrator technique, the fixed-time decentralized control method is presented. It is proved that the tracking errors converge to a small neighborhood of a fixed time. A simulation example is presented to confirm the validity of the developed control method.

This study deals with the output-feedback asymptotic tracking control problem for a class of nonlinear strict-feedback systems with actuator loss of effectiveness failure. To handle with the output-feedback control issue in the presences of nonlinearities, a new reduced-order observer design is presented, by utilizing the dynamic gain technique, which not only eliminates the limitation that the Lipchitz coefficients are required to be known in the existing output-feedback results, but makes full use of the measurable information. Furthermore, a new failure compensation mechanism is proposed to erase the effect of actuator failure, by introducing a cubic absolute-value Lyapunov function method and a novel (σ,σf)-modification technique. Compared with the existing output-feedback failure compensation results, our proposed method can not only relax the assumption requirement on nonlinear function, i.e., the nonlinear function with respect to output y can be extended to the nonlinear one with respect to state variable χ¯i in the means of asymptotic tracking, but also avoid the issue that the estimate for actuator efficiency indicator drifts to a large value suddenly. Further, within the framework of backstepping design, a new high-gain reduced-order observer based adaptive output-feedback failure compensation control is developed. Then, with the aid of Lyapunov analysis method, it is shown that all the signals in the closed-loop system are globally bounded, and the system output can asymptotically track a given reference signal. Finally, a simulation example is given to illustrate the efficiency of the proposed techniques.

This article presents an adaptive fuzzy output-feedback dynamic surface control (DSC) for the nonlinear systems with dead-zone output nonlinearity. First, a novel smooth approximation of the output dead zone is given to conveniently fuse with the backstepping technique. After that a nonlinear fuzzy state observer is constructed to estimate the unmeasurable states, by employing the fuzzy logic systems for identifying the unknown compounded nonlinear functions, which releases the limitation in the existing references that the state observer needs to be linear in the presence of dead-zone output nonlinearity. Then, to reduce the effect of unknown dead-zone coefficients, an adaptive compensation mechanism is introduced by using the characteristic of hyperbolic tangent function, which can replace the widely used Nussbaum-type function-based control strategy. Furthermore, a novel DSC method based on the nonlinear filters is proposed, which not only avoids the issue of explosion of complexity inherent in the backstepping procedure, but also improves the system control performance. Under the certain assumptions, the stability of the closed-loop system is proved by use of Lyapunov function stability theory. Finally, the applicability of the proposed control method is rigorously verified by a single-link robot arm simulation example.

This paper presents a novel adaptive fuzzy backstepping dynamic surface control (DSC) scheme for a class of single-input single-output strict-feedback fractional-order uncertain nonlinear systems. The controlled systems contain unknown nonlinear functions and unknown external disturbances. Fuzzy logic systems are employed for approximating the unknown nonlinear functions. Further, an auxiliary function is introduced into the control function to simultaneously compensate the unknown external disturbance and the approximation error caused by fuzzy approximation, which erases the possible chattering phenomenon in the existing results. Meanwhile, a new DSC method based on the fractional-order filter is proposed to avoid the issue of explosion of complexity inherent in the backstepping procedure, which releases the limitation that the fractional-order derivative of the intermediate control function needs to be completly known in the existing references. Under certain assumptions, the stability of the closed-loop system is proved by using the fractional-order Lyapunov function stability criterion. Finally, contrastive simulation results are provided to validate the effectiveness of our proposed control strategy.