This paper investigates tracking control of nonlinear systems in state-triggering setting. Since the trigger mechanism may result in system states being discontinuous throughout the whole control operation, the controller design process and stability analysis algorithms based on conventional backstepping design are not applicable for such discontinuous systems due to the non-differentiability of the virtual control. To tackle this dilemma, the first-order filter is introduced to smooth the discontinuous virtual control. Nonlinear impulse dynamics technique is adopted to verify the boundedness of the error variables generated by the filter in stability analysis. Furthermore, the adaptive parameters are solely updated at triggering points to alleviate the communication burden. Particularly, all states are required to be updated synchronously with the established concurrent state-triggering mechanism. It is indicated that with the developed backstepping-based state-triggering control scheme, the entire closed-loop signals are bounded, the system output tracks a prescribed reference trajectory and Zeno behaviour is absent. Finally, the simulation study illustrates the applicability of the proposed control scheme.Note to Practitioners—The study of tracking control for nonlinear systems is prevalent in various practical engineering applications, such as the gripping operation of a single-link manipulator, the orbiting of a spacecraft, and the cruising process of an unmanned aerial vehicle. Moreover, with the rapid development of networking, communication resources are often limited. Continuous information transmission in the sensor-to-controller channel can generate communication blockage and degrade transmission efficiency. Therefore, it is crucial to utilize the limited bandwidth resources to solve the tracking control problem of nonlinear systems. In this paper, we propose a backstepping-based state-triggering control scheme, where the triggered states and triggered adaptive parameters are used for the whole control design. The proposed scheme accomplishes the tracking control task well, especially in applications with limited bandwidth and energy. It is more consistent with requirements of the actual engineering.

This article is concerned with the event-triggered fault-tolerant control (FTC) for uncertain nonlinear cyber-physical systems (CPSs) by only exploiting the triggered faulty output. During the control design process, the unknown system dynamics, the time-varying sensor, and the actuator faults are considered simultaneously. Based on the event-triggered mechanism, the first-order filter technique and the nonlinear impulsive dynamics approach, an adaptive neural event-triggered output feedback FTC scheme is established. More specifically, one triggering condition is established for both the measurable output and the state estimations, with the adaptive parameters being triggered at the same instants. Another triggering condition is established for the controller, eliminating the need for real-time monitoring of control information and thereby reducing the computational burden. Then, a neural state observer is designed from triggered faulty output and triggered state estimations. The first-order filter technique is introduced to handle the non-differentiability of virtual controls stemmed from the event-triggered mechanism. The nonlinear impulsive dynamics approach is employed for stability analysis of the discontinuous error dynamics. It is proved that, with the proposed scheme, all the closed-loop signals are bounded, meanwhile the system output converges to the origin asymptotically, and the Zeno behavior is excluded. Finally, simulation results present the feasibility and effectiveness of the seeking schemes.

This paper investigates the event-triggered trajectory tracking control for unmanned surface vessels with prescribed performance subjected to asymmetric time-varying state constraints. In order to realize the prescribed-time control task, a performance function is introduced and the fuzzy logic systems are adopted to estimate the unknown nonlinearities of the USVs model. Subsequently, a flexible event-triggered mechanism is designed to minimize the wastage of communication resource, taking into account the continuous updating of the actual control input. A practical virtual control signal is devised and used for backstepping design. On this basis, an event-triggered adaptive fuzzy trajectory tracking control algorithm with prescribed performance is developed by combining the defined performance function and the barrier Lyapunov function approach, which not only solves the asymmetric time-varying state constraints of the USVs, but also ensures that the USVs achieve the predetermined tracking performance, meanwhile, all the closed-loop signals can be maintained bounded. Finally, the simulation results are performed to demonstrate the feasibility of the developed control scheme.

If a control input signal constructed by backstepping is used to generate an event-triggered control signal, then not only the real control input is discontinuous, but all the virtual control signals that make up the control signal are also discontinuous and have the same triggering instants with the control input. This fact makes the control design and stability analysis more difficult and complex via backstepping. This article focuses on this problem and works at coming up a new backstepping design procedure of event-triggered fuzzy adaptive control for nonlinear strict feedback systems. A concurrent event-triggered mechanism is proposed to ensure that the real control input and the virtual control signals have the same triggering instants. Then, a systemic backstepping design procedure is developed to construct event-triggered fuzzy adaptive controller. The Lyapunov technique for nonlinear impulsive system is employed to provide the closed-loop stability analysis. It is shown that the derived event-triggered fuzzy adaptive controller ensures that the closed-loop system output well follows the desired tracking trajectory and the other closed-loop signals remain bounded. Eventually, two examples are provided to further verify the applicability and reliability of the presented control scheme.

This article addresses event-triggered tracking control for nonlinear systems in strict-feedback form with triggered output signal. During control design process, system state variables are unavailable except for output variable. Fuzzy state observer is thus built by triggered output and triggered state estimations to produce the available state information. A compulsory event-triggered mechanism is established to determine the moment of online data transmission. Furthermore, only two triggering conditions are set up in sensor-to-controller and controller-to-actuator channels, respectively. Then a triggered output feedback adaptive controller, with the triggered adaptive laws, is designed via backstepping. Notice that both of the virtual and real control signals are discontinuous at each triggering time. So, nonlinear impulsive dynamics approach is used for the closed-loop stability analysis. It is strictly proved that the proposed triggered output feedback adaptive controller guarantees that all closed-loop signals are bounded and the Zeno phenomenon cannot occur. The illustrative simulation is used to authenticate the effectiveness of our proposed strategy.

Time delay has always been one of the main factors affecting the application performance of neural network (NN) systems, and dynamic performance research of NNs with time delays has been the focus of many scholars in recent years. This article enquires into the exponentially synchronous problem of switched delayed NNs with time delay in the leakage term. Adopting an unusual form from a common switched system, the switching modes of the switched delayed NNs system in this article are dependent on time delays. In the first place, the master, slave, and error NNs models are reconstructed into the switched form by introducing the switched delay idea. Then with the help of the admissible edge-dependent average dwell time (AED-ADT) method and delay-dependent switching adjustment indicators, a novel set of generalized delay-mode-dependent multiple Lyapunov-Krasovskii functionals (MLKFs) is built for analyzing the cases where a state-feedback controller exists and does not exist in the model, and where parts of LKFs may increase during the period when the corresponding subsystems are activated. For these cases, several effective exponential synchronization criteria and switching laws are presented accordingly. At last, the verification of the theoretical results is shown through a few examples.

It is known that a controller designed via backstepping contains all virtual control signals constructed at the previous design steps. When this controller is used to generate an event-triggered control signal which is discontinuous, those virtual control signals are simultaneously triggered and the corresponding triggered virtual signals are utilized for the real control operation, they are also discontinuous. This fact results in the noncontinuity of the virtual error dynamics. This problem is ignored in the existing backstepping-based event-triggered control schemes, in which control design and stability analysis are implemented according to the continuous virtual error dynamics, namely, all the virtual control signals are required to be continuous. Based on this observation, this article focuses on the backstepping-based event-triggered fuzzy adaptive control for nonlinear strict-feedback systems with unavailable states. To do so, a concurrent event generator is first introduced, furthermore, the backstepping-based output feedback control design is given according to the discontinuous virtual error dynamics. Then, the Lyapunov approach for nonlinear impulsive systems is used to provide the closed-loop performance analysis. The results show that with the event-based fuzzy adaptive controller, all the closed-loop signals meet bounded requirements and the Zeno phenomenon is prevented. Ultimately, the proposed theoretical results are supported by two simulation examples.

This article is concerned with adaptive fuzzy control design of nonlinear strict-feedback systems, in which system functions are unknown and system states are subjected to some constant constraints. The main control issue is to design an adaptive fuzzy controller such that the system output follows the reference signal, meanwhile, all the state variables abide by their constrained requirements. Differing from the existing way to build barrier function, the constructed virtual control signal is employed to build the barrier function. Furthermore, a backstepping-based adaptive fuzzy control design process is presented in this article. Compared to the control schemes in the existing literature on full-state constrained systems, the current control scheme has the following advantages: 1) the built virtual control signals in this article are ensured to satisfy corresponding state constraints, rather than assuming that they do, as in the existing articles; 2) the choice of initial values is independent of the norm of the virtual control signal. So, the conservatism of initial value selection is considerably reduced. It is shown that the proposed adaptive fuzzy controller not only ensures perfect tracking performance, but also ensures that all the states abide by the preassigned state constraints during the operation. At last, simulation study further verifies the efficacy of the proposed control strategy.

We study the exponential stability of discrete-time neural networks (NNs) with a time-varying delay which contains a few intermittent large delays (LDs). By modeling the considered discrete-time NN as a discrete-time switched NN which contains two subsystems and one of them may be unstable over the LD periods (LDPs), switching techniques are employed to analyze the problem. Delay-dependent exponential stability conditions to check the frequency and the length of the LDs allowed for guaranteeing the exponential stability are proposed by applying a novel Lyapunov-Krasovskii functional (LKF) with LDP-based terms, Wirtinger-based summation inequality, and reciprocally convex combination technique. Based on these conditions, associated evaluation algorithms are developed. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.

This article investigates the exponential stability of generalized neural networks (GNNs) with a time-varying delay. Different from the literatures on the similar topic, the considered time delay contains a few intermittent large-delay periods (LDPs). A new approach is proposed to determine how frequent and how long the LDPs are allowed for guaranteeing the exponential stability by using switching techniques. The GNN is first modeled as a switched time-delay system which may include an unstable subsystem. Then, based on a novel Lyapunov–Krasovskii functional with LDP-based terms, a delay-dependent exponential stability criterion and associated evaluation algorithm are developed. Finally, two numerical examples are provided to show the effectiveness of the proposed method.

In this paper, the stability problem of delayed neural networks is investigated by adopting some switching methods. First, the delay interval is divided into many smaller variable intervals, and when the smaller variable interval is regarded as a mode of the delay, delayed neural networks are modeled as switched systems. Then, by using some switching methods, less conservative stability criteria are derived to ensure the stability of delayed neural networks. Finally, one example is provided to show the effectiveness of the obtained criteria.

This paper investigates the problem of adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems with unstable inverse dynamics via time-dependent switching. Adaptive backstepping technique and fast average dwell time switching are exploited to set up the adaptive fuzzy control scheme for the systems under consideration. It is shown that the designed controllers and switching signals can ensure that all the signals remain bounded and the tracking error converges to a small neighborhood of the origin. Finally, simulation results are presented to demonstrate the effectiveness of the proposed design method.

Science of Neural Networks, and even much more so computing applications, have undergone developments beyond any predictions since McCullock-Pitts artificial neuron (1943) up via Hopfield's neurons (1982, 1984) to Kasabov spiking-neurons neucube (2014) and evolving connectionist systems (2003). Still computational functionality of all kinds of neural network implies guaranteed operating steady-state equilibrium is fast-reached first. On the other side of this spectrum Science of Neurophysiology yielded insights converging to Systems Biology approach Gayton-Hall (2006). It appeared, on the crossroad of these findings with Kolmogorov's representation superposition and Hilbert's Thirteen problem certain rater delicate subtle issues emerged Sprecher (2017). This paper gives one perception of these issues and suggested a revised view on the foundations of past developments, possibly by re-thinking own stability results for recurrent neural networks which possess time-varying delays.