Event-triggered adaptive neural network controller for uncertain nonlinear system
Introduction
Event-triggered control has gained considerable attention in the control community, especially in network control field. Traditional control method is based on continuous control input to guarantee the system stability which need much communication resource to transfer the control signal. Event-triggered strategy can be seen as a supervised control which can decide if control signal needs to be sent to the plant based on some predefined performance condition. In recent years, many results have been reported with different event-triggered conditions, for example see [1], [6], [11], [12], [13], [19], [20], [21].
Most of the existing event-triggered results are developed for known and linear systems, e.g. [12], [20], [24], [27], and the references therein. In [20], a very simple but useful event-triggered scheduling strategy is proposed to ensure system stability under the condition that the measurement error caused by the event-triggered mechanism. Furthermore, the event-triggered mechanism has been studied for robust control of linear uncertain system [23]. Nonlinear systems, which are more practical, also attract attentions from many researchers [1], [11], [19], [21]. Event-triggered schemes are developed in [15] based on hybrid systems tools for the stabilization of nonlinear system with ISS condition. Based on the decreasing property of Lyapunov function, results with static event-triggered conditions are established in [7], [11], [20]. A dynamic event-triggered condition is considered in [3] to reduce the communication burden which allows the Lyapunov function to increase at some points but decrease in the whole execution times. As a more realistic scenario, event-triggered design for unknown nonlinear systems is more meanful. However, because of the complicated system structure and unknown parameters, only limited results on event-triggered control of unknown nonlinear systems are available. Three different mechanisms are proposed in [29] to address the adaptive event-triggered problem of nonlinear systems. On the other hand, neural networks have been studied for decades and the universal property of neural network has been used in adaptive control process, see for example [4], [14], [25], [26], [30]. The application of on-line neural network for control and learning system need response rapidly, but the neural network need a lot of calculation and learning processes which cost too much time, so it is necessary to find a practical method to reduce the calculation burden of the neural network application. Considering the event-triggered mechanism, the adaptive neural controllers with event-triggered condition have been proposed in [17] and [16], in order to reduce the computational and communication burden of the closed-loop system for the continuous-time and discrete-time domains, respectively.
The objective of this paper is on designing event-triggered adaptive neural controllers for unknown nonlinear systems. We begin with designing a basic adaptive controller based on the backstepping technique [9] and neural network with low-frequency learning [31]. Then we propose static mechanism and dynamic mechanism for the obtained adaptive neural controller. In the static mechanism, we design an event-triggered condition based on a given fixed bound of the measurement error, which is the error between the control signals generated by the event-triggered controller and the designed controller. In order to ensure the stability of the system, the basic adaptive neural network controller with the addition of introducing an extra term. In the dynamic mechanism, a new variable which can be considered as a filtered value of the signal in the static mechanism is designed. The stability of the closed-loop system is then established. Both event-triggered mechanisms can guarantee that the lower bound of inter-event time is larger than zero, i.e., Zeno behavior is avoided. Compared with the event-triggered mechanisms in existing neural controllers such as those in [17] and [16], our event-triggered conditions are designed without the input-to-state linearizable condition while with simpler threshold. Furthermore, the established results are to solve the nonlinear tracking problems which are more practical than the stabilization problem in [17]. It is noted that ISS condition is also required in [17] where continuous-time systems are considered.
The remainder of the paper is organized as follows. Section 2 formulates the control problem. Static and dynamic event-triggered controllers are given in Section 3, followed by some simulation examples which illustrate our theories in Section 4. The paper is concluded in Section 5.
Section snippets
System description and problem formulation
In this section, we formulate the design problem and introduce the RBF neural network which will be used in next section.
Event-triggered neural network with adaptive backstepping
In this section, we present a basic adaptive backstepping control with low-frequency leanrning neural network firstly. Then static event-triggered mechanism and dynamic event-triggered mechanism are introduced based on the proposed adaptive neural network control method.
Simulation example
In this section, we consider the following system from a rigid robot of actuator dynamics [10], [22] to illustrate the established theoretical results.Define and which are unknown functions to designers. The initial states of the system are chosen to be . The parameters of the system are for simulation purpose. We choose, and . The event-triggered parameters are chosen as
Conclusions
In this paper, we consider the event-triggered adaptive control for unknown nonlinear system. Two different event-triggered mechanisms are proposed to ensure the stability of the closed-loop system with low communication burden. The static event-triggered method is designed based on the event-triggered error of the designed control signal and event-triggered control signal. And in dynamic mechanism, a filtered variable of the event-triggered condition in static mechanism is introduced to build
Declaration of Competing Interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript.
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