A Lyapunov‐based small‐gain approach on design of triggering conditions in event‐triggered control systems

Abstract

This paper studies a Lyapunov-based small-gain approach on design of triggering conditions in eventtriggered control systems. The event-triggered control closed-loop system is formulated as a hybrid system model. Firstly, by viewing the event-triggered control closed-loop system as a feedback connection of two smaller hybrid subsystems, the Lyapunov-based small-gain theorems for hybrid systems are applied to design triggering conditions. Then, a new class of triggering condition, the safe, adjustable-type triggering condition, is proposed to tune the parameters of triggering conditions by practical regulations. This is conducive to break the restriction of the conservation of theoretical results and improve the practicability of event-triggered control strategy. Finally, a numerical example is given to illustrate the efficiency and the feasibility of the proposed results.L 2 -gain analysis of event-triggered networked control systems with network-induced delays in a discontinuous Lyapunov functional approach.Recently, more and more studies (see [14][15][16] and the references therein) attempted to formulate the event-triggered control system as a hybrid system model (see [17][18][19][20] for more details on hybrid dynamical systems). The authors in [14] and [16] combined the event-triggered control and the time-triggered control by introducing an auxiliary hybrid clock variable. With this method, the positive lower bound of inter-event times in the case of output feedback is guaranteed. In [15], the authors proposed a unifying framework for the event-triggered stabilization from a hybrid system model point of view. They applied this framework to analyze several types of event-triggered control systems, such as that in [3] and [5]. The aforementioned works showed that the hybrid system model is closer to the actual operating process of event-triggered control systems. That is, during the intervals of two consecutive triggering instants, the dynamic flows according to some continuoustime rules, and at the triggering instants, the dynamic jumps depending on some discrete-time jump rules. Hence, by this more feasible description, one can expect to propose some essential results on event-triggered control systems. Meanwhile, a natural and familiar way to represent the dynamics of an event-triggered control system is to view the closed-loop system as a feedback connection of two smaller (hybrid) subsystems, that is, a subsystem of the state vector x and a subsystem of the state error e; which is the difference between the current state and the latest transmitted state. This kind of system with a feedback connection of two subsystems is pretty common and canonical in engineering fields. Also, it is well known that the small-gain theory is an important and powerful tool to analyze and design such a system. Hence, naturally, one can expect to apply the small-gain theory to analyze and design the event-triggered control systems. Some studies of the small-gain theory on traditional (purely continuous time or purely discrete...

Event‐triggered adaptive backstepping control for parametric strict‐feedback nonlinear systems

“…Remark Compared with the existing traditional adaptive backstepping schemes and the traditional event‐triggered methods,() our proposed scheme has the following advantages and disadvantages. Advantages Compared with traditional backstepping schemes with a continuous transmission,() the proposed method only uses the sampling states to update the controller and the adaptive law, which can largely reduce the transmission load and the computation burden. Different from the traditional event‐triggered methods() where the controllers are given a priori, and then the ISS property is used to handle the measurement errors, both the adaptive controller and the adaptive event‐triggering condition are designed at the same time such that the ISS assumption is no longer required. The threshold of the event‐trigger condition in this paper is designed as a function of both the parameters estimates and the system state vector.…”

“…Advantages Compared with traditional backstepping schemes with a continuous transmission,() the proposed method only uses the sampling states to update the controller and the adaptive law, which can largely reduce the transmission load and the computation burden. Different from the traditional event‐triggered methods() where the controllers are given a priori, and then the ISS property is used to handle the measurement errors, both the adaptive controller and the adaptive event‐triggering condition are designed at the same time such that the ISS assumption is no longer required. The threshold of the event‐trigger condition in this paper is designed as a function of both the parameters estimates and the system state vector. Thus, the threshold becomes adaptive unlike the traditional threshold conditions,() which are the functions of system state vector or constants alone. Since we have dynamics that evolve in continuous and discrete‐time, we write the closed‐loop system as an impulsive model and prove the closed‐loop stability for both the continuous and jump dynamics. Disadvantages In order to obtain the adaptive event‐triggering condition, the global Lipschitz continuity condition of the known function is needed, which can be relaxed in our future work.…”

“…Remark In our previous works,() some traditional event‐triggered methods have been presented, where the controllers are given a priori, and then, the input‐to‐state stability (ISS) property is used to handle the measurement errors. By comparison, in Equation , since the system contains unknown parameters, it is hard to check whether the ISS condition with respect to the state or input signal measurement errors holds or not .…”

“…As a result, the problem studied in this paper is quite different from and more challenging than that in our previous works. ()…”

“…Such updates or transmissions are, in general, aperiodic and depend on the system state, which is computationally better than the periodic sampled‐data control schemes. () Because of this advantage, significant contributions have been reported on the problem of ETC control for many different types of systems with different performance guarantees() and references therein. However, it is worth mentioning that the controllers are given a priori, and only the event‐triggered rules have to be designed in the aforementioned papers.…”