In this study, an integral-based event-driven mechanism is proposed for a general class of non-linear systems. The proposed scheme is less conservative than earlier work on the subject and achieves asymptotic stability without forcing the derivative of the Lyapunov function to be negative between samples. A rigorous proof is given, showing that the proposed triggering condition is more effective than the corresponding traditional approaches. Simulation results are provided to illustrate the effectiveness of the proposed solution. IntroductionMost digital control systems encountered today operate under the assumption that sampling takes place at regular intervals. 'Periodic' or 'time-driven' sampling is well understood and control design for such systems can be done following techniques analogous to the continuous time case. See for example [1,2]. In a time-driven system, the periodic nature of the information flow forces the controller to perform real-time computations and update information at every sample, consuming valuable computer time and demanding communication between different system components, even if the signals measured by the sensors experience no significant change. This issue makes time-driven systems potentially inefficient with respect to communication channel capacity and resource utilisation which may be significant when dealing with distributed and networked systems, where optimal usage of communication network capacity is of great preference, or embedded microprocessor systems with limited computational power and memory storage. Over the last several years, event-triggered systems have emerged as an attractive alternative to time-driven systems. In an event-driven system, information is exchanged between system components only if a pre-specified triggering condition (TC) is violated [3]. This condition can be defined in a number of ways, depending on the nature of the system. For example, in a power grid, a fault can be considered as an event, triggering a control action in the system [4]. Generally speaking, in most systems, an event occurs when changes in the sensor readings exceed a certain threshold [5]. Early works on event-based systems were proposed in late 1990s [6,7]. Because of use of different triggering strategies, event-triggered systems have also been called 'level crossing sampling' [8], 'event-driven systems' [9] and 'state-triggered feedback systems' [10].A fundamental contribution in the theory of event-driven systems was reported in reference [5]. In this paper, Tabuada formulated an input-to-state stable (ISS) Lyapunov function framework for event triggering design of a general class of non-linear systems. Assuming that a continuous-time feedback control has been already designed, Tabuada proposes an event generator condition to maintain the Lyapunov function V decreasing along the system trajectories. The work is important in that it provides a solid theoretical foundation to the fundamental stability problem of time-driven systems using classical tools. Following Tabuad...

In this work, γ-cyclodextrin-modified titanium dioxide nanoparticles (NPs) as an efficient nanoadsorbent were synthesized and used to remove methylene blue (MB), malachite green (MG), crystal violet (CV), disperse red 1 (DR1), acid blue 113 (AB113), and congo red (CR) dyes from synthetic wastewater. Characterization of cyclodextrin-functionalized TiO 2 nanoparticles using Fourier transform infrared, field-emission scanning electron microscopy, transmission electron microscopy, X-ray diffraction, energy-dispersive X-ray spectroscopy, and Brunauer−Emmett−Teller (BET) analysis revealed that the TiO 2 NPs were effectively modified with γ-CD. Furthermore, synthesized adsorbent with a large BET specific surface area of 383.131 m 2 /g showed excellent adsorption performance toward organic dyes. The dye adsorption kinetics, isotherms, and thermodynamics using TiO 2 /γ-CD NPs were also investigated. The results showed that pseudo-second-order model and Freundlich isotherm model well describe the dye adsorption process on the TiO 2 /γCD NPs. Additionally, the maximum adsorption capacity of MB, MG, CV, DR1, AB113, and CR onto TiO 2 /γCD NPs was found to be 134, 244, 213, 238, 157, and 5000 mg/g, respectively. On the basis of the obtained RL values, which were less than 1.0, the adsorption of all tested dyes was favorable. Furthermore, the calculated thermodynamic parameters showed that the adsorption process of the studied dyes was endothermic.

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