This paper is concerned with passivity of a class of delayed neural networks. In order to derive less conservative passivity criteria, two Lyapunov-Krasovskii functionals (LKFs) with delay-dependent matrices are introduced by taking into consideration a second-order Bessel-Legendre inequality. In one LKF, the system state vector is coupled with those vectors inherited from the second-order Bessel-Legendre inequality through delay-dependent matrices, while no such coupling of them exists in the other LKF. These two LKFs are referred to as the coupled LKF and the noncoupled LKF, respectively. A number of delay-dependent passivity criteria are derived by employing a convex approach and a nonconvex approach to deal with the square of the time-varying delay appearing in the derivative of the LKF. Through numerical simulation, it is found that: 1) the coupled LKF is more beneficial than the noncoupled LKF for reducing the conservatism of the obtained passivity criteria and 2) the passivity criteria using the convex approach can deliver larger delay upper bounds than those using the nonconvex approach.

This article is concerned with the extended dissipativity of discrete-time neural networks (NNs) with time-varying delay. First, the necessary and sufficient condition on matrix-valued polynomial inequalities reported recently is extended to a general case, where the variable of the polynomial does not need to start from zero. Second, a novel Lyapunov functional with a delay-dependent Lyapunov matrix is constructed by taking into consideration more information on nonlinear activation functions. By employing the Lyapunov functional method, a novel delay and its variation-dependent criterion are obtained to investigate the effects of the time-varying delay and its variation rate on several performances, such as $H_∞$ performance, passivity, and $l_2-l_∞$ performance, of a delayed discrete-time NN in a unified framework. Finally, a numerical example is given to show that the proposed criterion outperforms some existing ones.

This paper is concerned with event-triggered control of discrete-time systems with or without input saturation. First, an accumulative-error-based event-triggered scheme is devised for control updates. When the accumulated error between the current state and the latest control update exceeds a certain threshold, an event is triggered. Such a scheme can ensure the event-generator works at a relatively low rate rather than falls into hibernation especially after the system steps into its steady state. Second, the looped functional method for continuous-time systems is extended to discrete-time systems. By introducing an innovative looped functional that links the event-triggered scheme, some sufficient conditions for the co-design of control gain and event-triggered parameters are obtained in terms of linear matrix inequalities with a couple of tuning parameters. Then, the proposed method is applied to discrete-time systems with input saturation. As a result, both suitable control gains and event-triggered parameters are also co-designed to ensure the system trajectories converge to the region of attraction. Finally, an unstable reactor system and an inverted pendulum system are given to show the effectiveness of the proposed method.