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This paper deals with the problem of control synthesis of discrete-time Takagi-Sugeno fuzzy systems by employing a novel multiinstant homogenous polynomial approach. A new multiinstant fuzzy control scheme and a new class of fuzzy Lyapunov functions, which are homogenous polynomially parameter-dependent on both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions, are proposed for implementing the object of relaxed control synthesis. Then, relaxed stabilization conditions are derived with less conservatism than existing ones. Furthermore, the relaxation quality of obtained stabilization conditions is further ameliorated by developing an efficient slack variable approach, which presents a multipolynomial dependence on the normalized fuzzy weighting functions at the current and past instants of time. Two simulation examples are given to demonstrate the effectiveness and benefits of the results developed in this paper.

This paper focuses on the analysis and the design of event-triggering scheme for discrete-time systems. Both static event-triggering scheme (SETS) and adaptive event-triggering scheme (AETS) are presented for discrete-time nonlinear and linear systems. What makes AETS different from SETS is that an auxiliary dynamic variable satisfying a certain difference equation is incorporated into the event-triggering condition. The sufficient conditions of asymptotic stability of the closed-loop event-triggered control systems under both two triggering schemes are given. Especially, for the linear systems case, the minimum time between two consecutive control updates is discussed. Also, the quantitative relation among the system parameters, the preselected triggering parameters in AETS, and a quadratic performance index are established. Finally, the effectiveness and respective advantage of the proposed event-triggering schemes are illustrated on a practical example. 4105Similar ETC scheme can be found in [7]. A self-triggered scheme is given in [8] for continuoustime linear systems with disturbance, while guaranteeing the finite-gain L2 stability of the resulting self-triggered feedback control systems. In [9], a distributed relative-difference ETC scheme for distributed networked control systems with data dropout and transmission delays is proposed for linear and nonlinear subsystems. A decentralized relative-difference event-triggering mechanism is proposed for continuous-time linear systems in [10], where the closed-loop stability and L 1 performance and the bounds on the minimum time between two subsequent events generated by each node are also studied. More recently, a relative-difference discrete event-triggered communication scheme has been proposed in [11,12], where the term discrete refers to the fact that the approach needs the triggering condition to be checked only at discrete times, which is completely different from the preceding works where the implementation of the designed ETC schemes requires a continuous supervision of the system state to detect whether it violates the triggering condition. Also, an H 1 performance analysis and control synthesis of networked continuous-time linear systems with transmission delays are established in terms of linear matrix inequalities, which successfully solves the open problem in the previous published papers. Up to now, there are several interesting extensions to the research presented in [11], including event-triggered quantized control design problem for networked linear systems [13-15], event-based H 1 filter design problem for networked systems [16][17][18], L 2 -gain analysis of event-triggered networked linear systems [19], a co-design method of the ETC scheme and state feedback controller for networked T-S fuzzy systems and linear systems in [20] and [21], respectively, event-based H 2 and H 1 control design for discrete-time networked linear systems in [22] , output feedback control design for networked control systems [12,23], a decentralized state-dependent tr...

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