On Commutators of Fuzzy Multigroups

Fuzzy multigroup is an application of fuzzy multiset to group theory. Although, a lots have been done on the theory of fuzzy multigroups, some group's theoretic notions could still be investigated in fuzzy multigroup context. Certainly, the idea of commutator is one of such group's theoretic notions yet to be studied in the environment of fuzzy multigroups. Hence, the aim of this article is to establish the notion of commutator in fuzzy multigroup setting. A number of some related results are obtained and characterized. Among several results that are obtained, it is established that, if $A$ and $B$ are fuzzy submultigroups of a fuzzy multigroup $C$, then $[A, B]\subseteq A\cup B$ holds. Some homomorphic properties of commutator in fuzzy multigroup context are discussed. The notion of admissible fuzzy submultisets $A$ and $B$ of $C\in FMG(X)$ under an operator domain $\mathcal{D}$ is explicated, and it is shown that $(A,B)$ and $[A,B]$ are $\mathcal{D}$-admissible.

Optimal State Feedback Control in Operator Domain for Multi-Rate Networked Control Systems with Long Time Delay

By multi-rate networked control systems (NCS), we mean the sampling periods of the sensor, the controller and the actuator in networked control systems are not the same, that is to say there are more than one sampling rate in networked control systems. For the long time delay multi-rate NCS with event-driven controller and actuator, a stochastic discrete model is established under operator. The state feedback control laws for the multi-rate NCS in operator domain are designed by using a dynamic programming approach. The derived optimal LQG controller can be used as a delay-compensator for multi-rate NCS with long time delays. An example is given to verify the theory results of this paper.