Characterizations of quasitrivial symmetric nondecreasing associative operations

We provide a description of the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We also prove that associativity can be replaced with bisymmetry in the definition of this class. Finally we investigate the special situation where the chain is finite

PID, BFO-optimized PID, and PD-FLC control of a two-wheeled machine with two-direction handling mechanism: a comparative study

In this paper; three control approaches are utilized in order to control the stability of a novel five-degrees-of-freedom two-wheeled robotic machine designed for industrial applications that demand a limited-space working environment. Proportional–integral–derivative (PID) control scheme, bacterial foraging optimization of PID control method, and fuzzy logic control method are applied to the wheeled machine to obtain the optimum control strategy that provides the best system stabilization performance. According to simulation results, considering multiple motion scenarios, the PID controller optimized by bacterial foraging optimization method outperformed the other two control methods in terms of minimum overshoot, rise time, and applied input forces