The main motivation for this paper is to apply LQ and LQG methodologies for quadcopter control system. The developed control system is for both the rectangular position (xy) and altitude (z) as well as the orientation (attitude - angles around the axes) based on 6-Degree of Freedom (6DOF) mathematical model. 6DOF refers to the model with 3 linear and 3 angular motions. The altitude and attitude controllers are designed and the results presented in both the continuous and the discrete time cases. For the controller design, a nonlinear mathematical model was obtained first for 6DOF. The next step was to linearize the nonlinear model in hovering mode, and the final step was the reduction of the resulted linear model to be used as starting model for the controller design. The reduced linear model was tested for controllablity and observability. The control goal was to track a spatial trajectory with the quadcopter center of gravity under environment disturbances and sensor measurement errors. For this purpose, designed LQ controller was augmented by Kalman Filter state observer. The resultant controllers provide precise and robust performance for an input reference signal and for a regulation problem. After the transient response (of order of few seconds) the tracking error is acceptable which provides safe handling even under disturbances and measurement noises. The transient response can be further reduced by controllers fine tuning.

The objective of this paper is to present new and simple mathematical approach to deal with uncertainty transformation for fuzzy to random or random to fuzzy data. In particular we present a method to describe fuzzy (possibilistic) distribution in terms of a pair (or more) of related random (probabilistic) events, both fixed and variable. Our approach uses basic properties of both fuzzy and random distributions, and it assumes data is both possibilistic and probabilistic. We show that the data fuzziness can be viewed as a non uniqueness of related random events, and prove our Uncertainty Balance Principle. We also show how Zadeh’s fuzzy-random Consistency Principle can be given precise mathematical meaning. Various types of fuzzy distributions are examined and several numerical examples presented.

In this paper we extend our previous results in dual approach to analysis and simulation of a complex ecological system of preys and predators. We first define nonlinear dynamic equations Lotka-Volterra Model (LVM) with three preys and three predators and then simulate the equivalent situation with an Agent Based Model (ABM) which models a variety of species attributes and behaviors using NetLogo simulation environment for ABM model. The idea is that the LVM and ABM methods reinforce each other as the predator-prey models become more complex and their dimensionality rises. In particular LVM’s parameters, components of community matrix, can be fine tuned using ABM simulations. Dual approach may be able to answer and qualify some of the long standing ecological paradoxes.