On the abstraction of conventional dynamic systems: from numerical analysis to linguistic analysis

Linguistic dynamic systems (LDS) are dynamic processes involving mainly computing with words instead of numbers for modeling and analysis of complex systems and human-machine interfaces. The goal of studying LDS is to establish a methodology of design, modeling, and analysis of complex decision-making processes bridging the machine world in numbers and the human world in words. Specifically in this paper, conventional dynamic systems are converted to different types of LDS for the purpose of verification and comparison. The evolving laws of a type-I LDS are constructed by applying the fuzzy extension principle to those of its conventional counterpart with linguistic states. The evolution of type-I LDS represents the dynamics of state uncertainty derived from the corresponding conventional dynamic process. In addition to linguistic states, the evolving laws of type-II LDS are modeled by a finite number of linguistic decision rules. Analysis of fixed points is conducted based on point-to-fuzzy-set mappings and linguistic controllers are designed for goals specified in words for type-II LDS. An efficient numerical procedure called α-cuts mapping is developed and applied to obtain extensive simulation results.