Fundamentals of a Fuzzy-Logic-Based Generalized Theory of Stability

Stability is one of the fundamental concepts of complex dynamical systems including physical, economical, socioeconomical, and technical systems. In classical terms, the notion of stability inherently associates with any dynamical system and determines whether a system under consideration reaches equilibrium after being exposed to disturbances. Predominantly, this concept comes with a binary (Boolean) quantification (viz., we either quantify that systems are stable or not stable). While in some cases, this definition is well justifiable, with the growing complexity and diversity of systems one could seriously question the Boolean nature of the definition and its underlying semantics. This becomes predominantly visible in human-oriented quantification of stability in which we commonly encounter statements quantifying stability through some linguistic terms such as, e.g.,
absolutely
unstable,
highly
unstable,. . .,
absolutely
stable, and alike. To formulate human-oriented definitions of stability, we may resort ourselves to the use of a so-called precisiated natural language, which comes as a subset of natural language and one of whose functions is redefining existing concepts, such as stability, optimality, and alike. Being prompted by the discrepancy of the definition of stability and the Boolean character of the concept itself, in this paper, we introduce and develop a generalized theory of stability (GTS) for analysis of complex dynamical systems described by fuzzy differential equations. Different human-centric definitions of stability of dynamical systems are introduced. We also discuss and contrast several fundamental concepts of fuzzy stability, namely, fuzzy stability of systems, binary stability of fuzzy system, and binary stability of systems by showing that all of them arise as special cases of the proposed GTS. The introduced definitions offer an important ability to quantify the concept of stability using some continuous quantification (that is through the use of degrees of stability). In this manner, we radically depart from the previous binary character of the definition. We establish some criteria concerning generalized stability for a wide class of continuous dynamical systems. Next, we present a series of illustrative examples which demonstrate the essence of the concept, and at the same time, stress that the existing Boolean techniques are not capable of capturing the essence of linguistic stability. We also apply the obtained results to investigate the stability of an economical system and show its usefulness in the design of nonlinear fuzzy control systems given some predefined degree of stability.