© copyright FACULTY of ENGINEERING ‐ HUNEDOARA, ROMANIA 97 1. Anna JADLOVSKÁ, 2. Kamil HRUBINA, 3. Jozef MAJERČÁK APPLICATION OF STABILITY THEORY OF NONLINEAR SYSTEMS AND LYAPUNOV TRANSFORMATION IN CONTROL OF ARTIFICIAL PNEUMATIC MUSCLE 1. TECHNICAL UNIVERSITY IN KOŠICE, THE FACULTY OF ELECTROTECHNICS AND INFORMATICS, DEPARTMENT OF CYBERNETICS AND ARTIFICIAL INTELIGENCE, LETNÁ.9, 040 01 KOŠICE, SLOVAKIA 2,3. TECHNICAL UNIVERSITY IN KOŠICE, THE FACULTY OF MANUFACTURING TECHNOLOGIES WITH THE SEAT IN PREŠOV, DEPARTMENT OF MATHEMATICS, INFORMATICS AND CYBERNETICS, BAYEROVA 1, 080 01 PREŠOV, SLOVAKIA ABSTRACT: The following paper explores the concept of nonlinear systems stability and characteristic exponent. Definitions and theorems, necessary for solving the predefined problem of control of a nonlinear system, are included. The paper also deals with the Lyapunov transformation to carry out a linear system whose matrix elements are functions of a system with a constant matrix. The stability of systems with changeable parameters as well as the application of nonlinear systems control theory to the problems of "artificial pneumatic muscle - APM" control have also been investigated in the paper. KEYWORDS: cybernetics, nonlinear systems stability, Lyapunov transformation, artificial pneumatic muscle  INTRODUCTION The stability of a given system is often defined in the sense that the system is capable of returning to an equilibrium if a signal acting, which led the system out of this state, finished. This definition is sufficient for a linear system, its stability, however, can be defined in a different way, e.g. a linear system is stable if and only if its response to an arbitrary bounded input is bounded. There are several definitions of a nonlinear system stability. Many of them have a limited utilization and were defined for specific cases. In general, the processes going on in linear and nonlinear systems can be expressed by a mathematical model, which actually is a system of differential equations. Lyapunov stability theory enables to investigate the system stability without the necessity of solving either differential equations of the given order or a system of differential equations. A. M. Lyapunov proposed two methods in order to investigate the stability. Lyapunov first method enables to consider the nonlinear system stability according to an approximate linear model, (local stability). Lyapunov second method enables to consider the stability or the asymptotic stability in a certain area Ω, in general with the linear or nonlinear system, (of both excited and unexcited system). When solving the stability problem, the success of the method lies with the investigator’stability to find a suitable function (the so called Lyapunov function) as well as to determine its definiteness, [1, 3, 4, 5]. This paper will deal with the investigation of nonlinear systems stability described by a vector differential equation, a characteristic exponent and an asymptotic stability. It will also deal with the Lyapunov transformation as well as the stability of the systems with variable coefficients of the system of differential equations.  SYSTEM STABILITY AND CHARACTERISTIC EXPONENT We will consider a homogeneous linear vector differential equation (or a homogeneous linear system of differential equations) in the form ︶︵︶︵ t t (t) .x A x = & (1) where matrix individual elements (t)) (a (t) ij = A (2) are continuous functions in the interval ) (a, ∞ + . Theorem 1. A linear system described by the equation (1) is stable in the sense of Lyapunov in the interval , , 0 > +∞ < t if all solutions to the equation (1) are bounded functions in the interval > +∞ < , 0 t .