Stability conditions of pulse-width-modulated systems through the second method of Lyapunov

Abstract

PWM systems contain inherent nonlinearities which arise from their modulation scheme. Thus, for a legitimate study of stability, such systems must be treated as nonlinear sampled-data systems without initially resorting to linear approximations. For a nonlinear system whose dynamic behavior is described by a set of first-order difference equations, one of the theorems in the second method of Lyapunov gives, as a sufficient condition for asymptotic stability in the large, the existence in the whole space of a positive-definite Lyapunov's function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> , whose difference <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\DeltaV</tex> is negative definite. Hence, by choosing a positive-definite quadratic form as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> , the sufficient condition is reduced to the negative-definiteness in the whole space of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\DeltaV</tex> . Upon this basis, a systematic procedure of obtaining analytically a sufficient condition for asymptotic stability in the large is developed for various types of PWM systems; the condition is stated as the negativeness of all the eigenvalues of three matrices associated with the PWM system.