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

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
V
, whose difference
\DeltaV
is negative definite. Hence, by choosing a positive-definite quadratic form as
V
, the sufficient condition is reduced to the negative-definiteness in the whole space of
\DeltaV
. 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.