Generalized pulse-modulated feedback systems: Norms, gains, Lipschitz constants, and stability

Sufficient conditions for the asymptotic stability in the large of pulse-modulated feedback systems are developed from the operator theoretic viewpoint. Stability here requires that the pulse-modulated feedback system be a Lipschitz continuous operator on the extended space
L_{1e}
. This strong definition of stability is motivated by an examination of a first-order pulsewidth-modulated system. To provide a unified format for the main development two distinct general classes of pulse modulators are defined. Type I includes the pulsewidth modulator and more general pulsewidth frequency modulators that contain a sampler. Type II includes, for example, the integral pulse frequency modulator and its generalizations. For elements of Type I conditions are derived to bound the incremental gain (on
L_{1e}
) of the modulator in cascade with a linear element; a standard transformation of the feedback loop similar to that used in the derivation of the Popov criterion yields sufficient conditions for stability of the feedback system in the above strong sense. Type II modulators are discontinuous on any normed linear space and thus, only conditions for boundedness of the closed-loop system as an operator on L
1
are given for this case.