Systems with Lebesgue sampling

Sampling is normally done periodically in time. For linear time invariant systems this leads to closed loop systems that linear and periodic. Many properties can be investigated by considering the behavior of the systems at times that are synchronized with the sampling instants. This leads to drastic simplifications because the systems can be described by difference equations with constant coefficients. This is the standard approach used today when designing digital controllers. Using an analog from integration theory, periodic sampling can also be called Riemann sampling. Lebesgue sampling or event based sampling, is an alternative to Riemann sampling, it means that signals are sampled only when measurements pass certain limits. This type of sampling is natural when using many digital sensors such as encoders. Systems with Lebesgue sampling are much harder to analyze than systems with Riemann sampling, because the time varying nature of the closed loop system can not be avoided. In this paper we investigate some systems with Lebesgue sampling: Analysis of simple systems shows that Lebesgue sampling gives better performance than Riemann sampling.