Sampled-Data Observers: Scarce Arbitrarily Large Sampling Intervals

Stability and observer designs using new variants of Halanay’s inequality

“…Our key assumption in this section will be that s is small enough as compared with the other parameters, which can be interpreted to mean that during each time interval [t k + T, t k+1 ) that is outside the union (8) that defines the set E, the sampling points s i are close enough together, but this does not require any periodicity of the sampling interval lengths s i+1 − s i . On the other hand, we allow T and so also T to be arbitrarily large, which is a scarcity condition as described in Mazenc (2019) that allows the s i 's to be further apart during the time intervals that define the set E; see Figure 1 below. To specify our requirements, we use the constants…”

“…Then we apply Theorem 1 to a class of systems whose delays can switch between small and large values. Finally, we apply Theorem 1 to an observer design problem with sampled outputs, in which there are scarce arbitrarily large sampling intervals in the same sense that scarce was used in Mazenc (2019). However, unlike Mazenc (2019) where the systems did not contain delays, the systems in our observer design application are allowed to have arbitrarily long delays, and our assumptions are less restrictive than those of Mazenc (2019).…”

“…In this section, we revisit Mazenc (2019), where continuoustime systems with discrete measurements were studied using the technique of Karafyllis and Kravaris (2009). The work Mazenc (2019) designed converging observers in cases where the lengths of some intervals between the measurements can exceed the upper bound that ensures convergence of the observer that is provided in (Karafyllis and Kravaris, 2009, Equation (4.7)). This scarcity condition on the intervals in Mazenc ( 2019) is improved by the result that we give below, because our result below does not use the integral condition from (Mazenc, 2019, Assumption A3).…”

“…This scarcity condition on the intervals in Mazenc ( 2019) is improved by the result that we give below, because our result below does not use the integral condition from (Mazenc, 2019, Assumption A3). Moreover, by contrast with Mazenc (2019), the system we consider has a delay.…”

“…Our objectives differ significantly from other variants of Halanay's inequality, such as the notable works by Baker (2010) (which provides discrete time versions and nonlinear bounds) and Hien et al (2015) (which uses integral conditions involving time varying decay rates and time varying gains, which we do not use in this work). We also cover systems with sampled outputs with scarce arbitrarily long sampling intervals in the sense of Mazenc (2019), but our results give more easily checked sufficient conditions than the integral condition in (Mazenc, 2019, Assumption A3). Our less restrictive results allow the sampling to be more frequent outside those intervals where violations of the usual Halanay's conditions occur.…”