Convergence and Equivalence results for the Jensen's inequality - Application to time-delay and sampled-data systems
This work provides theoretical justification for fragmentation methods used in time-delay and sampled-data systems analysis, addressing a known conservatism issue.
The paper proves that the Jensen gap can be made arbitrarily small by increasing the order of uniform fragmentation, and shows that non-uniform fragmentation can accelerate convergence. It also characterizes a family of bounds equivalent to Jensen's inequality, which offer better numerical properties.
The Jensen's inequality plays a crucial role in the analysis of time-delay and sampled-data systems. Its conservatism is studied through the use of the Grüss Inequality. It has been reported in the literature that fragmentation (or partitioning) schemes allow to empirically improve the results. We prove here that the Jensen's gap can be made arbitrarily small provided that the order of uniform fragmentation is chosen sufficiently large. Non-uniform fragmentation schemes are also shown to speed up the convergence in certain cases. Finally, a family of bounds is characterized and a comparison with other bounds of the literature is provided. It is shown that the other bounds are equivalent to Jensen's and that they exhibit interesting well-posedness and linearity properties which can be exploited to obtain better numerical results.