A study on the spectrum of the sampled-data transfer operator with application to robust exponential stability problems

This paper studies the relation between robust L₂-stability and robust exponential stability in the context of sampled-data systems. It is shown that they are equivalent irrespective of the perturbation structures/sets to be considered, if the nominal sampled-data system is exponentially stable, if the perturbations are either finite-dimensional linear time-invariant or h-periodic (where h is the sampling period), and if the perturbations are internally stable. Thus, a solid theoretical basis is established for the robust stabilization/performance design for sampled-data systems, in which only L₂-stability conditions are taken care of. The arguments are mainly based on the spectral properties of the transfer operators of sampled-data systems. Through a ´nonasymptotic´ characterization of the transfer operator, its spectrum is determined in terms of finite-dimensional eigenvalue problems. The close connection with such eigenvalue problems and the exponential stability condition is a key that. enables us to deduce the above conclusion.