On ℓ 1 Performance and Robust Filtering for a Class of Bernoulli Switching Linear Systems

This paper addresses $\ell_{1}$ analysis and synthesis problems for a class of Markov jump linear systems, under the assumption that the jump process degenerates to a sequence of independent, identically distributed switches. This class of systems has received a great deal of attention in recent years, for instance, in the study of networked control systems. We consider here the scenario that comes up when all system coefficients are positive-to the best of our knowledge, this scenario has never been considered for Bernoulli jump systems. The main results include the tight characterization of two different metrics for $\ell_{1}$ performance, along with efficient methods for the analysis and synthesis of a positive $\ell_{1}$ filter that ensures robustness, in a quite general scenario where the system coefficients are subject to linear fractional uncertainty. The results are expressed in terms of linear programs of considerably smaller dimension than the existing methods that treat Markov jump linear systems with more general Markov chain structures.