Dissipativeness of dynamical systems is a crucial notion in control theory that consolidates the link with physics. It extends Lyapunov theory for autonomous systems to open ones and formalizes the relation between frequency domain conditions and matrix inequalities in state space representation. As emphasized in the limited and recent literature on this topic, dissipativeness of hybrid or continuous-time switched systems is a not intuitive and delicate notion. This paper copes with the dissipativeness analysis of discrete-time switched linear systems. Conditions in the form of linear matrix inequalities are provided to ensure dissipativeness of such systems with arbitrary switching law. The approach relies on modal storage functions. A second contribution is to design feedback switching laws, based on a min-switching strategy related to the modal storage functions, which ensures a dissipative behaviour of the closed-loop system. Implication in terms of passivity and stability of one single switched system, paving the way to the framework of interconnected switched sub-systems are discussed, before numerical illustrations.
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