  This work proposes a new a framework for determining robust periodic
invariant sets and their associated control laws for constrained uncertain
linear systems. Necessary and sufficient conditions for stabilizability by
periodic controllers are stated and proven using finite step Lyapunov functions
for the unconstrained case. We then introduce a scenario tree interpretation of
finite step Lyapunov functions for uncertain systems and show that this
interpretation results in useful criteria for the design of robust stabilizing
controllers. In particular, novel convex feasibility criteria for the synthesis
of simple static controllers and what we call linear interpolating tree
periodic controllers with memory are derived. It is proven that for a
sufficiently large length of the period, a stabilizing linear interpolating
tree periodic controller can always be found using the proposed criterion
provided that the uncertain system is stabilizable by such controllers. In this
sense, the presented synthesis method is non-conservative. The results are then
extended to constrained uncertain linear systems and conditions for controllers
that realize robust periodic invariant sets which are less conservative than
those that result from the known methods in the literature are derived.
