  In this work a theory is developed for unifying large classes of nonlinear
discrete-time dynamical systems obeying a superposition of a weighted maximum
or minimum type. The state vectors and input-output signals evolve on nonlinear
spaces which we call complete weighted lattices and include as special cases
the nonlinear vector spaces of minimax algebra. Their algebraic structure has a
polygonal geometry. Some of the special cases unified include max-plus,
max-product, and probabilistic dynamical systems. We study problems of
representation in state and input-output spaces using lattice monotone
operators, state and output responses using nonlinear convolutions, solving
nonlinear matrix equations using lattice adjunctions, stability and
controllability. We outline applications in state-space modeling of nonlinear
filtering; dynamic programming (Viterbi algorithm) and shortest paths (distance
maps); fuzzy Markov chains; and tracking audio-visual salient events in
multimodal information streams using generalized hidden Markov models with
control inputs.
