Dynamical Systems On Weighted Lattices: General Theory

TL;DR

This paper develops a unified mathematical framework for a broad class of nonlinear discrete-time dynamical systems on weighted lattices, encompassing max-plus, max-product, and probabilistic models, with applications in control, filtering, and pathfinding.

Contribution

It introduces the theory of complete weighted lattices for modeling nonlinear systems, unifying various existing models and providing tools for analysis and control.

Findings

Unified framework for nonlinear systems on weighted lattices.

Application to state-space modeling, dynamic programming, and filtering.

Demonstrated relevance to diverse fields like control, pathfinding, and multimodal data analysis.

Abstract

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.