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Dynamical systems on weighted lattices: general theory

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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 audiovisual salient events in multimodal information streams using generalized hidden Markov models with control inputs.

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Notes

  1. As explained in [31, 32], the adjunction is related to a concept in poset and lattice theory called ‘Galois connection.’ In [31, 59] an adjunction pair is denoted as \(({\large \varepsilon },{\large \delta })\), but in this paper we prefer to reverse the positions of its two operators, so that it agrees with the structure of a residuation pair \((\psi , {\psi }^\sharp )\).

  2. Minimax algebra [21] has been based on bands (idempotent semigroups) and belts (idempotent presemirings), whereas max-plus algebra and its application to DES [2, 15, 20, 28] is based on dioids (canonically ordered semirings). In [21], a semilattice is called a commutative band and a lattice is called band with duality. Further, a belt is a semilattice-ordered semigroup, and a belt with duality [21] is a pair of two idempotent predioids [28] whose ‘additions’ are dual and form a lattice. Adding to a belt identity elements for and \(\vee \), the latter of which is also an absorbing null for , creates an idempotent dioid [2, 20, 28]. More general (including nonidempotent) dioids are studied in [28]. Finally, belts that are groups under the ‘multiplication’ and as lattices have global bounds are called blogs (bounded lattice-ordered groups) in [21].

  3. In every clodum and clog, we have a pair of dual ‘additions’ and a pair of dual ‘multiplications.’ However, for brevity, we assign them shorter names that contain only one ‘addition’ (max) and one ‘multiplication,’ e.g., ‘max-plus clog’.

  4. It is simply a matter a convention that we selected to call \(\wedge \) and as ‘dual addition and multiplication’ (instead of \(\vee \) and ).

  5. In this paper, as ‘scalars’ we use numbers from \(\overline{\mathbb {R}}\) or its subsets, but the general definition of a weighted lattice allows for an arbitrary clodum as the set of ‘scalars.’

  6. If in our definition of a weighted lattice, one focuses only on one vector ‘addition,’ say the vector supremum, and its corresponding scalar ‘multiplication,’ then the weaker algebraic structure becomes an idempotent semimodule over an idempotent semiring. This has been studied in [19, 28, 41].

  7. A dimension theory for semimodules has been developed in [65]. Further, the concept of an upper basis has been used in [15] to define the dimension of finite-dimensional subspaces of max-plus matrix algebra.

  8. Despite its notation [15, 21], \(\varvec{M}^*\) is not the elementwise conjugate of the matrix \(\varvec{M}\) but is obtained via transposition and elementwise conjugation of \(\varvec{M}\). To avoid the above ambiguity, we prefer the terminology ‘adjoint’ which is based on some conceptual similarities with the adjoint of a linear operator in Hilbert spaces [21].

  9. Although the main results [15] of max-plus eigenvalue analysis in the max-plus semiring assume all scalars \(< +\infty \), in the more general max- eigenvalue analysis over a clodum we allow scalars to equal \(\top \); this has direct applications for cloda \(\mathcal {K}=[0,1]\) in fuzzy systems, like the max–min clodum, where \(1=e=\top \).

  10. For the max-plus clog \((\overline{\mathbb {R}},\vee ,\wedge ,+)\) the mean of a cycle \(\varvec{\sigma }\) is given by \(w(\varvec{\sigma }) / \ell (\varvec{\sigma })\), for the max-product clog \(([0,\infty ],\vee ,\wedge ,\times )\) it is given by \(w(\varvec{\sigma })^{1 / \ell (\varvec{\sigma })}\), whereas for the max–min clodum \(([0,1],\vee ,\wedge ,\min ,\max )\) the cycle mean is simply \(w\varvec{\sigma }\).

  11. The column (row) rank of a matrix over a clodum can be defined as the largest number of max- independent columns (rows). In [15, 21, 28] there are also weaker concepts of vector independence in minimax algebra.

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Acknowledgements

The author wishes to thank the anonymous Reviewers for their constructive comments. He also wishes to thank Petros Koutras at NTUA CVSP laboratory for producing Fig. 4 and Anastasios Tsiamis at the University of Pennsylvania for Example 6.

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  1. School of Electrical and Computer Engineering, National Technical University of Athens, Zografou Campus, 15773, Athens, Greece

    Petros Maragos

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Correspondence to Petros Maragos.

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This research was supported by the project “COGNIMUSE” under the “ARISTEIA” Action of the Operational Program Education and Lifelong Learning and was co-funded by the European Social Fund and Greek National Resources. It was also partially supported by the European Union under the projects MOBOT with Grant FP7-600796 and BabyRobot with Grant H2020-687831.

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Maragos, P. Dynamical systems on weighted lattices: general theory. Math. Control Signals Syst. 29, 21 (2017). https://doi.org/10.1007/s00498-017-0207-8

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