Dynamical systems on weighted lattices: general theory

  • Petros MaragosEmail author
Original Article


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.


Nonlinear dynamical systems Lattice theory Minimax algebra Control Signal processing 



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.


  1. 1.
    Avrachenkov KE, Sanchez E (2002) Fuzzy Markov chains and decision-making. Fuzzy Optim Decis Mak 1:143–159MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baccelli F, Cohen G, Olsder GJ, Quadrat J-P (1992) Synchronization and linearity: an algebra for discrete event systems (web edition, 2001). Wiley, New YorkzbMATHGoogle Scholar
  3. 3.
    Bellman R, Karush W (1961) On a new functional transform in analysis: the maximum transform. Bull Am Math Soc 67:501–503CrossRefzbMATHGoogle Scholar
  4. 4.
    Bellman R, Karush W (1963) On the maximum transform. J Math Anal Appl 6:67–74MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bertsekas DP (2012) Dynamic programming and optimal control. Athena Scientific, Belmont (vol. I, 3rd ed., vol. II, 4th ed.)zbMATHGoogle Scholar
  6. 6.
    Birkhoff G (1967) Lattice theory. American Mathematical Society, Providence, RIzbMATHGoogle Scholar
  7. 7.
    Bishop CM (2006) Pattern recognition and machine learning. Springer, BerlinzbMATHGoogle Scholar
  8. 8.
    Bloch I, Maitre H (1995) Fuzzy mathematical morphologies: a comparative study. Pattern Recognit 9(28):1341–1387MathSciNetCrossRefGoogle Scholar
  9. 9.
    Blyth TS (2005) Lattices and ordered algebraic structures. Springer, BerlinzbMATHGoogle Scholar
  10. 10.
    Blyth TS, Janowitz MF (1972) Residuation theory. Pergamon Press, OxfordzbMATHGoogle Scholar
  11. 11.
    Borgefors G (1984) Distance transformations in arbitrary dimensions. Comput Vis Graph Image Process 27:321–345CrossRefGoogle Scholar
  12. 12.
    Brockett RW (1970) Finite-dimensional linear systems. Wiley, New YorkzbMATHGoogle Scholar
  13. 13.
    Brockett RW (1994) Language driven hybrid systems. In: Proceedings of 33rd conference on decision and controlGoogle Scholar
  14. 14.
    Brogan WL (1974) Modern control theory. Quantum Publishers, New YorkzbMATHGoogle Scholar
  15. 15.
    Butkovič P (2010) Max-linear systems: theory and algorithms. Springer, BerlinCrossRefzbMATHGoogle Scholar
  16. 16.
    Cassandras CG, Lafortune S (1999) Introduction to discrete event systems. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  17. 17.
    Charisopoulos V, Maragos P (2017) Morphological perceptrons: geometry and training algorithms. In: Proceedings international symposium on mathematical morphology (ISMM-2017), Fontainebleau, FranceGoogle Scholar
  18. 18.
    Cohen G, Dubois D, Quadrat JP, Viot M (1985) A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans Autom Control 30:210–220MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cohen G, Gaubert S, Quadrat JP (2004) Duality and separation theorems in idempotent semimodules. Linear Alegbra Appl 379:395–422MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cohen G, Moller P, Quadrat JP, Viot M (1989) Algebraic tools for the performance evaluation of discrete event systems. Proc IEEE 77:39–58CrossRefGoogle Scholar
  21. 21.
    Cuninghame-Green R (1979) Minimax algebra. Springer, BerlinCrossRefzbMATHGoogle Scholar
  22. 22.
    Cuninghame-Green RA, Butkovic P (2003) The equation Ax=By over (max, +). Theor Comput Sci 293:3–12CrossRefzbMATHGoogle Scholar
  23. 23.
    Doustmohammadi A, Kamen EW (1995) Direct generation of event-timing equations for generalized flow shop systems. In: Modeling, simulation, and control technologies for manufacturing, volume 2596 of proceedings of SPIE, pp 50–62Google Scholar
  24. 24.
    Evangelopoulos G, Zlatintsi A, Potamianos A, Maragos P, Rapantzikos K, Skoumas G, Avrithis Y (2013) Multimodal saliency and fusion for movie summarization based on aural, visual, and textual attention. IEEE Trans Multim 15(7):1553–1568CrossRefGoogle Scholar
  25. 25.
    Felzenszwalb PF, Huttenlocher DP (2004) Distance transforms of sampled functions. Technical Report TR2004-1963, Cornell UniversityGoogle Scholar
  26. 26.
    Gaubert S, Katz RD (2007) The Minkowski theorem for max-plus convex sets. Linear Alegbra Appl 421:356–369MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gazarik MJ, Kamen EW (1999) Reachability and observability of linear systems over max-plus. Kybernetika 35(1):2–12MathSciNetzbMATHGoogle Scholar
  28. 28.
    Gondran M, Minoux M (2008) Graphs, dioids and semirings: new models and algorithms. Springer, BerlinzbMATHGoogle Scholar
  29. 29.
    Hardouin L, Cottenceau B, Lhommeau M, Le Corronc E (2009) Interval systems over idempotent semiring. Linear Alegbra Appl 431(5–7):855–862MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Heidergott B, Olsder GJ, van der Woude J (2006) Max plus at work: modeling and analysis of synchronized systems: a course on max-plus algebra and its applications. Princeton University Press, PrincetonzbMATHGoogle Scholar
  31. 31.
    Heijmans HJAM (1994) Morphological image operators. Academic Press, BostonzbMATHGoogle Scholar
  32. 32.
    Heijmans HJAM, Ronse C (1990) The algebraic basis of mathematical morphology. Part I: dilations and erosions. Comput Vis Graph Image Process 50:245–295CrossRefzbMATHGoogle Scholar
  33. 33.
    Ho Y-C (ed) (1992) Discrete event dynamic systems: analyzing complexity and performance in the modern world. IEEE Press, New YorkGoogle Scholar
  34. 34.
    Hori T, Nakamura A (2013) Speech recognition algorithms using weighted finite-state transducers. Morgan & Claypool, San RafaelGoogle Scholar
  35. 35.
    Kaburlasos VG, Petridis V (2000) Fuzzy lattice neurocomputing (FLN) models. Neural Netw 13:1145–1169CrossRefGoogle Scholar
  36. 36.
    Kailath T (1980) Linear systems. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  37. 37.
    Kamen EW (1993) An equation-based approach to the control of discrete even systems with applications to manufacturing. In: Proceedings of international conference on control theory and its applications, Jerusalem, IsraelGoogle Scholar
  38. 38.
    Kamen EW, Doustmohammadi A (1994) Modeling and stability of production lines based on arrival-to-departure delays. In: Proceedings 33rd conference on decision and controlGoogle Scholar
  39. 39.
    Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  40. 40.
    Lahaye S, Boimond J-L, Hardouin L (2004) Linear periodic systems over dioids. Discrete Event Dyn Syst 14:133–152MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Litvinov GL, Maslov VP, Shpiz GB (2001) Idempotent functional analysis: an algebraic approach. Math Notes 69(5):696–729MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Litvinov GL, Sobolevskii AN (2001) Idempotent interval analysis and optimization problems. Reliab Comput 7(5):353–377MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lucet Y (2010) What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev 52(3):505–542MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Maragos P (1994) Morphological systems: slope transforms and max–min difference and differential equations. Signal Process 38:57–77CrossRefzbMATHGoogle Scholar
  45. 45.
    Maragos P (2005) Lattice image processing: a unification of morphological and fuzzy algebraic systems. J Math Imaging Vis 22:333–353MathSciNetCrossRefGoogle Scholar
  46. 46.
    Maragos P (2013) Representations for morphological image operators and analogies with linear operators. In: Hawkes PW (ed) Advances in imaging and electron physics, vol 177. Academic Press, London, pp 45–187Google Scholar
  47. 47.
    Maragos P, Koutras P (2015) Max-product dynamical systems and applications to audio-visual salient event detection in videos. In: Proceedings IEEE international conference on acoustics speech and signal processing (ICASSP)Google Scholar
  48. 48.
    Maragos P, Schafer RW (1990) Morphological systems for multidimensional signal processing. Proc IEEE 78:690–710CrossRefGoogle Scholar
  49. 49.
    Maragos P, Stamou G, Tzafestas SG (2000) A lattice control model of fuzzy dynamical systems in state-space. In: Goutsias J, Vincent L, Bloomberg D (eds) Mathematical morphology and its application to image and signal processing. Kluwer, DordrechtGoogle Scholar
  50. 50.
    Maslov VP (1987) On a new superposition principle for optimization problems. Uspekhi Mat Nauk [Russ Math Surv] 42(3):39–48Google Scholar
  51. 51.
    McEneaney WM (2006) Max-plus methods for nonlinear control and estimation. Birkhauser, BostonzbMATHGoogle Scholar
  52. 52.
    Mohri M, Pereira F, Ripley M (2002) Weighted finite-state transducers in speech recognition. Comput Speech Lang 16:69–88CrossRefGoogle Scholar
  53. 53.
    Oppenheim AV, Schafer RW (1989) Discrete-time signal processing. Prentice-Hall, Englewood Cliffs, NJzbMATHGoogle Scholar
  54. 54.
    Pearl J (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufman Publishers, Los AltoszbMATHGoogle Scholar
  55. 55.
    Rabiner L, Juang B-H (1993) Fundamentals of speech recognition. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  56. 56.
    Ritter GX, Sussner P, Diaz de Leon JL (1998) Morphological associative memories. IEEE Trans Neural Netw 9(2):281–293CrossRefGoogle Scholar
  57. 57.
    Ritter GX, Urcid G (2003) Lattice algebra approach to single-neuron computation. IEEE Trans Neural Netw 14(2):282–295CrossRefGoogle Scholar
  58. 58.
    Rockafellar RT (1970) Convex analysis. Princeton University Press, PrincetonCrossRefzbMATHGoogle Scholar
  59. 59.
    Serra J (ed) (1988) Image analysis and mathematical morphology, volume 2: theoretical advances. Academic Press, LondonGoogle Scholar
  60. 60.
    Sternberg SR (1986) Grayscale morphology. Comput Vis Graph Image Process 35:333–355CrossRefGoogle Scholar
  61. 61.
    Tsitsiklis JN (1995) Efficient algorithms for globally optimal trajectories. IEEE Trans Autom Control 40(9):1528–1538MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    van den Boom TJJ, De Schutter B (2012) Modeling and control of switching max-plus-linear systems with random and deterministic switching. Discrete Event Dyn Syst 22:293–332MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Verbeek P, Dorst L, Verwer B, Groen F (1986) Collision avoidance and path finding through constrained distance transformation in robot state space. In: Proceedings of international conference on intelligent autonomous systems, AmsterdamGoogle Scholar
  64. 64.
    Verdu S, Poor HV (1987) Abstract dynamic programming models under commutativity conditions. SIAM J Control Optim 25(4):990–1006MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Wagneur E (1991) Moduloids and pseudomodules–1. Dimension theory. Discrete Math 98:57–73MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Yang P-F, Maragos P (1995) Min–max classifiers: learnability, design and application. Pattern Recognit 28(6):879–899CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Electrical and Computer EngineeringNational Technical University of AthensAthensGreece

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