A LATTICE CONTROL MODEL OF FUZZY DYNAMICAL SYSTEMS IN STATE-SPACE ∗ PETROS MARAGOS, GIORGOS STAMOU and SPYROS TZAFESTAS National Technical University of Athens, Dept. of Electrical & Computer Engineering, Zografou, 15773 Athens, Greece. Abstract. Lattice control can unify nonlinear control systems where the basic vector and signal superpositions or transformations are based on the lattice supremum and infimum. In this paper we introduce a special case of lattice control that can model fuzzy dynamical systems in state space. Vector and signal transformations are represented as lattice dila- tions or erosions. The state and output responses are computed via supremal convolutions based on fuzzy norms. Causality and stability issues are studied. Finally, solutions to the controllability and observability problem are found using lattice adjunctions. Key words: Nonlinear control, lattice morphology, fuzzy systems, minimax algebra. 1. Lattice Model for Max and Min Control In [7, 8, 6] a unified model was proposed based on lattice theory for large classes of nonlinear control systems, such as discrete event dynamical systems, recur- sive morphological filters, and fuzzy dynamical systems. Lattice morphology [9, 3] is ideally suited to studying such systems because all vector and signal op- erations and mappings involved can be expressed as morphological operators, and solutions to important control issues such as responses, stability and con- trollability are obtained using simple lattice-theoretic morphological concepts. In this paper we examine a special case of lattice control systems applicable to fuzzy dynamical systems. In classical linear control the state vectors, the input/output signals, and the system matrices take values from the field of reals equipped with standard addition and multiplication. In lattice control we take the set V of scalars to be a complete sublattice of R and equip it with the standard real number ordering ≤ and four binary operations: (A). A generalized ‘addition’, which will be the supremum ∨ on reals. (A ′ ). A ‘dual addition’, which will be the infimum ∧ on reals. (M). A commutative generalized ‘multiplication’ ⋆ under which: (i) V is a monoid (i.e., semigroup possessing an identity) with identity V id and null element V inf =  V, and (ii) ⋆ is a scalar dilation, i.e., distributes over any * This paper was published in: Mathematical Morphology and Its Application to Image and Signal Processing, Edited by J. Goutsias, L. Vincent and D. Bloomberg, Kluwer Aca- demic Publishers, Boston, 2000, pp.61-70 (Proc. ISMM 2000). This research was supported by the Greek Secretariat for Research and Technology under Grants EΠET - 98ΓT 26 and ΠENE∆ - 99E∆164. 61