Models of Continuous and Discrete Dynamical Systems

The authors have introduced in [KSW97a], KSW97b] two complementary algebras for systems with boundaries. The first is loops in the suspension of a monoidal category ⊗) , (C , denoted ⊗)) , ( Σ ( Ω C , an algebra of deterministic systems in which the natural feedback operation has delay. The second is Span(Graph), an algebra of specifications of systems (nondeterministic transition systems), which is a discrete cartesian bicategory, in which the natural feedback is instantaneous. This algebra provides a compositional calculus of transition systems with interfaces, which generalizes the operation of product of automata of [A94], [Z87], and Kurshan. Further, to an expression in each algebra there is a graphic representation, which corresponds to the distributed nature of the system. We show here that in addition to the discrete examples described earlier also continuous linear systems (including RLC circuits) are examples of the algebra. The main difference between this approach and the traditional one [Rugh96] is that our theory is completely compositional, and includes in a common framework the continuous and the discrete. This approach provides a clear algebraic setting in which to develop the theory of hybrid systems.