Feedback Control and Hierarchical Modeling of Discrete-Event Dynamic Systems
Cüneyt M. Özveren, Alan S. Willsky · Birkhäuser Boston eBooks · 1991
This paper is concerned with the development of a servo theory for discrete-event dynamic systems (DEDS) The class of systems we consider are defined over G = (X,∑ , ϕ, Γ,≡), where X is the set of states, with n = ∣X∣,∑ is the finite set of possible events, ϕ ⊂ ∑ is the set of controllable events, Γ ⊂ ∑ is the set of observable events, and ≡ ⊂ ∑ is the set of tracking events. Also, U = 2ϕ denotes the set of admissible inputs. The dynamics on G are: 1 $$ x[k + 1] \in f(x[k],\sigma [k + 1]) $$ 2 $$ \sigma [k + 1] \in (d(x[k]) \cap u[k]\, \cup (d(x[k]) \cap \,\bar \Phi ) $$ The set-valued function d specifies the set of possible events defined at each state, and the state transition function f is also set-valued. Whenever an event in Γ occurs, we observe it; otherwise, we see nothing. Thus, our output equation is 3 $$ {\rm{\gamma [}}k + {\rm{1] = }}h{\rm{(}}\sigma {\rm{[}}k{\rm{ + 1])}} $$ where h is the projection map from ∑* to Γ*, obtained by deleting all events not in Γ. The set ≡ denotes the tracking alphabet, and t : ∑* → ≡*, denotes the projection of strings over ∑ into ≡*. A = (G,f, d,h, t) represents our system.