Model Predictive Control

14.3 Discrete Hybrid Automata 293
e(k) d
u(k) u
e(k) d
FFinite
State
Machine
(FSM)
EEvent
Generator
(EG)
y(k)
x(k) x
u(k) u
e(k) d
MMode
Selector
(MS)
xc(k) y
uc(k) v
uc(k) v
Switched
Affine
System
i(k) i
1
...
s
(SAS) S
yc(k)
Figure 14.8 A discrete hybrid automaton (DHA) is the connection of a finite
state machine (FSM) and a switched affine system (SAS), through a mode
selector (MS) and an event generator (EG). The output signals are omitted
for clarity
additional continuous and autonomous state variable, τ(t + 1) = τ(t) + Ts, where
Ts is the sampling time, and by letting [δe(t) = 1] ↔ [tTs ≥ τ0], where τ0 is a given
time. By doing so, the hybrid model can be written as a time-invariant one. Clearly
the same approach can be used for time-varying events δe(t) = h(xc(t), uc(t), t), by
using time-varying event conditions h : R nc × R nc × T → {0, 1} ne .
14.3.3 Boolean Algebra
Before dealing in detail with the other blocks constituting the DHA and introduce
further notation, we recall here some basic definitions of Boolean algebra 1 .
A variable δ is a Boolean variable if δ ∈ {0, 1}, where “δ = 0” means something
is false, “δ = 1” that is true. A Boolean expression is obtained by combining
Boolean variables through the logic operators ¬ (not), ∨ (or), ∧ (and), ← (implied
by), → (implies), and ↔ (iff). A Boolean function f : {0, 1} n−1 ↦→ {0, 1} is used
to define a Boolean variable δn as a logic function of other variables δ1, . . .,δn−1:
δn = f(δ1, δ2, . . .,δn−1) (14.13)
Given n Boolean variables δ1, . . . , δn, a Boolean formula F defines a relation
F(δ1, . . . , δn) (14.14)
1 A more comprehensive treatment of Boolean calculus can be found in digital circuit design
texts, e.g. [76, 129]. For a rigorous exposition see e.g. [184].