Page 140 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

A New Do11ain and lore Process Models ll5

where "int" means taking the integer value of L/(vh).
To get a feel for how Eq. (4-22) might work, let's put in some num-
bers. Let L = 100.0, and h = 1. Let the nominal value of the speed be
v = 10.0. Consequently, the nominal delay vector decrement n = 10.
Therefore, at every instant t ,t , ••• the index i is incremented, the
1 2
speed is placed in the delay vector V(i), the decrement n is calculated
and the delayed speed V(i- n)= V(i-10) is fetched. Assume that at
some point in time t' the speed is decreased from 10.0 to 5.0 because
of a control move. The index is incremented to i', the new speed is
placed in the delay vector and a new decrement is calculated from

n'= inf~ )=20
0

The delayed speed is fetched from V(i'- 20) = V(i + 1-20) =
V(i- 19) which is a problem. On the previous instant the delayed
value was fetched from V(i- 1 0) but on the very next instant of time
the delayed value is fetched from V(i- 19) which contains speed that
is older than the one just fetched. This violates our common sense
and more importantly could cause problems if the fetched value is
being used in a simulation, which in tum is being used for control
purposes.
To solve this problem one must realize that Eq. ( 4-22) represents a
steady-state model and is not valid when the speed is varying. In fact,
a common sense (and rigorous) definition of the dead time requires
the use of the integral to take account of the varying speeds, as in

t
L = I v(u)du (4-23)
t-D(t)

which says that the integral of the conveying speed v(t) over the
period of the dead time D(t), which is now a variable, equals the dis-
tance over which the buckets are conveyed. In the special case where
the speed is constant, v(t) = vc' then Eq. (4-23) gives

t
L = I v du = v D = v hn
c
c
c
t-D(t)
which is the same as Eq. (4-22).
To actually solve Eq. (4-23) online, the integral can be approxi-
mated, as in

L = h[v(t;)+ v(t;- h)+ v(t;- 2h) + ··· + v(t;- hn)] (4-24)