Page 139 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

114 C h a p t e r F o u r

4·6 A Few Comments about Simulating Processes
with Variable Dead Times
Consider the process idealized in Fig. 4-33 where the describing
equation is
dy
-r-+ y = gU(t- D)
dt
We will discuss the discrete time domain in Chap. 9, but assume
for the time being that the time domain is broken up into discrete
points, t ,t , ••• that are separated by a constant interval, h, as in
1 2
t; = t;_ +h. To determine the value of y at time t we need the value of
1
U at time t- D. Assume that we have an infinitely long delay vector
V(i), i = 1, 2, . . . available for the storage of U. At every discrete
moment of time t;, we increment the index i, to the vector V and insert
the value U(t;) as in

i+-i+1
(4-20)
V(i) +- U(t;)

Further, assume that the dead time D is an even multiple of the
constant interval, as in D = nh .
To simulate the process we need U(t- D); how do we get it? One
way is to augment the simple algorithm in Eq. (4-20) as follows:

i+-i+1
V(i) +- U(t;)
(4-21)
j +-i-n
U(t- D)+- V(j)

In words, Eq. (4-21) says "increment the delay index, place the
current value of U in the delay vector, decrement the delay index by
the number of increments in the dead timeD, and fetch the delayed
value of U."
If the speed of the belt v is the controller output and is therefore
variable, how can the correct value of U be obtained? The simplest
and in my experience the most common approach uses a variable
index calculated from the speed. The distance over which the belt has
to carry the buckets L, is related to the dead time and the belt speed
according to

L=vD=vnh
(4-22)