Page 74 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 49

dY
-r-+Y=gU
dt
I
U(t) = ke(t)+ I J due(u)
0
To combine these two equations, and in the process get rid of the
unwieldy integral, we have to take the derivative of each equation
and replace e with its definition of S - Y. The derivative of the first
equation is

The derivative of the second equation is

dU =kde +le=kd(S-Y) +l(S-Y)
dt dt dt
dS dY
=k--k-+15-IY
dt dt

In taking the derivative of U(t) we used the fact that differentiat-
ing the integral simply releases the integrand. For more on this check
App.A.
Now, do some minor algebra to eliminate dU I dt between the two
equations and get

(3-21)

To avoid some difficulties that we will deal with later on, let's
assume that the set point S is constant and has been so for all time,
hence dS I dt = 0 and

(3-22)

How could we use Eq. (3-22) to show that Y ultimately goes to
Yss? We could try to solve Eq. (3-22) and then let t -+ oo. This would
take some effort and at this point it probably is not worth it. Instead,
let's just suggest that as t -+ oo, things do settle down to a final
steady state where

and dy -+0
dt