Page 148 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

I atrices 11 ~ Hig~e r-0 rd er Prtce s s I ode Is l23

(pA,s+ ~)X,(s) = _i i 1(s)
1 (5-3)

(pA,s+ ~)X,(s)= ~ i 2(s)

Y(5)=X (5)
3
These equations can be combined in the Laplace domain, eliminat-
ing the X;· First, rewrite Eq. (5-3) slightly, introducing three time
constants.

(~p~5 + 1)X (5) = U(5) 1' = ~p~
1
1
-r 2 =~p~
(~p~5+1)X2 (5)= ~ X 1 (5)
R3 -
-
3
(~pA 5 + 1)X (5) = ~ X (5) -r = R pA 3
3
2
3
3
Starting with the last equation, eliminate the three X; and develop
an expression for the process transfer function G.
Second, eliminate X and X to get X :
2
3
1
1 ~ 1 ~-
= 1'35+ 1 ~ 1'25+ 1 ~ X1(5)

1~1~~-
5
= 1' 5+1 ~ 1' 5+1 ~ 1' 5+1 U( )
3 2 1
Next, use the fact that Y(5) = X (5) and find the transfer function:
3

(5-4)

- ~
- (1'11'21'3) 53 + (1'11'2 + 1'11'3 + 1'21'3) 52 + (1'1 + 1'2 + 1'3)5 + 1