Page 65 - Modern Control of DC-Based Power Systems

30 Modern Control of DC-Based Power Systems

V c 1
v in
5 5 (2.25)
^ v out ðsÞ D LC
^ CLs 1 L s 1 1 s 1 1 s 1 1 1
2
2
dðsÞ R RC LC
2 V out1
s ðÞ 5 ðC 1 L e s Z1L e s1ZÞV m 1 ZHðsÞG c s ðÞ D (2.26)
Z IN CL 2
V m D ð1 1 ZC 1 sÞ 2 HðsÞG c sðÞV out1 D

K D 2 K P K I
G c sðÞ 5 s 1 s 1 (2.27)
s K D K D
Assuming:
• A constant sensor gain;
• A PWM gain of 1;
• Using a general PID controller (2.27) for regulating the closed-loop
converters, with proportional gain K P , integral gain K I , and derivative
gain K D .
Those assumptions yields the following generalized expression:
3 2
ZC 1 Ls 1 L 1 ZK D v in Þs 1 Z 1 ZK P v in1 Þs 1 ZK I v in
s ðÞ 5
ð
ð
2 2 2 2 2 2
Z IN CL
ZC 1 D 2 K D D v in Þs 1 D 2 K P D v in1 Þs 2 K I D v in1
ð ð
3 2
n 3 s 1 n 2 s 1 n 1 s 1 n 0
5
2
d 2 s 1 d 1 s 1 d 0
(2.28)
By assuming that Z L sðÞ 5 R the numerators are defined as:
n 3 5 RC 1 L; n 2 5 R 1 ZK D v in ; n 1 5 R 1 RK P v in1 ; n 0 5 RK I v in while the
denominators are defined as:
2 2 2 2 2
d 2 5 RC 1 D 2 K D D v in ; d 1 5 D 2 K P D v in1 ; d 0 52K I D v in1 (2.29)
The steady-state error for a step input can be calculated with the final
value theorem and corresponds to the DC gain which will yield for
Z L sðÞ 5 R:
1 n 0 R
s ðÞ 5 52 (2.30)
e ss 5 lim s Z IN CL 2
s-0 s d 0 D
This behaves like a negative destabilizing resistance. The authors in
of a POL is
[19] and [20] state that the closed-loop input resistance Z IN CL
. The negative resistance of (2.30) pushes an
approximately 2 Z IN open
open-loop pole into the right-half plane and destabilizes the transfer func-
tion of (2.25). Therefore, often in literature the approximation of a com-
plex load is performed, which exhibits constant power behavior (2.31).