Page 69 - Modern Control of DC-Based Power Systems

34 Modern Control of DC-Based Power Systems

As in this case the Hurwitz necessary conditions of stability are not
is observed. It has to be noted that while instabil-
met: instability in Z IN CL
ity can be observed its cause is different from the CPL case.

2.4.4 Nonminimum Phase Impedance
Nonminimum phase systems are quite seldom found as time continuous
systems, while in DC DC the most prominent case is the boost and
buck boost converters which exhibit right half plane zeros. They merit a
consideration due to the wide implementation of digital controllers,
where the minimum phase property can be lost due to the sampling.
Their response to changes in the input signal are typically in opposite
directions, i.e., when the input signal increases, the system output drops
briefly before rising again, in contrast to systems with negative reinforce-
ment. Due to the presence of zeros in the right complex half plane, a
high feedback system will exhibit instability. Considering a first-order
nonminimum phase load Z Load sðÞ:
s 2 b 1
Z Load sðÞ 52 ;a; bAR (2.38)
s 1 a
4 3 2
n 4 s 1 n 3 s 1 n 2 s 1 n 1 s 1 n 0
s ðÞ 5 (2.39)
Z IN CL
3 2
d 3 s 1 d 2 s 1 d 1 s 1 d 0
where in (2.38):
n 4 52C 1 L; n 3 5 L1bC 1 L 2 K D v in ;
n 2 5 La 1 bK D v in 2 1 2 v in1 K P ; n 1 5 b 1 bv in1 K P 2K I v in ;

2
2
n 0 5 bK I v in ; d 3 52C 1 D 2 K D D v in1 ;
2
2
d 2 5 D ðbC 1 2 aK D v in1 1 1 2 K P v in1 Þs ;
2 2 2
d 1 5 aD 1 D v in1 ðK I 2 aK P Þ; d 0 52 aK I D v in1
With the previously performed analysis it has been shown that a
stability analysis has to include suppositions about the dynamic behavior
on top of the CPL assumption. Load impedances which have either zeros
or poles in the right half plane will exhibit a drastically different behavior
than the nondynamic CPL assumption.