Page 131 - Modern Control of DC-Based Power Systems

Background 95

error, the larger will be also the compensation term. If y 5 ^y, then the
state estimation is perfect and there will be no compensation term.
In order to determine how quickly the estimated state converges to
the real state, let us define the error vector as:

e 5 x 2 ^x (3.20)
By substituting the variables x and ^x from (3.6) and (3.19) and deriva-
tion, the error vector dynamics can be obtained:
_ e 5 A 2 GCÞe (3.21)
ð
Therefore, the dynamic of the observer are determined by the eigen-
values of the term A 2 GCÞ.
ð
Impact of observer in a closed-loop system with state-feedback: Using the esti-
mated state of the system (^x) rather than the actual system state (x) in the
pole placement technique affects the behavior of the overall closed-loop
system. Using Eqs. (3.4) and (3.19), it can be shown that the dynamics of
the closed-loop estimated-state feedback system is governed by the fol-
lowing equation:

_ x A 2 BK BK x
5 (3.22)
_ e 0 A 2 GC e
Consequently, the characteristics equation can be written as:
(3.23)
j
j sI 2 A 1 BKj sI 2 A 1 GCj
This means that the closed-loop poles of a system with state feedback
and a state observer are composed of the poles from the state feedback
when no observer is present plus the poles from the state observer design.
In other words, the dynamics of the closed-loop system is determined by
both the poles of the observer and the new poles from pole placement.
From a practical point of view, the above conclusion means that the
observer and the state-feedback can be designed separately. Furthermore,
in order to minimize the impact of the presence of the state observer on
the overall dynamics of the systems, its poles are usually placed farther
from the imaginary axis on the left-hand side plane compared to the
desired closed-loop poles, which are placed using the state feedback.

3.6 DROOP

Paralleling a number of converters typically offers advantages over a
single high power converters, such as low component stresses or increased