Page 154 - Modern Control of DC-Based Power Systems

118 Modern Control of DC-Based Power Systems

In this way the system (5.16) is transformed to be represented in the
so-called nonlinear controllable canonical form [6]:

_ z 1 z 2 0
5 2 1 Uu (5.21)
_ z 2
a L b L a c xðÞ
L c xðÞ
with the following states:

y
z 1 c xðÞ
5 5 (5.22)
z 2 _ y L a c x ðÞ
A noteworthy detail that should be pointed out is that after the exact
input output linearization the linearized system has the same order as
the nonlinear system. This means that there are no internal dynamics and
the property of observability remained in this case intact. In order to line-
arize a SISO system, the input signal has to be of the following form:

T
2
L c xðÞ 1 k z V
u 52 a 1 w (5.23)
L b L a c xðÞ L b L a c xðÞ
with
T
k 5 a 0 a 1 (5.24)
Then, the resulting system can be written in the linear controllable
canonical form:

_ z 1 0 1 z 1 0
5 1 Uw (5.25)
_ z 2 2a 0 2a 1 z 2 V
where w is the new input of the system.
The system considered here, on the contrary, is a MIMO system.
That is why the control law, i.e., the duty cycle d i , has to be slightly mod-
ified to divide the linearizing first term between the weighted number of
participating LRC (S i ).
T
2
1 L c xðÞ 1 k z V i
d i 52 a 1 w i (5.26)
S i L b L a c xðÞ i L b L a c xðÞ i
The overall control vector is

T
u 5 d 1 d 2 d 3 : (5.27)