Cascaded Current-Voltage H Repetitive Control - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 129

where P u is the generalised plant and C u is the voltage controller to be designed. The
generalised plant

P u consists of the original plant P u together with the low-pass filter

, which is part of the internal model for repetitive control. The details of
how to select W u can be found in (Hornik and Zhong 2010b, 2011). A weighting parameter
ξ u is added to adjust the relative importance of

A w u B w u
C w u D w u

W u =

v u with respect to w u and another weight-

ing parameter μ u is added to adjust the relative importance of u u with respect to b u . The

parameters μ u also play a role in guaranteeing the stability of the system; see more

details in (Hornik and Zhong 2010b, 2011). It can be found out that the generalised plant P u is
realised as

ξ u and

⎡ ⎤

A u 0 0 B u 1 B u 2

⎣ ⎦

B w u C u 1 A w u B w u ξ u B w u D u 1 B w u D u 2

P u =

D w u C u 1 C w u D w u ξ u D w u D u 1 D w u D u 2 . (6.5)

μ u
--------------------------------------------------------
C u 1

0 0 0 0

ξ u

0 D u 1 D u 2

P u using the H ∞

The controller C u can then be found according to the generalised plant

control theory, e.g. by using the function hinfsyn provided in MATLAB R .

6.4 Design of the Current Controller
As explained before, when designing the outer-loop current controller, it can be assumed that
the inner voltage loop tracks the reference voltage perfectly, that is u o = u ref . Hence, the

control plant for the current loop is simply the grid inductor, as shown in Figure 6.3. The
formulation of the H ∞ control problem to design the H ∞ compensator C i is similar to that in
the case of the voltage control loop shown in Figure 6.6, but with a different plant P i and the
subscript u replaced with i .

6.4.1 State-space Model of the Plant P i
Since it can be assumed that u o = u g from Figures 6.3

and 6.4, that is, u i is actually the voltage dropped on the grid inductor. The feed-forwarded
grid voltage u g provides a base local load voltage for the inverter. The same voltage u g appears
on both sides of the grid interface inductor L g and does not affect the controller design. Hence,
the feed-forwarded voltage path can be ignored during the design process. This is a very
important feature. The only contribution that needs to be considered during the design process
is the output u i of the repetitive current controller.
The grid current i 2 flowing through the grid interface inductor L g is chosen as the state
variable x i =

u ref , then u o = u g + u i or u i = u o −

w i =

i 2 . The external input is i ref and the control input is u i . The output

signal from the plant P i is the tracking error e i =

−

i 2 , i.e. the difference between the
current reference and the grid current. The plant P i can then be described by the state
equation

i ref

x i = A i x i + B i 1 w i + B i 2 u i