Voltage and Current H Repetitive Control - Control of Power Inverters in Renewable Energy and Smart Grid Integration - page 94

~

a b

v 1

ξ

~

~

W

v 2

μ

W

e

w

+

~

P

+

i o

u

C

C = [ C 1 C 2 ]

Formulation of the H ∞ control problem

Figure 4.3

difficult to stabilise (a stabilising compensator may not exist, or it may need an unreasonably
high bandwidth).
In order to find a controller C , a standard H ∞ problem is formulated as shown in

] T

Figure 4.3, with ˜ w = v 1 v 2 w and

[

z
y

P ˜

u

= , u = C y .

The nonzero parameter ξ is introduced to offer more freedom in the design process. The

small positive parameter μ is introduced to satisfy a rank condition needed to make the

A u

D u is a weighting function whose value at infinity,

B u

H ∞

problem solvable and W u =

C u

0, is also needed to meet a rank condition. The problem formulated here is a
slight improvement over the one in (Weiss and Hafele 1999), where W u was a constant. The
fact that W u is a frequency-dependent high-pass filter has the effect of reducing the controller
gains at high frequencies.
According to (Weiss and Hafele 1999), the closed-loop system in Figure 4.2 is exponentially
stable if the finite-dimensional closed-loop system shown in Figure 4.3 is stable and its transfer
function from a to b , denoted T ba , satisfies

D u = W u ( ∞ ) =

1. The intuitive explanation for this is
that, in the control system of Figure 4.2, a delay line is connected from the output b to the input
a appearing in Figure 4.3. To make this interconnected system stable, it is sufficient to make
the gain from a to b less than 1 at all frequencies according to the small gain theorem (Zhou
et al. 1996). Thus, the controller C needs to be designed so that the above two conditions are
satisfied. Moreover, in order to obtain a small steady-state error, it is necessary to minimise
γ 0

T ba ∞ <

, where γ 0 = T e w ∞ , γ = T ba ∞ < 1, according to (Weiss and Hafele 1999).

1 − γ

-analysis and synthesis toolbox from MATLAB R (the routine hinfsyn ), a
controller C which nearly minimises the H ∞ -norm of the transfer matrix from

Using the μ

w ˜ to z ,