= k
v k
i k
x
rect L
is the state vector, and
LC 1
= ω
is the cut-off frequency of the low-pass filter.
From 3 and 4, the discrete-time equations can be rewritten:
1 1
1 1
1 12
1 11
1
k i
B B
k v
B A
k i
B A
k i
B k
u
or d
or L
L
− −
− +
=
5
1 1
21 2
21 2
21 22
21
k i
A B
k u
A B
k v
A A
k v
A k
i
or d
or or
L
− −
− +
=
6 From 5 and 6, it can be seen that additional
disturbance terms appear because of model discretization. As compared to the continuous-time model, there exist
two disturbances instead of one, acting on the inductor current and output voltage. The current and voltage
disturbance terms can be written as:
1 1
1 12
k i
B B
k v
B A
k i
or d
or d
− −
= 7
21 2
21 2
k i
A B
k u
A B
k v
or d
d
− −
=
8 Based on the discrete-time equations, the digital model of
the system can be represented by the block diagram in Fig. 6, where
1 −
z
denotes a unit delay.
III. C
ONTROLLER
D
ESIGN
Fig. 7 shows the proposed Deadbeat controller for the BHFL inverter. It consists of inner current loop controller,
outer voltage loop controller and a feed-forward controller. The feed-forward controller, which imposes a
gain scheduling effect according to the reference signal, is used to compensate the steady-state error of the system.
From the discrete-time model of the plant in Fig. 6, it is known that there are disturbance terms acting on the
inductor current and output voltage. These disturbances are compensated using additional decoupling networks in
7 and 8. With that, the Deadbeat controller has good disturbance rejection capability and improved robustness
towards load variations.
A. Current Loop Controller
Fig. 8a shows the inner current loop controller. The current disturbance decoupling network is added to
compensate the disturbances acting on the inductor current. Canceling the current disturbance coupling
allows a simple gain, K
i
to be applied in forming the inner current loop. Referring to Fig. 8a, the current loop
control law can be derived:
[ ]
k i
k i
k i
K k
u
d L
ref i
+ −
=
9 where
k u
is the control signal applied to the PWM modulator,
k i
ref
is the inductor current reference generated by the outer voltage loop, and
k i
d
is the current disturbance decoupling network from 7. Fig.
8b shows the simplified current loop. The discrete-time open-loop transfer function is:
1 11
1 1
1
− −
− =
z A
z B
z G
i
[ ]
1 1
cos 1
sin
− −
− =
z T
L z
T
s s
ω ω
ω
10 The corresponding discrete-time closed-loop transfer
function of the current loop is:
1 z
G K
z G
K k
i k
i z
C
i i
i i
ref L
i
+ =
=
[ ]
1
1 11
1 1
1
+ −
=
− −
z A
B K
z B
K
i i
[ ]
L z
T L
T K
z T
K
s s
i s
i
ω ω
ω ω
ω +
− =
− −
1 1
cos sin
sin
11 From 11, the characteristic equation of the closed-loop
current controller can be written as:
[ ]
1 11
= −
− B
K A
z
i
sin 1
cos =
− −
s i
s
T L
K T
z ω
ω ω
12 In discrete-time control systems, the closed-loop poles or
the roots of the characteristic equation must lie within the unit circle in z-plane for the system to be stable [10].
Therefore, the range of K
i
for the system to be stable is:
[ ]
1 1
1 11
− −
B K
A
i
,
1 11
1 11
1 1
B A
K B
A
i
+ −
and
[ ]
[ ]
sin 1
cos sin
1 cos
s s
i s
s
T T
L K
T T
L
ω ω
ω ω
ω ω
+ −
13 To achieve Deadbeat response, the root is to be placed at
the origin of the z-plane
= z
. Hence, the current loop gain, K
i
is designed as:
1 11
B A
K
i
=
sin cos
s s
T T
L
ω ω
ω
=
14 Substituting 14 into 11 yields:
1 11
k i
z A
k i
ref L
−
= cos
1
k i
z T
ref s
−
=
ω 15
When
s
T
ω
is sufficiently small,
s s
T T
ω ω
≈ sin
and 1
cos ≈
s
T ω
. Therefore, 15 can be written as
1
k i
z k
i
ref L
−
=
, which is the Deadbeat response.
B. Voltage Loop Controller
Fig. 9a shows the outer voltage loop controller. The voltage disturbance decoupling network is added to
compensate the disturbances acting on the output voltage. This will improve the robustness of the system towards
load variations, enabling various types of loads to be connected. Besides, it also acts as an additional current
loop command to produce the needed load current without waiting for errors in voltage to occur.
The design procedure of the voltage loop controller is similar to the current loop controller. The voltage loop
gain, K
v
is applied to achieve Deadbeat response. From Fig. 9a, the voltage loop control law is derived:
[ ]
k v
k v
k v
K k
i
d or
ref v
ref
+ −
= 16
where
k i
ref
is the generated current loop command for the inner current loop,
k v
ref
is the rectified sinusoidal voltage reference, and
k v
d
is the voltage disturbance decoupling network from 8. Fig. 9b
shows the simplified voltage loop. It can be noted that the inner current loop is viewed as a constant gain, with the
condition of current loop is well designed. The corresponding discrete-time open-loop transfer function
is:
1047
1 22
1 21
1
− −
− =
z A
z A
z G
v
[ ]
1 1
cos 1
sin
− −
− =
z T
C z
T
s s
ω ω
ω
17 The discrete-time closed-loop transfer function of the
voltage loop is:
1 z
G K
z G
K k
v k
v z
C
v v
v v
ref or
v
+ =
=
[ ]
1
1 22
21 1
21
+ −
=
− −
z A
A K
z A
K
v v
[ ]
C z
T C
T K
z T
K
s s
v s
v
ω ω
ω ω
ω +
− =
− −
1 1
cos sin
sin
18 From 18, the characteristic equation of the closed-loop
voltage controller can be written as:
[ ]
21 22
= −
− A
K A
z
v
,
sin 1
cos =
− −
s v
s
T C
K T
z ω
ω ω
19 For the system to be stable, the range of K
v
is:
[ ]
1 1
21 22
− −
A K
A
v
,
21 22
21 22
1 1
A A
K A
A
v
+ −
[ ]
[ ]
sin 1
cos sin
1 cos
s s
v s
s
T T
C K
T T
C ω
ω ω
ω ω
ω +
−
20 Similar to the current loop gain, the voltage loop gain, K
v
is designed such that the root of the system can be placed at the origin of z-plane:
21 22
A A
K
v
=
sin cos
s s
T T
C ω
ω ω
=
21 Substituting 21 into 18 yields:
1 22
k v
z A
k v
ref or
−
= cos
1
k v
z T
ref s
−
=
ω 22
When
s
T ω
is sufficiently small,
s s
T T
ω ω
≈ sin
and
1 cos
≈
s
T ω
. Therefore, 22 can be written as
1
k v
z k
v
ref or
−
=
, which is the deadbeat response.
C. Feed-forward Controller
