In the following sections, the design procedures for application of. H-Infinity Control on .... Bode plot of the non-ideal boost converter is depicted in Fig. 3, and the ...
Application of H-Infinity Control on Boost Converter Prepared by Omar Abdel-Rahim Ph. D Student, Electrical and Electronic Engineering Department, Utsunomiya University 147102H Submitted to Prof. Mitsu Hirata
Introduction During
the
past
decade,
there
have
been
many
theoretical
advances in the field of robust control, and, in particular, h-infinity control.
A
problem
breakthrough
in
feasible
1988
algorithms
in
led
to
for
of
weighting outputs input
input
functions and
reflect
disturbances
solution
the
finding
of the key steps in the formulation
the
of
the
development
h-infinity
h-infinity of
control
computationally
optimal
controllers.
One
h-infinity control design approach is
and
are
output
utilized
the
spatial
and
the
weighting
functions.
normalize
the
to and
frequency
performance
the
These
inputs
dependency
and
of
specifications
the
of
the
output variables [1]-[4]. The
weighting
performance
(i.e.,
time
function
should
tracking
performance
overshoot.
In
frequency
gain,
one
would
function
have
the
expect,
selected
response) low
and
general,
is
the
more
at
gain
limiting
low at
larger the
the
overshoot the
reflect
characteristics.
gain
high
to
the
The
sensitivity
frequencies
high
is
overshoot
for
frequencies
magnitude
of
limited. is
desired
good
to
limit
the
high
However,
accomplished
as by
adding more damping at the expense of response speed, which sets up a design tradeoff between overshoot and response speed. Pulse-Width-Modulation nonlinear average describes
and model,
(PWM)
time-variant we
systems.
obtain
approximately
the
a
dc–dc After
linear behavior
converters linearization
time-invariant of
the
are of
the
model
that
system
for
frequencies up to half of the switching frequency. The tasks of the controller in dc–dc converters are to [5]: 1) Assure stability of the closed-loop system; 2)
Minimize
sensitivity
to
load
changes,
i.e.,
reduce
the
output
impedance; 3) Attenuate input–output transmission (low audio susceptibility). Traditional
designs
systems
are
linearization).
of
feedback
based A
on
major
loops
in
dc–dc
frequency-domain
problem
is
switched-mode
analysis
(after
however,
when
encountered,
the transfer function of the power stage (from the control input to the
output
cases RHPZ
voltage)
(e.g.,
boost
severely
has or
a
fly
restricts
right-half-plane back
the
Zero
converters),
closed-loop
(RHPZ).
the
band
In
presence
width
that
such
of
the
can
be
obtained by the classical frequency domain approach, [5]-[7]. H-∞ Infinity Control: SOME GENERAL FACTS For a stable and proper transfer function G(s), the norm is defined as
‖()‖ = Where
()
the
maximum
singular
is
(1) value
of
a
complex
matrix A and is defined as
() = (∗ ) Note
that
maximum
∗ is
value
the of
conjugate eigenvalues.
transpose If
G(s)
(2)
of
A,
is
a
Output (SISO) system, (1) takes on a simpler form as
and Single
Input
is
the
Single
|()|
‖()‖ = i.e.,
the
norm
corresponds
to
the
maximum
Furthermore, the following inequality
‖()‖ < 1 ⟺ | ()| < 1, ∀ Obtained
from
(3)
is
commonly
used
substituted for the norm condition.
(3) of
gain
G(s).
(4)
when
the
gain
condition
is
Using (4), the robust stability condition can be represented by the
norm condition of
‖& '‖ < 1
H-Infinity Control for Boost DC-DC Converter
In
the
following
H-Infinity details.
sections,
Control
Design
on
procedure
(5)
the
design
procedures
for
application
of
dc-dc
boost
converter
are
described
in
and
design
equations
all
are
based
on
the IEEE Transaction in [8]. Most of the results in this report are based on the design equations developed in [8]. A. THE BOOST CONVERTER
As converter
a
design
operated
example, in
we
took
continuous
a
typical
conduction
low-power
mode,
shown
boost
in
Fig.
1, which transforms a nominal 12-V input voltage into 24 V at its output.
The
current
source
switching in
the
frequency
is
240
output
to
simulate
is
(changes
in
the
output
current).
inductor,
switch,
diode,
and
The
capacitor
kHz.
changes
series are
The
role in
resistance all
of
the
the
load
of
the
shown.
The
controller imposes the duty cycle), which is the ratio between the
time '*+ when the switch is closed and the period'. Applying the
well-known a
state-space
averaging
that,- = ,. = ,/ = 0,
moment
procedure
[8]
we
the
find
and
assuming
following
for
transfer
function from the duty cycle to the output voltage 1234 .
() =
6 ( 5234 789)6
Recall
5:; -
?6 234 A(7BC H
=78
)
and
N*LM
see
and
that
and O
the
denote
fixed
denote
above
(6) values
deviations
transfer
at
from
function
an this
exhibits
Right Hand Pole or Zero (RHPZ) 6 Q5:;
PB = -5 6 234
If we do not neglect ,- , ,. STO ,/ function
discussed
above
(7)
then the formula of the transfer
becomes
more
complicated,
of
course,
but it retains the same features. Transfer function from the control input O to the output voltage N*LM were found to be:
1234 .
() =
8 .77V(BWXWY )(8WZWZ ) 6 BW[77BXZ
(8)
Input-to-output voltage transfer function of boost converter is described as follow:
G^_ (S) =
b c (d) a f b e (d) b a g(d)h
=
ZZV(BWXWY )
6 BW[77BXZ
(9)
Output current-to- output voltage transfer function of boost converter is described as follow: P* (j) =
b (S) V o p Ino (S) db rSs=0
= −.12rj+45460srj+4100s j +4311j+5200000
(10)
Figure 1: DC-DC Boost Converter
B. H-∞ Infinity Control for the boost converter
ω{
v in
PO
iout
W
vout
z
d
C
Wd
y
Figure 2. The control of the converter as a standard H∞ control problem.
In the diagram, depicted in Fig. 2, v* is the converter, w is
the controller to be designed, and x is a stable weight function.
The vector contains the perturbations (NK+ andy*LM ), the output z is
the
weighted
measurement
error
outputs
of
signal, the
the
vector
(N*LM andNK+ ),
plant
{
contains
and
the
the
control
input | is the duty cycleO. We denote by '}~ the transfer function
from w to z (in closed loop, i.e., with the controller connected to the
plant).
controller
The C,
standard
which
problem
stabilizes
of
the
H∞
control
is
plant
(i.e.,
the
to
find
a
closed-loop
system
is
stable)
weight
function
frequencies
(it
perturbations the
in
weight
that
the
bigger
the
minimization
the
is
can
frequencies
the W.
weighted
where
more
and
at
Thus, output
of
the
for
the
different
By
adjusting
example,
worse
way
around.
Note
input–output
voltage-
both
design
objective
impedance.
and
W
weight
function
can
be
combined
to
The
state-space
equations
of
the
averaged
and
which is a system with inputs &, | and outputsz, {.
better
of
impedance,
our
of
expense
other
the
The
presence
disturbing).
the
output
of'}~ .
norm
importance
obtain,
'}~ are
of
function
H∞
relative
frequencies
and
the
frequencies,
signal we
attenuation
of
the
for
error
lower higher
weight
minimizes
expresses
components
disturbance by
at at
the
is
also
function,
performance attenuation
x
and
The
multiplied includes
converter
form
v*
the
plant
P,
linearized
plant
v*
in our design example are
= + 7 + Z |
N*LM = w + )7 + )Z |
(11)
Where q is the state of the converter (the inductance current and
capacitor
voltage).
The
matrices
appearing
in
equations are
−4208 −2283 2086 −103.1 4975 228.3 7h 0 −4535 119540 Z = −5370 =
w =
0.046 1 )7 =
0
−0.1
(12) (13) (14) (15) (16)
the
state
)Z =
−0.118
(17)
For our design example, we chose the weight function
x () =
BZ∗K∗[X BZ∗K∗X
(18)
By combining v* with a realization of W, we obtain the plant P, which is of order three as follow:
w ()
−24300000(j − 102. 10Y )(j + 12140)(j + 4120) 1.87. 107[ = (j − 4.47. 107W )(j + 3140)(j + 45460) j − 4.47. 107W (19) The
raw
extremely
w has
controller big
positive
an
pole,
unstable
which
cannot
pole—actually be
realized
by
an any
means. (The closed-loop system would be stable, even if we could
w realize and use it.) To overcome this problem, the unstable pole was approximated by a constant
8
≈
8
, For|j| ≪ |S|.
(20)
A similar approximation was used to get rid of the very big zero lying these
in
the
right-half
approximations
controller
over
the
plane. do
Further
not
frequency
affect
range
of
investigation the
has
shown
performance
interest
(up
to
that
of
the
about
100
kHz).
The approximate controller is
w() =
8X.XY(BW7Z )(B7Z7W ) (B[7W )(BWXWY )
− 0.0417
(21)
We remark that the controller C has a low dc gain compared to
the
voltage-mode
controller,
feed
forward
controller,
or
current-
mode
controller,
which
will
be
derived
later.
With
the
controller
C, the voltage-attenuation curve looks like that in Fig. 4. In the same
figure,
the
corresponding
curves
are
shown
for
two
other
possible choices of the weight function, namely x7 () =
xZ () =
BZ∗K∗[X BZ∗K∗X
(22)
BZ∗K∗[X BZ∗K∗X
(23)
The curves of the output impedance for the three weight functions, when
using
a
logarithmic
scale
for
the
impedance,
look
very
similar, so we omit them. Bode plot of the non-ideal boost converter is depicted in Fig. 3, and the bode plot of the designed controller is shown in Fig. 4. Bode
plot
of
the
closed
loop
system
with
different
weighting
factor are depicted in Fig. 5, 6 and 7, while Fig.8 shows the bode plot of the closed loop system without using any weighting factor. We
remark
compared
to
that the
the
h-infinity
voltage-mode
or current-mode controller
controller controller,
has feed
a
low
forward
dc
gain
controller,
Figure 3. Bode diagram of open loop boost converter.
Figure 4. Bode diagram of controller.
Figure 5. Bode diagram of the closed loop system with weighting factor W.
Figure 6. Bode diagram of the closed loop system with weighting factor W1.
Figure 7. Bode diagram of the closed loop system with weighting factor W2.
Figure 8. Bode diagram of the closed loop system without any weighting factor.
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Y.
flexible
Qian,
C.
structures
Hai, on
H. the
dong,” H-infinity
Tracking control
control method,
research “
2nd
of
high-order International
Conference
on
Advanced
Computer
Control
(ICACC),
F.
Novazzi
and
of
a
exchanger
27-29
March,
2010,
Pp. 111-115. [2]-F.
Delatore,
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model
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Leonardi,
matching
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Power
storage
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