R
2
2 ( ¯
μ 1 s) C 2
C 2
⎦
+
( |v
dt
s| − Vs)
˜ i
˜
1
2
− 1 (1 − ¯ μ
L
2 s)
− r 2
i 2
2
L 2
L 2
0
+
− ¯ i 2 s
− 1
C 2
+
+ ˜
C 2
¯ v
˜ v 2
i 2
˜
μ 2
(13)
2 s
1
0
L 2
L 2
=
p f c p f c
p f c
p f c
p f c
p f c
p f c
: Ap xp + B
w
p 1
p
+ Bp + xp Np
u 2
(14)
where − ¯ μ 1 s < ˜ μ 1 < 1 − ¯ μ 1 s and − ¯ μ 2 s < ˜ μ 2 < 1 − ¯ μ 2 s. The following discussion constructs controllers on the basis of the averaged models (Σ f c) and (Σ p f c) around the set point.
A
A
3.2 Linear controller design for FC
A robust controller for the FC is given by linear H∞ control technique because the averaged
model (Σ f c) is a linear system. The control technique incorporates weighting functions
A
Wv 1( s) := kv 1/( s + ε 1) to keep an output voltage constant against variations of load resistance and apparent input voltage as shown in Fig. 3, where kv 1, ε 1 are synthesis parameters. Then,
the linear H∞ control technique gives a linear gain K 1 of the form
f c T
f c
f c T
u 1 = K 1 x f c := −( D
D ) − 1 B
Y f c− 1 x f c
(15)
12
12
2
T
T T
f c
T
T
f c
T
f c
where x f c :=
f c
f c
f c
x
, B
:=
, D
:= 0 T WT
and x
p
xw
2
Bp
0 T
12
u 1
w denotes state
of weighting function Wv 1( s) as shown in Fig.3. In the figure, matrices We 1 and Wu 1 are
weighting coefficients of a performance index in the linear H∞ control technique and used
only in the controller synthesis.
The matrix Y f c that constructs the gain K 1 is given as a positive-definite solution satisfying a
Lyapunov-based inequality condition
26
Applications of Nonlinear Control
6
Nonlinear Control
ze 1
zu 1
We 1
Wu 1
A f c
p
+
x f c
u 1
f c
v 1 r
W
˙ f c
x
x
v 1( s)
+
p
p
−
f c
f c
x
1
w
K 1
Bp
+
s
˜ v 1
Linear
[
Gain
1 0 ]
controller
Fig. 3. Block diagram of linear controlled system for FC
T
T
T
T
A f cY f c + Y f c A f cT − B f c( D f c D f c) − 1 B f c + γ f c− 2 B f cB f c Y f cC f c 2
12
12
2
1
1
1
< 0
(16)
f c
C Y f c
−I
1
f c
f c
where the matrix A f c, B , C
are coefficients of a generalized plant given by Fig.3 and γ f c is
1
1
a synthesis parameter in the linear H∞ control technique.
3.3 Nonlinear controller design for PFC
A robust nonlinear controller design for the PFC, that is a nonlinear system, consists of the
following two steps. First, a nonlinear gain is given to guarantee closed loop system stability
and reference tracking performance for source current and output voltage against variations of
the apparent load resistance. Second, a source current reference generator is derived to adjust
an amplitude of current reference to variations of source voltage and load resistance. At the
first step, as shown in Fig. 4 incorporated are a weighting function Wi 2( s) := ki 2 ω 2 /( s 2 +
i 2
2 ζωi 2 s + ω 2 ) for a source current to be sinusoidal and be in phase with a source voltage and
i 2
a function Wv 2( s) := kv 2/( s + ε 2) for an output voltage to be kept constant against those variations, where ki 2, ζ, ωi 2, kv 2, ε 2 are synthesis parameters. Then, a nonlinear H∞ control technique in a work by Sasaki & Uchida (1998) gives a nonlinear gain of the form
T
K 2( xp f c
p
) := −( Dpfc Dpfc) − 1 Bpfc( xpfc) TYpfc− 1
(17)
12
12
2
p f c
T
T T
p f c
T
p f c
as shown in Fig.4, where xp f c =
p f c
p f c
p
x
, D
= 0 T WT , B ( xpfc) = B
+
p
xw
12
u 2
2
0
Bp f c
p f c
B
T
T
xp f c
p 21
+ xpfc
p 22 , Bp f c = 0 1
, Bp f c =
− 1 0
and xp f c
1
w
denotes state of
0
2
0
p 21
L 2
p 22
C 2
weighting functions. In the figure, matrices We 2 and Wu 2 are weighting coefficients of a
performance index in the nonlinear H∞ control technique. A block named as Generator is
not used for the synthesis and is discussed at the following second step.
Lyapunov-Based Robust and Nonlinear Control
for Two-Stage Power Factor Correction Converter
27
Lyapunov-Based Robust and Nonlinear Control for Two-Stage Power Factor Correction Converter
7
vs
ze 2
z
p f c
u 2
wp
We 2
Wu 2
Bp f c
p 1
√
1/ 2 Ve
Feedforward
Nonlinear
+
√
i 2 r+
I
Gain
e( v 2 r )
2
W
Ap f c
p
+
i 2( s)
−
p f c
x
xp f c
p f c
u 2
˙ p f c
x
x
w 2,3
K( s)
p f c +
+
p
p
1
Feedback
Bp
p f c
+
+ +
s
Generator
p f c
K
+
x
2( xp
)
v
w 1
p f c
2 r
W
p f c
p f c
v 2( s)
−
N
{x
p
p
Np }u 2
˜ i 2
˜ v 2
Controller
Fig. 4. Block diagram of nonlinear controlled system for PFC
The matrix Yp f c is given as a positive-definite solution satisfying a state-depended
Lyapunov-based inequality condition
⎡
⎤
p f c
p f c T
p f c
p f c
T
p f cT
⎢ Ap fcYp fc + Yp fcAp fcT − B
( x)( D
D
) − 1 B ( x) YpfcC
2
12
12
2
1
⎢
⎥
⎢
⎥
T
⎢
p f c p f c
⎥ < 0
(18)
⎣
+ γpfc− 2 B B
⎥
1
1
⎦
p f c
C
Yp f c
−I
1
p f c
p f c
p f c
p f c
where x is a state of a generalized plant given by Fig.4, matrices Ap f c, B
, B
( x), C , D
1
2
1
12
are coefficients of the plant and γp f c is a synthesis parameter in the nonlinear H∞ control
technique.
For any state x, which is current and voltage in a specified domain, the matrix Yp f c satisfying
the inequality (18) is concretely given by solving linear matrix inequalities at vertices of a
convex hull enclosing the domain as shown in a work by Sasaki & Uchida (1998).
Next, given is a mechanism to generate a source current reference i 2 r. As shown in Fig.4
the reference generator consists of a feedforward loop given by steady state analysis in
Section 2.3 and a feedback loop with a voltage error amplifier. The amplifier K( s) is given
by K( s) = kP + kI / s + kDs where kP, kI and kD are constant parameters decided by system designers as discussed in a work by Sasaki (2009). The feedback loop is the same structure
as a conventional loop used in many works (e.g., Redl (1994)). Note that the amplifier K( s)
works only for variations from the nominal values in the circuit, because the feedforward loop
gives the effective value of the source current as shown in Section 4.4.
4. Computer simulations
This section finally shows efficiencies of the approach through computer simulations. It
is also clarified that consideration of nominal load resistance for each part characterizes a
performance of the designed controlled system. A software package that consists of MATLAB,
Simulink and LMI Control Toolbox is used for the simulations.
28
Applications of Nonlinear Control
8
Nonlinear Control
Parameters of the circuit shown in Fig.1 are given by Table 1.
r 1 1 [mΩ] r 2 1 [mΩ]
L 1 10 [mH] L 2 1 [mH]
C 1 450 [ μ F] C 2 1000 [ μ F]
N
1/36
Table 1. Circuit parameters of two-stage power factor correction converter as shown in Fig.1
4.1 Design specification
Control system design specification for the two-stage power factor correction converter is
given by
(1) An effective value of source voltage is 82 ∼ 255 volts, and its frequency is 45 ∼ 65 hertz ;
(2) An output voltage is kept be 5 volts, and its error of steady state is within ± 0.5%;
(3) An output current is 0 ∼ 30 amperes ;
(4) A source current is approximately sinusoidal and is phase with source voltage.
The above specifications are treated for controller design as the following respective
considerations ;
(1) A nonlinear gain for PFC is designed for a source voltage whose nominal effective value
is Ve = (82 + 255)/2 = 168.5 volts (then, average full-wave rectified voltage is Vs = 151.7
volts). Moreover, the gain is designed to be robust against source voltage variations by
p f c
treating that as a disturbance wp as shown in Fig.4.
(2) Incorporated is a weighting function Wv 1( s) whose magnitude is high at low frequencies.
(3) A load resistances R varies between 1/6 ∼ ∞ ohms at an output voltage of 5 volts.
(4) Incorporated is a weighting function Wi 2( s) whose magnitude is high at frequencies in
the range of 45 ∼ 65 hertz for the source current to be approximately sinusoidal at the
frequencies.
4.2 Nominal load resistance
Now, a nominal value of load resistance R need be decided to design controllers. The nominal
value influences a performance of closed-loop system controlled by the designed controllers.
Root loci of linearized converters for the two parts as shown in Figs. 5 and 6 , which are
pointwise eigenvalues of the system matrices for variations of load resistance R, show that the
larger the load resistance R is, the more oscillatory the behavior of FC is and the slower the
transient response in PFC is. Controllers, here, are constructed in consideration of undesirable
behavior of the converter system. Therefore, a FC controller is designed for a system with
oscillatory behavior, that is for a large load resistance, so that an output voltage does not
oscillate. A PFC controller is designed for a system with fast response, that is for a small load
resistance, so that a source current is not distorted by the controller responding well to source
voltage variations.
Lyapunov-Based Robust and Nonlinear Control
for Two-Stage Power Factor Correction Converter
29
Lyapunov-Based Robust and Nonlinear Control for Two-Stage Power Factor Correction Converter
9
500
R = 500
400
Oscillatory
300
200
100
0
Im
R = 1/6
R = 1/6
−100
Asymptotic
−200
−300
−400
R = 500
−500
−14000
−12000
−10000
−8000
−6000
−4000
−2000
0
Re
f c
Fig. 5. Root locus of coefficient Ap of linear model for FC
1000
800
R = 1/6
R = 500
600
400
200
0
Im
Fast
Slow
−200
−400
−600
−800
R = 1/6
R = 500
−1000
−1
−0.8
−0.6
−0.4
−0.2
0
Re
p f c
Fig. 6. Root locus of coefficient Ap of linearized model for PFC