Design of a Low-Order Sensorless Controller by Robust H∞ Control for Boost Converters

Journal of Power Electronics.
2016.
May,
16(3):
1025-1035

- Received : March 10, 2015
- Accepted : November 28, 2015
- Published : May 20, 2016

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Luenberger observer (LO)-based sensorless multi-loop control of a converter requires an iterative trial-and-error design process, considering that many parameters should be determined, and loop gains are indirectly related to the closed-loop characteristics. Robust H∞ control adopts a compact sensorless controller. The algebraic Riccati equation (ARE)-based and linear matrix inequality (LMI)-based H∞ approaches need an exhaustive procedure, particularly for a low-order controller. Therefore, in this study, a novel robust H∞ synthesis approach is proposed to design a low-order sensorless controller for boost converters, which need not solve any ARE or LMI, and to parameterize the controller by an adjustable parameter behaving like a “knob” on the closed-loop characteristics. Simulation results show the straightforward closed-loop characteristics evaluation and better dynamic performance by the proposed H∞ approach, compared with the LO-based sensorless multi-loop control. Practical experiments on a digital processor confirmed the simulation results.
The following definitions are used in this study:
I
:
Unit matrix
A
^{-1}
:
Inverse of matrix A
A^{T}
:
Transpose of matrix A
Im(
A
) :
Range space of matrix A
Ker
(
A
) :
Kernel space of matrix A
V
^{-}
(Σ ) :
Weakly unobservable subspace of system Ʃ
S
^{-}
(Σ ) :
Strongly controllable subspace of system Ʃ
T_{v}
and an inner loop
T_{i}
. The outer loop provides a reference current for the inner current loop. Current
î_{LO}
in the inner loop is an estimated inductor current.
F
_{1}
,
F
_{2}
,
F
_{3}
,
F
_{4}
,
F
_{5}
, and
Z_{p}
are the transfer functions from the input voltage, load current, and duty ratio to the output voltage and inductor current of the power stage;
F_{m}
and
F_{v}
are the inner and outer compensators; and
G
_{3}
,
G
_{4}
, and
G
_{5}
are the transfer functions from the input voltage, duty ratio, and output voltage to the estimated inductor current, respectively. The LO expressed in Eq. (1) is effective in estimating the inductor current of a switching converter for the sensorless multi-loop control.
Small signal block diagram of a sensorless multi-loop control.
where
is the estimated system state and
is the system input. Matrices
A
and
B
are derived from the small signal model of the converter, whereas
L
is the parameter of the LO. Transfer functions
G
_{3}
,
G
_{4}
, and
G
_{5}
can be obtained through Laplace transformation of Eq. (1). From
Fig. 1
, the closed-loop characteristics, namely, audio susceptibility and output impedance, are written in Eqs. (2) and (3), respectively.
Transfer functions
F
_{1}
,
F
_{2}
,
F
_{3}
,
F
_{4}
,
F
_{5}
, and
Z_{p}
are fixed for a given converter. The closed-loop characteristics expressed in Eqs. (2) and (3) are dominated not only by compensators
F_{m}
and
F_{v}
but also by transfer functions
G
_{3}
,
G
_{4}
, and
G
_{5}
. The design of an LO-based sensorless multi-loop control has the following process: select parameter
L
of the LO; design compensators
F_{m}
and
F_{v}
; evaluate closed-loop stability by examining loop gains at points
T
_{1}
and
T
_{2}
in
Fig. 1
; evaluate the closed-loop characteristics expressed in Eqs. (2) and (3); and repeat the design process until the desired dynamic performance is maintained. The design process shows the difficulty in designing a conventional sensorless multi-loop control for a converter because of the need to determine many parameters and because the closed-loop characteristics are indirectly related to the loop gains
T_{i}
=
F_{m}G
_{4}
and
T_{v}
=
F_{m}F_{v}F
_{2}
+
F_{m}
G
_{5}
F
_{2}
.
w
∈
R^{l}
, the controlled output is
z
∈
R^{q}
, the controller output is
u
∈
R^{m}
, and the measured output is
y
∈
R^{p}
. A weight matrix
W
is used to adjust disturbance attenuation on the controlled output
z
. The definition for the state space equation of the converter is expressed in Eq. (4). A compact sensorless controller, also called measurement feedback controller, is written in Eq. (5).
where
x
∈
R^{n}
is the system state including the inductor current and capacitor voltage.
Block diagram of robust H∞ control.
Generally, the order of the controller in Eq. (5) is the same as that in the system in Eq. (4) with an order
n
. However, the order of the controller can be reduced to
n − rank
[
C
_{1}
D
_{1}
] +
rank
(
D
_{1}
) ≤
n
. A low-order controller has the advantage of less computational volume or simpler circuit and is more suitable for real-time control.
Mainly, designing a lower-order controller has two H∞ synthesis approaches. One approach is to solve the AREs as introduced in
[14]
[15]
, and the other is to solve the LMIs as introduced in
[16]
[17]
. These synthesis approaches have several common drawbacks. One drawback is that an exhaustive solution search procedure is required to solve the AREs or LMIs. The second drawback is that the closed-loop evaluation cannot be simplified to demonstrate its advantages over the conventional sensorless multi-loop control. Therefore, in this study, a novel H∞ synthesis approach is proposed to design a low-order sensorless controller for boost converters. The proposed H∞ approach need not solve any ARE or LMI, and most importantly, performs a straightforward closed-loop characteristics evaluation by parameterizing the controller with an adjustable parameter that behaves like a “knob” on the dynamic performance.
G_{cl}
(
s
) of the system in Eq. (4) with the controller in Eq. (5), that is, from disturbance
w
to controlled output
z
, is written as follows:
where:
The main objective of robust H∞ controls is to minimize
G_{cl}
(
s
) according to the following H∞ standard:
The basis of the proposed robust H∞ synthesis approach is to decompose the system in Eq. (4) into a special coordinate basis (SCB)
[18]
[21]
. Through SCB decomposition, checking solvability conditions and designing the controller via a step-by-step procedure become easy, as will be presented in the subsequent sections.
_{P}
:= (
A, B, C
_{2}
,
D
_{2}
) and Ʃ
_{Q}
:= (
A, E, C
_{1}
,
D
_{1}
). The solvability conditions of the system in Eq. (4) with the controller in Eq. (5) are as follows:
I) (
A, B
) is stabilizable;
II) (
A, C
_{1}
) is detectable;
III) Ʃ
_{P}
and Ʃ
_{Q}
have no invariant zero on the imaginary axis;
IV) Im(
E
) ⊂
V
^{-}
(Ʃ
_{P}
) +
S
^{-}
(Ʃ
_{P}
) ;
V)
Ker
(
C
_{2}
) ⊃
V
^{-}
(Ʃ
_{Q}
) ∩
S
^{-}
(Ʃ
_{Q}
) .
I and II are the necessary conditions, whereas III, IV, and V are the sufficient conditions.
γ
* of the system in Eq. (4).
Step 3: Set any
γ
>
γ
* and define an auxiliary system in Eq. (8):
where:
Through SCB decomposition of Eq. (4),
P, Q, C
_{1p}
,
D
_{1pq}
,
C
_{2}
_{p}
, and
D
_{2pq}
can be obtained.
Step 4: Transform the system in Eq. (8) to the following form:
Step 5: Design a full state feedback controller
u
=
F_{p}
(
γ, ɛ
)
x
for the following subsystem of the system in Eq. (9):
Step 6: Design a full state feedback controller
u
=
K_{Q}
(
γ, ɛ
)
x
for the following subsystem of the system in Eq. (9):
Step 7: Denote
F_{P}
(
γ, ε
) = [
F
_{P1}
F
_{P2}
] and
K_{Q}
(
γ, ε
) = [
K
_{Q1}
K
_{Q2}
]. A low-order controller in Eq. (5) is expressed as follows:
where:
Symbol
γ
represents the desired disturbance attenuation level satisfying
γ
>
γ
*.
γ
* is the H∞ infimum of the system in Eq. (4) and can be computed as introduced in
[22]
[23]
. Parameter
ε
> 0 is tunable.
ε
* > 0 exists, such that, for all 0 <
ε
<
ε
*, the closed-loop system becomes internally stable and
γ
-suboptimal ║
G_{cl}
(
s
)║∞ <
γ
is satisfied.
where
, and
and
A sensorless controlled boost converter.
Symbol
D
denotes the duty ratio at a given operating point and
D'
= 1 −
D
, and
represents the duty ratio adjustment from the given operating point when a disturbance occurs. Ignoring the equivalent series resistor of the output capacitor,
is equivalent to
. Here,
and
are not scaled by the corresponding dividing resistors in
Fig. 3
for the convenience of evaluating the practical closed-loop characteristics. The coefficients of a controller will be scaled at the execution stage on a digital processor.
The matrices
A, B
, and
E
are the same as that in Eq. (13), and
The measurable output is
, the controlled output is
, and
w_{o}
is the weight on the output voltage, as shown in
Fig. 2
. Then, we denote:
_{P}
:= (
A, B, C
_{2}
,
D
_{2}
) can be obtained as follows:
and
The SCB decomposition of subsystem Σ
_{Q}
:= (
A^{T}, C
_{1}
^{T}, E^{T},D
_{1}
^{T}
) can be obtained as follows:
and
Evidently,
Ker
(
C
_{2}
)
R
^{2}
; thus, solvability condition V is also satisfied. Although the SCB decomposition of subsystem Σ
_{P}
shows a state
x_{b}
, condition IV is unsatisfied, which does not mean that the controller in Eq. (5) for the system in Eq. (15) is unsolvable. It is still solvable; however, a complicated computation of the H∞ infimum
γ
* is required, as will be presented in the subsequent sections.
_{Q}
is
γ_{Q}
* = 0. Thus, the H∞ infimum of the system in Eq. (15) is determined only by subsystem Σ
_{P}
. Referring to
[23]
, for a given
γ
> 0, a positive real symmetric solution
s_{x}
to the ARE in Eq. (18) should exist.
where:
According to the existing condition of the previously presented solution, the H∞ infimum of Eq. (15) can be obtained as follows:
T_{x}
= [0 1; 1 0] on system state
x
is performed to transform Eq. (15) into the form in Eq. (9). Following the design procedure described in subsection III.B, a low-order controller in Eq. (5) can be obtained as follows:
where:
λ
in Eq. (20) can be set to any negative value, for example, −1. Parameter
ɛ
> 0 is tunable.
PARAMETERS OF A BOOST CONVERTER
By setting
λ
= −1, the low-order sensorless controller for the boost converter in
Table 1
is obtained in Eq. (21).
First, we set
w_{o}
= 1 to examine the closed-loop characteristics. From Eq. (19), the H∞ infimum of the system in Eq. (15) is
γ
* = 2.1341. We set
γ
= 2.2; then,
s_{x}
= 0.0053 can be obtained from Eq. (18).
Figs. 4
(a) and
Figs. 4
(b) show the closed-loop audio susceptibility and output impedance, respectively, by substituting
γ
and
s_{x}
into Eq. (21). Lower than approximately
ε
= 1/6,000, the disturbance on the output voltage begins to be attenuated. The disturbance attenuation increases as long as
ε
decreases. Lower than
ε
= 1/100,000, disturbance attenuation will clearly not change, particularly for output impedance.
Bode plots of closed-loop characteristics (w_{o} = 1).
Second, we set
w_{o}
= 5 to augment the effect of disturbance attenuation on the output voltage. Similarly, the H∞ infimum of the system in Eq. (15) is
γ
* = 2.1628. We set
γ
= 2.2; then,
s_{x}
= 0.0022 is obtained. The closed-loop characteristics are shown in
Fig. 5
. Lower than approximately
ε
= 1/3,000, the disturbance on the output voltage begins to be significantly attenuated. Lower than
ε
= 1/20,000, disturbance attenuation will evidently be unchanged.
Bode plots of closed-loop characteristics (w_{o} = 5).
Then, we set
w_{o}
= 10 to further augment the effect of disturbance attenuation on the output voltage. The H∞ infimum of the system in Eq. (15) is
γ
* = 2.1640. We set
γ
= 2.2; then,
s_{x}
= 0.0013 is obtained. The closed-loop characteristics are shown in
Fig. 6
. Lower than approximately
ε
= 1/2,000, the disturbance on the output voltage begins to be significantly attenuated. Lower than
ε
= 1/8,000, disturbance attenuation will evidently be unchanged, which means that the closed-loop characteristics have reached its limit with
w_{o}
= 10; thus, this value can be selected as the suitable value for
ε
.
Bode plots of closed-loop characteristics (w_{o} = 10).
Although a greater
w_{o}
value can further enhance the disturbance attenuation on the output voltage, has been obtained. The closed-loop dynamic responses of the output voltage to a step change in the disturbance are shown in
Fig. 7
for wo = 10.
Step responses of .
For the sensorless controller in Eq. (21), p is equivalent to the estimated inductor current. Through Laplace transformation of Eqs. (21) and (15), transfer functions
are obtained. After the parameters listed in
Table I
are substituted,
and
î_{L}
(
s
)/
î_{o}
(
s
) =
p
(
s
)/
î_{o}
(
s
) are derived. For
w_{o}
= 10 and
ε
= 1/8,000, the step responses of
î_{L}
and
p
are the same (
Fig. 8
), which means that
p
is a complete estimation of the inductor current
î_{L}
.
Step responses of î_{L} and p .
T_{i}
=
F_{m}G
_{4}
and the voltage loop
T_{v}
=
F_{m}F_{v}F
_{2}
+
F_{m}
G
_{5}
F
_{2}
in
Fig. 1
. The overall loop gain at point
T
_{1}
is written as
T
_{1}
=
T_{i}
+
T_{v}
, and the outer loop gain at point
T
_{2}
is written as
T
_{2}
=
T_{v}
/1 +
T_{i}
. From the expressions of
T
_{1}
and
T
_{2}
, the crossover frequency of current loop
T_{i}
should be as high as possible to provide a 90° phase boost for voltage loop
T_{v}
, whereas its gain should be as small as possible at low frequencies. Loop gain
T_{v}
should be as large as possible to attenuate the disturbance on the output voltage.
For the boost converter with parameters listed in
Table I
, the LO in Eq. (1) is written as follows:
where
. The PI controllers in Eq. (23) are used as compensators
F_{m}
and
F_{v}
:
In Eqs. (22) and (23), the parameters to be determined are
L
= [
l
_{1}
l
_{2}
]
^{T}
,
K
_{P1}
,
K
_{I1}
,
K
_{P2}
, and
K
_{I2}
. An iterative trial-and-error design process involves assigning the eigenvalues of
A
−
LC
to determine
L
= [
l
_{1}
l
_{2}
]
^{T}
, where
C
= [0 1]; tuning
K
_{P1}
,
K
_{I1}
,
K
_{P2}
, and
K
_{I2}
by examining the bode plots of loop gains
T
_{1}
=
T_{i}
+
T_{v}
and
T
_{2}
=
T_{v}
/1 +
T_{i}
until a stable control system is maintained; further tuning
K
_{P1}
,
K
_{I1}
,
K
_{P2}
, and
K
_{I2}
by examining the closed-loop characteristics in Eqs. (2) and (3) to obtain good dynamic performance; and repeating these steps until the desired dynamic performance is obtained. Through this iterative process, the best eigenvalues {−0.0093, −7.5003} × 10
^{5}
are determined, and correspondingly,
L
= [0.01 0.75]
^{T}
× 10
^{6}
. After the LO is determined, compensators
F_{m}
and
F_{v}
are used to show the design of compensators
F_{m}
and
F_{v}
:
The bode plots of loop gains
T
_{1}
and
T
_{2}
are shown in
Fig. 9
, with the stability characteristics given in Eq. (25).
Bode plots of T _{1} and T _{2}.
Fig. 10
shows the closed-loop characteristics in Eqs. (2) and (3), and
Fig. 11
shows the step responses of the output voltage. From curves I and II in
Figs. 9
,
10
, and
11
, increasing the gain of inner compensator
F_{m}
can increase gain crossover frequency and decrease peak output voltage; however, recovery time is prolonged. Curves I and III show that increasing the gain of outer compensator
F_{v}
can improve dynamic performance; however, phase margins are reduced. Curves I, II, and III demonstrate that I in Eq. (24) is the best.
Bode plots of closed-loop characteristics.
Step responses of .
The closed-loop characteristics of the inductor current
î_{L}
and estimated inductor current
î_{LO}
can also be easily derived from
Fig. 1
. For the previously obtained LO and compensator I, the step responses of
î_{L}
and
î_{LO}
are shown in
Fig. 12
. The estimated inductor current
î_{LO}
perfectly estimated inductor current
î_{L}
when the input voltage is disturbed. Meanwhile, a slight error occurs between
î_{L}
and
î_{LO}
when the load current is disturbed.
Step responses of î_{L} and î_{LO} .
s
= 2(
z
− 1)/
T_{s}
(
z
+ 1), where
T_{s}
is equivalent to the switching period, the digital counterpart of the controller in Eq. (21) obtained by the proposed H∞ synthesis approach is written in Eq. (26) for
w_{o}
= 10 and
ε
= 1/8,000. The practical dynamic response of output voltage is shown in
Fig. 13
.
Dynamic response of output voltage (H∞ approach).
For LO-based sensorless multi-loop control, the discrete counterpart of Eq. (22) is obtained in Eq. (27) by the zero-hold discretization method. The discrete counterpart of Eq. (23) is obtained in Eq. (28) by the backward difference
s
= 1 −
z
^{−1}
/
T_{s}
.
where
, and
Fig. 14
shows the practical dynamic response by the LO and PI controllers.
Dynamic response of output voltage (LO-based).
Fig. 15
presents the practical experimental environment. A digital 16 bit DSC NJU20010 produced by the NJRC is used to execute the aforementioned digital controllers. The clock frequency of DSC is 62.5 MHz. ADC and PWM are integrated into the DSC. The limit of the duty ratio is set to 0.05–0.88. The slew rates of the load and input voltage are 250 mA/μs and 2.0 V/μs, respectively. A 25 Ω resistor is used as the normal load. An electronic load PLZ164W is used to generate the load current disturbance of 0.8 A. Input voltage is alternated by a switch.
Experimental environment.
γ
. The closed-loop characteristics evaluation becomes straightforward because this parameter is directly related to the closed-loop characteristics.
Figs. 6
,
7
, and
13
show that better dynamic performance is maintained in the proposed H∞ synthesis approach than in the conventional LO-based sensorless multi-loop control.
Xutao Li received his B.S. degree in Automation from Wuhan Technology University, China in 1999 and M.S. degree in Precise Instrument and Machinery from Shanghai Jiaotong University, China in 2006. He worked for several years in Japan. He received his Ph.D. degree in the Graduate School of Information, Production and Systems, Waseda University, Japan in 2016. His research interests include switching mode power suppliers and control theory.
Minjie Chen received his B.S. degree on Electrical Engineering and Automation from Shanghai Jiaotong University, China in 2009 and M.S. degree from the Graduate School of Information, Production and Systems, Waseda University, Japan. He is currently a Ph.D. candidate in the same school. His research interest includes power converter technology.
Hirofumi Shinohara received his B.S. and M.S. degrees in Electrical Engineering and Ph.D. degree in Informatics from Kyoto University in 1976, 1978, and 2008, respectively. He has engaged in the administration of collaborative studies on VLSI circuits between industry and academy in the Semiconductor Technology Academic Research Center. He is currently a professor at the Graduate School of Information, Production and Systems, Waseda University, Japan. His research interests include advanced SRAM, low-power circuits, and variation aware design.
Tsutomu Yoshihara received his B.S. and M.S. degrees in Physics and Ph.D. degree in Electronic Engineering from Osaka University, Osaka, Japan in 1969, 1971, and 1983, respectively. In 1971, he joined the ULSI laboratory of Mitsubishi Electric Corporation, Hyogo, Japan, where he has been engaged in the research and development of MOS LSI memories. Since April 2003, he has been a professor at the Graduate School of Information, Production and Systems, Waseda University, Japan and is currently involved in research on system LSI. He is a member of the IEEE Solid-State Circuits, IEICE of Japan, and Institute of Electrical Engineers of Japan.

I. INTRODUCTION

The controller of a switching converter must guarantee that power conversion is stable under all operating conditions and that the desired dynamic performance is maintained when a disturbance occurs in the input voltage or load. The dynamic performance of a switching converter, whether it has a single-loop or multi-loop control, is determined by its closed-loop characteristics, including audio susceptibility and output impedance
[1]
. For a single-loop output voltage-controlled buck or boost converter in discontinuous conduction mode, stability and dynamic performance can be guaranteed by making the loop gain as large as possible with high crossover frequency and adequate phase and gain margins because loop gain and crossover frequency are directly related to the closed-loop characteristics. Nevertheless, the transfer function has a right-half-plane zero (RHPZ) from the duty ratio to the output voltage for boost, buck–boost, and flyback converters in continuous conduction mode (CCM)
[2]
. RHPZ significantly restricts the crossover frequency of the open-loop gain, which results in poor dynamic performance if single-loop voltage control is adopted. Multi-loop control is extensively adopted to improve dynamic performance. However, a current sampling circuit, such as a shunt resistor with an amplifier, a transformer, or an active filter
[3]
, is required to obtain the inductor or switch current, which causes an increase in cost, size, and weight of the circuit. Sensorless multi-loop control solves these problems by estimating the inductor current
[4]
,
[5]
,
[6]
. The Luenberger observer (LO)
[7]
is effective in estimating the inductor current for the sensorless control. However, evident drawbacks exist, including many parameters to be determined in the LO-based sensorless multi-loop control. Moreover, the closed-loop characteristics are indirectly related to the loop gains for the multi-loop control
[8]
,
[9]
. Consequently, an iterative trial-and-error process is needed to design the LO-based sensorless multi-loop control.
Modern control directly handles the inductor current and capacitor voltage in the time domain. However, a state feedback controller with a state observer, as presented in
[10]
, provides no more benefits to the closed-loop characteristics evaluation than the sensorless multi-loop control. Robust H∞ control, which directly considers disturbance attenuation as the target, provides an approach to designing a compact sensorless controller. Two robust H∞ synthesis methods are mainly adopted in previous studies, namely, algebraic Riccati equation (ARE)-based
[11]
,
[12]
and linear matrix inequality (LMI)-based methods
[13]
. However, these methods have several common drawbacks. One drawback is that an exhaustive search procedure is needed to solve the ARE or LMI; in particular, obtaining a solution for the low-order controller is difficult. The second drawback is that the closed-loop evaluation cannot be simplified to demonstrate its advantages over the conventional sensorless multi-loop control. Moreover, no previous study has introduced the design of a sensorless controller, particularly a low-order controller, which has the advantage of less computational volume or simpler circuit, for switching converters by the robust H∞ control. Therefore, in this study, a novel H∞ synthesis approach is proposed to design a low-order sensorless controller for boost converters. This approach need not solve any ARE or LMI, and most importantly, it parameterizes the controller by an adjustable parameter, which behaves like a “knob” on the dynamic performance. Simulations show a straightforward closed-loop characteristics evaluation and better dynamic performance by the proposed H∞ approach, compared with the LO-based sensorless multi-loop control. Practical experiments on a digital processor confirmed the simulation results.
II. PROBLEM FORMULATION

- A. Difficult Design of Conventional Sensorless Multi-Loop Control of Boost Converters

The topology-independent block diagram of the conventional sensorless multi-loop control of a switching converter is shown in
Fig. 1
. The control system consists of an outer loop
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- B. Problems of Previously Proposed Robust H∞ Control for Boost Converters

The block diagram of the robust H∞ control of a switching converter is shown in
Fig. 2
. The disturbance from the input voltage and load is
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III. INTRODUCTION OF THE PROPOSED ROBUST H∞ SYNTHESIS APPROACH TO THE SENSORLESS CONTROL OF BOOST CONVERTERS

The closed-loop transfer function
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- A. Solvability Conditions of the Proposed Robust H∞ Synthesis Approach

We denote subsystems Ʃ
- B. Design Procedure of a Low-order Controller in Eq. (5)

Referring to
[19]
[20]
[21]
, the design procedure of a low-order controller in Eq. (5) is as follows:
Step 1: Decompose the system in Eq. (4) into SCB and check the solvability conditions.
Step 2: Compute the H∞ infimum
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IV. DERIVATION OF A LOW-ORDER SENSORLESS CONTROLLER FROM THE PROPOSED ROBUST H∞ SYNTHESIS APPROACH FOR BOOST CONVERTERS

- A. Construction of the AC Small Signal Average Value Model of Boost Converters

Fig. 3
shows the boost converter used in this study, in which several parasitic components are considered. Referring to
[24]
, the AC average value model of the boost converter in CCM is written in Eq. (13):
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- B. Construction of the State Space Equation for the Proposed Robust H∞ Control

From the AC small signal model in Eq. (13), the state space equation for the proposed robust H∞ control is written as follows:
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- C. SCB Decomposition of the System in Eq. (15)

The SCB decomposition of subsystem Σ
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- D. Solvability Verification

Proving that solvability conditions I, II, and III in subsection III.A are satisfied is easy. The following expressions can be obtained after SCB decomposition:
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- E. Computation of the H∞ Infimum of the System in Eq. (15)

Through SCB decomposition, the H∞ infimum of subsystem Σ
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- F. Derivation of a Low-order Controller in Eq. (5)

Although the system in Eq. (9) can be constructed through SCB decomposition of the system in Eq. (15), the system in Eq. (15) is already similar to the system in Eq. (9). Therefore, a transformation
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V. SIMULATIONS AND EXPERIMENTS

In this study, a boost converter, with parameters listed in
Table I
is used to show the straightforward closed-loop characteristics evaluation by the proposed robust H∞ synthesis approach, compared with the conventional LO-based sensorless multi-loop control.
PARAMETERS OF A BOOST CONVERTER

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- A. Simulations of the Sensorless Controller Derived from the Proposed H∞ Synthesis Approach

By substituting the parameters in
Table 1
into the system in Eq. (13), the following matrices are obtained:
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- B. Simulations of the Conventional LO-based Sensorless Multi-loop Control

We denote the current loop
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- C. Practical Experiments of the Sensorless Control of the Boost Converter

The previously presented continuous controller should be discretized to execute the controller on a digital processor. Through bilinear transformation
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- D. Summary

For conventional LO-based sensorless multi-loop controls, six parameters should be determined. An iterative trial-and-error process is needed to determine these parameters. Comparatively, the sensorless controller derived from the proposed H∞ synthesis approach has only one adjustable parameter
VI. CONCLUSION

For multi-loop control of a boost converter in CCM, closed-loop characteristics and loop gains are generally indirect. The conventional observer-based sensorless multi-loop control severely aggravates the closed-loop characteristics evaluation because more parameters are related to the closed-loop characteristics. The proposed robust H∞ synthesis approach performs a straightforward closed-loop characteristics evaluation by parameterizing the controller with an adjustable parameter. Simulations show the significant advantages of the closed-loop characteristics evaluation, and practical experimental results confirmed the simulation results. The sensorless controller derived by the proposed H∞ synthesis approach is suitable for boost converters. The proposed H∞ synthesis approach is also suitable for the sensorless controller design of other converters, such as buck–boost and quadratic converters.
BIO

Ahmadi R.
,
Paschedag D.
,
Ferdowsi M.
“Closed-loop input and output impedances of DC-DC switching converters operating in voltage and current mode control,”
in 36th Annual Conference on IEEE Industrial Electronics Society(IECON)
2010
2311 -
2316

Choi H.
“Practical feedback loop design considerations for switched mode power supplies,”
Fairchild Semiconductor Power Seminar
2010-2011
489 -
498

Forghani-zadeh H. P.
,
Rincon-Mora G. A.
“Current-sensing techniques for DC-DC converters,”
in 45th Midwest Symposium on Circuits and Systems
Aug. 2002
Vol. 2
577 -
580

Midya P.
,
Philip T. K.
,
Matthew F. G.
“Sensorless current mode control—an observer-based technique for DC–DC converters,”
in 28th Annual IEEE Power Electronics Specialists Conference(PESC)
Jun. 1997
Vol. 1
197 -
202

Cimini G.
,
Ippoliti G.
,
Orlando G.
,
Pirro M.
“Current sensorless solution for PFC boost converter operating both in DCM and CCM,”
in 21st Mediterranean Conference on Control & Automation(MED)
2013
137 -
142

Beccuti A. G.
,
Mariethoz S.
,
Cliquennois S.
,
Wang S.
,
Morari M.
2009
“Explicit model predictive control of DC–DC switched-mode power supplies with extended kalman filtering,”
IEEE Trans. Ind. Electron.
56
(6)
1864 -
18740
** DOI : 10.1109/TIE.2009.2015748**

Luenberger D. G.
1971
“An introduction to observers,”
IEEE Trans. Autom. Control
16
(6)
596 -
602
** DOI : 10.1109/TAC.1971.1099826**

Rozman A. F.
,
Boylan J. J.
“Band pass current control,”
in Applied Power Electronic Conference and Expression
Feb. 1994
Vol. 2
631 -
637

Ridley R. B.
,
Cho B. H.
,
LEE F. C. Y.
1988
“An analysis and interpretation of loop gains of multi-loop controlled switching regulators,”
IEEE Trans. Power Electron
3
(4)
489 -
498
** DOI : 10.1109/63.17971**

Cho H.
,
Yoo S. J.
,
Kwak S.
2015
“State observer based sensor less control using Lyapunov's method for boost converters,”
IET Power Electron.
8
(1)
11 -
19
** DOI : 10.1049/iet-pel.2013.0920**

Naim R.
,
Weiss G.
,
Ben-Yaakov S.
1997
“H∞ control applied to boost power converters,”
IEEE Trans. Power Electron.
12
(4)
677 -
683
** DOI : 10.1109/63.602563**

Vidal-Idiarte E.
,
Martinez-Salamero L.
,
Valderrama- Blavi H.
,
Guinjoan F.
,
Maixe J.
Oct
“Analysis and design of H∞ control of nonminimum phase-switching converters,”
IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.
50
(10)
1316 -
1323
** DOI : 10.1109/TCSI.2003.816337**

Olalla C.
,
Leyva R.
,
Aroudi A. E.
,
Garces P.
,
Queinnec I.
2010
“LMI robust control design for boost PWM converters,”
IET Power Electron.
3
(1)
75 -
85
** DOI : 10.1049/iet-pel.2008.0271**

Moran A.
,
Hayase M.
“Design of reduced order H∞ controllers,”
in Proc. the 36th International Session Papers(SICE) Annual Conference
1997
1013 -
1018

Yung C. F.
2000
“Reduced-order H∞ controller design: an algebraic riccati euqation approach,”
Automatica
36
(6)
923 -
926
** DOI : 10.1016/S0005-1098(99)00219-8**

Yu J.
,
Sideris A.
“H∞ control synthesis via reduced order LMIs,”
in 36th Conference on Decision & Control
Dec. 1997
Vol. 1
183 -
188

Gahinet P.
1994
“Explicit controller formulas for LMI-based H∞ synthesis,”
Automatica
32
(7)
1007 -
1014
** DOI : 10.1016/0005-1098(96)00033-7**

Berg M. C.
1998
“Introduction to a special coordinate basis for multivariable linear systems,”
IEE Proceedings – Control Theory and Applications
145
(2)
204 -
210
** DOI : 10.1049/ip-cta:19981580**

Saberi A.
,
Chen B. M.
,
Lin Z. L.
1994
“Closed-form solutions to a class of H∞-optimization problems,”
International Journal of Control
60
(1)
41 -
70
** DOI : 10.1080/00207179408921451**

Stoorvogel A. A.
,
Saberi A.
,
Chen B. M.
1994
“A reduced order observer based controller design for H∞-optimization,”
IEEE Trans. Autom. Control
39
(2)
355 -
360
** DOI : 10.1109/9.272333**

Chen B. M.
2010
Robust and H∞ control
Springer

Chen B. M.
,
Saberi A.
,
LY U. L.
1992
“A non-iterative method for computing the infimum in H∞-optimization,”
International Journal of Control
56
(6)
1399 -
1418
** DOI : 10.1080/00207179208934370**

Scherer C.
1990
“H∞-control by state-feedback and fast algorithms for the computation of optimal H∞-norms,”
IEEE Trans. Autom. Contr.
35
(10)
1090 -
1099
** DOI : 10.1109/9.58551**

Marian K. K.
2008
Pulse-width modulated DC–DC power converters
John Wiley & Sons
New Jersey

Citing 'Design of a Low-Order Sensorless Controller by Robust H∞ Control for Boost Converters
'

@article{ E1PWAX_2016_v16n3_1025}
,title={Design of a Low-Order Sensorless Controller by Robust H∞ Control for Boost Converters}
,volume={3}
, url={http://dx.doi.org/10.6113/JPE.2016.16.3.1025}, DOI={10.6113/JPE.2016.16.3.1025}
, number= {3}
, journal={Journal of Power Electronics}
, publisher={The Korean Institute of Power Electronics}
, author={Li, Xutao
and
Chen, Minjie
and
Shinohara, Hirofumi
and
Yoshihara, Tsutomu}
, year={2016}
, month={May}