This paper considers the robust controller design problem for a boost DCDC converter. Based on the TakagiSugeno fuzzy modelbased approach, a fuzzy controller as well as a fuzzy load conductance observer are designed. Sufficient conditions for the existence of the controller and the observer are derived using Linear Matrix Inequalities (LMIs). LMI parameterizations of the gain matrices are obtained. Additionally, LMI conditions for the existence of the fuzzy controller and the fuzzy load observer guaranteeing αstability, quadratic performance are derived. The exponential stability of the augmented fuzzy observercontroller system is shown. It is also shown that the fuzzy load observer and the fuzzy controller can be designed independently. Finally, the effectiveness of the proposed method is verified via experimental and simulation results under various conditions.
1. Introduction
Precise control of DCDC converters is not easy due to the nonlinearities of converters, unavoidable uncertainties, parameter or load variations. To get around this problem various control design methods have been proposed in the literature such as
[1

7]
. Most of the previous control design methods require the good knowledge of the parameter values and the control performances can be severely degraded in the presence of load or parameter variations. On the other hand, the TakagiSugeno (TS) fuzzy modelbased control theory has been successfully applied to control of complex nonlinear or illdefined uncertain systems
[8

12]
.
Considering these facts, a nonlinear control design method for a boost DCDC converter is proposed based on the TS fuzzy approach. This paper first gives a TS fuzzy modelbased control design method. This paper introduces an error vector associated with the desired output voltage, and using this error vector a TS fuzzy error dynamics model of a boost DCDC converter is obtained to construct a fuzzy controller. In the fuzzy error dynamics model for controller design a boost DCDC converter is represented as an average weighted sum of simple local linear subsystems. Local linear controllers are designed for each local subsystem and a global nonlinear controller from the local linear controllers is derived via a standard fuzzy inference method. An LMI condition for the existence of the fuzzy controller is derived and an explicit parameterization of the fuzzy controller gain is obtained in terms of the solution matrices to the LMI condition. LMI existence conditions of the fuzzy speed controller guaranteeing
α
stability or quadratic performance are additionally derived. Secondly, the fuzzy controller method is applied to design a TS fuzzy load conductance observer. It should be noted that the previous fuzzy control systems of
[13

18]
can suffer from lack of systematic and consistent design guidelines to determine design parameters such as fuzzy partition of the input and output spaces, membership function shapes, the number of fuzzy rules
[11
,
18]
. However, in the proposed TS modelbased approach, one can systematically design a fuzzy controller as well as a load observer guaranteeing the asymptotic stability of the closedloop control system. And one can handle various useful convex performance criteria such as
α
stability, quadratic performance. Through simulations and experiments, it is shown that the proposed method can be successfully used to control a boost DCDC converter under load variations and it is very robust to the model parameter variations.
2. System Description
A boost converter shown in
Fig. 1
can be represented by the following nonlinear equation
[3
,
4]
:
where
i_{L}
,
v_{C}
,
u
represent the input inductor current, the output capacitor voltage, the discretevalued control input taking values in the set{0, 1}, and
E
,
L
,
C
,
R
are the external source voltage value, the inductance of the input circuit, the capacitance of the output filter, the output load resistance, respectively.
Topology of boost converter
Following assumptions are used in this research:

A1 :iL,vCare available.

A2 :can be neglected and it can be set as= 0 whereY=R1.

A3 : The inductor current is never allowed to be zero, i.e. the converter is in continuous conduction mode.
By introducing the duty ratio input function
u_{c}
(∙) ranging on the interval [0, 1] and the following error terms
[1
,
3
,
5]
.
the following approximate averaged model can be derived from (1)
where
z
= [
z
_{1}
,
z
_{2,}
,
z
_{3}
]
^{T}
and
V_{r}
is the desired reference output voltage such that
V_{r}
>
E
> 0. After all, our design problem can be formulated as designing a fuzzy controller for the system (2).
Fig. 2
shows a schematic diagram of the control system.
Schematic diagram of proposed control system
3. Fuzzy Controller Design
Based on the TS fuzzy modeling approach
[8

12]
, the model (2) can be approximated by a second order
r_{c}
rule fuzzy model to design a fuzzy controller. The
i
th rule of the TS fuzzy model is of the following form:
Plant Rule i
: IF
x
is
F_{ci}
, THEN
where
F_{ci}
(
i
= 1,∙∙∙,
r_{c}
) denote the fuzzy sets,
r_{c}
is the number of fuzzy rules,
x
= [
i_{L}
,
v_{c}
]
^{T}
,
A_{c}
and
B_{ci}
is given by
Each fuzzy set
F_{ci}
is characterized by a membership function
m_{ci}
(
x
) and the
i
th operating point
x
= (
X
_{1i}
,
X
_{2i}
). Via a standard fuzzy inference method, the following global nonlinear model can be obtained.
where
m_{ci}
:
R
→ [0,1],
i
= 1,∙∙∙,
r_{c}
is the membership function of the model with respect to plant rule
i
,
h_{ci}
can be regarded as the normalized weight of each IFTHEN rule and it satisfies
h_{ci}
(
x
) ≥ 0 and
Let the local controller be given by the following linear controller
Controller Rule i
: IF
x
is
F_{ci}
, THEN
v
=
K_{i} z
where
K_{i}
∈
R
^{1×3}
are gain matrices. Then the final fuzzy controller is given by
and the closedloop control system is given by
Assume that the following LMI condition is feasible
where
P_{c}
∈
R
^{3×3}
and
Y_{ci}
∈
R
^{1×2}
are decision variables.
And assume that the controller gain matrices
K_{i}
are given by
Then there exists a matrix
Q_{c}
> 0 such that
Let us define the Lyapunov function as
V_{c}
(
z
) =
z^{T}X_{c}z
where
Its time derivative along the closedloop system dynamics (6) is given by
which implies that the origin
z
=
0
is exponentially stable.
Theorem 1
Assume that the LMI condition (7) is feasible for (
P_{c}
,
Y_{ci}
) and the gain matrices
K_{i}
are given by (8). Then,
x
converges exponentially to zero.
Remark 1
The LMI parameterization of the controller gain enables one to handle various useful convex performance criteria such as
α
stability, quadratic performance, and generalized
H
_{2}
/
H
_{∞}
performances
[12
,
19]
. For example, if the controller gain is set as
K_{i}
of (8) with
P_{c}
and
Y_{ci}
satisfying for some
α_{c}
> 0
then by referring to (9) the following can be obtained
which implies that
x
converges to zero with a minimum decay rate
α_{c}
. On the other hand, if the controller gain is set as
K_{i}
of (8) with
P_{c}
and
Y_{ci}
satisfying
Then the followings can be obtained
where the Shur complement lemma of
[19]
is used. After all, it can be seen that the fuzzy controller (5) guarantees the quadratic performance bound constraint
4. Fuzzy Load Conductance Observer Design
In this section, a fuzzy observer to estimate the load conductance
Y
=
R
^{1}
will be designed. By applying the TS fuzzy modeling methods
[9
,
10]
to (2) or the equivalent error dynamics (2), the boost converter and the dynamics of
, can be approximated by a second order
r_{o}
rule fuzzy model. The
i
th rule of the TS fuzzy model is of the following form:
Plant Rule i
: IF
v_{c}
is
F_{oi}
, THEN
where the assumption A2 is used,
F_{oi}
(
i
= 1,∙∙∙,
r_{o}
) denote the fuzzy sets,
r_{o}
is the number of fuzzy rules,
x_{o}
= [
v_{c}
,
Y
]
^{T}
is the state,
y_{o}
=
v_{c}
is the output,
,
u_{o}
=
i_{L}
(1 −
u
)/
C
, and
Each fuzzy set
F_{oi}
is characterized by a membership function
m_{oi}
(
v_{c}
) and the
i
th operating point
V_{i}
. Via a standard fuzzy inference method, the following global nonlinear model can be obtained :
where
m_{oi}
∶
R
→ [0,1],
i
= 1,∙∙∙,
r_{o}
is the membership function of the system with respect to plant rule
i
,
h_{oi}
is the normalized weight of each IFTHEN rule and it satisfies
h_{oi}
(
v_{c}
) ≥ 0 and
Let the local observer given by the following linear observer
Observer Rule i
: IF
v_{c}
is
F_{oi}
, THEN
where
L_{i}
∈
R
^{2×1}
are gain matrices,
Then the final fuzzy observer induced as the weighted average of the each local observer is given by
which gives the following error dynamics.
where
Theorem 2
Assume that the following LMI condition is feasible for (
P_{o}
,
Y_{oi}
)
where
P_{o}
∈
R
^{2×2}
,
Y_{oi}
∈
R
^{2×1}
are decision variables. And assume that the observer gain
L_{i}
is given by
Then, the estimation error converges exponentially to zero.
Proof:
Assume that (15) is feasible. Then there exists a matrix
Q_{o}
> 0 such that
Let us define the Lyapunov function as
Its derivative with respect to time is given by
which implies that
is exponentially stable.
Remark 2
The LMI parameterization of the observer gain (16) also provides some degrees of freedom which can be used to handle various useful convex performance criteria such as
^{α}
stability, quadratic performance, and generalized
H
_{2}
/
H
_{∞}
performances
[12
,
19]
. For example, if the observer gain is set as
L_{i}
of (16) with
P_{o}
and
Y_{oi}
satisfying for some
α_{o}
> 0
then
converges to zero with a minimum decay rate
α_{o}
. On the other hand, if the observer gain is set as
L_{i}
of (16) with
P_{o}
and
Y_{oi}
satisfying for some
Q_{o}
≥ 0
Then the followings can be obtained
After all, it can be easily shown that the fuzzy observer (14) guarantees the quadratic performance bound constraint
5. Separation Property and Design Algorithm
This section illustrates the exponential stability of the augmented control system containing the fuzzy controller and the fuzzy load observer. The following theorem implies that the separation property holds.
Theorem 3
Assume that the LMIs (7) and (15) are feasible, and the controller (5) is replaced with the following load observerbased control law
where
and
is the estimated output conductance via the fuzzy observer (14). Then
z
and
converge exponentially to zero.
Proof
: It should be noted that because
the vector
can be rewritten as
where
Let us define the Lyapunov function as
where
η
is a sufficiently large scalar,
P_{c}
and
P_{o}
satisfy the LMIs (7) and (15). Its derivative with respect to time is given by
where
If
η
is large enough to guarantee
, then
for all
This proves the exponential stability of
Remark 3
From the standard results
[19]
, it can be shown that if one of the pairs (
A_{c}
,
B_{ci}
) is not stabilizable then the LMI condition (7) is not feasible. And it can be easily shown that (
A_{c}
,
B_{ci}
) are stablilzable as long as
X
_{1i}
≠ 0 or
X
_{2i}
≠ 0. It can be also shwon that if one of the pairs (
A_{oi}
,
C_{o}
) is not detectable then the LMI condition (7) is not feasible. And it can easily shown that (
A_{oi}
,
C_{o}
) are detectable as long as
V_{i}
≠ 0. These facts imply that the LMI condition (7) and (15) is always feasible for an appropriately chosen set of the operating points.
Remark 4
Theorems 12 imply that our design problem is a simple LMI problem which can be solved very easily via various powerful LMI optimization algorithms. Theorem 3 implies that the controller gains and the load conductance observer gains can be independently designed. And Remark3 implies that our design problem is always feasible for an appropriately chosen set of the operating points. Our results can be summarized as the following LMIbased design algorithm.

[Step1] Choose an appropriate set of {V1,∙∙∙,Vro} and obtain the fuzzy model (13).

[Step2] Solve the LMIs (15), obtain the gain matricesLi, and construct the fuzzy observer (14).

[Step3] Choose an appropriate set of and obtain the fuzzy model {(X11,X21),∙∙∙, (X1rc,X2rc)} and obtain the fuzzy model (4).

[Step4] Solve the LMIs (7), obtain the gain matricesKi, and construct the load observerbased fuzzy control law (20)
Remark 5
Via extensive numerical simulations and experimental studies, it has been found that fuzzy models with r
_{o}
= 2 and r
_{c}
= 2 are enough to obtain load observerbased fuzzy control laws with satisfactory performances. As can be seen in the next section a tworule fuzzy model (13) with the following normal membership functions is enough to design a fuzzy load observer with satisfactory performances
where
ε_{oi}
> 0 . A fuzzy controller with satisfactory performances can also be obtained by using a fuzzy model (4) with
r_{c}
= 2 and
where
ε_{ci}
> 0, and the nominal values of
E
,
L
,
C
,
R
are used.
Remark 6
To design a fuzzy load observer under the several performance specifications, one has only to gather LMI conditions corresponding to each design performance specification, and form a system of LMIs as a subset of (15), (18), (19), and solve the system of LMIs under the assumption that the Lyapunov matrices ‘
P_{o}
’ are common. Similarly, one can design a fuzzy controller under the several performance specifications.
6. Simulation and Experiment
Consider a boost converter (1) with
L
=200[uH],
C
= 220[uF],
v_{c}
= 20[V], the PWM switching frequency 60[kHz]. Assume that the nominal load resistance is
R_{O}
= 100[Ω] and desired reference output voltage
V_{c}
is
V_{c}
= 20[V]. The parameters values used for the simulations and experiments are summarized in
Table 1
. Let us first design a fuzzy observer guaranteeing the minimum decay rate
α_{o}
= 10. Here, the following tworule fuzzy model to design a fuzzy observer is used.
Utilized components and parameters
Utilized components and parameters
Plant Rule 1
: IF
v_{c}
is about
V_{r}
, THEN
Plant Rule 2
: IF
v_{c}
is about 0.5
V_{r}
, THEN
where
And
h
_{o1}
=
m
_{01}
/(
m
_{01}
+
m
_{02}
),
h
_{o2}
=
m
_{02}
/(
m
_{01}
+
m
_{02}
),
m
_{o1}
=
e
^{(vcvr)2}
,
m
_{o2}
=
e
^{(vc0.5vr)2 }
. By solving (18) with
α_{o}
= 100 the following fuzzy observer (14) with the following gain
Now, let us design a fuzzy controller guaranteeing the minimum decay rate
α_{o}
= 20. In order to design a fuzzy controller, the following tworule fuzzy model is used.
Plant Rule 1
: IF (
i_{L}
,
v_{c}
) is about (0.4,
V_{r}
), THEN
Plant Rule 2
: IF (
i_{L}
,
v_{c}
) is about (0.4, 0.25
V_{r}
), THEN
where
and
h
_{c1}
=
m
_{c1}
/(
m
_{c1}
+
m
_{c2}
),
h
_{c2}
=
m
_{c2}
/(
m
_{c1}
+
m
_{c2}
),
m
_{c1}
=
e
^{(vcvr)2}
,
m
_{c2}
=
e
^{(vc0.25vr)2}
. By solving (10) with
α_{c}
= 100 , the following controller gain can be obtained
After all, the following observerbased fuzzy controller can be obtained
where
Figs. 3
and
4
show the proposed fuzzy load observer and fuzzy controller, respectively.
Block diagram of the proposed fuzzy load observer.
Block diagram of the proposed fuzzy controller.
A conventional cascade PI controller shown in
Fig. 5
is also considered for performance comparisons. The gains of the above PI controller are designed based on the method
[21]
. The P and I gains of the voltage PI controller are
K_{pV}
=0.1 and
K_{iV}
=4. The P and I gains of the current PI controller are
K_{pI}
=0.8 and
K_{iI}
=1.
Fig. 6
illustrates the Matlab/Simulink simulation model of the proposed controller system.
PI control block diagram
Simulation model of the proposed control system implemented with Simulink.
In order to verify the effectiveness of the proposed method, the following four cases are considered :

C1) The input voltageEchanges from 10[V]→15[V]→10[V] while the load resistorRis constant at the nominal valueR=100 [Ω].

C2) The input voltageEchanges from 10[V]→5[V]→10[V] while the load resistorRis constant at the nominal valueR=100 [Ω].

C3) The reference voltageVrchanges from 20[V]→13[V]→20[V] whileEandRare kept constant atE=10 [V] andR=100 [Ω].

C4) The load resistorRchanges from 100[Ω]→20[Ω]→100[Ω] whileVrandEare kept constant atVr=20 [V] andE=10 [V].
Fig. 7
shows the time responses under the case C1.
Fig. 7(a)
shows the time histories of
E
,
v_{c}
, and output current
i_{o}
by the conventional PI control method. The PI gain values are computed based on the methods given in
[1

2]
.
Fig. 7(b)
depicts the time histories of
E
,
v_{c}
, and output current
i_{o}
by the proposed load observerbased fuzzy controller (23).
Fig. 8
shows the time responses under the case C2.
Fig. 8(a)
shows the time histories of
E
,
v_{c}
, and output current
i_{o}
by the conventional PI control method. The PI gain values are computed based on the methods given in
[1

2]
.
Fig. 8(b)
depicts the time histories of
E
,
v_{c}
, and output current
i_{o}
by the proposed load observerbased fuzzy controller (23).
Fig. 9
shows the time responses under the case C3.
Fig. 9(a)
shows the time histories of
E
,
v_{c}
, and output current
i_{o}
by the conventional PI control method. The PI gain values are computed based on the methods given in
[1

2]
.
Fig. 9(b)
depicts the time histories of
E
,
v_{c}
, and output current
i_{o}
by the proposed load observerbased fuzzy controller (23).
Fig. 10
shows the time responses under the case C4.
Fig. 10(a)
shows the time histories of
E
,
v_{c}
, and output current
i_{o}
by the conventional PI control method.
Fig. 10(b)
depicts the time histories of
E
,
v_{c}
, and output current
i_{o}
by the following load observerbased fuzzy controller.
Figs. 7
,
8
,
9
and
10
imply that our method gives a faster recovery time as well as a less overshoot.
Simulation results under C1.
Simulation results under C2.
Simulation results under C3.
Simulation results under C4.
The conventional PI control algorithm as well as our method is implemented on a Texas Instruments TMS 320F28335 floatingpoint DSP. A Tektronix TDS5140B digital oscilloscope is used to measure and plot the signals
v_{C}
,
i_{L}
,
E
.
Fig. 11
shows the circuit scheme of power stage.
Fig. 12
illustrates the experimental setup.
Figs. 13
,
14
,
15
and
16
show the experimental results. It can be seen that our method outperforms the conventional PI method.
Power stage circuit scheme of the boost converter
Experimental setup.
Experimental results under C1.
Experimental results under C2.
Experimental results under C3.
Experimental results under C4.
7. Conclusion
A simple fuzzy load observercontroller design method was proposed for a boost converter under an unknown load resistance. Explicit parameterizations of the fuzzy controller gain and the fuzzy load conductance observer gain were given in terms of LMIs. LMI existence conditions of the fuzzy controller and the fuzzy observer guaranteeing
α
stability or quadratic performance were also derived. Finally, the robust performance of the proposed method was verified via numerical simulations and experiments.
Acknowledgements
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2014R1A2A1A11049543). This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology under Grant 2012R1A1A2001439.
BIO
SangWha Seo received the B.S. degrees in electronic engineering from Seokyeong Univ., Seoul, Korea, in 2007 and M.S. degrees in electrical engineering from Dongguk Univ. Seoul, Korea, in 2010. Since 2013, he has been with Dongguk university research institute for industrial technology. He is currently working toward the Ph.D. degree in the Div. of EEE, Dongguk Univ.Seoul.
Han Ho Choi received the B.S. degree in Control and Instrumentation Eng. from SNU, Seoul, Korea, and the M.S. and Ph.D. degree in Electrical Engineering from KAIST in 1988, 1990, and 1994, respectively. He is now with the Div. of EEE, Dongguk Univ.Seoul.
Yong Kim received the B.S., M.S. degrees and Ph.D degrees in Department of Electrical Engineering from the Dongguk University in 1981, 1983, and 1994. Since 1995 He has been a professor in the Div. of EEE, Dongguk Univ. Seoul. his teaching and research interests include switch mode power supply and electrical motor drives.
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