Frequency / Duty Cycle Current-Mode Fuzzy Control
for LCC Resonant Converter
Manli Hu, Joachim Böcker, Norbert Fröhleke
DEPARTMENT OF POWER ELECTRONICS AND ELECTRICAL DRIVES (LEA),
PADERBORN UNIVERSITY
Warburger Str. 100, D-33098 Paderborn, Germany
Tel: +49/(0)5251-60-3145
Fax: +49/(0)5251-60-3443
E-Mail: {hu, boecker, froehleke}@lea.upb.de
URL: http://wwwlea.uni-paderborn.de
Acknowledgements
The authors would like to thank European Commission for funding this project.
Keywords
«Fuzzy control», «Adaptive control», «Resonant converter»
Abstract
A novel current-mode nonlinear fuzzy controller with adaptive gain is developed for the LCC resonant
converter applied in very low frequency high voltage generators using switching frequency and duty
cycle as control variables. The designed fuzzy controller is verified through simulations. Comparison
between the fuzzy controller and a standard linear controller indicates the correctness and
effectiveness of the novel approach.
I. Introduction
Mobile very low frequency (VLF) high-voltage (HV) generators are required for examination of high
voltage cables during commissioning or services. Such a test generator has to provide the full voltage
of some tens to hundreds of kV at a relatively small power of few to tens of kW. Compared with
50 Hz-voltage test systems, the VLF-generators have considerable advantages in terms of volume,
weight and energy consumption [1]. The well-known LCC resonant converter is an attractive solution
for such VLF HV generators due to its large voltage conversion ratio, natural soft-switching feature,
preferable wide load and voltage operating region and ability to make use of parasitics such as leakage
inductance, winding capacitance of HV-transformer and capacitors for controlling the static and
dynamic voltage stress across HV-rectifiers [2] [3] [4]. In high-voltage DC applications, the
Cockcroft-Walton (CW) multiplier is usually adopted, which shows lower losses and size as an
equivalent transformer if the CW stage number is limited to 3-5 [5]. As an example, Fig. 1 shows a
circuit diagram of the LCC resonant converter: the zero-voltage switched LCC circuit is connected
Fig. 1 Circuit diagram of high-voltage generator with LCC resonant converter [3]
1
with a symmetrical three-stage CW multiplier via a step-up transformer.
Based on the derived large-signal and small-signal models of the LCC resonant converter, a
conventional linear PI controller was developed and implemented in prototypes due to its simple
structure and ease of design [3] [6]. Since the LCC resonant converter is an evident nonlinear system,
the conventional linear PI controller has limitations on its control performance. As an alternative, a
novel current-mode fuzzy proportional-derivative (PD) controller with adaptive gain is developed for
the LCC resonant converter in this paper. As another novelty, due to the wide range of output voltage
and load, and the allowed operation margins of switching frequency fs and duty cycle d, both of them
are employed as the control variables and integrated in the fuzzy control strategy.
This paper is organized as follows: Section II briefly introduces the characteristics of the LCC
resonant converter and the limitations using standard linear PI control. Section III represents a currentmode fuzzy PD control design, including the introduction of rule base, membership functions and
control block diagram. Section IV shows simulation results of the fuzzy control in comparison with a
conventional linear control. Section V summarizes main results of the paper and gives an outlook.
II. Limitation of Linear PI Control
The steady-state voltage gain of the LCC resonant converter (Fig. 1) is shown in Fig. 2. The voltage
gain M depends both on switching frequency fs and duty cycle d. However, due to specified output
voltage range, load range and other operating limits, the converter is operated only within the shaded
operating region that is spanned by boundary operating points OP1-OP8. The voltage conversion ratio
M, normalized switching frequency fsn, characteristic impedance Z and the quality factor Q of the
resonant tank are also denoted in Fig. 2 (the capital letter indicates the steady-state). Among which, n
is transformer turns ratio, k is stage number of CW-generator, f0 is the natural resonant frequency, and
R0 is the equivalent resistor in each steady-state operating point (refer to [3] for details).
The large-signal and small-signal models of the LCC resonant converter were derived in previous
papers [3] [6], which are used for the design of a conventional linear PI controller. However, since the
VLF HV generator is naturally a nonlinear system, the linear controller has inevitable drawbacks such
as lower dynamics and variable control bandwidth for different operation points. Fig. 3 shows the loop
gains of the boundary operating points with linear PI controller: the control bandwidth with frequency
control for different Q is nearly the same: fn=0.01 (fn is the normalized modulation frequency
regarding f0); while the control bandwidth with duty cycle control are different for various Q, the
Fig. 2 Voltage gain M LCC resonant converter
left: M vs. normalized switching frequency f sn = f s / f 0
right: M vs. duty cycle d:
M=
Uo
2 ⋅ n ⋅ k ⋅ Uin
,
f0 =
1
Cs ⋅ C p
Ls
R0
Z=
Cg =
2π Ls Cg
Cg Q = Z
Cs + C p
,
,
,
2
Fig. 3 Loop gains of boundary operating points with linear PI controller
(left) OP1-OP3 with frequency and (right) OP5-OP7 with duty cycle control
maximal crossover frequency is nearly same as fn=0.7, while the minimal is fn=0.02. It can be
concluded that with such small-signal characteristics, a conventional linear controller cannot get
satisfying performance for a wide operating range.
Some nonlinear control scheme is expected to yield enhancement. Fuzzy logic control is successfully
implemented over the past two decades in many industrial control applications and has been proven to
be a successful control approach to many complex nonlinear systems. Considering the simple structure
and easy design of linear control, it is reasonable to combine fuzzy logic control and conventional
linear control together to design a nonlinear controller with simple structure and high efficiency.
It is well known that an inner current-mode control is beneficial to cancel the gain variation of the
outer voltage loop [4]. As a result, a current-mode fuzzy proportional-derivative (PD) control using
switching frequency fs and duty cycle d is developed in this paper. This control approach is further
supplemented by an adaptive scaling gain with respect to the reference resonant effective current in
order to get improved control performance for various operating points.
III. Current-Mode Fuzzy PD Control
The basic structure of a fuzzy control system applied in LCC resonant converter is shown in Fig. 4.
The fuzzy controller consists of four conceptual components: knowledge base, inference mechanism,
fuzzification interface and defuzzification interface. The knowledge base contains all the controller
knowledge and it comprises a fuzzy control rule base and a data base. The data base is the declarative
part of the knowledge base which describes definition of objects and definition of membership
Fig. 4 Basic structure of fuzzy control system applied in LCC resonant converter
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functions. The fuzzy control rule base is the procedural part of the knowledge base which contains
information on how these objects can be used to infer new control actions. The inference mechanism
is a reasoning mechanism which performs inference procedure upon the fuzzy control rules and gives
conditions to derive reasonable control actions, which is the central part of a fuzzy control system. The
fuzzification interface (or fuzzifier) defines a mapping from a real-valued space to a fuzzy space, and
the defuzzification interface (or defuzzifier) defines a mapping from a fuzzy space over an output
universe of discourse to a real-valued space [7].
As a model-free control, the knowledge base of fuzzy control is mostly dependent on a good
understanding of the behaviour of the LCC resonant converter. From [3] and [4], the low frequency
gain of control-to-output transfer functions of LCC resonant converter: Gf0 (regarding fsn) and Gd0
(regarding d) are proportional to the respective slope of the DC conversion ratio curve at the given
operating point. From Fig. 2, it is clear that in preferable operating region, Gf0 is negative and Gd0 is
positive. Also, due to the similar change pattern of the gain of control-to-resonant-current transfer
function and that of control-to-output-voltage transfer function, the following formulas can be
obtained:
Gf 0 ∝
∂i
∂M
∝ L <0
∂f sn ∂f sn
(1)
Gd 0 ∝
∂M ∂iL
∝
>0
∂d
∂d
(2)
Based on above principle and practical experiences, the corresponding rule base for current-mode
fuzzy control is represented in Table I. The error signal e(t) is the difference between the reference
resonant effective current iL_eff* and the real resonant effective current iL_eff; its derivative is de(t)/dt,
denotes the change-in-error. Both of them are selected as the fuzzy controller inputs. The ranges of e(t)
and de(t)/dt are divided into nine subsections and indicated as “0” to “8”, respectively. Among which,
“0” corresponds to negative maximal error or negative maximal error derivative; “4” corresponds to
zero error or zero error derivative; “8” corresponds to positive maximal error or positive maximal
error derivative, respectively. According to e(t) and de(t)/dt, the corresponding fuzzy logic control
actions u(t) can be generated, which is also divided into nine parts, as “0” denotes zero control output
and “8” denotes the maximal control output. For example, as indicated in Table I, negative maximal
(0) e(t) and negative maximal (0) de(t)/dt will generate minimal (0) u(t), and positive maximal (8) e(t)
and positive maximal (8) de(t)/dt will generate maximal (8) u(t).
The corresponding membership functions of e(t), de(t)/dt and u(t) are given in Fig. 5. Each of them
adopts triangle form and is decomposed in nine sections. The nine subsections of the membership
functions marked with 0 ~ 8 correspond with the nine subsections 0 ~ 8 in Table I, respectively. The
peak of each triangle has certainty ratio “1”, while the bottom of each triangle has certainty ratio “0”.
The centre value of each triangle for e(t), de(t)/dt and u(t) are also given in Fig. 5. For example, the
triangle centre value of e(t) is specified as -4A, -3A, -2A, -1A, 0A, +1A, +2A, +3A, +4A, respectively.
In order to regulate the scaling, e(t) and de(t)/dt have their respective scaling gains: ke and kse. Studies
show that fuzzy controller with constant ke have poor control performance considering different
operating points, such as large steady-state error with small reference resonant effective current iL_eff*,
or obvious ripple for large reference effective current iL_eff*. In order to solve this problem, an adaptive
gain ke is developed according to iL_eff*.
Since u(t) is an intermediate fuzzified controller output, it should be further transferred to real control
variables: fs(t) and d(t). According to Eq.(1), Eq.(2) and Table I, it can be concluded that d(t) has a
positive proportional relationship with u(t), while fs(t) has a negative proportional relationship with
u(t). The corresponding transition diagram between u(t) and fs(t), d(t) can be obtained, as shown in
Fig. 6.
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Table I Rule base of fuzzy PD controller
0
1
2
error e(t)
3 4 5
6
7
8
0
0
0
0
0
1
2
2
3
8
1
0
0
1
1
2
2
3
4
8
2
0
1
2
2
2
3
4
5
8
3
4
5
6
7
0
0
0
0
0
1
2
2
3
4
2
2
3
4
5
2
3
4
5
6
3
4
5
6
6
4
5
6
6
7
5
6
6
6
7
6
6
7
7
8
8
8
8
8
8
8
0
5
6
6
7
8
8
8
8
output u(t)
error
derivative
de(t)/dt
Fig. 5 Membership functions of inputs and output of the fuzzy PD controller
The resulting current-mode fuzzy PD control diagram is depicted in Fig. 6: it comprises a cascaded
two-closed-loop control for the LCC resonant converter. The outer is a conventional linear control,
which regards the output voltage uo as the control object. This content will not be discussed here. The
inner is the current-mode fuzzy control loop, which regards the resonant effective current iL_eff as the
control object, which is the emphasis of this paper. As shown in Fig. 6, the reference current iL_eff*
compares with the measured current iL_eff, the difference between them is the error signal e(t), which is
further deduced for its derivative de(t)/dt. After regulation by adaptive scaling gain ke and constant
scaling gain kse, the error signal e(t) and derivative signal de(t)/dt are fed into the fuzzy controller.
According to the membership functions, the fuzzification procedure is executed and both inputs in
real-valued space are transferred to a fuzzy space. Next, referring to the rule base, the inference
mechanism is activated and the reasonable control actions are derived. Adopting the “centre of gravity
(COG)” defuzzification method, the fuzzy controller output u(t) is generated from a fuzzy space to a
real-valued space, which is then further transferred to actual control variables: fs(t) and d(t). As
denoted in Fig. 6, with a specified modulation limit, fs(t) is negative proportional to u(t) with a positive
5
Fig. 6 Current mode fuzzy control block diagram of LCC resonant converter
offset, while d(t) is positive proportional to u(t) without offset. The generated fs and d are provided to a
pulse width modulator, the resulting transistor gate drive signals are given to transistors of the fullbridge inverter. The resonant effective current iL_eff is measured and feedback via appropriate filter and
compared with the reference signal iL_eff* to complete the current-mode fuzzy closed-loop.
IV. Simulation Studies
In order to ensure the correctness and effectiveness of the current-mode fuzzy PD controller, some
simulation and comparison are executed. As shown in Fig. 7, the blue curve is the reference sinusoidal
resonant effective current iL_eff* with magnitude from 1 A to 19 A with increasing frequency from
10 Hz to 100 Hz in 0.14 s, while the red curve is the measured resonant effective current iL_eff from
LCC resonant converter in MATLAB. From Fig. 7, it can be concluded that the plant resonant current
agrees well with the reference current in the complete magnitude range and frequency range besides a
little spikes at peak reference value.
Fig. 8 shows the simulation results with a square form reference resonant current. The blue is the
referenced 50 Hz pulsing iL_eff* with magnitude from 4 A to 19 A, while the red is the measured
simulated resonant current iL_eff with the developed fuzzy PD controller. It can be seen that the red
follows the step-change blue in about 2 ms without overshoot. The exclusive shortage is that there
exists some vibration (about 5%) around the steady-state large reference current. But since a cascaded
two-closed-loop is adopted for the LCC resonant converter, such vibrated signal can be attenuated
through the additional outer voltage controller. Such estimation should be verified through later study
and experiment. In order to show the advantages of the fuzzy PD controller, a comparison between the
fuzzy controller and a linear PI controller is implemented and the results are also shown in Fig. 8.
Among which, the light blue is the measured resonant effective current iL_eff with a conventional PI
controller. From the comparison, it can be seen that the plant resonant effective current iL_eff with the
fuzzy controller has smaller ripple than that with a linear PI controller, which demonstrates the
benefits of the novel fuzzy control approach.
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Fig. 7 Referenced sinusoidal increasing frequency current and plant response with fuzzy controller
Fig. 8 Referenced pulse current and plant response with fuzzy and linear controller
V. Conclusions
A novel current-mode fuzzy PD controller with adaptive gain is developed for the LCC resonant
converter applied in very low frequency high voltage generators using switching frequency and duty
cycle control variables. Through simulation studies, the developed fuzzy control shows satisfying
performance for various operating points. The comparison between the fuzzy PD control and a
conventional linear PI control shows performance improvement with smaller ripple of the novel fuzzy
control approach.
The advantage of the developed fuzzy controller is not so dominant when compared with a standard
linear controller. In the other aspect, such model-free fuzzy control always suffers from criticism of
lacking of systematic design with consistent and guaranteed performance. Due to such shortage, a
model-based fuzzy control with systematic stability analysis and controller design is expected in the
future.
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