Contoured Bode Plot based Robust Decentralised Controller for Three-
Input Integrated Dc-Dc Converter
M. MANOGNA, B. AMARENDRA REDDY, KOTTALA PADMA
Department of Electrical Engineering,
Andhra University College of Engineering,
Visakhapatnam, Andhra Pradesh,
INDIA
Abstract: - Designing controllers for multi-input multi-output (MIMO) integrated Dc-Dc converter is
complicated due to shared elements, integrated structure, and relation between the input and output variables of
the converter. In this work, a robust PID controller based on a Contoured Robust Controller Bode Plot
(CRCBP) is designed for control of the three-input integrated Dc-Dc (TIID) converter. This method combines
robust control with classical loop-shaping. In this procedure, the outlines of the robust metric are drawn on the
Bode charts of the controller, and the controller is adjusted till its frequency response does not cross the
contours of the robust metric to meet the stability and performance goals. The TIID converter is modeled using
state-space analysis and a Transfer Function Matrix (TFM) is acquired from the small signal continuous time
model. The interactions between the inputs and outputs of the converter are quantified and input-output pairing
is identified by Relative Gain Array (RGA). The input-output pairing suggested by RGA decides the controller
structure. Further, the weight functions (loop-shaping filters) are designed based on the TFM which represents
the desired robustness and performance of the controller. These weight functions are used to define the robust
metric for the controller design. Based on this, the CRCBP controller is designed iteratively. A standard TIID
converter of power rating 288 W with input voltage levels of 24V, 30V, and 36V is considered to show the
effectiveness of the proposed controller under varying operating conditions. The real-time simulation results
disclose the proposed controller's superiority over the existing approaches in the literature.
Key-Words: - Hybrid Electric Cars, Renewable Energy Systems, Contoured Robust Controller Bode Plot
(CRCBP), Three-Input Integrated Dc-Dc (TIID) converter, state-space modeling, small-signal
analysis, Transfer Function Matrix (TFM), multivariable PID controller.
Received: April 22, 2024. Revised: October 25, 2024. Accepted: November 11, 2024. Published: December 16, 2024.
1 Introduction
It has been demonstrated that switched-mode
MIMO converters are more adaptable, affordable,
dependable, and efficient than single-input single-
output (SISO) converters, [1], [2] and [3]. MIMO
converters are used in the design of power
electronic applications such as hybrid electric cars,
[4], locomotives, and other systems, [5], powered
by renewable energy sources, [6], [7]. Therefore,
building multi-variable PID controllers for MIMO
converters is more complex than for SISOs because
of the integrated structure, shared components, and
interactions between the input and output variables
of the converter. With just three tuning parameters
and a multitude of accessible methods, designing a
SISO PID is quite easy, [8]. However, compared to
a SISO scenario, the MIMO system has a
significantly higher number of variables due to an
increase in the process's control inputs and outputs.
This makes the problem more difficult to solve.
Numerous studies have also been conducted on
MIMO PID controllers. Even though PIDs make up
over 90% of the controllers used in the market,
MIMO PID controller design still has a lot of
issues. Therefore, to construct MIMO PID
controllers with improved performance, effective
tuning techniques must be developed.
The recent research reports on many
approaches to constructing resilient MIMO PID
controllers for MIMO systems. For a two-input dc-
dc converter system, a decentralized controller is
designed using the effective transfer function
method, [9], while a centralized controller is
created using the equivalent transfer function
methodology, [10]. For the two-input Buck-SEPIC
dc-dc converter system, diagonal controllers are
recommended using individual channel design
(ICAD), [11]. A decoupler network is designed for
a three-port dc-dc converter suitable for a satellite
application to minimize control-loop interactions,
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[12]. An interaction independent robust controller
for a two-input fourth-order integrated (TIFOI) dc-
dc converter is designed using the Loop Shaping
design approach, [13]. Even if there are several
methods for developing a TIID converter’s
controller, they nonetheless have drawbacks like:
(i) Equivalent and effective transfer function
methodologies require SOPDT models, [14],
(ii) ICAD needs an initial controller, [15],
(iii)
H
loop shaping design procedure requires
uncertainty representation in co-prime factor
form, [16], [17]
(iv) Design of a suitable decoupling network is a
tedious process, [18].
To address these limitations, the CRCBP
method is implemented in [19]. This approach
combines robust control with classical loop-
shaping, [20], [21]. Using this procedure, the
outlines of the robust metric are drawn on the Bode
charts of the controller. The controller is adjusted
till its frequency response does not cross the
contours of the Robust Performance Metric (RPM)
to meet the stability and performance goals.
Novelty of the work:
The CRCBP approach provides a novel
controller design method that permits finite
structured uncertainty and is based on the simple
H
- norm, without raising the amount of
complexity in the design process, [19]. The
available literature only provides controller design
for single-input and dual-input dc-dc converters,
[19], [21]. This paper attempts to implement
CRCBP based multivariable controller for TIID
converter. The detailed iterative controller design
process is explained along with forbidden regions
of RPM contours. At each stage of the iterative
process, the relation between contoured plots and
sensitivity plots is graphically shown. All the PI
controllers in the iteration process that satisfy and
violate RPM criteria and their relation with
sensitivity function are explained graphically.
Contribution of the paper:
The following contributions are made from this
work:
(i) A fourth-order TIID converter is proposed in
[22]. Here, the guidelines from [23], are
applied to merge two boost converters with a
buck-boost converter. The converter
operation and dynamics are represented by a
mathematical model. State space analysis
along with the small-signal averaging method
is performed in each mode of operation to
obtain the TFM.
(ii) Tor to determine the controller structure,
interaction analysis is carried out to
determine the converter's input-output
pairing. Further, CRCBP based multi-
variable controller is proposed for the TIID
converter, which is a major contribution of
the present work.
(iii) Different operating situations, such as
fluctuating source voltages, loads, or both,
are simulated in real time, and the effects of
parameter alterations on the dynamics of the
converter system are examined.
Following this, the paper is organized as
follows: Section 2 describes the mathematical
modeling of the TIID converter and derivation of
the TFM of the converter. Sections 3 and 4 depict
the procedure for designing CRCBP controller and
its implementation to control the TIID converter.
Further, the real-time simulation results are
disclosed in Section 5. Finally, the results drawn
are shown as conclusions in Section 6.
2 Mathematical Modelling of TIID
Converter
Based on the knowledge provided in [21], two
boost converters as well as a buck-boost converter
are combined in this study. Figure 1 depicts a
traditional fourth-order TIID converter. Three
separate input voltage sources
1g
V
,
2g
V
and
3g
V
are suggested for this integrated converter: and. In
addition to controlling the output voltage
o
V
, Low
Voltage Source (LVS) currents
1g
i
and
2g
i
are also
controlled to ensure appropriate load distribution
and uninterrupted power supply. The three switches
on the TIID converter are each individually
controlled by the relevant duty ratios
1
d
,
2
d
and
3
d
. As a result, power can flow to the load from
three separate sources either concurrently or
individually, and the duty ratios serve as the
converter's controlling inputs. As a result, TIID
operates in four separate modes, as seen in Figure
2.
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Load
1g
V
2g
V
3g
V
1
S
2
S
3
S
1
L
2
L
3
L
o
C
o
R
1
D
2
D
3
D
Fig. 1: Schematic of the TIID converter
MODE 1
MODE 2
MODE 3 MODE 4
1
d
2
d
3
d
Fig. 2: TIID converter gating signals
This paper uses state space equations in each
mode to analyze the converter dynamics and
operations. As a result, a collection of transfer
functions put together in TFM form serves as a
model for the functional dependency between
output and input variables. In all operating modes,
the state-variable model and small-signal modeling
are used to obtain the TFM.
Equations (1) and (2) provide the state-space
equations for the four operating modes, where
i
=1,2,3,4. By averaging these state equations as
shown in (3) and applying a small change
k
in
each of the state variables as in (4), the small-signal
modeling of the converter can be obtained, from
there the TFM as given in (5) is developed in
MATLAB environment. The detailed and complete
derivation aspects of small-signal modeling of the
TIID converter are given in [22].
00
,
i i i i
x Ax Bu y E x F u
(1)
0
1
2
g
g
v
i
y
i
1
1
0
2
i
i
i
E
P
E
P
1
1
01
2
i
i
F
F
F
F
(2)
1 1 2 1 2 3 2 3 3 4
1 1 2 1 2 3 2 3 3 4
1 1 2 1 2 3 2 3 3 4
1 1 2 1 2 3 2 3 3 4
( ) ( ) (1 )
( ) ( ) (1 )
( ) ( ) (1 )
( ) ( ) (1 )
d A d d A d d A d A
d B d d B d d B d B
d E d d E d d E d E
d F d d F d d F d F
A
B
E
F
(3)
1 1 1
1
2 2 2 3 3 3 3 3 3
ˆ
ˆ ˆ ˆ
( ) , ( ) , ( ) , ,
ˆ ˆ ˆ
, , (1 )
x t X x u t U u y t Y y d D d
d D d d D d d D d
(4)
01
11 12 13
121 22 23 2
31 32 33
23
ˆ
ˆ() ()
( ) ( ) ( )
ˆˆ
() ( ) ( ) ( ) ()
( ) ( ) ( )
ˆˆ
() ()
g
g
vs ds
G s G s G s
is G s G s G s ds
G s G s G s
is ds
(5)
11 12 13
21 22 23
31 32 33
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
G
G s G s G s
G s G s G s
G s G s G s
(6)
3 Decentralised Controller Structure
Using the TFM
G
given in Appendix Eq. (A1)-
(A9), identify the input-output pairing. It describes
which input controls which output predominantly
than others. The pairing problem is addressed by
performing Interaction Analysis using RGA as
given in [24]. The TIID converter's computed
matrix is provided in (7). TFM is diagonally
dominating, as can be seen from this matrix
(0.9505, 0.8920, and 0.9059). As a result, RGA
recommends matching the input-output variables of
the TIID converter diagonally i.e.,
1o
dV
,
21g
di
and
32g
di
. This leads to the decentralized or
diagonal controller topology seen in Figure 3.
0.0139 0.0
0
356
( ) 0.0 0.8920
5
495 0.0
.
585
09.0 0
0.950
900 0 0.0 5941
RGA G s
(7)
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+-
+-
+-
11
22
33
00
00
00
c
cc
c
G
GG
G
11 11 11
21 22 23
31 32 33
G G G
G G G G
G G G
11
22
33
00
00
00
W
WW
W
WFM W
TFM GController
1g ref
i
2g ref
i
1g
i
2g
i
o
V
1
2
o
g
g
V
yi
i
oref
V
Fig. 3: Schematic of the TIID converter's closed-
loop system
The controller
c
G
is said to be the robust if it
rejects disturbances and noises injected at the plant
output. The robustness is acquired by direct loop
shaping of singular value plots of a closed loop
system. The system performance and robustness are
described in terms of sensitivities: Sensitivity (
S
)
and complementary sensitivity (
T
) and controller
sensitivity (
KS
). The required performance
objectives of system (
S
,
KS
and
T
) are
represented with weight functions (loop-shaping
filters)
1
W
,
2
W
and
3
W
respectively. These are
incorporated into the system before designing the
controller
c
G
. Weight Function Matrix (WFM)
(loop-shaping filters) is given in (8) and is designed
using the principles given in [25]. WFM is the
frequency dependent and represents the frequency
response upper bound of the
S
,
KS
,and
T
.
1
2
3
()
WS
W s W KS
WT
(8)
4 CRCBP Controller For TIID
Converter
For the TIID converter, any deviation in line
voltages
1g
V
,
2g
V
,
3g
V
and load will reflect in
converter dynamics and its characteristics. A robust
controller is required to govern the three output
variables of the TIID converter to handle this
unpredictable scenario. is considered resilient
when there is uncertainty present but the closed-
loop system remains stable, [26]. As was
previously mentioned, the WFM is used to indicate
the controller's performance and resilience. In the
CRCBP method, RPM
is defined based on
the mixed sensitivity-
H
control of WFM i.e.,
is the maximum singular value (
) of
WFM as given in (9). In this method, the sufficient
condition for a closed-loop converter system to be
robust is the RPM
1
.
1
2
3
()
WS
W s W KS
WT
(9)
The CRCBP are contours of RPM
,
which are set over the controller's bode magnitude
and phase charts. The contours of RPM where
1
are termed as forbidden regions. The
controller is not robust in the frequency range if the
controller frequency response crosses these
contours. Thus, the objective is to choose the
controller
c
G
iteratively, such that its frequency
response doesn’t intersect with the forbidden
regions at all frequencies. As a result, the CRCBP
approach makes it easier to optimize the controller
iteratively since the plots make it evident where the
RPM contours are in relation to the prohibited
regions. Additionally, this method allows for
flexibility in choosing the controller parameters
without changing the weighting functions. This
approach works better with converters that have
fluctuating operating points and unclear
parameters.
4.1 Implementation of CRCBP Controller
for TIID converter
In this section, the procedure to tune the PI
controller parameters of
1c
G
to control
11
G
of
TIID converter using CRCBP is presented and
these are iteratively obtained in MATLAB
environment. In each iteration, the bode plots of the
controller are observed so as not to intersect with
forbidden regions of RPM
1
. If the
intersections are found, then the controller is not
robust so the iterations are repeated till a robust
controller is obtained. Before designing
1c
G
, the
WFM is to be constructed first.
1
W
and
3
W
are
selected as given in (10). The controller effort is not
limited so
20W
.
1
20
21
s
Ws
20W
3
500
0.005 1000
s
Ws
(10)
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4.1.1 Iteration 1
Choose
1c
G
as a Proportional Controller (
1
P
K
).
Figure 4 displays the CRCB charts for the closed
loop system. The area of
1c
G
where the forbidden
region intersects is indicated by the red part (
1
), implying that the designed controller is
not robust. Figure 5 shows the Bode Magnitude
Frequency Response (BMFR) of
1
1W
,
1
S
and
3
1W
,
1
T
and indicates that
3
1W
is below
1
T
in
high frequency region. Thus, closed-loop
complementary sensitivity function is violating the
constraint in the high frequency region indicating
that the system is exhibiting poor disturbance
rejection characteristics which caused the
interaction with the forbidden region in high
frequency region as shown in Figure 4. So, the
controller needs to have integral action to
compensate this.
Fig. 4: CRCBP of
1c
G
and
11
G
Fig. 5: BMFR of
1
1W
,
1
S
and
3
1W
,
1
T
4.1.2 Iteration 2
Choose
1c
G
as a combination of Proportional and
Integral Controller (
1, 1
PI
KK
). The CRCB
plots of the closed loop system are plotted in Figure
6. The BMFR of
1
1W
,
1
S
and
3
1W
,
1
T
is shown
in Figure 7. The Figures are identical to that in
iteration 1. So, the controller parameters are to be
adjusted further.
Fig. 6: CRCBP of
11c
G
and
11
G
Fig. 7: BMFR of
1
1W
,
1
S
and
3
1W
,
1
T
4.1.3 Iteration 3
1c
G
is taken as solely Integral Controller (
0.95
I
K
). The CRCB plots and BMFR are
shown in Figure 8 and Figure 9 respectively and
show that there are no interactions with the
forbidden region implying the robustness of the
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