Impact of Weight Functions on Performance of Three-Input Integrated

Dc-Dc Converter in

H

Control

MANOGNA M., AMARENDRA REDDY B., PADMA KOTTALA

Department of Electrical Engineering,

Andhra University College of Engineering,

Visakhapatnam, Andhra Pradesh,

INDIA

Abstract: - Many power electronic systems applications namely locomotives, hybrid electric vehicles, and all

renewable energy sourced systems are shifting towards Multi-Input Multi-Output (MIMO) integrated DC-DC

converters due to their reliability and flexible nature. Designing controllers for MIMO integrated DC-DC

converter is complicated due to its integrated structure, presence of common elements, and interactions between

the input and output variables of the converter. In this work, a Three-Input Integrated DC-DC (TIID) converter

is modeled using state-space analysis, and a Transfer Function Matrix (TFM) is acquired from the small signal

continuous time model. A robust



H

controller based on the loop shaping method is designed for the TIID

converter. In this loop-shaping method, the desired robustness and the performance of the controller are

represented with weight functions i.e., loop-shaping filters. These weight functions are designed using TFM

and are frequency-dependent. The robustness of the controller depends on the weight function parameters. The

effect of varying the parameters of the weight functions on system dynamics, robustness, and performance are

studied and plotted. TIID converter of 288 W, 24V-30V-36V to 48 V is considered and the impact of weight

function on closed-loop system dynamics and sensitivity characteristics under varying parameter conditions are

analyzed in MATLAB Environment.

Key-Words: - Three-Input Integrated Dc-dc (TIID) converter, state-space modeling, small-signal analysis,

Transfer Function Matrix (TFM), Weight Function Matrix (WFM),

H

controller.

1 Introduction

The MIMO converters are proven to be more

flexible, efficient, reliable, and economical, [1], [2],

[3], [4], in many power electronic systems

applications namely locomotives, hybrid electric

vehicles, and all renewable energy-sourced systems.

Thus, designing a MIMO PID controller is more

complicated than a Single-Input Single-Output

(SISO) PID controller. Since the number of tuning

parameters is limited to three in the SISO case, the

problem of designing a SISO PID is rather simple

and a wide variety of methods are available for this

purpose, [5], [6].

Different methodologies to design MIMO PID

controllers for MIMO systems that ensure stability

and performance are reported in the recent literature,

[7], [8]. Diagonal controllers are proposed for a

Two-input Buck-SEPIC dc-dc converter system

using individual channel design (ICAD), [9]. A

decoupler network is designed to minimize the

control-loop interactions for a three-port dc-dc

converter which is suitable for a satellite

application, [10]. The interaction -independent

robust controller is designed for a two-input fourth-

order integrated (TIFOI) dc-dc converter, [11],

using

H

Loop Shaping design procedure.

H

loop-

shaping controllers are extensively studied and

applied to a MIC, [12]. According to

H

control,

[13], the infinity norm of the closed-loop system is

minimized in designing a controller which is related

to the robust stability margin of the closed-loop

system. The advantage of a



H

controller over other

controllers is that it can be directly used to handle

single-input as well as multi-input systems with

cross-coupling between channels, [14], [15].

The controller design using an H∞ loop-shaping

procedure is proposed for a MIMO system and

presented in [16], [17] and [18]. Robust controller

5HFHLYHG0DUFK5HYLVHG-DQXDU\$FFHSWHG0DUFK3XEOLVKHG$SULO

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Manogna M., Amarendra Reddy B., Padma Kottala

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design through the H∞ loop-shaping method is

extensively studied and applied to a MIC, [19].

Here, the robustness of a closed-loop system is

ensured by the controller designed using a loop-

shaping technique. Weight function formulation is

difficult for MIMO systems because each input

signal may affect many controlled variable output

signals. It is important to study the impact of

interactions between inputs and outputs beforehand.

Novelty/ Contribution of the work:

In this work, a systematic way of identifying the

weight functions for TIID converter using



H

methods is presented. According to



H

control

theory, [20], [21], [22], the infinity norm of the

weighted sensitivity function is minimized in

controller design, and infinity norm is related to the

robust stability margin of the closed-loop system.

The following contributions are made from this

work:

(i) A fourth-order TIID converter is proposed

in [23]. Here, the guidelines from [24] 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.

(iii) Further, the required performances of the

TIID converter are represented using the

weight functions

1

W

,

2

W

and

3

W

respectively. These are employed to design

the robust



H

controller.

(iv) The impact of variation of each parameter on

dynamics, sensitivity functions, and inverse

of weight functions are analyzed and plotted.

The paper is divided into the following sessions:

(i) mathematical modeling of the TIID Converter,

(ii) quantifying the interactions and identifying i/o

pairing to determine the controller structure, (iii)

synthesis of the



H

controller, (iv) designing of

different weight functions, (v) illustrates the impact

of variation of each parameter of weight functions

on dynamics, sensitivity functions and inverse of

weight functions followed by Conclusions.

2 Modeling of TIID Converter

In Figure 1, the converter for TIID is shown. For

this integrated converter, three distinct voltage

sources (

1g

V

,

2g

V

and

3g

V

) are suggested.

Furthermore, for proper load sharing and power

continuity, the output voltage

o

V

and the LVS

currents (

1g

i

,

2g

i

are the input currents of

1g

V

,

2g

V

)

are regulated. Three duty ratio control signals

1

d

,

2

d

and

3

d

are used in the proposed converter to

independently regulate each of the three switches.

As a result, three different sources of power can

supply the load simultaneously or separately. The

duty ratios function as the governing inputs of the

converter. This enables four different modes of

operation to be possible as shown in Figure 2. As a

result, each mode of operation's state space

equations evaluates the converter's dynamics and

performance. The input and output variables are

hence functionally dependent on one another.

Hence, this functional dependency is modeled by a

set of transfer functions assembled in TFM form. To

derive the TFM, a state-variable model along with

small-signal modeling is implemented in all the

operating modes.

C0

Vg1

Vg2

Vg3

L1

L2

L3

S2

S3 R0

S1

D1

D2

D3

Load

Fig. 1: Circuit diagram of the TIID converter

Eqs (1), and (2) give the state-space equations

for the four operational modes, where

i

=1,2,3,4.

The small-signal modeling of the converter can be

generated by averaging these state equations as

indicated in (3) and applying minor change

k



to

each of the state variables as indicated in (4). From

there, the TFM

G

as indicated in (5) is developed

in a MATLAB environment. The detailed modeling

is given in [23].

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00

,

i i i i

x Ax Bu y E x F u   

(1)

0

1

2

g

v

i

y

i











1

0

2

i

E

P

E

P











1

01

2

i

F











(2)

Fig. 2: PWM gating signals for the TIID Converter

1 1 2 1 2 3 2 3 3 4

( ) ( ) (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

ˆˆ

() ()

( ) ( ) ( )

ˆˆ

() ( ) ( ) ( ) ()

( ) ( ) ( )

ˆˆ

() ()

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











(6)

3 Decentralised Controller Structure

Using the TFM

G

in (6), identify the input-output

pairing. It describes which input controls which

output predominantly than others. Pairing problem

is addressed by performing Interaction Analysis

using

RGA

as given in [25]. The TIID converter is

designed with the specifications given in Table 1.

Using these parameters, the TFM of the TIID

converter (6), considering all the modes is obtained

in MATLAB and is given from (7)-(15).

Table 1. Specifications and Parameter Values

Parameters

Value

Vg1, Vg2, Vg3

36V,30V,24V

o

V

, Load power

o

P

,

o

R

48V, 288W

1L

i

,

2L

i

2.5A,2A

1

L

,

2

L

,

3

L

150µH, 250µH, 20µH

o

C

200µF

switching frequency

s

f

50KHz

L

i

,

o

V

10%, 5%

4 4 3 9 2 12 15

11 4 3 7 2 11 13

0.3488 2.493 10 1.051 10 5.608 10 1.796 10

6195 6.126 10 1.3 10 2.885 10

    

   

s x s x s x s x

Gs s x s x s x

(7)

4 5 3 9 2 12 13

12 4 3 7 2 11 13

0.6379 1.293 10 6.573 10 2.308 10 6.963 10

6195 6.126 10 1.3 10 2.885 10

   

   

s x s x s x s x

Gs s x s x s x

(8)

4 4 3 8 2 12 13

13 4 3 7 2 11 13

0.4423 4.073 10 3.755 10 2.48 10 8.226 10

6195 6.126 10 1.3 10 2.885 10

    

   

s x s x s x s x

Gs s x s x s x

(9)

5 3 9 2 13 16

21 4 3 7 2 11 13

3.249 10 2.093 10 1.442 10 1.382 10

6195 6.126 10 1.3 10 2.885 10

  

   

x s x s x s x

Gs s x s x s x

(10)

3 8 2 13 16

22 4 3 7 2 11 13

4253 7.506 10 3.238 10 1.029 10

6195 6.126 10 1.3 10 2.885 10

   

   

s x s x s x

Gs s x s x s x

(11)

3 8 2 12 16

23 4 3 7 2 11 13

2949 1.943 10 1.899 10 1.215 10

6195 6.126 10 1.3 10 2.885 10

  

   

s x s x s x

Gs s x s x s x

(12)

7 2 12 16

31 4 3 7 2 11 13

4.446 10 2.665 10 1.243 10

6195 6.126 10 1.3 10 2.885 10



   

x s x s x

Gs s x s x s x

(13)

3 8 2 13 15

32 4 3 7 2 11 13

2552 4.15 10 1.577 10 1.4 10

6195 6.126 10 1.3 10 2.885 10

   

   

s x s x s x

Gs s x s x s x

(14)

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5 3 9 2 13 16

33 4 3 7 2 11 13

1.967 10 1.237 10 1.061 10 1.592 10

6195 6.126 10 1.3 10 2.885 10

  

   

x s x s x s x

Gs s x s x s x

(15)

The TIID converter's computed

RGA

matrix is

given in (16). 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.,

1o

dV

,

21

g

di

and

32

g

di

. This

leads to the decentralized or diagonal controller

topology seen in Figure 3. This controller is

designed using



H

control as described below.

 

0.892

0.0139 0.0356

(

9

.

0.9505

) 0.0495 0 058

0

00. 059

5

0.00 0.0941











RGA G s

(16)

+-

11

22

33

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

W

WW

W











WFM W

TFM GController

1g ref

i

2g ref

i

1g

i

2g

i

o

V

1

2

o

g

V

yi

i











oref

V

Fig. 3: Schematic of the closed-loop TIID converter

4 Synthesis of Mixed Sensitivity

H

Controller

For the TIID converter, any deviation in inputs (

1g

V

,

2g

V

,

3g

V

) and load will reflect in converter dynamics

and its characteristics. A robust controller can

address this uncertain situation to regulate the three

output variables of the TIID converter. The robust

controller can be designed by applying loop-shaping

along with the



H

technique, [26]. This technique

allows the designer to shape the frequency response

of the converter system and then optimize the

response of the system to achieve robustness. Figure

4 shows the LTI model of a system with a plant

P

and controller

K

. Equation (17) represents a

dynamic model of plant

P

with its inputs(

u

,

w

) and

outputs(

z

,

y

).

*

   



   

   

zw

P

yu

(17)

w

u

z

y

K

P

Fig. 4: LTI model of

P

and

K

-

+

e

w

y

u

3

W

z

G

P

1

W

2

W

1

z

2

z

3

z

K

Fig. 5: The augmented plant

P

with

K

To synthesize the controller

K

for

P

, the

robust controller must reject disturbances and noises

injected at the plant output. In



H

control, this

robustness is acquired by direct loop shaping of

singular value plots of a closed loop system. The

required performance objectives of the system are

represented along with weight functions (loop-

shaping filters). The plant performance and

robustness can be specified in terms of

S

and

T

.

S

ensures disturbance rejection,

KS

is the controller

effort and

T

represents tracking and noise

attenuation characteristics. Hence, these system

performances are represented using the weight

functions

1

W

,

2

W

and

3

W

respectively. These are

incorporated into the system before designing the

controller

K

as shown in Figure 5, where

G

is the

TFM of the converter and

W

is the Weight

Function Matrix (WFM) of

1

W

,

2

W

and

3

W

.

W

represents the TFM from

w

to

z

as given in

equation (18).

In mixed sensitivity



H

control method, the

controller

K

which stabilizes the system

G

is

designed such that it minimizes the



H

norm of the

closed-loop system i.e.,

W



, where

1



.

1

2

3

()











WS

W s W KS

WT

(18)

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5 Designing of Weight Functions

The weight functions

1

W

,

2

W

and

3

W

are written in

the form of transfer functions that represent

frequency response upper bounds for

S

,

KS

and

T

. WFM parameters are selected for this integrated

converter using the standard methodology, [27] and

these weight functions are designed as follows:

5.1 Design of

1

W

To track the reference signal with high accuracy and

to subdue the external disturbances, the sensitivity

function

S

should be small enough in the desired

frequency range. Choose

1

W

large inside the control

bandwidth to obtain small

S

.

1

W

is a function of

desired steady-state error (

s

e

), Desired bandwidth (

s



) and Maximum allowed sensitivity peak of the

system (

s

M

) as given in (19).

1





s

ss

s

M

Wse



(19)

5.2 Design of

2

W

2

W

is specified in terms of the desired controller

specifications. To limit control effort in a particular

frequency band, increase the magnitude of W2 in

this frequency band to obtain small KS. The transfer

function of W2 is given in (20).

2





bc

bc bc

sM

Wes



(20)

5.3 Design of

3

W

If

S

is small in the desired bandwidth, then

T

is

large in the desired bandwidth. Therefore,

choose

3

W

to be large outside the control bandwidth

to obtain small

T

for good robustness and noise

attenuation characteristics. The transfer function

3

W

is given in (21).

3





b

bb

sM

Wes



(21)

The weight function transfer functions given

using equations (19- 21) are used for loop shaping

to get the desired performance of the closed-loop

system thereby designing the controller.

6 Results & Discussion

In this section, the impact of varying the parameters

on the dynamics of the system is studied. The

parameters of all three weight functions are varied

and the impact of each parameter on the dynamics

of the system

11

G

are studied. The impact of

variation of each parameter on dynamics, sensitivity

functions, and inverse of weight functions are

analyzed and plotted. Thus, the choice of the

parameters is justified and validated from these

plots, [28], [29].

To design a controller for

11

G

, the WFM

111

W

,

112

W

and

113

W

of

11

G

are obtained and are given in

(22).

111 112 113

0.5 100 500 6667

,,

20 0.01 100 0.001 10000

  

  

  

s s s

W W W

s s s

(22)

6.1 Impact of Parameter Variation of

111

W

6.1.1 Varying

s

e

of

111

W

Consider

111

W

, the parameter

s

e

is varied from 0.01

to 1 while keeping

s



and

s

M

constant. The other

two weight functions

112

W

and

113

W

are also kept

constant. The variations of dynamics of the step

response of the closed loop system of

11

G

are shown

in Figure 6, Figure 7, Figure 8 and Figure 9. From

Figure 6, it is evident that as

s

e

is increased the

settling time (

s

t

) decreases implying that a higher

value of

s

e

is suitable for best closed-loop

performance. From Figure 7, it is evident that there

is no Peak overshoot (

o

P

) and it doesn’t change

(NC) with

s

e

. From Figure 8, it is observed that

with an increase in

s

e

, the steady-state error (

ss

e

)

also increases. When

0

s

e

, the deviation in

closed-loop step response is zero. From Figure 9,

the Peak magnitude (

m

P

) decreases with increase in

s

e

. The step response with varying

s

e

is given in

Figure 10. As

s

e

increases, the step response

deviates indicating the increase in

ss

e

. Thus, from

these plots, it can be concluded that a lower value of

s

e

is suitable to have better closed-loop

performance. Similarly, the variation in other

parameters and their impacts are studied.

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Fig. 6: Variation of

s

t

with

s

e

Fig. 7: Variation of

o

P

with

s

e

6.1.2 Arbitrary Variation of Three Parameters

on

111

1W

:

The bode plots of

111

1W

with varying parameters are

plotted from Figure 11, Figure 12 and Figure 13.

Figure 11, shows that when desired

s

e

is varied

while keeping

s

w

and

s

M

constant, the values of

gains at low frequency varies (

lf

g

) i.e., when

s

e

increases

lf

g

also increases while the gain at high

frequency(

hf

g

) is constant. In Figure 12, when

s

M

is varied while keeping

s

e

and

s

w

constant,

hf

g

varies

i.e., when

s

M

increases,

hf

g

also increases while

lf

g

is constant. In Figure 13, when

s

w

is varied while

keeping

s

e

and

s

M

constant,

lf

g

and

hf

g

are constant

but the cross-over frequency (

c



) varies i.e., with

increase in

s

w

,

c



is decreased. The Impact of all

the parameter variations of WFM on system

characteristics is tabulated in Table 2.

Fig. 8: Variation of

ss

e

with

s

e

Fig. 9: Variation of

m

P

with

s

e

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Fig. 10: Variation of Step response with

s

e

Fig. 11: Bode plot of

111

1W

with varying

s

e

,

Fig. 12: Bode plot of

111

1W

with varying

s

M

Table 2. Impact of Parameter Variation on System

Characteristics

Weight

function

parameters

Characteristics of Step

Response

Remarks on

performance

characteristics and

inverse weight

functions

s

t

o

P

ss

e

m

P

111

W

s

e

↓

none

↑

↓

lf

g

of

1

S

↑

lf

g

of

111

1W

↑

s



↓

none

↑

↓

lf

g

of

1

S

↑

c



of

111

1W

↓

s

M

↓

none

↓

↑

lf

g

of

1

S

↑

hf

g

of

111

1W

↑

112

W

bc

e

N

C

none

N

C

NC

NC in

1

KS

hf

g

of

112

1W

↑

bc



N

C

none

N

C

NC

NC in

1

KS

c



of

112

1W

↑

bc

M

N

C

none

N

C

NC

NC in

1

KS

lf

g

of

112

1W

↑

113

W

b

e

↑

none

↑

NC

NC in

1

T

hf

g

of

113

1W

↑

b



↑

none

N

C

NC

NC in

1

T

c



of

113

1W

↑

b

M

↓

none

↓

NC

hf

g

of

1

T

↑

lf

g

of

113

1W

↑

6.2 Simulation Results

The designed Controller with the WFM The test

bench is shown in Figure 14, and MATLAB

connected with the OPAL4510-RT simulator using

RT-LAB simulation software confirms the closed-

loop performance of

c

G

for TIID converter, [27],

[28]. The real-time observations are made using a

digital storage oscilloscope. Under various operating

scenarios, the robust controller's performance is

validated.

Fig. 13: OPAL4510-RT Simulator test bench

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Figure 14 displays the simulation of the nominal

settings given in Table 1. Figure 15 displays the HIL

Simulation results of the OP4510 measured in DSO.

Fig. 14:

o

V

,

o

I

of TIID converter

Fig. 15:

o

V

,

o

I

of TIID converter

6.2.1 Under Varying Both Load and Sources

In this case, the load and the source voltages are

varied and the corresponding

o

V

and

o

I

are

observed.

o

R

is varied from 8Ω to 12Ω at t=25msec.

At t=40msec,

3g

V

is varied from 24V to 20V. At

t=60msec,

2g

V

is varied from 30V to 25V and at

t=80msec,

1g

V

is varied from 36V to 30V. The

simulation results of the corresponding

o

V

and

o

I

are given in Figure 16. Hence, the

H

controller

can regulate

o

V

at 48V with variations in load and

all the source voltages as shown in Figure 16.

Fig. 16:

o

V

,

o

I

of TIID converter with varying load

and source voltages

The HIL simulation results using the Data

Logger Method (DLM) are given in Figure 17 (a)-

(d) and Figure 18 (a)-(d). The HIL simulation

results measured in CRO are given in Figure 19,

Figure 20, Figure 21, Figure 22 and Figure 23.

Fig. 17: a)

o

V

from DLM, b)

1g

V

from DLM, c)

2g

V

from DLM, d)

3g

V

from DLM

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Fig. 18: a)

o

I

from DLM, b)

1

d

of switch

1

S

,c)

2

d

of

2

S

d)

3

d

of

3

S

from DLM

Fig. 19: Input voltage

1g

V

and

3g

V

of TIID converter

Fig. 20: Input voltage

2g

V

of TIID converter

Fig. 21: Duty ratios of

1

S

and

2

S

of TIID converter

Fig. 22: Duty ratios of

2

S

and

3

S

of TIID converter

Fig. 23:

o

V

,

o

I

of TIID converter

7 Conclusion

The TIID converter is modeled using state-space

analysis and a Transfer Function Matrix is acquired

from the small signal continuous time model. The

desired robustness and the performance of the

controller are represented with weight functions

(loop-shaping filters). These weight functions are

designed using the obtained Transfer Function

Matrix. The robustness of the controller depends on

the weigh function parameters. The impact of

parameter variations of the weight functions on

system dynamics and performance is studied and

plotted. TIID converter of 288 W, 24V-30V-36V to

48 V is considered and the impact of weight

function on closed-loop system dynamics and

sensitivity characteristics under varying parameter

conditions are analyzed in MATLAB Environment.

The Future scope of this research work is (i) to

implement a Dc Microgrid with three different

Renewable energy sources as three inputs for the

proposed TIID converter of the designed



H

controller, (ii) to regulate the output voltage of DC

Microgrid with PI controllers, that are to be

designed by



H

loop-shaping method and (iii) the

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relation between WFM and Sensitivities at different

stages of controller design are to be graphically

studied.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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