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AbstractZeta converters are the fourth-order DC-DC

converters capable of operating in both step-up and step-

down modes and do not suffer from the polarity reversal

problem. There are many applications which require a

variable output voltage commanded by an external reference

signal. So, the Zeta converters can be particularly useful for

such applications. To achieve a Zeta converter with

adjustable output voltage capable of following an external

reference signal smoothly and accurately, there will be a need

for a suitable control system. Since the Zeta converter model

that is used in this paper is nonlinear, we propose a

combination scheme of model reference adaptive control

(MRAC) with neural networks (NN). In this paper, we

propose and design a neural network adaptive model

reference controller to control the output voltage of Zeta

converter. Simulation results show the effectiveness of the

proposed scheme for the Zeta converters with adjustableoutput voltage.

Key words: Zeta Converter, Neural Network, Model Reference

Adaptive Control.

I. INTRODUCTIONDC-DC converters are widely used as power supply in

electronic systems. Some of the main DC-DC converters

are Buck, Boost, Buck-boost, Cuk, SEPIC, and Zeta

converters. However, the Buck-boost and Cuk converters,

in their basic form, produce the output voltage, whose

polarity is reversed from the input voltage. The problem

can be corrected by incorporating an isolation transformer

into the circuits, but this will inevitably lead to theincreased size and cost of the converters. Zeta converters

are the fourth-order DC-DC converters capable of

operating in both step-up and step-down modes and do not

suffer from the polarity reversal problem. Therefore, Zeta

converters are attractive for the afore-mentioned

advantages [1].

There are many applications which require a variable

output voltage commanded by an external signal. The

signal is usually a voltage referenced to the output return.

Applications for variable output converters include

adjustable RF amplifiers, lighting controls, capacitor

charging power supplies, battery chargers and constantcurrent sources. Because of the mentioned features, the

Zeta converters with the adjustable output voltage can be

useful in these applications. To achieve a Zeta converter

with such a capability which can follow an external

reference signal smoothly and accurately, it is needed to

use a suitable control system.

This paper presents a control scheme to develop a step-

down Zeta converter with the variable output voltage. The

main steps in the control system design of a plant are

Modeling, Identification and Control. Modeling plays a

key role in revealing the insight of the converters dynamicbehavior. In the last two decades, there has been a

continually active research on DC-DC converters; as a

consequence, several modeling methods have been

proposed [2]. Among them, the State-Space Averaging

(SSA) technique is one of the best-known methods. The

SSA technique is commonly used to model the second-

order converters such as buck, boost, and buck-boost

converters. Also this technique is recently used to model

the Zeta converter which is a fourth-order converter [3].

Although in many applications, linearization is applied to

extracted model, but it cannot be applicable for the Zeta

converters with the adjustable output voltage due to the

required wide range of its output voltage. Therefore we

face with a nonlinear model for the plant to be controlled.

After the modeling of the Zeta converter, it is required

to do the identification and control steps for it. In practice,

PID controllers have been used widely for DC-DC

converters. However, PID controllers have limitations in

characteristic changes of plant during operation and also in

the case of nonlinear models. It is well-known that a model

reference adaptive control (MRAC) is very effective to

compensate characteristic changes of the plant. However,

it is not useful for nonlinearity of the plant. Many

researchers have employed NNs in their control design in

the case of nonlinear plants. Using neural networkstechnologies to the MRAC structure was found to provide

a greatly improved performance over conventional

approaches [5,6].

Since the Zeta converter model that is used in this paper

is nonlinear, we propose a combination scheme of MRAC

with neural networks (NN). Simulation results show the

effectiveness of the proposed scheme for the Zeta

converters with adjustable output voltage.

This paper is organized as follows. Section II introduces

the model of a Zeta converter through SSA method.

Section III presents the proposed neural network based

adaptive control scheme for the Zeta converter with theadjustable output voltage. The simulation results are

demonstrated in section IV. Finally, the paper is concluded

in Section V.

II. ZETA CONVERTER MODELLING THROUGH SSAMETHODThere exist two different operating modes for most of

the DC-DC converters, zeta converter included; continuous

conduction mode (CCM) and discontinuous conduction

mode (DCM). The CCM is of practical interest, since in

this mode the relationship between input and output

voltage is manifestly specified by the duty cycle of

switching.

Fig. 1 shows circuit configuration of a typical zeta

converter. The zeta converter comprises the power

Adjustable Output Voltage Zeta Converter Using Neural Network

Adaptive Model Reference Control

B. Moaveni1, H. Abdollahzadeh

2, and M. Mazoochi

3

IranUniversityofScienceand Technology, b_moaveni@iust.ac.ir2Department of Electrical Engineering, East Tehran Branch,Islamic Azad University, h.abdollahzadeh@gmail.com3IranTelecommunications Research Centre,mazoochi@itrc.ac.ir

2011 2nd International Conference on Control, Instrumentation and Automation (ICCIA)

978-1-4673-1690-3/12/$31.002011 IEEE 552

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electronic switch S with high speed switching capability,

diode D, the capacitors C1and C2, and inductors L1and L2.

The resistor R and current source IZmodel the load. The

capacitors and inductors in the Zeta converter are assumed

to be non-ideal, and they have been represented by their

corresponding ESR (equivalent series resistance).

Fig. 1. Zeta converter circuit

Fig. 2. Zeta converter configurations: (a) 0

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(21)

)(11100

0001

)(

100

01

0)(1

22

22

2

2

11

22

1

22

11

RrCRr

R

C

C

RrL

R

Rr

Rrr

L

Lrr

L

A

CC

CC

C

L

CL

b

(23)00

(22)1

00

0001

21

1

22

2

t

2

1

Rr

R

Rr

RrC

Rr

R

CRr

Rr

LB

CC

C

b

CC

Cb

(24)0

2

2

Rr

RrD

C

C

b

B.Averaging these equations to attain a single averagedstate-space equationIn the second step, based on SSA method, the state

spaces in each interval are weighted by the corresponding

duration. The ultimate state space equations of the plant,

Zeta converter, are the average of the weighted equations

over one cycle of switching, Ts:

(25)))1(())1((

)1()(()((1

u(t)BBx(t)AA

t)uBx(t)At)uBx(t)A(t)x

BA

.

avav

baba

SbbSaa

S

dddd

TddTT

(26)))1(())1((

)1()()(1

)(

u(t)DDx(t)CC

u(t)Dx(t)Cu(t)Dx(t)Cy

DC

avav

baba

SbbSaa

S

dddd

TddTT

t

Where:

baav dd AAA )1(

(27)

)(

11100

001

)(

)1()1(

10

01

0)1(

22

22

2

2

2

2

12

11

22

11

222

11

RrCRr

R

C

C

d

C

d

RrL

R

Rr

Rrdrdd

LRr

Rrrr

L

d

L

d

L

rdr

CC

CC

C

L

C

C

CL

CL

(28)100

0001

)1(

t

2

1

22

2

Rr

R

CRr

Rr

L

dd

CC

Cbaav BBB

(29)00)1(

21

1

Rr

R

Rr

Rrdd

CC

C

baav CCC

(30)0)1(

2

2

Rr

Rrdd

C

C

baav DDD

It should be noted that in addition to the input vector

u(t), containing input voltage Vd and load current Iz, the

duty cycle d is also an input parameter. Moreover, in terms

of d, the Zeta converter model is a nonlinear one.

Linearization cannot be applied to the model, since the

objective of the research demands a relatively wide rangeof duty cycle variations.

III.PROPOSED NEURAL NETWORK BASED ADAPTIVECONTROL SCHEME FOR THE ZETA CONVERTER WITH THE

ADJUSTABLE OUTPUT VOLTAGE

Model reference adaptive control (MRAC) is one of the

most popular methods in the growing field of adaptive

control to overcome the difficulties of creating a controlledsystem that could work over a wide range of operating

conditions because of the effects of process variations and

disturbances. Fig. 3 illustrates the general block diagram of

a MRAC.

Fig. 3. General block diagram of a MRAC

Usually an MRAC is implemented by a reference

model, an adjustment mechanism, a controller and a plant.

In the conventional MRAC scheme, the controller is

designed to realize a plant output convergence to reference

model output based on the assumption that the plant can be

linearized. Therefore, this scheme is effective for

controlling a linear plant with unknown parameters in the

ideal case, but it may not be assured to succeed in

controlling a nonlinear plant with unknown structures in

the real case [5,6].As it was mentioned in the previous section, the Zeta

converter model in terms of the duty cycle (d) is a

nonlinear plant. Linearization cannot be applied to the

model, since the objective of the research demands a

relatively wide range of duty cycle variations.

The Neural Networks (NNs) are the prime candidates to

be utilized in the area of nonlinear control systems. Fig. 4

illustrates a typical control of a plant using NN.

Fig. 4. A typical control scheme of a plant using NN

The feed-forward equations for the control scheme in

Fig. 4 can be written as follows:

u1= w

1y

0

y1

= f1(u1)y

2= w

2y

1

u(k) = y2= f (u

2) (31)

To determine the weights of the NN, a learning

algorithm is required. The back-propagation (BP)

algorithm is the most well-known and widely used among

other learning algorithms. It is based on steepest descent

technique expanded to each of the layers in the network by

the chain rule. This algorithm computes the partial

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derivative of the error function with respect to the weights

[8,9]. Based on this concept, the weights of the various

layers of the NN are updated by the following equations:

22

2 22

2 2 2 2

( ) ( ) ( ) (32)

1 1( ) ( ) ( ) ( ) (33)

2 2

( ) ( ) ( ) ( ) ( ) (34)

( ) ( ) ( )

d p

d p

e k y k y k

E k e k y k y k

E k y k u k y k u kw e

w u k y k u k w

Plant Jacobi

2 2 1

2

2 2 2 1 1 1

( ) ( ) ( ) ( ) ( ) 1( ) ( ) (35)

( ) ( ) ( ) ( ) ( ) ( )

( ) (36)

( )p

E k y k u k y k u k y k u kw e

w u k y k u k y k u k w k

y kJ

u k

Plant Jacobi

From equations above, it is apparent that in the

procedure of updating the weights, the plant Jacobian Jpisrequired. Jp acquisition would be problematic since it is

inaccessible priori to evaluating the plant output. The

following methods have been suggested in literature to

tackle such a problem [10]:

1stmethod:

(37))1()(

)1()(

)(

)(

)(

)(

kuku

kyky

ku

ky

ku

kyJp

The drawback of the 1st method is the Jp approaches

infinity as u(k) approaches u(k-1).

2nd

method:

(38))1()(

)1()(

)(

)(

kukusign

kykysign

ku

kyJp

The main drawback of the 2ndmethod is the possibilityof oscillating behavior in the learning process.

3rd

method: Using the NN identifier (Fig.5)

Fig. 5. Plant Jacobian computation using the NN identifier

As seen in the Fig. 5, the plant Jacobian can be easily

computed by the following equation without the previously

mentioned drawbacks:

(39))(

)(

)(

)(

ku

ky

ku

kyJ ip

Fig. 6. Proposed combination scheme of MRAC with neural networks for

the Zeta converter plant with variable output voltage

In this paper, the 3rd method is chosen to compute the

plant Jacobian, and thereby the weights of the NNs.Therefore we propose a combination scheme of MRAC

with neural networks (NN) for the Zeta converter plantwith variable output voltage, as shown in Fig. 6.

In the proposed control scheme (see Fig. 6), two

individual NNs are used for identification and control of

the intended plant.

Identification consists of adjusting the weights of the

related NN to optimize the performance functions using

the back-propagation algorithm based on the error between

the plant and the NN outputs [7].

The NN controller is designed in a way that the plant

outputypfollows the reference model output yrexactly for

a given reference input r. Based on the dynamics of the

plant, an appropriate reference model is to be selected. The

output of the NN identifier is used to cancel out the

nonlinearity of the plant so that the closed loop error

dynamics follows the dynamics of the reference model. It

is clear that for control to be accurate, the identification

model should imitate the plant precisely.

IV.SIMULATION RESULTSIn order to assess the performance of the proposed

neural network based adaptive control scheme for the Zeta

converter with the adjustable output voltage, computer

simulations are conducted using MATLAB software. Thesimulations consist of two main steps: 1) Identification of

the plant and 2) Control of which. The parameters of the

plant model (Fig. 1) which are used in the simulations

listed in table I.

TABLEIPARAMETERS OF THE PLANT MODEL

Circuit Parameters Values Circuit Parameters Values

Vd 15V L1 100H

R 1 L2 55H

C1 100F rL1 1m

C2 200F rL2 0.55m

rC1 0.19 Iz 0ArC2 0.095

A. The identification of the plant using NN

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The neural network which is used for identification of

the plant has the architecture 2-50-1. The input u(t) of the

plant is duty cycle (d) which is generated randomly in the

interval [0,0.6], following normal distribution as shown in

Fig. 7. Then 6000 samples of the random input are applied

to the plant and these samples with the correspondingoutputs are recorded as the training data to learn the NN

identifier.

0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Amplitude

Fig. 7. The random input signal for identification step

To update the weights of the NN, the Levenberg-

Marquardt back-propagation algorithm, which is of the

fastest back-propagation algorithms, is used. The initial

value for the learning rate is chosen equal to 0.001, and

also decreasing and increasing factors of which is selected

as 10 and 0.1, respectively. The adaptive value of learning

rate is increased by the increasing factor until the change

above results in a reduced performance value. The change

is then made to the network and the learning rate isdecreased by the decreasing factor.

The NN identifier is trained for 1000 epochs. Fig. 8

illustrates the result of the identification step.

0 0.1 0.2 0.3 0.4 0.5 0.6

-10

-50

5

10

15

20

25

30

35

40

Time (s)

Amplitude(V)

NN output

Plant output

Fig. 8. The result of the identification step

As seen in Fig. 8, the NN identifier output is a very

close approximation of the plant output. The result shows

that the NN identifier is suitable to be employed in the

control step.

B. The control of the plant using NNIn this step, the NN controller is to be trained in a way

that the plant output mimics the reference model outputexactly for a given reference input. According to the

dynamics of the Zeta converter, a first order transfer

function with time constant of 0.001sec is considered as

the reference model of the plant:

(40)1

)()(

Ts

kGsTr

where

(41)1 r(k)

r(k)G(k)

The neural network used for control of the plant has the

architecture 3-10-1. The input signal r(t) of the NN

controller has the same form as defined in the

identification step but with the interval [0,0.3] as shown in

Fig. 9. Then 3000 samples of the random input are applied

to the reference model and these samples with the

corresponding outputs are recorded as the training data to

learn the NN controller.

0 0.05 0.1 0.15 0.2 0.25 0.3

0.4

0.5

0.6

Time (s)

Amplitud

e

Fig. 9. The random input signal for control step

The initial value and the decrease and increase factors of

learning rate in training of the NN controller are the same

as they was considered in the identification step but

controller training is significantly more time consumingthan plant model training because the controller must be

trained using dynamic back-propagation.

The NN controller is trained for 450 epochs. Fig. 10

illustrates the result of the control step.

0 0.05 0.1 0.15 0.2 0.25 0.3

10

15

20

25

Time (s)

Amplitude(V)

Reference Model Output

Plant Output

Fig. 10. The result of the control step

As shown in Fig. 10, the NN controller is trained such

that the plant output follows the desired reference model

output.

In order to verify the main objective of the research, the

Zeta converter with the adjustable output voltage, the

proposed combination scheme of MRAC with neural

networks (NN) is applied to the plant model. After feeding

the proposed controller with a random input r(t), the

controller

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time(s)

DutyCycle

(a) Random input r(t)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

Time (s)

u(t)

(b) Controller output u(t)Fig. 11. The controller input/output signals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

2

4

6

8

10

12

14

15

Time (s)

Amplitude(V)

Plant output with controller

Plant output without controller

Reference model output

Fig. 12. The verification of the proposed control scheme

adjusts the plant input u(t) in a way that its output follows

the reference model output (Fig. 11). The result of

employing the proposed control scheme for the plant is

illustrated in Fig. 12.

Fig. 12 shows that the proposed control scheme achieves

properly the desired objective to produce a Zeta converter

with the adjustable output voltage.

V. CONCLUSIONSIn this paper, a step-down Zeta converter with adjustable

output voltage has been developed based adaptive model

reference control using neural network. To find the model

of the Zeta converter in Continuous Conduction Mode

(CCM), the state-space equations of the converter were

extracted in the various switching intervals, and then the

State-Space Averaging (SSA) technique was applied to

attain the overall converter model. Since the Zeta converter

model in terms of the duty cycle (d) was a nonlinear plant,

and also linearization cannot be applied to the model dueto the research objective, a controlling scheme taking

advantage of MRAC (Model Reference Adaptive Control)

as well as neural networks was proposed for the Zeta

converter plant with variable output voltage.

The proposed controlling scheme was comprised of two

steps of plant identification and controller design, each

using individual neural networks. The identification step

was done in a way that the performance functions were

optimized using the back-propagation algorithm based on

the error between the plant and the NN outputs. Also the

controller was designed in such a way that the plant output

followed the output of the specified first order reference

model exactly.

In order to assess the performance of the proposed

neural network based model reference adaptive control

scheme, computer simulations were carried out in two

steps of identification and control of the plant using

MATLAB software. The results demonstrated that the NN

identifier output was a very close approximation of the

plant output, and also the NN controller was trained such

that the plant output could mimic the reference model

output exactly. Fig. 12 showed that the proposed control

scheme properly achieved the desired objective of

developing a Zeta converter with the adjustable output

voltage.

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