Design of controller for Cuk converter using Evolutionary ...In this paper, the modelling and design...

Design of controller for Cuk converter

using Evolutionary algorithm via Model

Order Reduction S.Suguna1

, *M. Siva Kumar2

1 ,2 Dept. Of EEE, Gudlavalleru Engineering

college,

E-mail:[email protected]

Abstract

In this paper, the modelling and design of controller

for Cuk converter operating in continuous conduction

mode (CCM) is proposed. The Cuk converter is a DC-

DC converter, operating in step-up as well as step-

down modes based on a switching buck-boost

topology. By using the State Space Averaging (SSA)

technique, the mathematical model of this converter

is carried out and yields to a fourth order system. The

feedback compensator design for higher order system

is very difficult. In this proposed paper, the fourth

order system is reduced to a second order model using

Evolutionary algorithm based Particle Swarm

Optimization via model order reduction by

minimizing the Integral Square Error (ISE)and the

controller designed by this proposed method gives the

satisfactory results.

Keywords: Model Order Reduction, Cuk Converter, State-Space

Averaging, Compensator, Integral Square Error (ISE).

1. Introduction Now a days the switched mode dc-dc converters,

which converts electrical voltage from one level to another

by using the switching action, are mostly used because of

International Journal of Pure and Applied MathematicsVolume 114 No. 8 2017, 297-307ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

297

their greater efficiency, lighter weight and small size. The

Cuk converter is considered as a series combination of both

boost and buck converters. Cuk converter consists of

excellent properties like negative output voltage, low

output current ripple and switching ripple, capacitive

energy transfer, smooth input and output currents. Due to

the fourth order characteristics the dynamic response is

usually affected, which automatically calls for the

limitations in band width of the closed-loop system.

Moreover, to decouple the stages of input and output, big

energy capacitors are required by stability, which involves

complexity in both the control and theoretical

implementation. In order to avoid these problems, the

order of the transfer function is reduced then after that

controller is designed. A Cuk converter comprises of two inductors, two

capacitors, a diode and power switch, hence it is a fourth order

system which is nonlinear. A linear model is required to design a

feedback controller. The linear model is derived by replacing the

switch and diode by small signal averaged model. The state space

averaging (SSA) technique [2] to model the power stages is used

to obtain the analytical description of Cuk converter which is a

linear model. By depending on the control to output transfer

function, the output voltage is regulated by using PWM controller

[6] is designed. Since the feedback compensator design for higher

order system is very difficult, to avoid these complexities, the

original fourth order system is reduced to a second order model by

evolutionary algorithm via Model Order Reduction [6-17].The

proposed paper is organized as, The section 2 consists of State

Space Averaging (SSA) technique, section consists 3 of Cuk

converter Analysis, section 4 consists of controller design of Cuk

converter and simulation results and section 5 consists of

conclusion and references.

2. SSA Technique The closed loop system's power stage is a non-linear

system, which are quite complex to model and also difficult

to predict their nature. So it is preferable to approximate it

as a linear one. Bode plot is mostly used to design the

compensator in feedback loop for the desired response. For

this purpose the state space averaging technique is used.

The dc-dc converters which are operating in

continuous condition mode have mainly two states, one

during the switch is on and other when the switch is off.

During the switch on;

𝑋 = 𝐴1𝑋+𝐵1𝑉𝑑 0<t<dT (2.1)

During switch off;

𝑋 = 𝐴2𝑋+𝐵2𝑉𝑑 0<t<(1-d)T (2.2)

International Journal of Pure and Applied Mathematics Special Issue

298

𝑉0 = 𝐶1𝑋 during dT (2.3)

𝑉0 = 𝐶2𝑋 during (1-d)T (2.4)

The averaged model for the Cuk converter can be

produced over above mentioned switching period, the state

space equations corresponding to the two states are time

weighted and time averaged, resulting in below mentioned

equations

𝑋 = [𝐴1𝑑 + 𝐴2(1 − 𝑑)]𝑋+[𝐵1𝑑 + 𝐵2(1 − 𝑑)]𝑉𝑑 (2.5)

𝑉0= [𝐶1𝑑 + 𝐶2(1 − 𝑑)]𝑋 (2.6)

3. Cuk Convert Analysis

3.1 Cuk converter modelling by state space

technique: The Cuk converter comprises two capacitors

C1 and C2 with equivalent series resistances rC1, rC2

respectively, two inductors L1 and L2 with equivalent

series resistances rL1 and rL2 respectively, switch S , diode

D and resistance R is represented as load. The converter

mainly exchanges the energy between capacitors and

inductors to achieve the conversion from one level of

voltage to another. From the voltage source, input voltage

Vd is applied to the converter circuit through L1.During

the ON position of switch S, the current flowing through

L1, iL1 increases at the same instant the voltage across the

capacitor VC1, turns off the diode by reverse biasing it. The

capacitor C1, discharges its energy into the circuit C1, C2,

L2 and R. During the OFF position of switch S, in order to

produce the uninterrupted current the voltage across the

inductor L1 will reverse its polarity. The diode D is forward

biased, the capacitor C1 is charged by Vd, and the energy

stored in input conductor. The load current is supplied by

the energy stored in the inductor L2 and also the capacitor

C2. Under the assumption that the voltage VC1 is constant,

the Sum of the currents iL1 and iL2 must be equal to zero in

the steady state. The relation between Vc and Vd for the

ideal converter is given 𝑉𝑐

𝑉𝑑 =

𝑑

1−𝑑 Where d is the duty cycle. From these equations

the output voltage V0 can be controlled by controlling the

duty cycle. The duty cycle of the converter can be varied by

using a controller and the circuit can also be made to reject

disturbances.


299

3.2 State space equation of Cuk converter:

The state space equations for Cuk converter during

switch on and off are

During switch is ON 𝑑𝑖𝐿1

𝑑𝑡 = −

𝑟𝐿1𝑖𝐿1

𝐿1+

𝑉𝑑

𝐿1

(3.1) 𝑑𝑖𝐿2

𝑑𝑡 =

𝑉𝐶1

𝐿2 =

(𝑟𝐿2+𝑟𝐶1+𝑟𝐶2) 𝑖𝐿2

𝐿2 – (

𝑟𝐶2

𝑟𝐶2+𝑅-1)

𝑉𝐶2

𝐶2

(3.2)

𝑑𝑉𝐶1

𝑑𝑡 =−

𝑖𝐿2

𝐶1

(3.3)

𝑑𝑉𝐶2

𝑑𝑡 = −

𝑅𝑖𝐿2

(𝑟𝐶2+𝑅)𝐶2−

𝑉𝐶2

(𝑟𝐶2+𝑅)𝐶2

(3.4)

When the switch is off

𝑑𝑖𝐿1

𝑑𝑡 =

(𝑟𝐿1+𝑟𝐶1)𝑖𝐿1

𝐿1 -

𝑉𝐶1

𝐿1 +

𝑉𝑑

𝐿1

(3.5)

𝑑𝑖𝐿2

𝑑𝑡=

(𝑟𝐿2+𝑅 𝑟𝐶2)𝑖𝐶2

𝐿2 +

(𝑟𝐶2

𝑅+𝑟𝐶2−1)𝑉𝐶2

𝐿2

(3.6) 𝑑𝑉𝐶1

𝑑𝑡 =

𝑖𝐿1

𝐶1

(3.7) 𝑑𝑉𝐶2

𝑑𝑡 =

𝑅𝑖𝐿2

(𝑟𝐶2+𝑅)𝐶2 -

𝑉𝐶2


(3.8) According to the above mentioned

equations, we can write the averaged matrices for the

steady-state and linear small-signal state-space equations

𝐴1=

−𝑟𝐿1

𝐿10 0 0

0 −(𝑟𝐿2+𝑟𝐶1+𝑟𝐶2 𝑅)

𝐿2

1

𝐿2

−(𝑟𝐶2

𝑅+𝑟𝐶2−1)

𝐶2

1

𝐶10 0 0

0𝑅


−1


(3.9)


300

𝐴2=

−𝑟𝐿1+𝑟𝐶1

𝐿10

−1

𝐿20

0 −(𝑟𝐿2+𝑟𝐶2 𝑅)

𝐿20

(𝑟𝐶2

𝑅+𝑟𝐶2−1)

𝐶2

1

𝐶10 0 0

0𝑅


−1


(3.10)

B1 = B2 = B =

1

𝐿1

000

C1 = C2 = C = 0 0 0 1

E1= E2 = E = (0)

3.3 Transfer function:

With the state space matrices defined above, the

control to output transfer function can be calculated as

Gvd = C(𝑆𝐼 − 𝐴)−1𝐵𝑑 + 𝐸𝑑

(3.11)

Where Bd = (𝐴1 − 𝐴2)X+(𝐵1 − 𝐵2)𝑉𝑑

(3.12)

Output to input transfer function

Gvg = C(𝑆𝐼 − 𝐴)−1𝐵

(3.13)

X = -C𝐴−1𝑉𝑑

(3.14)

3.4 Particle Swarm Optimization: PSO[5] is an evolutionary algorithm that can be

used for solving the nonlinear equations. It is a kind of

swarm intelligence that is based on social-psychological

principles and provides insights into social behaviour, as

well as contributing to engineering applications. In PSO,

the velocity and position are randomly chosen for a set of

particles. During the start, the initial position is taken as

the best position and the velocity is updated.

The main purpose of this optimization method is

a) A global optimum for the nonlinear system may be

found,

b) It can produce a many number of solutions,

c) There are no mathematical limitations on the

formulation of the problem,

d) Comparatively very simple in execution and

e) Numerically strong.


301

The parameters c1 and c2 determine the relative pull of

the pbest and gbest and r1 and r2 helps to stochastically

unreliable these pulls.

𝑣𝑖𝑗𝑡+1 = 𝑤𝑣𝑖𝑗

𝑡 + 𝑐1𝑟1𝑗𝑡 𝑝𝑏𝑒𝑠𝑡 ,𝑖

𝑡 − 𝑥𝑖𝑗𝑡 + 𝑐2𝑟2𝑗

𝑡 [𝐺𝑏𝑒𝑠𝑡 − 𝑥𝑖𝑗𝑡 ]

(3.15)

�χ� .𝑔(𝑡+1)

= �χ� .𝑔(𝑡)

+ 𝑣𝑗 .𝑔(𝑡+1)

(3.16)

With j =1,2,3,………..,n and g =1,2,3,…….,m

Where

n = number of particles in the group, m = number of

components for the vectors and, t = pointer of iterations

(generations),

𝑣𝑗 .𝑔(𝑡+1)

= velocity of 𝑔𝑡ℎ component of particle j at iteration t,

w = inertia weight factor,

w = 𝑤𝑚𝑎𝑥 − 𝑤𝑚𝑎𝑥 − 𝑤𝑚𝑖𝑛 ∗𝑘−1

𝑁−1

(3.17)

Where K=current iteration number and N= maximum number

iterations.

c1, c2 = acceleration constant and r1, r2 = random numbers between

0 and 1

�χ� .𝑔(𝑡)

=current position of 𝑔𝑡ℎ component of particle j at iteration t,

pbest j = Best previous position of 𝑗𝑡ℎparticle. ,

gbest = Best particle among all the particle in the

population.

In Table 1, the typical parameters for PSO optimization

routines, used in the present study are given.

TABLE1

Typical parameters used by PSO

Name Value(type)

Number of generations 100

Population size 50

Maximum Particle

velocity

2

Epoch 100

Termination method Maximum

Generation

3.5 Performance index:

The arrangement of the lower order system is

established by the performance index principle. In the

present study, PSO is applied to minimize the Integral

square error between the transient part of step response of

original system. ISE is frequently employed for the

performance evaluation because of ease of achievement.


302

𝐼𝑆𝐸 = 𝑦 𝑡 − 𝑦𝑟(𝑡) 2𝑑𝑡∞

0

(3.18)

Mathematically, the integral square error can be

represented as

𝐼𝑆𝐸 = 𝑦 𝑡 − 𝑦𝑟 𝑡 2𝑀

𝑖=0

(3.19)

where, y (t) represents the step response of higher order

yr (t) represents the step response of reduced order model

at the 𝑡𝑡ℎ instant in the time interval0 ≤ t ≤ M, where M is

to be chosen.

4. Control of Cuk Converter

4.1 PWM feedback control:

The PWM control for the converter [3-4] is shown in

fig 2(a). The voltage at the output V0 is compared with

Vref. The error voltage Ve between V0 and Vref, is fed to the

compensator, Gc(s) to generate the control signal Vc, and

then to be compared with saw tooth voltage VM by using

PWM comparator. As shown in Fig. 2(b), the Switch S is

turned on when the control signal Vc is larger than saw

tooth voltage Vsaw, and it is turned off when VC is smaller

compared to Vsaw.

If V0 is varied, the feedback control will adjust Vc and

then duty cycle until V0 is again equal to Vref

Fig. 2.a PWM control Fig. 2.b waveform of PWM comparator

Fig 3 shows the closed loop block diagram of the converter.

Gvd represents the power stage transfer functions. The

PWM comparator transfer function is given by

FM = 1

𝑉𝑀 Where VM is the amplitude of saw tooth voltage.

The open loop transfer function is determined as T(s) =

GC(s) Gvd(s) FM


303

Fig.3 Block diagram of converter

T(s) is defined as the product of the small signal gain in the

forward and feedback paths of the feedback loop.

4.2. Numerical Example:

The transfer function is derived from (11) is as follows:

Gvd = −814𝑆3+2.456×107𝑆2−1.232×1012𝑆+2.154×1016

𝑆4+149.4𝑆3+4.922×108𝑆2+6.25×1010𝑆+2.02×1014 (4.1)

This is a fourth order transfer function which consists of and three

zero in the RHP and two pair of complex pole Poles

Zeros are

-63.5356 +637.7841i 20156.5294

-63.5356 - 637.7841i - 4995.0229 +35864.4541i

-11.1685 +22175.8678i 4995.0229 - 35864.4541i

-11.1685 - 22175.8678i

4.3 Model Order Reduction: Using Particle Swarm Optimization [5], the

reduced order model for the converter is obtained as

follows

G1vd = −2512𝑆+4.378×107

𝑆2+126.2𝑆+4.105×105

(4.2)

Zeros and poles of reduced order system are

Zeros Poles

17425.388 -63.1+637.5879i

-63.1-637.5879i

Fig. 4 Step response of original and reduced order model


304

Integral Square Error (ISE) between the original and

reduced order model is 0.0003463.

5. Conclusion This paper proposes the design of the controller

for Cuk converter. By applying the SSA technique the

linear model for the Cuk converter in terms of the ratio of

duty cycle to output voltage (Gvd) is determined and it

yields to a higher order system. Since the feedback

compensator design for the fourth order system is quite

complex, the fourth order function of the Cuk converter is

reduced to a second order model by using PSO technique

via model order reduction and it is observed that the

reduced and original system’s step response is almost

similar. By comparing with the original system, the

controller designed for the reduced order system gives the

satisfactory results.

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