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Multiple Controllers for Boost Converters

under Large Load Range:

a GA and Fuzzy Logic Based ApproachFabrcio Hoff Dupont1, Vincius Foletto Montagner2, Jos Renes Pinheiro2, Humberto Pinheiro2,

Srgio Vidal Garcia Oliveira1 and Adriano Pres11Universidade Regional de Blumenau (FURB)

2Universidade Federal de Santa Maria (UFSM)

fhd@ieee.org

Abstract- This paper presents a new methodology to designcontrollers for boost converters operating in a wide range ofload variation. The proposed technique uses genetic algorithmto find local LQR controllers providing optimal performance interms of overshoot and ITSE criteria. Fuzzy logic strategy isthen used to combine these local controllers, yielding a strategythat guarantees good performance for a large set of loadconditions. The proposed approach is compared with a

conventional technique based on a single controller and with astrategy of switching controllers. Simulation results illustratethe superior performance of the converter with the proposedcontroller.

I. INTRODUCTIONBoost converters are used in large number of applications

being of fundamental importance to industry as well as the

subject of studies by the academia [1-4]. One difficulty to

control these converters comes from the nonlinearity of the

model. A way to overcome the nonlinearity is to linearize the

model assuming small perturbations around some operating

point. Controllers designed for this situation ensure stability

and performance characteristics only locally. However, oftenone has to guarantee good performance for a wide load range,

and in this case, controllers designed for a single operating

point may present poor responses.

The main goal of this paper is to propose a methodology to

design a control system that guarantees good transient and

steady responses for a boost converter operating in a large

load range. The proposed methodology is based on the

design of local linear quadratic regulators (LQR [5]) for

several load conditions. For each load condition, genetic

algorithm (GAs) [6-8] is used to select the LQR with better

performance in terms of overshoot and ITSE criteria. These

optimal local controllers are combined using fuzzy logic [9-12] to provide a global controller with good performance for

the load range under consideration. Extensive simulations

results demonstrate the good performance of the proposed

controller which ensures a smooth transition between control

actions thanks to the fuzzy logic combination of the multiple

controllers

In the sequence, Section II presents the model and the

parameters of the boost converter used here. It also shows a

conventional design of an LQR (derived for the nominal load

condition) and the improvement of LQR performance with

the aid of a GA. Section III presents the use of multiple

controllers to obtain good system response under load

variations. Finally, the results are compared and the

performance superiority of the proposed strategy is

demonstrated.

II. BOOST CONVERTER AND LQRCONTROLLERThe design of a control system depends on a mathematical

model that describes its dynamic behavior. Usually thesemodels are described in the form of differential equations and

often exhibit nonlinear behavior. Assuming that the system

will be perturbed by small signals around a given operating

point, these models can be linearized, allowing to apply linear

control techniques [13].

A. State-Space Average ModelFig. 1 illustrates the circuit of the boost converter. The

losses related to the parasitic resistance of the components are

considered, except to the diode, which is considered as an

ideal switch. When the boost converter operates in

continuous conduction mode, CCM, there are two circuits,

the first one represents the stage when the switch S is onwhile the second when it is off. Since that the capacitor

voltage ripple and the inductor current ripple are usually kept

small by a suitable converter design, it is reasonable for the

controller design to consider the average value of variables in

a switching period, which can be obtained by:

1 t

t Tss

x t x t dtT

(1)

where x is the average value of a given variable x and sT is

the PWM carrier period [1-2]. By Fig. 1, the description of

the circuit for the period in which the switch S is on

(0 t DTs is given by:

0 1

10 0

0

L DS

L Li

C C

L C

LLo

CL C

r rx xL

VLx x

C R r

xRv

xR r

x x

N1M1

yxT1

(2)

where Lx and Cx are the states related to the inductor current

and capacitor voltage, respectively. The rest of the symbols

978-1-4244-5697-0/10/$25.00 2010 IEEE 105

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are represented in Fig. 1. For the period that the switch is off

(DTs t< Ts), the description obtained is:

2

2

1

10

L L C L C L

L C L CL Li

C CL

L C L C

LL C Lo

CL C L C

R r r r r R

L R r L R rx xVL

x xR

C R r C R r

xR r Rv

xR r R r

ux x

N

M2

yxT

(3)

From (2) and (3), the state-space average model is given in (4):

iV

y

x M x N

T x

(4)

where the average matrices are defined by:

1 2

1 2

1 2

D D

D D

D D

M M M

N N N

T T T

(5)

being D the nominal duty cycle and 1D D . Thus, the

equilibrium point of the converter is determined from (6),where the sub index q highlights the operating point:

1q iV

X M N (6)

For the model linearization, it is assumed that all the

disturbances caused in the system are negligible, so that any

product of the disturbance variables can also be neglected.

Thus, the linearized model for small perturbations is obtained

in (7):

1 2 1 2

1 2

d

y d

BA F

CE

x M x M M x N N u N u

T x T T x

(7)

where d is the duty cycle perturbation which represents the

control signal to be applied, and ivu is the input voltage

perturbation. In this analysis the input voltage is considered

constant, so 0u . This equation describes the dynamic

behavior of the circuit, representing the deviation of the

variables with respect to the equilibrium point obtained in (6).

B. Converter ParametersFor simulation and performance analysis in the following

sections, a boost converter of 1.5 kW whose parameters are

presented in Table I will be used.

For this configuration, the converter will enter in the DCM

only for power levels less than 2.5% of nominal value, case

that is not of interest here. This paper will address the

problem of controlling this converter with a load varying in alarge range of values.

C. State space controlThere are several possible control techniques for systems,

each one with its own complexity. In the state feedback, the

control signal is a function of the state variables, which are

supposed to be measured or estimated. Moreover, this type of

strategy is very useful for optimal control techniques [13].

Since the system described in (7) does not have integrators,

the output voltage steady state error cannot be driven to zero

with a simple state feedback control law. To ensure zero

steady state error, an additional state ex is created, defined as:

0 0

t t

ex t e d y d (8)which corresponds to the integral of error of the output

voltage. Note that in this description the error of the output

voltage is described as a disturbance in the output voltage

with opposite sign ( e y ). Thus, the additional state can be

written as:

ex d C x E (9)

Including (9) in (7), one has that the augmented system of

state equations given by (10):

0

0 ee

dxx

y d

A B

x xA B

C E

C x E

(10)

where A and B denote the augmented matrices A e B ,

respectively. For the circuit parameters shown in Table I, the

system is controllable by means of the control law

e

dx

xK

,

(11)

where K is the gain vector

1 2 3K K KK , (12)

to be determined [13].

III. SINGLE CONTROLLERIn order to improve the system response, optimal control

techniques are often used, where a given cost fucntion is

minimized. The LQR (also known in the literature as 2H optimal control problem [5]) is a very popular optimal

controller that can provide good stability margins and it will

be used in this work.

A. Conventional LQRIn case of system (10) be rewritten as

d A B (13)

+

-

+

-

i

rL L

S

rDS

rC

C

RL

xC

vo

DB

+ -

vL

xL

Fig. 1 Circuit of the boost converter with losses.

TABLE I

PARAMETERS OF THE BOOST CONVERTER USED.

Parameter Value Parameter Value Parameter Value

Vi 56 V L 602.11 H C 27 F

Vo 200 V rL 5 m rC 50 m

Fs 50 kHz rDS 10 m RL 26.666

D 0.72

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the LQR cost function to be minimized is given by

0

T TLQRJ d d dt

Q R , (14)

where the superscriptT denotes the matrix transpose and the

weighting matrices Q and R are real, symmetric and

positive defined, i.e. TQ Q , TR R , 0Q e 0R . This

index is minimized when the equality (15) is obtained.1 T K R B P , (15)

being the matrix P the solution of the Riccatis equation:1 0T T A P P A P B R B P Q (16)

The computational solution of LQR is available in

specialized programs such as MATLAB (lqr function), being

a useful and easy to apply synthesis tool. However, the

design of a good LQR is fundamentally based on the

appropriate choice of Q and R matrices, that depends on the

knowledge of the control system. In general, one does not

have a specific method for the determination of them [8].

The control of the proposed boost converter is

accomplished from the block diagram illustrated in Fig. 2,

where refV is the reference voltage and d t is the control

signal and the control gains are obtained from the LQR As is

conventionally done, the converter controller is designed for

the nominal load condition, considered here as 100% of

output power. In this situation, the operating point is defined

in (17).

100

100

26.503A

197.9V

Lq

Cq

X

X

(17)

An acceptable response can be obtained using the

following weighting values:

100

6

3100

1 0 0

0 1 0

0 0 1 10

10 10R

Q(18)

which result in the control gain vector in (19):3 3

100 46.7925 10 2.9557 10 10LQRK (19)

Fig. 3 illustrates the simulation results for the controller

with gains in (19). The reference voltage is set to 200 V and

the converter starts up at maximum load until 20 ms, when

the load is switched to 25% of nominal power, returning to

100% in 35 ms. The observed overshoot in P1C is 41.8%, at

20.77 ms. In P2C the undershoot is 29.55%, at 35.56 ms.

B. Refinement of LQR Using Genetic AlgorithmsAs mentioned above, one of the main advantages of the

state feedback control is the possibility of optimization

through various methods. It was also highlighted that one of

the difficulties in design of a LQR is the determination of the

weighting matrices, usually obtained after many attempts.

Even with the effort and knowledge of the designer, one

cannot guarantee that the set of weights found is the best for

the situation.

The genetic algorithm (GA) is a technique of search and

optimization based on the natural selection principle and may

be the evolutionary computation technique most widelyknown and applied today [6, 8]. The algorithm starts with a

random population of individuals (chromosomes), and

through genetic processes similar to those occurring in nature,

evolve under specified rules in order to minimize a cost

function. Since the population is generated randomly, the GA

is able to virtually search the entire solution space, and

provide simultaneous searches at different points in this

space. During the algorithm execution, the chromosomes that

possess the best characteristics (lowest cost) generate

offspring, improving the average cost value of the population

as a whole.

Due to the quantity of parameters and the stochastic nature

of the process, the algorithm can converge to different resultseach run or be confined in a local minimum point. To avoid

this problem, sufficiently large population values, rates of

mutation and crossover can be adjusted to ensure a good

population diversity. Moreover, elitism can be used to ensure

the survival of the best chromosomes for the next generation.

In this article, the GA objective is to determine Q and R

matrices so that the LQR presents small overshoot in the

event of a load disturbance. For this, each chromosome is

composed of the genetic structure defined in Table II, which

corresponds to elements of the weighting matrices given by

11

22

33

11

0 0

0 00 0

GA

GA

q

qq

R r

Q . (20)

The cost function to be minimized by the genetic algorithm

is defined by

25 5te e

GA tb b

tJ ovr t e t dt ovr ITSE

t , (21)

where ovr is the percent overshoot, e the error with respect

to the reference, bt and et are the times of beginning and end

of the load disturbance, respectively.

The solution process of the algorithm follows the steps:

i. Randomly creates the initial population;ii. Determines the LQR gains for each chromosome;

iii. Simulates the closed loop system for eachchromosome, reducing the load by 25% in 20 ms;

iv. Calculates the cost function (21) for each simulationresult;

v. Sort results based on their fitness values. If the stopcondition is met, skip to step vii;

vi. Creates the next generation from changes in thechromosome of a single parent (mutation) or

combination of genes from a pair of chromosomes

(crossover). Return to step ii;

Input current

Output voltage

K1

K2

K3

~d

D

XLq

XCq

Vref

+

+

+++

d(t)+

+

Fig. 2 Block diagram of the controller.

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vii. Determines the best weighting matrices based on thebest chromosome;

viii. Terminates the algorithm.In this article the population is kept fixed in 300

chromosomes, limited to the range [1106; 1107], and the

two best chromosomes of each generation have ensured its

perpetuation (elitism). Of the remainder, excluding the

elitism, each new generation is composed of 55% ofcrossover between pairs of chromosomes and the other 45%

are generated by mutations.

After about 150 generations the best chromosome found

was the presented in Table III, leading to the gain vector

defined in (23).3 3

100 80.1262 10 9.4583 10 -29.433GAK (22)

Fig. 3 presents the results using the gain vector (22) for the

same conditions of the previous simulation. In P1G is

observed an overshoot of 34.52% at 20.53 ms. In P2G the

undershoot is 27.04% at 35.48 ms. As shown in Fig. 3, the

LQR optimized by GA provided a superior performance

when compared with the controller designed previously.IV. MULTIPLE CONTROLLERS

As can be observed by the system model, the variation of

load resistance directly changes the dynamics of the circuit

and also the equilibrium point. Although shown above that

the controllers were able to maintain a good performance

even with a load variation of 100% to 25%, it can be expected

that controllers designed to work in specific ranges of the

power load will ensure a better performance, when properly

selected or combined.

The equilibrium points of the model, as well the load

resistance and output current for different output power are

given in Table IV.These four load situations will be used to design multiple

controllers and to verify the control performance for the boost

converter analyzed. Since the load resistance variation

directly reflects the output current of the converter, this

current will be used as decision variable for choosing the

most appropriate controller for each situation.

To design the LQR for each load condition, the augmented

matrices in (10) are recalculated with the appropriate values

of LR , and using the genetic algorithm exposed above, the

control gain vectors from (24) were obtained:3 3

100

3 375

3 350

3 325

80.126 10 9.458 10 29.433

62.290 10 9.400 10 21.585

85.903 10 20.263 10 20.509

48.212 10 15.715 10 10.108

GA

GA

GA

GA

K

K

K

K

(23)

As will be seen in the following sections, besides a suitable

design of the controllers, another crucial factor for the proper

operation of a system with multiple controllers is the

technique used to provide the selection of one of them or to

do the combination of more than one of the controllers.

A. Switching ControllersThe simplest technique for the application of multiple

controllers is a direct switch from one controller to another.

In this case, based on the decision variable, the signal from

the most appropriate controller is selected to be used.

For the case studied here, the output current is divided into

four sections, one for each of the controllers obtained (23).

The thresholds between each section are presented in Table V.

The selection strategy of one of the controllers is carried

out with comparators, as shown in Fig. 4. The signals 1Ctrl ,

2Ctrl , 3Ctrl and 4Ctrl assume values 1 and 0 for active and

inactive, respectively. The signals from each controller are:

1d , 2d

, 3d and 4d

and the resulting control signal d is given

by:1 1 2 2 3 3 4 4d Ctrl d Ctrl d Ctrl d Ctrl d

(24)

observe the signal 1 100GAd K , being obtained as the

variables deviation in relation to the equilibrium point for the

100% load condition. The signals 2d to 4d

are obtained in a

similar way.

The results of this strategy are presented and compared to

the other strategies investigated here in subsection C.

B. Controllers Combined with Fuzzy LogicOne of the problems of switching between controllers is the

discontinuity of the control signal caused by the sudden

transition between different sets of gains. To minimize this

problem and ensure a smooth control signal, a fuzzy logic

20 25 30 35 40 45

150

200

250

300

Outputvo

ltage

(V)

Time (ms)

LQR (conventional)

LQR (GA)P1G

P1C

P2G P2C

Fig. 3 Comparison between the conventional LQR and the LQR obtained byGA, both designed for 100% load situation.

TABLE IIPARAMETERS OF GENES REPRESENTED BY CHROMOSOME.

Gene 1 2 3 4

Parameter 11q 22q 33q 11r

TABLE III

BEST CHROMOSOME FOUND FOR THE LQRFOR THE NOMINAL LOAD CONDITION.

Gene 1 2 3 4

Value 462.66103 155.85103 913.81103 1.054103

TABLE IV

EQUILIBRIUM POINTS OF THE MODEL FOR DIFFERENT OUTPUT POWER.

Outputpower Lq

X (A) CqX (V) LR () oI (A)

100% 26.503 197.89 26.666 7.421

75% 19.930 198.41 35.555 5.580

50% 13.321 198.94 53.333 3.730

25% 6.678 199.46 106.666 1.870

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based supervisor is used.

As shown in Fig. 5, the supervisor control uses two input

variables. The first one, the output current ( oi ) is divided into

four membership functions denominated L (low), ML

(medium low), MH (medium high) and H (high), whichcorresponds to the four ranges of the output power used for

each LQR 25%, 50%, 75% and 100% of the output power,

respectively. The second one, the derivative of the output

current ( oi ), is divided into five membership functions,

denominated NB (negative big), NM (negative medium), Z

(zero), PM (positive medium) and PB (positive big). The oi

variable acts to anticipate the choice of most appropriate

controller in the occurrence of significant load variations.

Before being used by the inference process, the signals oi and

oi are scaled, to adjust them to the variables universe of

discourse. Such scaling gains are defined by:

6

0.1333

16 10i

i

K

K

(25)

being iK just a normalization factor for the maximum output

current, and with iK is determined by trial and error.

The inference process of Takagi-Sugeno has been used

here. The defuzzification method is the weighted average of

the rules activation, expressed in (27).

1

n

i ii

i

w z

Ctrlw

(26)

where represents one of the four outputs of the fuzzy

supervisor, iz is a constant obtained from the rule base and

iw is the value of the activation rule, given by:

AND ,i o ow i i (27)

The AND method uses the product of the membership

values ( ) of the output current ( oi ) and its derivative ( oi ).

One advantage of using a fuzzy supervisor is that the

signals 1Ctrl , 2Ctrl , 3Ctrl and 4Ctrl now can assume any

value between 0 and 1, and the control signal is now obtained

by a combination of each LQR controller action in (24),

resulting in d without sudden changes.

The performance of this strategy compared to others is

shown next.

C. Simulation results and comparisonIn order to verify the results of each control strategy,

simulations have been carried out for load variations

according specifications given by (29):

20 100%

20 30 75%

30 40 50%

40 50 25%50 100%

o

o

o

o

o

t ms P

ms t ms P

ms t ms P

ms t ms P t ms P

(28)

The simulation results are shown from Fig. 7 to Fig. 10, In

these figures, LQRGA identifies the system output when a

single controller (genetic algorithm based controller designed

for 100% of power output) is applied, LQRGAsw represents the

system output for switching between controllers, and

LQRGAfuzzy is the system output for the fuzzy logic controllers

combination.

Despite of the simplicity of using a single controller, the

limitation of this technique (LQRGA) becomes clear for large

load disturbances. The multiple controller based techniques

LQRGAsw and LQRGAfuzzy provide better performance than thesingle controller LQRGA and have similar performance when

directly compared, as shown from Fig. 7 to Fig. 10. The

superiority of the multiple controller based techniques is

confirmed by Fig. 11, which presents a comparison between

the ITSE performance criteria for each situation. In this

figure, the ITSE is shown normalized with respect that of the

single controller LQRGA.

However, the strategy of switching between controllers

present a serious problem when the decision variable takes

values close to the thresholds. To illustrate this problem,

simulations were performed considering load disturbances

according specifications given by (30) and the results are shown

in Fig. 12:20 100%

20 30 87.5%

30 40 62.5%

40 50 37.5%

50 100%

o

o

o

o

o

t ms P

ms t ms P

ms t ms P

ms t ms P

t ms P

(29)

It is clear that the combination of multiple controllers using

fuzzy logic overcomes an important difficulty which affects

the switching controller. In all cases, the fuzzy logic

combination of multiple controllers demonstrates good

TABLE VTHRESHOLDS FOR SWITCHING CONTROLLERS

Thres

ho

ld

Output power (%) Loadcurrent (A)

87.5

62.5

37.5

6.562

4.687

2.812

Output

current

6.562

4.687

2.812

Ctrl1

Ctrl2

Ctrl3

Ctrl4

+

+

+

Fig. 4 Selection strategy used for switching the controllers.

Ctrl1

Ctrl2

Ctrl3

Ctrl4

Fuzzy

Superv

isorio

io

Ki

Ki

1 0.8 0.5 0 0.5 0.8 10

0.5

1

NB NM Z PM PB

(io)

io

0 0.25 0.5 0.75 1

0

0.5

1L ML MH H

(i

o)

io

Fig. 5 Block diagram and membership functions of fuzzy supervisor control.

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behavior for the entire range of operation, with transients with

fast accommodation, few oscillations (even in the case of

operation near thresholds) and small overshoots and

undershoots.

V. CONCLUSIONSThis paper presented as a first contribution the use of a

genetic algorithm to find LQR controllers (local controllers)

suitable to optmise the performance of a boost converter in

terms of overshoot and ITSE criteria for each load condition.

Another contribution is the implementation of a fuzzy logic

that combines the local controllers. This combination

provides a global controller with good transient and steady

state performances. A comparison of this controller with a

conventional LQR controller and with a switched LQR

controller showed the advantages of the proposed strategy,

which can be extended to control other power converters.

ACKNOWLEDGEMENTS

This work is supported by PROCAD (Academic

Cooperation Program) from CAPES (Brazilian Commissionfor Higher Education).

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20 21 22 23 24 25190

195

200

205

210

215

220

225

Time (ms)

Outputvo

ltage

(V) LQRGA

LQRGAswLQRGAfuzzy

Fig. 6 Transient response to load disturbance from 100% to 75%.

30 31 32 33 34 35195

200

205

210

215

220

225

Time (ms)

Outputvo

ltage

(V) LQRGA

LQRGAswLQRGAfuzzy

Fig. 7 Transient response to load disturbance from 75% to 50%.

40 41 42 43 44 45190

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205

210

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220

225

Time (ms)

Outputvo

ltage

(V) LQRGA

LQRGAswLQRGAfuzzy

Fig. 8 Transient response to load disturbance from 50% to 25%.

50 51 52 53 54 55120

140

160

180

200

220

240

Time (ms)

Outputvo

ltage

(V)

LQRGALQRGAsw

LQRGAfuzzy

Fig. 9 Transient response to load disturbance from 25% to 100%.

100% t o 75% 75% t o 50% 50% to 2 5% 25% to 1 00%0

0.2

0.4

0.6

0.8

1

Noma

lize

dITSE

0.3

74

0.4

04

0.1

47

0.1

63

0.0

54

0.0

59

0.7

10

0.6

06

LQRGALQRGAswLQRGAfuzzy

Fig. 10 Normalized ITSE criteria for each load disturbance.

20 25 30 35 40 45 50 55 60150

160

170

180

190

200

210

220

230

Time (ms)

Outputvo

ltage

(V)

LQRGAswLQRGAfuzzy

Fig. 11 Load perturbations to the threshold regions.

110