Download - Pinheiro_05472661
-
7/28/2019 Pinheiro_05472661
1/6
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)
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
-
7/28/2019 Pinheiro_05472661
2/6
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
106
-
7/28/2019 Pinheiro_05472661
3/6
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.
107
-
7/28/2019 Pinheiro_05472661
4/6
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
108
-
7/28/2019 Pinheiro_05472661
5/6
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.
109
-
7/28/2019 Pinheiro_05472661
6/6
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).
REFERENCES
[1] J. G. Kassakian, M. F. Schlecht, and G. C. Verghese, Principles ofpower electronics. Addison-Wesley, Reading, Mass., 1991.
[2] R. W. Erickson and D. Maksimovic, Fundamentals of powerelectronics, 2nd ed. Kluwer Academic, Norwell, 2001.
[3] B. K. Bose,Power Electronics and Motor Drives: advances and trends.Academic Press, Burlington, 2006.
[4] G. Liping, J. Y. Hung, and R. M. Nelms, "Evaluation of DSP-BasedPID and Fuzzy Controllers for DC/DC Converters," in IEEETransactions on Industrial Electronics, vol. 56, pp. 2237-2248, 2009.
[5] P. J. Antsaklis and A. N. Michel,A linear systems primer. Birkhuser,
London, 2007.[6] R. L. Haupt and S. E. Haupt,Practical Genetic Algorithms, 2nd ed. JohnWiley and Sons, Hoboken, 2004.
[7] C. Wongsathan and C. Sirima, "Application of GA to design LQRcontroller for an Inverted Pendulum System," inIEEE InternationalConference on Robotics and Biomimetics, 2008. ROBIO 2008.,pp. 951-954, 2009.
[8] M. B. Poodeh, S. Eshtehardiha, A. Kiyoumarsi, and M. Ataei,"Optimizing LQR and pole placement to control buck converter bygenetic algorithm," inInternational Conference on Control, Automationand Systems, 2007. ICCAS '07,pp. 2195-2200, 2007.
[9] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design andAnalysis: a linear matrix inequality approach. John Wiley and Sons,New York, 2001.
[10] P. Mattavelli, L. Rossetto, G. Spiazzi, and P. Tenti, "General-purposefuzzy controller for DC/DC converters," in Tenth Annual AppliedPower Electronics Conference and Exposition, 1995. APEC '95.
Conference Proceedings 1995,vol. 2, pp. 723-730, 1995.[11] A. R. Ofoli and A. Rubaai, "Real-time implementation of a fuzzy logic
controller for switch-mode power-stage DCDC converters," in IEEETransactions on Industry Applications, vol. 42, pp. 1367-1374, 2006.
[12] V. F. Montagner, R. C. L. F. Oliveira, and P. L. D. Peres, "ConvergentLMI Relaxations for Quadratic Stabilizability and H Control ofTakagi-Sugeno Fuzzy Systems," in IEEE Transactions on FuzzySystems, vol. 17, pp. 863-873, 2009.
[13] R. C. Dorf and R. H. Bishop, Modern control systems, 11th ed.Pearson/Prentice Hall, Upper Saddle River, NJ, 2008.
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
195
200
205
210
215
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