Analysis and Comparison of Extremum SeekingControl Techniques
Carlos Olalla, Maria Isabel Arteaga, Ramon Leyva, Abdelali El Aroudi
Departament d’Enginyeria Electrònica, Elèctrica i AutomàticaUniversitat Rovira i VirgiliTarragona, SpainEmail: carlos.olalla@urv.cat, mariaisabel.arteaga@urv.cat, ramon.leyva@urv.cat, abdelali.elaroudi@urv.cat
Abstract
—Two nonperturvative extremum seeking controlapproaches are analyzed; the ﬁrst approach needs the sensingof the function’s gradient while the second one does not. Relationships between the algorithms parameters and their dynamicbehavior are found. Also expressions for the steady state errorof both approaches are derived. Finally, these results are used toverify and to compare, by means of simulation, the performanceof both methods.
I. I
NTRODUCTION
Extremum Seeking Control (ESC) considers the problem of tracking an input
x
which optimizes an unknown and usuallytimevariant function
y
(
x
)
,
x
∗
=
arg max
x
∈
R
y
(
x
)
(1)or also
x
∗
=
arg min
x
∈
R
y
(
x
)
(2)where
x
∗
is the argument which maximizes or minimizes thefunction
y
(
x
)
. It is assumed that there only exist a singlemaximum or minimum point in
y
(
x
)
, i.e. that
y
(
x
)
is concaveor convex.Since the srcinal work from Leblanc [1], ESC has beensubject of numerous researches. Reference [2] differentiatedbetween ESC methods which require an external perturbationof the input variable (also known as
perturb and observe
)[3], [4], and ESC methods which do not require an external
perturbation [5], [6], [7] and [8].The present work considers the algorithms which do notneed the external perturbation signal, and states the differencesbetween two of them. Reference [6] is one of the best knownalgorithms in the ﬁeld of Maximum Power Point Tracking(MPPT). It requires sensing the function gradient to switcha relay which sets the direction to the optimum point; thealgorithm is usually called
relay ESC
. Gradient sensors (or differentiators) tend to amplify noise and suffer from instabilityproblems at high frequencies. Usually the differentiator onlyacts for a short known safe bandwidth, which constraints thedinamic performance of the sensor. Nevertheless, references[7] and [8] do not sense the output gradient and are based inthe introduction of Sliding Modes (SM); we will refer to themas
SM ESC
.Both approaches will be analyzed and simulated in order toseek the maximum point (
y
(
x
)
is assumed to be concave )
x
y
t1 t2 t3 x*(t1) x*(t2) x*(t3)
Fig. 1. Timevariant concave function
y
(
x
)
for different time values.
of a function which is similar to the output power of a solarpanel array, and therefore the results could be applied to MPPTsystems (See ﬁg. 1).This work is organized as follows. Section II shows adetailed analysis which results in relationships between thesystem parameters and its dynamic behavior. Section III veriﬁes the results to set up and compare the algorithms by meansof simulation. Section IV presents the performance differencesbetween both methods and gives some conclusions.II. ESC W
ORKING
P
RINCIPLE
A
NALYSIS
This section analyzes the relay ESC and the SM ESC shownin [5], [6] and [7], [8] respectively.For both approaches there exist algorithms using constantand non constant seeking ratio. In this work we considerconstant seeking ratio.
A. Relay ESC
Relay ESC systems change the direction of the seeking inputdepending on the sign of the gradient
dy/dx
, that is obtainedfrom the derivative
g
=
dy/dt.
(3)Fig. 2, which has been adapted from [6], shows the blockdiagram of a relay ESC system. Note the gradient detector
Fig. 2. Block diagram of the relay ESC algorithm.Fig. 3. Block diagram of the relay ESC algorithm with memory.
g
=
dy/dt
and the actuator, where the relay is located andwhose response is described by
= +1
if
sign
dydt
·
sign
dxdt
>
0
=
−
1
if
sign
dydt
·
sign
dxdt
<
0
.
(4)We can describe the dynamic behavior of the state variable
x
as follows
dxdt
=
k
·
=
k
·
sign
dy dt
·
sign
dxdt
,
(5)where
sign
dy dt
=
sign
dy dx
·
dxdt
.
(6)Expression (5) can be simpliﬁed using (6) as follows
dxdt
=
k
·
sign
dy dx
.
(7)An equivalent algorithm can be also implemented using amemory element by
if
dy/dt >
0
Keep the sign of
if
dy/dt <
0
Change the sign of
.
(8)The memory can be implemented with a latch and theresulting block diagram is shown in Fig. 3In [6] it is demonstrated that the algorithm is stable for aconcave function and that the input
x
converges to
x
∗
witha slope
k
. Once the algorithm reaches the optimum pointits control variable
switches at a very high frequency. Tolimit the relay switching frequency several strategies have beenshow in the literature.1) To include a hysteresis in the control law as follows
if
dy/dt
−
∆
>
0
Keep the sign of
if
dy/dt
+ ∆
<
0
Change the sign of
.
(9)Note that the switching frequency depends on
k
,
∆
andthe function to optimize. The maximum error to theoptimum point depends on
∆
.2) To include a constant delay
T
d
to maintain the sign of
for a minimum time. The parameters to set up inthe algorithm are
T
d
and
k
. The switching frequencyis deﬁned by
f
relay
= 12
·
T
d
,
(10)and the maximum error between
x
and
x
∗
is found tobe
e
relay
=
k
·
T
d
,
(11)
B. Sliding Mode ESC
Sliding Mode (SM) ESC block diagram has been depictedin ﬁg. 4, extracted from [8]. The main advantage of this ESCalgorithm is that it does not require gradient sensors. Thebehavior of the block diagram is described by
u
=
dx/dt
(12)
u
=
U
0
sign
(
σ
1
σ
2
)
(13)
σ
1
=
(14)
σ
2
=
+
δ
(15)
=
g
−
y
(16)
dg/dt
=
ρ
+
Mυ
(
σ
1
, σ
2
)
(17)where
U
0
,
δ
,
ρ
,
M
are positive constants and
υ
is a three statesfunction of
σ
1
and
σ
1
as is depicted in ﬁg. 5.If the following inequality holds
M > U
0
df dx
+
ρ,
(18)
Fig. 4. Block diagram of the Sliding Mode ESC algorithm.Fig. 5. Function of the switching element
υ
(
σ
1
σ
2
)
.
then, assuming the initial state
(
σ
1
−
∆)(
σ
2
+ ∆)
>
0
, thechange of
σ
1
and
σ
2
is
˙
σ
1
= ˙
σ
2
=
df dxU
0
+
ρ
+
Mυ.
(19)Thanks to (19),
g
reaches a value close to
y
where
(
σ
1
−
∆)(
σ
2
+ ∆)
<
0
and
υ
= 0
. The required time to reach thissecond state is easily reduced by enlarging
M
. Once
υ
= 0
,the change ratio of
σ
1
and
σ
2
is
˙
σ
1
= ˙
σ
2
=
ρ
−
df dxU
0
sign
(
σ
1
σ
2
)
,
(20)while the following condition holds
df dx
U
0
> ρ.
(21)In this second state (
υ
= 0
), depending on the sign of
df/dx
,
σ
1
or
σ
2
will tend to zero, because in this state theirsign is opposite. Furthermore, any change in the sign of
σ
1
or
σ
2
changes the sign of its derivative, and therefore a slidingmode is srcinated in the system. In an ideal framework,
u
switches at inﬁnite frequency and the value of
σ
1
or
σ
2
iszero. If
υ
= 0
then we can state that the reference
g
increasesmonotonically with ratio
ρ
and therefore we can say that
y
follows the increasing reference, since
σ
1
=
g
−
y
≈
0
,
(22)
σ
2
=
g
−
y
+
δ
≈
0
.
(23)Therefore the output
y
increases to a certain point close tothe maximum, where (21) does not hold.In this third state the output
y
does not increase monotonically with
ρ
, and (23)(22) is not true anymore. However,
υ
maintains
σ
1
and
σ
2
inside the range
σ
1
<
∆
and
σ
2
>
−
∆
.Hence it has been demonstrated that the input
x
is keeps closeto the maximum point
x
∗
.In [7] it is shown that under these conditions
σ
1
or
σ
2
reaches
+∆
or
−
∆
during
t
1
and
t
2
, which are deﬁned by
t
1
= ∆
ρ
−
U
0
df dx
,
(24)
t
2
= ∆
ρ
+
U
0
df dx
.
(25)If we assume that
ρ
U
0

df/dx

in the vicinity of
x
∗
,then the time period is described by
t
12
=
t
1
+
t
2
= 2∆
ρ .
(26)If we approximate (20) by
˙
σ
1
= ˙
σ
2
=
ρ,
(27)then
σ
1
(or
σ
2
) will reach
+∆
(or
−
∆
) at the time
t
12
, when
υ
will be again different from zero and
σ
1
will return to
+∆
in a time close to zero, as
M
is very large.Therefore we can consider that when
x
is close to
x
∗
thereexist a sliding mode whose switching frequency is
f
sm
=
ρ
2∆
,
(28)
05101520250102030405060708090x
y
t1 t2 t3
Fig. 6. Timevariant
y
=
f
(
x
)
used for simulation.
and that the maximum error between
x
and
x
∗
is
e
sm
= ∆2
ρU
0
,
(29)The required parameters to set up the SM ESC algorithmare
∆
,
δ
,
M
,
U
0
and
ρ
. Note that the following relationshipsmust hold,
∆
< δ,
(30)
M
U
0
,
(31)
M
ρ,
(32)in other words,
U
0
must be chosen sufﬁciently large to meet(21) and sufﬁciently small to meet (23).III. S
IMULATION
R
ESULTS
In the previous section we have derived the steady stateswitching frequency and error of the two ESC algorithms of interest. Now we will ﬁx a switching frequency of 10 kHz toset up the simulation and to compare the performance of themethods.The objective function to optimize is shown in ﬁg. 6. Itsresponse is very similar to the one that could be found in apower limited current source, having a variable power limit.These are the usual conditions in photovoltaic applications andtherefore the work could be ported to MPPT systems.
A. Relay ESC
We set a relay ESC system with constant seeking slope
k
and delay time
T
d
. If we desire a steady state switchingfrequency of 10 kHz, then
T
d
= 50
µs.
(33)At the same time, if we consider that the minimum value of the input
x
is
x
min
= 10
, then we can set a maximum steadystate error of 0.1% as follows
K
=
T
d
e
relay
= 200
.
(34)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.021010.51111.51212.51313.514t (s)
x
Relay ESCSM ESC
Fig. 7. Seeking transient for
x
0
< x
∗
.
B. Sliding Mode ESC
For the SM ESC, we must fulﬁll
∆
< δ
∆ = 0
.
02
, δ
= 0
.
05
(35)Using (28) we meet the steady state switching frequency
ρ
= 400
.
(36)Then we choose
M
ρM
= 1
·
10
5
.
(37)Finally we must fulﬁll (21) and (23)
U
0
= 900
,
(38)which yields a steady state error of
e
SM
= 0
.
0225
→ ≈
0
.
2%
(39)
C. Results
Fig. 7 shows the transient response when the initial condition of
x
is
x
0
< x
∗
. Similarly, ﬁg. 8 shows the transientwhen
x
0
> x
∗
.The steady state error has been depicted in ﬁg. 9. Note thatthe switching frequency is equal in both cases.Finally, ﬁgs. 10, 11 and 12 show the transient response tochanges in
f
(
x
)
. Fig. 10 shows the response to step changes inthe maximum power point. Fig. 11 shows the transient whenthe maximum power point increases with a constant slope andﬁg. 12 shows a
x
−
y
detail under these conditions.From the simulation results we conclude that1) Due to the working principle of the SM ESC, it isconﬁgured to run in a single direction (up or down)depending on
ρ
, and its performance is degraded whenit runs in the opposite direction.2) In the SM ESC, the slope of the input signal increasesas
df/dx
decreases, while the slope is constant in therelay ESC.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0512.51313.51414.51515.5t (s)
x
Relay ESCSM ESC
Fig. 8. Seeking transient for
x
0
>x
∗
.
0.01 0.0101 0.0102 0.0103 0.010412.9512.9612.9712.9812.991313.0113.0213.0313.0413.05t (s)
x
Relay ESCSM ESC
Fig. 9. Steady state error.
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0512.51313.51414.5t(s)
x
Relay ESCSM ESC
Fig. 10. Seeking transient for a step perturbation.
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.112.91313.113.213.313.413.5
x
t(s)Relay ESCSM ESC
Fig. 11. Seeking transient for a ramp perturbation.
12.9 12.92 12.94 12.96 12.98 13 13.02 13.04 13.06 13.08 13.158.358.3258.3458.3658.3858.458.4258.4458.4658.48x
y
Relay ESCSM ESC
Fig. 12. xy detail of the seeking transient for a ramp perturbation.
3) Due to the previous facts, the relay ESC performance isbetter for the initial transient, when the initial conditionsare far from the maximum point.4) The relay ESC steady state error is lower than theobserved in the SM ESC.5) The SM ESC is able to track the ramp perturbation,maintaining the error close to the steady state value.However the relay ESC shows larger oscillations whenthe perturbation appears. This is due to the fact thatthe gradient detector can show positive readings whenthe maximum power point is moved, despite that thegradient of
f
(
x
)
may be negative.IV. C
ONCLUSIONS
This paper showed 2 nonperturbative ESC techniques andcompared their performance for the same switching frequency.We observed that the relay ESC has lower steady state errorand better performance under the initial conditions. Alsothe relay ESC showed large oscillations for changes in themaximum point, due to the gradient detector limitations. Onthe other hand, we observed that the SM ESC has the samesteady state error during the ramp perturbation. However thereare not
signiﬁcant
differences and both methods are verysimilar in terms of performance.The implementation of the SM ESC is constructed with alarger number of relays and integrators, but it does not requirea differentiator to sense the gradient. Nevertheless the relayESC block diagram is simpler, but it requires a differentiator,which may present several difﬁculties. Therefore SM ESCare a suitable alternative to avoid the use of differentiatorsin MPPT applications.R
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