Analysis and Comparison of Extremum Seeking Control Techniques

Analysis and Comparison of Extremum Seeking Control Techniques
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  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:,,,  Abstract  —Two non-perturvative extremum seeking controlapproaches are analyzed; the first approach needs the sensingof the function’s gradient while the second one does not. Rela-tionships 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 usuallytime-variant 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 dif-ferentiators) 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. Time-variant 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 veri-fies 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 mono-tonically 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. Time-variant  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 con-dition 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. x-y 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 non-perturbative 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 EFERENCES[1] M. Leblanc, “Sur la electrification des chemins de fer au moyen decoraunts alternatif de frequence elevee,”  Revue Generale de l’Electricite ,vol. XII, pp. 275–277, 1922.[2] I. S. Morosanov, “Methods of extremum control,”  Automatic and RemoteControl , vol. 18, pp. 1077–1092, 1957.[3] M. Krstic and H. Wang, “Design and stability analysis of extremumseeking feedback for general nonlinear systems,” in  Decision and control,Proceedings of the 36th IEEE Conference , 1997.[4] M. Krstic and K. B. Ariyur,  Real Time Optimizationby Extremum SeekingControl . Wiley Interscience, 2003.[5] J. L. Abatut, “Etude de rergulation extremale en vue notamment del’optimization d’une unité industrialle de reforming catalytique,” Ph.D.dissertation, Université de Toulouse, 1965.[6] R. Leyva, C. Alonso, I. Queinnec, A. Cid-Pastor, D. Lagrange, andL. Martinez-Salamero, “Mppt of photovoltaic systems using extremum- seeking control,”  Aerospace and Electronic Systems, IEEE Transactionson , vol. 42, no. 1, pp. 249–258, 2006, 0018-9251.[7] S. K. Korovin and V. I. Utkin, “The use of the slip mode in problems of static optimization,”  Automatic and Remote Control , pp. 50–60, 1972.[8] S. K. Korovin and V. I. Utkin, “Using sliding modes in static optimizationand nonlinear programming,”  Automatica , vol. 10, pp. 525–532, 1974.
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