ANFIS Hybrid Reference Control for Improving Transient Response of Controlled Systems Using PID Controller

Transcription

1 Ths artcle can be cted as E. Joelanto, D. C. Anura and M. P. Pryanto, ANFIS Hybrd Reference Control for Improvng Transent Response of Controlled Systems Usng PID Controller, Internatonal Journal of Artfcal Intellgence, vol. 10, no. S13, pp , Copyrght 2013 by CESER Publcatons ANFIS Hybrd Reference Control for Improvng Transent Response of Controlled Systems Usng PID Controller Endra Joelanto 1, Deddy Candra Anura 1 and Muhammad Pratkto Pryanto 1 1 Instrumentaton and Control Research Group Faculty of Industral Technology Bandung Insttute of Technology, Bandung 40132, Indonesa E-mal: ejoel@tf.tb.ac.d The standard PID controller s known to have performance lmtatons as t must trade off the transent response performance and the dsturbance attenuaton level. The paper proposes a hybrd reference control (HRC) wth adaptve neuro-fuzzy nference system (ANFIS) to mprove transent response performance of PID controller. The ANFIS s used to manpulate the set-pont of the PID controller n a specfc manner such that the transent response s mproved by learnng from expermental data. In ths structure, the ANFIS-HRC deals wth the achevng good transent response performance, whereas the PID controller stablzes the closed loop system and defnes the dsturbance attenuaton level. As a result, the transent response performances and the dsturbance attenuaton can be desgned ndependently. KEY WORDS: ANFIS; Hybrd reference control; Subtractve clusterng; PID controller; transent response MATHEMATICS SUBJECT CLASSIFICATION (MSC): 00A72, 68T05, 93C05, 93C42, 93C83, 93D99 COMPUTING CLASSIFICATION SYSTEM (CCS): B.1.1 B.4.3, C.5.3, D.1.7, G.1.6, I.2.3

2 1. INTRODUCTION PID (proportonal ntegral dervatve) controller s known to have nherent lmtatons n resultng smultaneously conflctng control desgn objectves. A PID controller cannot be tuned optmally to satsfy both requrements,.e. faster transent response to set-pont changes and good dsturbance rejecton. Many methods have been developed to deal wth the lmtatons, such as fuzzy PID selftunng, fuzzy PID swtchng, fuzzy precompensator PID, etc. Each method clams mprovement over the conventonal PID controller and superorty over other methods. Many publshed papers manly focused on the selecton of the three parameters of the PID controller as the applcaton of fuzzy system n mmckng the knowledge of the operators (Msr and Malk, 2006), (Mohan and Snha 2008). However, there has been lttle attenton to mplement fuzzy logc to perform a smlar way to an expert operator who suppresses overshoot by ether ncreasng or decreasng the nput of the controlled process. Other successful examples of the applcaton of fuzzy logc n control system desgn as can be found n (Chu, 1998), (Devasenapat and Ramachandran, 2011), (Precup et al., 2012), (Davd et al., 2012). A combnaton of PID controller usng a fuzzy expert control technque to produce the response stays at a predefned curve wth mnmal overshoot was consdered n (Yasuda et al., 1990). A dfferent method by usng fuzzy logc was nvestgated n (Km et al., 1994). The method s called as a fuzzy precompensator that compensates the set-pont of the conventonal closed loop feedback control usng PID controller. The method uses the error, the change of error, and a correcton term to compensate the set-pont by smply takng the sum of the nonlnear mappng of the the error and the change of error wth the correcton term. The fuzzy logc rule s obtaned by tral and error to compensate the overshoots and undershoots present n the output response when the plant has unknown nonlneartes that can yeld sgnfcant overshoots and undershoots n a conventonal PID controller. Further extenson of the method was developed n (Pratumsuwan and Thongchasuratkrul, 2011) that combned the advantages of both fuzzy pre-compensated PID controller (Km et al., 1994) and fuzzy precompensated fuzzy controller (Pratumsuwan and Thongcha, 2009) to form a precompensaton of a hybrd fuzzy PID controller. The method was amed to result n a fast rse tme, to produce a small overshoot, and to correct the poston wth respect to the set pont. Motvated by the method for mprovng transent response performances called hybrd reference control (HRC) developed n (Joelanto and Wllamson, 1997), a fuzzy logc based hybrd reference control (FHRC) to mprove transent responses of PID controller was consdered n (Joelanto and Tansr, 2007). The man advantage of the HRC method s the capablty to yeld deadbeat response at any predefned tme as the optmal soluton. The FHRC method s amed to control the reference sgnal (set-pont) nto a partcular temporary reference sgnal for transent response performances mprovement durng dsturbances or smply the error wth respect to the default reference sgnal s consdered very bg. In ths method, the FHRC manpulates the set-pont of the PID controller n a prelearnng manner that wll mprove the transent response performance when the dsturbance rejecton

3 propertes have been establshed by PID parameters tunng. Further extenson of a combnaton of PID controller wth fuzzy hybrd reference control (FHRC) was carred out n (Joelanto and Stanggang, 2009) by adaptng substractve clusterng n order to determne the number of membershp functons and membershp functons n a short tme. The fuzzy membershp functons and the rule base were obtaned by usng the substractve clusterng method (Chu, 1994), (Chu, 1997), (Hammouda and Karray, 2000). The combnaton of PID controller wth fuzzy hybrd reference control (FHRC) offers sgnfcant mprovement as the set-pont performance can be ndependently acheved wthout affectng the dsturbance rejecton capabltes. The stablty analyss of FHRC was also derved n (Joelanto and Stanggang, 2009) based on the state space representaton of the PID controller, detal descrpton can be found n (Joelanto, 2011) and the hybrd reference control analyss (Joelanto and Wllamson, 1997), (Joelanto and Wllamson, 2009) wth the help of the nternal model prncple (Francs and Wonham, 1976). On the other hand, adaptve neuro-fuzzy nference system (ANFIS) has been known to have good features from the fuzzy logc and neural networks. ANFIS as developed by (Jang et al., 1997) s a class of adaptve networks that s functonally equvalent to fuzzy nference systems, where the parameters of fuzzy nference systems are updated by neural networks from a set of tranng data. ANFIS has the advantages clamed by neural networks (NNs) and the lngustc nterpretablty of Fuzzy Inference Systems (FIS), wheren both NNs and FIS play actve roles n an effort to reach specfc goals. ANFIS has been successfully mplemented n ranfall-runoff predcton of ntermttent rver (Keskn et al., 2006), (Aql et al., 2007), (Jothprakash et al., 2009), evapotranspraton from weather forecast (Ca and Mu, 2005), stock market and fnancal decson (Patel and Marwala, 2006), (Cheng et al., 2007), tme seres predcton of earthquake events (Joelanto et al, 2009), complex large scale systems (Buragohan and Mahanta, 2008), etc. The adaptve capablty of ANFIS makes t almost drectly applcable to adaptve control and learnng control. The nonlnearty and structural knowledge representaton of ANFIS are the prmary advantages over classcal lnear approaches such as n control systems. The paper consders development of ANFIS-HRC to the PID controller n order to mprove transent response performances and ts applcaton to speed control of AC-motor. The prevous verson of the paper has appeared n (Joelanto and Anura, 2011). Closed loop stablty propertes are also brefly dscussed. 2. ANFIS HRC The structure s referred as ANFIS Hybrd Reference Control abbrevated as ANFIS-HRC.

4 Fgure 1. Block Dagram of ANFIS-HRC-PID Controller. Fgure 1 shows the dagram block of ANFIS-HRC-PID controller whch has the same structure as FHRC n (Joelanto and Transr, 2007), (Joelanto and Stanggang, 2009). In Fgure 1, the sgnal d (t) s the output of Fuzzy system that changes temporary the default set-pont r(t) durng transent response. The acton of the Fuzzy system s defned by an enable event t k s detected by the performance observer embedded n the fuzzy system. Ths event nforms the performance observer that the devaton of the closed loop system output ( y (t) ) to the default reference sgnal ( r (t) ) s bgger than the prescrbed tolerance δ such that e ( t) = y( t) r( t) δ (1a) The fuzzy system sends the reference sgnal ( d (t) ) ether contnuously or n a predefned tme nterval (τ ) untl the performance observer detects another dsable event where the error of the closed loop system s now enterng the allowable tolerance defned by e ( t) = y( t) r( t) < δ (1b) When ths event s detected, the fuzzy system then stops sendng reference sgnals ( d (t) ) and return to the default reference sgnal ( r (t) ) by sendng the sgnal d ( t) = 0. The closed loop stablty propertes of the control system wth FHRC have been derved n (Joelanto and Stanggang, 2009) based on the analytcal results on hybrd reference control (HRC) developed n (Joelanto and Wllamson, 2009). The stablty condtons for ANFIS-HRC follow drectly by employng the same arguments as n (Joelanto and Stanggang, 2009) by replacng fuzzy logc wth ANFIS and by consderng samplng tme as an event. The asymptotc stablty of ANFIS-HRC-PID controller s guaranteed f the reference sgnals generated by the ANFIS are admssble reference sgnals then the closed loop system s asymptotcally stable. The admssble reference sgnals refers to a condton that the reference sgnal eventually becomes zero such that the output of the controlled system wll track the orgnal reference sgnal. Hence, the asymtotc stablty follows from the asymtotc stablty of the closed loop system controlled by the PID controller.

5 The PID controller n Fgure 1 s descrbed by the followng equaton t 1 de u( t) = K c e + edt + Td (2) T dt 0 where u (t) denotes the manpulated varable of the plant or represents the control sgnal. The three parameters of the PID controller are denoted by K c (the controller gan), T (the ntegral tme) and T d (the dervatve tme). The mplementaton of the PID algorthm (2) n Labvew makes modfcatons, especally n the ntegral part by usng trapezodal ntegraton to avod sharp changes and n the dervatve part by takng the dervatf to the process varable to prevent dervatve kck (Natonal Instrument, 2001) ANFIS The block dagram of the ANFIS that was frst proposed by (Jang et al.,1997) s shown n Fgure 2. Each layer conssts of several nodes denoted by square and crcle. Nodes n the same layer l have the same output functon at node s denoted by O l. In prncple, ANFIS s an adaptve network conssts of nodes and drectonal lnks whch form nodes connectons. Based on the type of network, all or some of the nodes are adaptve and t s the task of the learnng rules to tune the nodes accordng to an error measure. Fgure 2. Block Dagram of ANFIS.

6 For smplcty, t s assumed that the consdered fuzzy nference system has two nputs x and y and one output f. A common rule set for a frst-order Sugeno fuzzy wth two fuzzy f-then rules s gven by Rule 1: If x s A1 and y s B 1, then f 1 = p1x + q1 y + r1 (3) Rule 2: If x s A2 and y s B 2, then f 2 = p2 x + q2 y + r2. (4) The mechansm n the forward pass can then be explaned as follow: Layer 1 Nodes n ths layer are adaptve nodes whch represent the membershp grade of nputs x and y of fuzzy sets A and B respectvely. The output of the node s denoted by a node functon O 1 μ A ( x), = μ ( x), B( 2) = 1,2 = 3,4 (5) The varables A and B are the lngustc labels such as small, medum, large, etc. The membershp functons of A and B are Gaussan membershp functons defned by μ 1 x c ( x) = exp 2 σ 2 Gaussan (6) The premse parameters { c, σ} are adaptve and determne the shape of the membershp functon. They represent the varous types of the membershp functon of fuzzy sets A and B. Layer 2 Each node n ths layer s not an adaptve node whch s denoted by Π. The output of the product layer at node s gven by the followng equaton O 2 = w = μ ( x) μ ( y), = 1,2 (7) A B The output of ths layer performs as the weght of each fuzzy rule usng the t-norm fuzzy operator. Layer 3 Each node n ths layer s a non adaptve node and s denoted by Ν. Each node normalzes the weght functons ( w ) obtaned from the product layer. The normalzaton s carred out usng the followng equaton 3 w O = w =, = 1,2 (8) w + w 1 2

7 Layer 4 Each node has defuzzfed output whch s computed usng the followng equaton 4 O = w f = w ( px + q y + r ), = 1,2 (9) where w denotes normalzed actvaton functon from the layer 3. Parameters p, q, r } are { consequent parameters n fuzzy rules of the correspondng node. Nodes n ths layer are adaptve n nature. Layer 5 The layer denoted by Σ s non adaptve and produces output functon by addng all nputs from the prevous layers. The outputs are calculated usng the equaton w f 5 O = w f =, = 1,2 (10) w Although the Mackey-Glass s often used to develop model (Jang et al., 1997), there are no establshed rules to buld an ANFIS model. Tral error method s the most and popular way n the model development. The objectve of an ANFIS s to obtan a relatonshp of the form: Y = f ( X n ) (11) m where X n denotes an n-dmensonal nput vector consstng of varables { x1, x2, L, xn} and Ym s an m-dmensonal output vector of nterest y, y, L, y }. { 1 2 m ANFIS uses hybrd learnng algorthm whch s a combnaton of Error Backpropagaton (EBP) learnng algorthm to update the nonlnear prems parameters and Recursve Least Square Error (RLSE) to update lnear consequent parameters. By usng hybrd learnng algorthm, ANFIS has been known to produce good model closed to the system of nterest. Detal hybrd learnng process n ANFIS can be found n (Jang et al., 1997). In the ANFIS-HRC structure, ANFIS represents the relaton between the error, the ncrement of error and the reference sgnal as a functon of the dynamcs of the plant and the PID controller. To obtan the dynamcs between the error, the delta error and the requred set-pont compensaton, ANFIS uses system dentfcaton process by means of numercal learnng conssts of two stages,.e. structure dentfcaton and parameter dentfcaton. Structure of the dentfcaton determnes an optmal f-then rules of fuzzy nference system, whle parameter dentfcaton s related to fndng the system parameters such as membershp functons, lnear coeffcents, etc.

8 2.2. Subtractve Clusterng Clusterng s a method of groupng data n order to determne ther structure where data wth same characterstcs wll be n the same group (Chu, 1994). Suppose there are n data x, x, L, x } n M dmenson that has been normalzed. Let * x 1 and { 1 2 n * P 1 be the locaton of the frst cluster center and ts correspondng potental value respectvely. The potental of each data pont x * 1 s then revsed by usng the potental equaton P 2 2 * 2 ( ) x x1 * rb 1 e P P (12) where r > 0 s a constant. Equaton (12) shows an amount of potental from each data pont as a b functon of ts dstance from the frst cluster center. The data ponts near the frst cluster center wll have greatly reduced potental, and t wll not be selected as the next cluster center. The constant r b denotes the radus defnng the neghborhood whch wll have measurable reductons n potental. To avod obtanng closely spaced cluster centers, r b s set to be greater than r a, a good choce s rb = 1. 5r a (Chu, 1994) or rb = 1. 25ra (Chu, 1997). Next, the potentals of all data ponts are revsed by usng equaton (4), the data pont wth the hghest remanng potental s chosen as the second cluster center. The process s then contnued untl the th k, = 1, L, n cluster center have been selected, the potental of each data pont s revsed by the followng formula P 2 2 * 2 ( ) x xk * rb 1 e P P (13) * where x k s the locaton of the k th cluster center and P * k s ts correspondng potental value. The process of acqurng new cluster center and revsng potentals s repeated untl the remanng * potental of all data ponts falls below some fracton of the potental of the frst cluster center P 1. Although the number of clusters (or rules) s automatcally determned by the method, t should be noted that the user specfed parameter r a (the radus of nfluence of a cluster center) strongly affects the number of clusters that wll be generated. A large r a generally results n fewer clusters wth a coarser model, whle a small r a can produce excessve number of clusters and a model that does not generalze well (by over-fttng the tranng data). Therefore, the constant r a acts as a tunng parameter of the desred resoluton of the model, whch can be chosen based on the resultant complexty and generalzaton ablty of the model. It s clear

9 that choosng r a very small or very large wll result n a poor accuracy because f r a s chosen very small the densty functon wll gnore the effect of neghborng data ponts; whle f taken very large, the densty functon wll take nto account all the data ponts n the data space. Chu (1997) suggests that the good value of r a s between 0.2 and 0.5, whle Hammouda and Karray (2000) show that a value of r a between 0.4 and 0.7 s adequate. In ths paper, the substractve clusterng approach s used to fnd the ntal membershp functons of the ANFIS-HRC system wth less number of rules and mnmum amount of computatonal tme. The desgn steps of ANFIS-HRC are shown n the flowchart presented n Fgure 3.

10 START Intalze δ Collect data & Normalzaton Intalze, s a f,ε, ε r Substractve clusterng n MATLAB Membershp functons and FIS n MATLAB ANFIS n MATLAB Develop ANFIS n LABVIEW Smulaton n MATLAB Experment n LABVIEW Transent response NO Transent response NO YES YES STOP STOP Fgure 3. ANFIS-HRC Desgn Step.

11 3. ANFIS HRC DESIGN AND SIMULATION To develop ANFIS-HRC, the followng steps are requred: Generate data tranng set consst of error, delta error and ncrement set-pont. Apply clusterng method. Tran ANFIS usng data tranng set and the obtaned membershp functon from clusterng method. Fgure 4. Archtecture of ANFIS wth 4 membershp functons and 4 rules. For smulaton, let consder a seond order system wth delay tme gven by the followng transfer functon. G( s) = 2 s 30s 5e + 9s + 20 (14) The parameters of the PID controller are selected as K = 0. 05, T = 0. 6 seconds, dan T = 15 seconds and the range of data are as follow: e ( k) [ 1,1 ], Δe ( k) [ 1,1 ], and o ( k) [ 3.3]. Frst, t s necessary to generate data by smulatng the closed loop system controlled by the PID controller wth manpulate the set-pont. Next, the membershp functons and the number of the membershp functons are found usng the followng parameters: r = 0. 2 ( r = r Δ = r ); s = 1. 5 (squash factor); = 0. 5 ε (accept rato); ε = 0. 3 a c ae (reject rato). Fgure 5 shows the data used to tranng ANFIS and the respectve cluster centers obtaned from substractve clusterng. In (Matlab 2012), the squash factor s to specfy that the cluster wll be far from each other gven by the followng equaton r = s r. b f a a e ao f d

12 Fgure 5. Sgnal data from error, ncrement error and set-pont.

13 Fgure 6. Membershp functons error sgnal and delta error sgnal. Intal and fnal rule base of the fuzzy nference system (FIS) obtaned from substractve clusterng and from ANFIS error backpropagaton (EBP) learnng are gven n the followng. Intal FIS Fnal FIS If error s E1 and ncrement error s DE1 then reference1 s 1.648(E1) (DE1) If error s E2 and ncrement error s DE2 then reference2 s 1.012(E2) (DE2) If error s E3 and ncrement error s DE3 then reference3 s 1.054(E3)+1.269(DE3) If error s E4 and ncrement error s DE4 then reference4 s (E4) 1.235(DE4) If error s E1 and ncrement error s DE1 then reference1 s 1.521(E1) (DE1) If error s E2 and ncrement error s DE2 then reference2 s 1.083(E2) (DE2) If error s E3 and ncrement error s DE3 then reference3 s 1.428(E3)+2.927(DE3) If error s E4 and ncrement error s DE4 then reference4 s (E4) 0.948(DE4) Fgure 7 depcts the 3D plot of the fuzzy nference system of the ANFIS-HRC after EBP learnng.

14 Fgure 7. 3-D plot of FIS after EBP learnng. Fgure 8 shows the transent response performances of the closed loop system wth the PID controller, Fuzzy HRC-PID and the ANFIS-HRC-PID controller. It can be seen that the ANFIS-HRC-PID results n faster transent response wth smaller overshoot and control sgnal magntude compare wth the PID controller. Table 1 shows the value of transent response crtera (rse tme T r, settlng tme Ts and maxmum overshoot % M p ) and ntegral error crtera (ntegral squared error (ISE), ntegral absolute error (IAE)). Ether transent response or ntegral error crtera of the ANFIS-HRC-PID s better than the PID controller. Fgure 9 shows the comparson of the response when the orgnal setpont s ncreased and decreased over a perod of tme. In ths case, the ANFIS-HRC-PID s stll outperform the PID controller.

15 Fgure 8. Transent response performance and the control nput. Table 1. Transent Response and Integral Crtera. T r (s) T s (s) %M p u p 2 u dt ISE IAE PID ANFIS- PID

16 Fgure 9. Transent response performance and the control nput wth changng the orgnal set-pont 4. IMPLEMENTATION ON SPEED CONTROL OF AC-MOTOR Ths part presents an mplementaton of ANFIS-hybrd reference control (HRC) for speed control of AC-motor. Fgure 10 shows the block dagram of the mplementaton of ANFIS-HRC to control ACmotor, the analog to dgtal and dgtal analog converter and an nverter. The hardware and the wrng are shown n Fgure 11. The actvaton condton δ (n the equaton (1a) and the equaton (1b)) to ntate the reference sgnal changes n HRC s set as δ = 2 % of the steady state error. Fgure 10. Block Dagram of Blok ANFIS-HRC-PID controller for Speed Control of AC-motor

17 Fgure 11. Hardware Confguraton of Expermental Setup. Devce specfcatons n the hardware confguraton are as follow: AC-motor: 3-phases nducton motor, wth poles:4, Output : 0.25 HP/0.18 KW, Volt: 220/330 V, AMP: 1.1/0.64, RPM: 1345 rpm Inverter: Altvar 31, ATV31HU11M2 type, Input power: 0,5 HP or 0,37 KW Analog tp Dgtal/Dgtal to Analog: Natonal Instrument (DAQ NI-USB 6008), Input: 0-10 V DC, Output: 0-5 V DC LabVew Software Next, the membershp functons and the number membershp functons are found usng the followng parameters: r = ( a r = r Δ = r ); s = (quash factor); ε = 0. 4 (accept rato); ae a e ao f ε = 0.05 (reject rato). These parameters mply that a center can be a new cluster center f has densty value compared to the hghest densty value s greater than 0.4. A new canddate of cluster center wll be rejected f t has densty rato compared to the hghest densty s less than Fgure 12 and Fgure 13 show the data used to tranng n substractve clusterng whch are obtaned by makng varous paterns of reference sgnal changes that yeld good speed control transent response performances of the AC-motor. Ths substractve clusterng produces 18 cluster ponts ether n the error or n the delta error to ntate the Fuzzy Inference System (FIS). The subtractve clusterng

18 method yelds RMSE It gves Gaussan parameters σ (the varance) and c (the mean) as follow: The parameter σ : L = [ ,0325] The cluster center matrx (C ): C = Fgure 12. Cluster Centers of Error and Reference Sgnal Changes

19 Fgure 13. Cluster Centers of Delta Error and Reference Sgnal Changes. After workng on the substractve clusterng to estmate the number of clusters and the cluster centers n a set of data, t s carred out ANFIS learnng by usng the same data n order to mprove the ntal FIS formed n the substractve clusterng step. The mprovement s done by correctng the premse parameters and the consequent parameters. The ANFIS learnng uses 3175 nput/output par data, 0.1 learnng rate and 100 tranng epochs. Ths ANFIS tranng reduces the RMSE from to Both error varables and delta error varables have 18 membershp functons. The value range of the error varables and the delta error varables are selected as [ 1.703,1.739] and [ 1.731,1.741] respectvely. The FIS obtaned from the ANFIS tranng has 18 rules. The resulted membershp functons of the error varables and the delta error varables are shown n Fgure 14 and Fgure 15 respectvely. Table 2 and Table 3 descrbe the consequent parameters and the rule of ANFIS found after the tranng process. Fgure 14. Membershp Functons of Error.

20 Fgure 15. Membershp Functons of Delta Error. Table 2. Consequent Parameters of ANFIS. p q r out1cluster1 0,1279 0,393 0,151 out1cluster2 3,189 0,1076 0,1662 out1cluster3 0,4487 2,069 0,3167 out1cluster4 0, ,0319 0,5301 out1cluster5 15,24 1,587 4,865 out1cluster6 0, ,0134 0,6219 out1cluster out1cluster8 10,62 0,1674 4,884 out1cluster9 0, , ,6231 out1cluster out1cluster11 0, , ,6342 out1cluster12 13,2 1,79 3,543 out1cluster13 0, , ,5969 out1cluster14 0,1294 0, ,4896 out1cluster15 0, , ,652 out1cluster16 0, ,0213 0,619 out1cluster17 30,35 0, ,82 out1cluster18 9,229 2,134 3,972

21 If Table 3. ANFIS Rules. Error Delta Error Output n1cluster1 n2cluster1 out1cluster1 n1cluster2 n2cluster2 out1cluster2 n1cluster3 n2cluster3 out1cluster3 n1cluster4 n2cluster4 out1cluster4 n1cluster5 n2cluster5 out1cluster5 n1cluster6 n2cluster6 out1cluster6 n1cluster7 n2cluster7 out1cluster7 n1cluster8 n2cluster8 out1cluster8 n1cluster9 n2cluster9 out1cluster9 And then n1cluster10 n2cluster10 out1cluster10 n1cluster11 n2cluster11 out1cluster11 n1cluster12 n2cluster12 out1cluster12 n1cluster13 n2cluster13 out1cluster13 n1cluster14 n2cluster14 out1cluster14 n1cluster15 n2cluster15 out1cluster15 n1cluster16 n2cluster16 out1cluster16 n1cluster17 n2cluster17 out1cluster17 n1cluster18 n2cluster18 out1cluster18 Next, the dentfed ANFIS-HRC s then mplemented as n Fgure 10 by testng on a range value of the reference sgnal (set-pont). The parameters of the PID controller are chosen as K = 0. 4, T = mnutes and T = mnutes that results n good transent response. Fgure 16 d shows the transent response of the PID controller and the ANFIS-HRC-PID controller at set-pont 1,64 V 1,70 V or 10 Hz.. It can be seen that the ANFIS-HRC-PID controller s outperform the transent response of the PID controller by producng less overshoot and faster settlng tme. It s also shown the reference sgnal produced by means of ANFIS-HRC. At the start, the reference sgnal leads to large error n order to speed up the response and then t s followed by reducng the error n order to suppress the overshoot. The control sgnal s shown n Fgure 17. The maxmum control sgnal of the ANFIS-HRC-PID controller s a bt hgher compare to the PID controller but the ntegral squared control sgnal s smaller than the PID controller. Table 4 shows that the performances of the ANFIS-HRC-PID controller are better than the PID controller except for maxmum control sgnal. Fgure 18 shows the transent response performances of the ANFIS-HRC-PID controller for changes set-pont n the value range of the set-pont used to tran the ANFIS-HRC. It can be observed that the performances of th ANFIS- HRC-PID controller are stll outperform the PID controller. c

22 Fgure 16. Comparson of Transent Response. Fgure 17. Comparson of Control Sgnal. Table 4. Performances. Controller u 2 dt Ts Tr % Mp RMSE IAE Up (V) PID 0, ms 140 ms 16,83% 0,46 0,1914 1,39 ANFIS HRC PID 0, ms 100 ms 5,00% 0,35 0,0993 1,47

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