Simulation of the adaptive neuro-fuzzy inference system (ANFIS) inverse controller using Matlab S- function

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1 Vol. 8(1), pp , 4 June, 013 DOI /SRE ISSN Academc Journals Scentfc Research and Essays Full Length Research Paper Smulaton of the adaptve neuro-fuzzy nference system (ANFIS) nverse controller usng Matlab S- functon Tarek Benmloud Unversty of Scence and Technology of Oran, 1505 Elmnaouar Oran-Algera. Accepted 3 Aprl, 013 In ths paper, for the purpose of smulatng the mathematcal model of adaptve neuro-fuzzy nference system (ANFIS), we use Matlab/Smulnk envronment wth ts powerful S-functons. The smulated model of ANFIS network can be then used to make the smulaton of the dentfcaton and the control of lnear or nonlnear systems. Created Smulnk block of ANFIS gve flexble explotaton of parameters of ANFIS network lke learnng rates and ntal local parameters. We use the S-functon of ANFIS to make the drect-nverse adaptve control of DC-motor. The obtaned results of Drect-nverse control of DCmotor are compared wth these of a classcal controller whch s the smplest type of controller, selected only to check the effectveness of the proposed ntellgent controller n terms of control performances and dsturbance rejecton. Key words: Drect nverse control, adaptve neuro-fuzzy nference system (ANFIS) controller, S-functon of Matlab. INTRODUCTION Jang (1993) proposed the famous adaptve neuro-fuzzy nference system (ANFIS), whch s one of the best n functon approxmaton among the several neuro-fuzzy models (Hrok et al., 009). It has been successfully appled n varous felds (dentfcaton, predcton and control). To desgn ANFIS based controllers Matlab provdes several tools, Smulnk based ANFIS toolbox (Howard and Mark, 1997). These tools support offlne smulaton process. It means pror to use the ANFIS model we need to smulate and tran the ANFIS by presentng the nput and target data set. The man dea of ths paper s to provde a custom tool to nvestgate better usage of ANFIS Networks. Most of the smulaton tools suggest ther own archtecture and tranng algorthms whch are fxed and uses offlne tranng. The proposed Smulnk model s mplemented usng S-functon whch allows makng onlne tranng of systems. It can be appled to any lnear or non-lnear system. Moreover learnng rates and step tme can be easy modfed. The created S-functon of ANFIS s used to make the control of DC-motor. ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM (ANFIS) ARCHITECTURE ANFIS (Fgure 1) (Jang, 1993) makes use of a hybrd learnng rule to optmze the fuzzy system parameters of a frst order Sugeno system. The output of the nodes n each respectve layer s represented by O, where s the th node of layer l. The followng s a layer by layer descrpton of a two nputs frst order Sugeno system (Jang, 1993; Hongxng et al., 001). The 1 st layer s for fuzzfcaton of the nput varables. It generates the E-mal: tarek_bm004@yahoo.fr.

2 876 Sc. Res. Essays A 1 x 1 A B 1 x B Π w 1 w 1 x 1 x N Π N w w x 1 x Layer 1 Layer Layer 3 Layer 4 Layer 5 Fgure 1. ANFIS archtecture Σ y The learnng procedure of ANFIS network has got two steps (Jang, 1993; Carrano et al., 008): In the 1 st step the nput patterns are propagated, and the optmal consequent parameters are estmated by an teratve least mean square procedure (Xuan, 006; Farzad et al., 005). In the nd step the patterns are propaged agan, and back-propagaton s used to modfy the local parameters (the values whch compose each membershp functons A1, A, B1, B). Ths procedure s then terated untl the error crteron s satsfed (Jang, 1993). The consequent parameters thus dentfed, are optmal under the condton that the local parameters are fxed. The form of the last layer s: u(k) e u + - u ˆ ( k) Plan ANFIS Network y(k+1) Fgure. Block dagramme for on-lne nverse learnng usng ANFIS. membershp grades: O 1 = g ( x) (1) Where g; s the membershp functon (MFs) of adaptve neuro-fuzzy system. The second layer generates the frng strength: O m = w = g( x) () j= 1 Y = X W (6) Where X s a vector of predctors, and W s the vector of regresson parameters to be estmated. To make the correcton of local parameters of ANFIS network, ANFIS network uses the sum of the gradent of the error e y of the output. The sgnal of error s back-propagated and local parameters are updated. We have the formula of modfcaton of the frst local parameter of the frst membershp functon of ANFIS network (Hongsheng and Feng, 007); h e a ( t + 1) = a ( t) p a y Where h; s the learnng rate for local parameter a The followng rule s used to calculate partal dervatves, employed for update of the parameters of membershp functon. E E y y w g = a y y w g a (7) (8) The thrd layer normalzes the frng strengths. O 3 w = w = (3) w + w 1 Layer 4 calculate rule outputs based on the consequent parameters. 4 O = y = w f = w ( p x + q y + r ) (4) Layer 5 sums all the nputs from layer 4. Ths s the overall output of the ANFIS system. (5) y = O = y = w f = w ( p x + q y + r ) 5 DIRECTE-INVERSE ADAPTIVE CONTROL ANFIS NETWORK Identfcaton and control usng adaptve neuro-fuzzy nference system (ANFIS) The tranng mode that wll be used s the nverse model (Toha and Tokh, 009; Gonzalez-Gomez et al., 011; Kasuan et al., 011) of the DC-motor. In ths case, The nput of the ANFIS network wll be the output y(t) of the DC-motor (ts speed) and the output of the ANFIS network wll be the estmaton of the sgnal of the control of the motor e u (t), as shown n Fgure. We know that the adaptve control of systems s consttuted by two loops; one loop of control havng a regulator wth adjustable parameters, and a second loop whch acts on the

3 Benmloud 877 y d (k) y(k+1) ANFIS Inverse controller u(k) + - eu eu(k) Plan Inv ANFIS Identfer Fgure 3. Block dagram of the drect-nverse adaptve control y d (t) y(t+1) y(t+) u(t+1) A w 1 N w 1 w B N w w 3 C N w 3 D N w 4 w 4 x 1 x x 1 x x 1 x x 1 x Σ y(k) u(t) Fgure 4. Topology of the ANFIS Network used n the DC Motor control. ref Ω Control by nverse model based ANFIS tranng Ω m Ω u( t ) DCM Fgure 5. Speed control of DCM wth onlne dentfcaton of the nverse model based ANFIS Network. parameters of the regulator, to mantan parametrc varatons. The structure of ths drect nverse control s shown by Fgure 3. Choce of the model of the adaptve neuro-fuzzy nference system (ANFIS) controller To represent non-lnear processes, several structures of models of type non-lnear black boxes were developed as: FIR, NARX, NOE, NARMAX. These models can be used n order wth the neuro-fuzzy network (Azeem et al., 000; Dena et al., 004; Toha and Tokh, 009; Kasuan et al., 011). To dentfy the nverse model of the MAS, we are gong to use the model NARX (non-lnear, autoregressve wth exogenous nput, whch s the most used for ts smplcty and ts not-recursve structure. The predctor s gven by the followng formula: y(t) ˆ = φ (y(t-1), ˆ, y(t-ny), ˆ u(t-1),, u(t-nu)) (9) ŷ : output of ANFIS network, u : nput of the ANFIS network The regresson conssts of outputs and the past nputs. The functon ϕ (.) s the nonlnear functon whch we want to approxmate. The choce of the number of nputs of ANFIS s not deducted of a man rule, but by successve tres (Fgure 4). The used ANFIS network for the Drect-nverse adaptve control contans 4 neurones n the nput layer wth trangular functons as membershp functon. The output of the ANFIS network s the sgnal of control u(t) of the DC-motor. The frst nput of the ANFIS network s the reference y d (t), the other nputs are; the output of the DCmotor wth a delay y(t+1), the output wth two delay y(t+), and the sgnal of control u(t+1). Adaptve neuro-fuzzy nference system (ANFIS) control of the DC-motor The control based on the adaptve ANFIS network uses the nverse model of the DC-motor (dentfed by an ANFIS), to control the speed of ths motor. It s drect nverse control; the dentfed nverse model s drectly used as controller. It can be a real-tme dentfcaton or separate tme dentfcaton. When the dentfcaton s real tme made the control s adaptve. To control DC-motor MAS, we wll use an adaptve control wth dentfcaton of the nverse model of the system (Fgure 5). The stablty of the control of the DC-motor usng the ntellgent controller s ensured snce the model DC-motor has a stable model, so, f the nverse model of the nducton motor s a good estmaton of ths model, the total system (DC-motor and the nverse controller) must be a stable system. SIMULATION OF ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM (ANFIS) NETWORK USING S- FUNCTION For the smulaton purpose, MATLAB/SIMULINK verson 6.5 s used. For the smulaton ode15 s (stff/ndf) solver s used. The smulaton results were taken for 80 s. For each step tme, ANFIS provded the value whch make the learnng of nverse model of the DC-motor to obtan the error e u (t), from whch the sgnal of control u(t) s calculated by ANFIS controller. ANFIS s traned va back

4 878 Sc. Res. Essays Table 1. Parameters of DC-motor. Parameter Symbol Value Resstance Ra 0.65 Ω Inductance La H Inerta mass J 0.65 kg/m Frcton F N Refs PID DC Motor S-fun ANFIS DC Motor Perturb Fgure 6. Smulnk model of drect-nverse adaptve control based ANFIS Learnng. propagaton algorthm (Jang, 1993; Eleftheros, 01). The drect-nverse control of the DC-motor based on ANFIS learnng s desgned to make the output speed track the reference Ω. ref and at the same tme acheve the desred dynamc performance. To verfy the effectveness of the ANFIS controller we use the PID controller. The am of usng the PID controller s to valdate the results of the ANFIS controller n terms of rapdty and precson as well as the dsturbance rejecton. The parameters of the DC-motor are gven n Table 1. The parameters of the PID controller are arbtrarly chosen to have rapd response of the speed of the DC-motor. We take: K p = 150, K = 7, K d =13. Fgures 7 and 8 gve the response of the speed of the DC-motor wth drect-nverse adaptve control based ANFIS tranng and PID controller. Fgure 8 shows the zoomed verson of the Fgure 7. Fgures 9, 10 and 11 show the evoluton of consequent and local parameters durng the control (and the tranng of the model) of the DC-motor. Fgures 1 and 13 gve the response of the speed of the DC-motor wth drect-nverse control and PID controller wth applcaton of load torque T L = 5 Nm at the nstant t =47 s. Fgures 14 and 15 gve the response of the load torque of the DC-motor usng the two controllers; drect-nverse controller and the PID controller. The convergence rate s manly affected by the four Learnng rates parameters, n the Smulnk block (Fgure 6) we can change these by smply changng the gan values of the S-functon block n Appendx. In present context t s set to 0.0. The smulaton result obtaned shows that the speed of DC-motor tracks rapdly ts reference (Fgures 7 and 1) wth statc error of trackng almost zero for the drectnverse control based ANFIS tranng, what s not the case of the PID controller whch, n spte of we have chooses hgh gans does not allow the cancelaton of the statc error. Drect-nverse control based ANFIS learnng keep very small statc error (Fgure 13) despte the applcaton of the load torque. We noted small varaton of local parameters of ANFIS (Fgure 10 and 11) durng the control of the DC-motor whch s not the case of consequent parameters (Fgure 9). Fgures 14, 15, 16 and 17 show that drect-nverse control based ANFIS learnng gve best response of load torque that these obtaned by PID controller whch present hgh values of pcks durng the varaton of speed references. CONCLUSION A drect-nverse adaptve control based ANFIS learnng s proposed as Matlab-Tool for dentfcaton and control of dynamc systems. The proposed Smulnk block s mplemented as S-functon of MATLAB software. The applcaton of ths block to control of speed of DC-motor gves good results of trackng of speed and load torque.

5 Benmloud Ref w PID w ANFIS 10 Speed [rad/sec] Speed (rad/s) Tme [s] Tme (s) Fgure 7. Matlab response of speed wth drect-nverse control based ANFIS learnng and PID controller Ref w PID w ANFIS Speed [rad/sec] Speed (rad/s) Tme [s] Tme (s) Fgure 8. Speed of the DC-motor wth drect-nverse control and PID controller (zoomed).

6 880 Sc. Res. Essays Consequent parameters Consequent parameters Tme [s] Tme (s) Fgure 9. Evoluton of consequent parameters of ANFIS durng the drect-nverse control of the DC-motor Local parameter a3 Local parameter a Tme [s] Tme (s) Fgure 10. Evoluton of the local parameters a3 of ANFIS durng the control of the DC-motor.

7 Benmloud Local parameter c Local parameter c Tme [s] Tme (s) Fgure 11. Evoluton of the load parameters c of ANFIS durng the control of the DC-motor. Refs PID DC Motor S-fun ANFIS DC Motor Load Torque Perturb Fgure 1. Smulnk model of drect-nverse adaptve control based ANFIS Learnng applcaton of load toruqe

8 88 Sc. Res. Essays Fgure 13. Adjustment of parameters of the S-functon Matlab block of ANFIS. 1 w ref w PID w ANFIS 10 Speed [rad/sec] Speed (rad/s) Tme [s] Tme (s) Fgure 14. Speed of the DC-motor wth drect-nverse control based ANFIS learnng and PID controller -applcaton of load torque at t=47 s.

9 Benmloud w ref w PID w ANFIS 7.8 Speed (rad/s) [rad/sec] Tme Tme (s) [s] Fgure 15. Speed of the DC-motor wth drect-nverse control and PID controller applcaton of load torque at t=47 s. (zoomed). 10 T wth ANFIS T ref 8 Torque [Nm] Torque (Nm) Tme [s] Tme (s) Fgure 16. Torque response of the DC-motor wth drect-nverse control based ANFIS learnng applcaton of load torque at t=47 s.

10 884 Sc. Res. Essays 60 T wth PID T ref 40 0 Torque (Nm) Tme Tme [s] (s) Fgure 17. Torque response of the DC-motor wth PID controller -applcaton of load torque at t=47 s. The smulated results show mprovement n the statc error speed wth mantenance of performances n the presence of load torque. The future work s to buld an S- functon of ANFIS to control real systems. REFERENCES Azeem MF, Hanmandlu M, Ahmad N (000). Generalzaton of adaptve neuro-fuzzy nference systems. IEEE Trans. Neur. Netw. 11: Carrano EG, Takahash RHC, Camnhas WM, Neto OM (008). A Genetc Algorthm for Multobjectve Tranng of ANFIS Fuzzy Networks. Congr. Evol. Comput. pp Eleftheros G (01). Study of Dscrete Choce Models and Adaptve Neuro-Fuzzy Inference System n the Predcton of Economc Crss Perods n USA. Econ. Anal. Pol. 4(1): Farzad R, Babak NA, Caro L (005). An Evolutonary Fuzzy Modelng Approach for ANFIS archtecture. IEEE Congress on Evolutonary Computaton, Ednburgh -Scotland, pp Gonzalez-Gomez JC, Ruz-Hernandez JA, Garca-Hernandez R, Sanchez EN (011). Real-Tme Neuro-Fuzzy Inverse Control Appled to a DC Motor. 8th Internatonal Conference on Electrcal Engneerng Computng Scence and Automatc Control (CCE). Hrok T, Koch T, Hsas T, Catherne V, Zheng T (009). Recurrent Type ANFIS Local Search Technque for Tme Seres Predcton. Proceedngs, IEEE Asa Pacfc Conference on Crcuts and Systems, pp , Nov 30-Dec 3, Macao, Chna. Hongsheng S, Feng Z (007). A Novel Learnng Method for ANFIS Usng EM Algorthm and Emotonal Learnng. Internatonal Conference on Computatonal Intellgence and Securty. Hongxng L, Phld Ghen CL, Han-Pang H (001). Fuzzy Neural Intellgent System: Mathematcal foundaton and the applcatons n engneerng. CRC Press LLC. Howard D, Mark B (1997). Neural Network toolbox for use wth Matlab. Users Gude verson 4; Mathworks. Jang JSR (1993). ANFIS/ Adaptve Neuro-fuzzy Inference System. IEEE Transactons on Systems Man, and Cybernetcs, Vol. 3, N3 May/ June1993. Kasuan N, Ismal N, Tab MN, Rahman MHF (011). Recurrent adaptve neuro-fuzzy nference system for steam temperature estmaton n dstllaton of essental ol extracton process. 7th IEEE Internatonal Colloquum on Sgnal Processng and ts Applcatons (CSPA). Toha SF, Tokh MO (009). Dynamc nonlnear nverse-model based control of a twn rotor system usng adaptve neuro-fuzzy nference system. Thrd UKSm European Symposum on Computer Modelng and Smulaton. Xuan FZ (006). Artfcal Intellgence and Integrated Intellgent Informaton Systems Emergng Technologes And applcatons. Idea Group Inc (IGI).