ANFIS Control Double-Inverted Pendulum

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1 895 A publcaton of CHEMICAL ENGINEERING TRANSACTIONS VOL. 46, 05 Guest Edtors: Peyu Ren, Yancang L, Hupng Song Copyrght 05, AIDIC Servz S.r.l., ISBN ; ISSN The Italan Assocaton of Chemcal Engneerng Onlne at DOI: /CET54650 Peng Shen ANFIS Control Double-Inverted Pendulum Unversty of Scence and Technology Laonng, Chna. sp-98@63.com For double nverted pendulum multvarable, strong couplng and nonlnear proposed adaptve fuzzy neural nference system (ANFIS) s appled nverted pendulum stablzaton control process. Adaptve control algorthm, fully able to meet the requrements of double nverted pendulum control, ANFIS system after tranng, wll be appled to the nverted pendulum system controller has better control accuracy and control performance, fully embodes the ANFIS system control nonlnear Advanced systems.. Introducton Over the past decade, these two areas began to ntegrate wth each other, formng a promsng control method - based Fuzzy Inference System (ANFIS) [44]. Because ANFIS fuzzy nference system can the concept of control and neural network together, not only has the advantage of neural networks, such as learnng ablty, the ablty to optmze the connecton structure and the control object model does not rely on, there are the advantages of fuzzy control system, If-Then rules as thought humanod and easy placement expertse. Thus, we can lower the neural network learnng and computng capabltes to fuzzy control system, but can also be advanced, thnkng and reasonng mechansms humanod If-Then rules provded to the neural network fuzzy control system, whch mproves the system ablty to learn and mprove the sklls of knowledge. Therefore, ANFIS nference system to acheve a certan human ntellgence, s an atypcal ntellgent control, s very popular wth people of attenton research.. ANFIS structural descrpton Takag and Sugeno proposed n 985 the TS fuzzy model T. Takag (985) reported, and later researchers called Sugeno fuzzy model. It s a non-lnear model, t s approprate to express the dynamc characterstcs of complex systems, t s also commonly used fuzzy reasonng model. Sugeno fuzzy model of a typcal fuzzy nference rules: f x s A, and y s B then z = f ( x, y) = the exact number of conclusons. Usually f ( x, y) s x and y polynomal. Model s called frst-order Sugeno fuzzy nference system. Correspondngly frst order Sugeno fuzzy model functonal equvalent of fuzzy nference system based on adaptve neural network (ANFIS) Jyh-Shng (993) reported, t s used to acheve the learnng process Sugeno fuzzy model. ANFIS Sugeno fuzzy model can be consdered as a neural network, the network s a mult-layer feed forward network structure as follows L (000) reported: As the premse of fuzzy numbers, z f ( x, y) Fgure : ANFIS level schematc Please cte ths artcle as: Shen P., 05, Anfs control double-nverted pendulum, Chemcal Engneerng Transactons, 46, DOI:0.3303/CET54650

2 896 Fgure nput vector, weghts, and a normal value from the premse of the product obtaned from the membershp functon, the output s a weghted average of the output of each rule, and the proporton of the total weght of the focuses for the rghts. The same level of output nodes have the same functons. As shown n Fgure, ANFIS model structure s dvded nto: fuzzy membershp computer nput of each rule apples for the degree of normalzaton, the output of each rule, the entre output of the fuzzy system and other fve, the nput layer drectly the output value to the next level, the output fuzzy layer s a membershp functon, fuzzy reasonng t a prerequste of basc fuzzy state corresponds to the fuzzy nference layer contact wth the premse and the concluson of the fuzzy nference, defuzzfcaton layer let the system determne the output s gven to perform. Frst layer: fuzzy layer, s responsble for the nput sgnal fuzzy, node havng an output functon: O = μ x () A ( ) among them x s the output of, x degree belong to μ A or μ A ( x) x c = exp a ( x) = b { a b, c } x c + a A s the fuzzy set. O s Membershp values of A, representaton A. Usually choose μ A ( x) bell-shaped functon and has a maxmum value, mn 0. E.g: (), are premse parameters, a s the wde, b s the slope, c s the central locaton. By adjustng these parameters, the shape of the membershp functon wth the change of these parameters change. In fact here s the membershp functon can take any pece wse contnuous functon, such as trapezodal or trangular functon functon. Second layer: Ths layer s a layer of release ntensty rule, ths layer node s responsble for the nput sgnal s multpled. Such as: ω = μ ( x) μ ( y), =, (4) A B One representatve of each output node relablty rules. Here s the AND operator of T specfcaton. By ths calculaton, to determne the actvaton strength of each fuzzy rule. Thrd layer: all the rules ntensty normalzaton layer, the node calculaton the rule normalzed credblty: ω ω =, (5) =, ω + ω Ths layer node are crcular nodes (fxed nodes), calculate the weghts of fuzzy rules, fuzzy rules for the actve strength of normalzed operatons. Fourth layer: output layer of fuzzy rules, the layer to calculate the output of the fuzzy rules, where each node s an adaptve node.the node output s: 4 O = ω f = ω ( p x + q y + r ) (6) The ω s the output of the thrd layer, ( p, q, r ) are the conclusons parameters. Ffth floor: defuzzfcaton layer whch has a crcular node, ts output s sad to have the nput sgnal and the result s that the fuzzy reasonng: Σω f Ol = Σω f = (7) 5 Σω Thus, gven the premse parameters (ntal membershp functons), the output ANFIS can be expressed as a lnear combnaton of parameters concluson: ω ω (8) f = f + f = ω f + ω f ω + ω ω + ω = ( ω x) p + ( ω y) q + ω r + ( ω x) p + ( ω y) q + ω r Corollary seen, for havng m the nput varables, every nput have k fuzzy sets of Sugeno fuzzy model, Accordng to the method descrbed above can be converted nto neural network structure, whch controls the total number of rules: m (9) n = k (3)

3 ANFIS learnng algorthm The man role of neuro-fuzzy controller s the applcaton of neural learnng technques, adjustng parameters and structure neuro-fuzzy control system. Fuzzy controller requres two types of adjustments: Restructurng and parameter adjustment; structural adjustments nclude the number of varables, on the nput and output varables of the doman partton number and other rules. Once you get a satsfactory structure, t s necessary to adjust the parameters. Parameter adjustment ncludng parameters related to the membershp functons, such as the center, wdth, slope, etc. adjustments. As the network structure has been determned, ANFIS learnng algorthm parameters of the controller s really just to learn, only we need to adjust the parameters of the premse and concluson parameters. 3. Updated four methods ANFIS parameters: () All parameters are updated wth the gradent descent; () The ntal concluson parameters obtaned by the least squares method, then gradent descent update all parameters; (3) Gradent descent and least squares method combned; (4) Only recurrence (approxmate) method of least squares; Whch method of selectng specfc crcumstances, should be consdered complexty and performance to be acheved by calculaton Jyh-Shng (995) reported. 3. ANFIS learnng process parameter optmzaton parameters nto the premse and concluson parameters learnng learnng Hybrd learnng algorthm summarzed as the followng two steps: () Concluson parameter learnng: Frst, determne the ntal value of the premse parameters, then the least squares method to calculate the concludng arguments. whch s: f ( ω x) p + ( ω y) q + ω r + ( ω x) p + ( ω y q + ω r = A X (0) = ) In the formula, X conclusons column vector elements set of parameters{ p, q, r, p, q, r }. If have p set of nput and output data, and the gven premse parameters, the matrx A, X, F, the medan p 6,6 and p. In general, the sample data s much larger than the number of the number of unknown parameters ( p >>6),Usng the least squares method to obtan the mnmum mean square error (mn AX f ) Concluson The best estmate of the vector n the sense of x : X T T = ( A A) A f () () Premse parameter learnng: Gven the premse parameters, output ANFIS can be expressed as a lnear combnaton of the concluson parameters, conclusons parameters calculated on a step error calculaton, usng an algorthm of the error from the output termnal nverted spread nput decrease wth the gradent method to update the premse parameters, thereby changng the shape of the membershp functon. 3.3 ANFIS structures and algorthms to establsh learnng as follows: () Extract fuzzy rules f then obtaned by the experts at, Average dvded nput varable space (or accordng to expert opnon) to establsh ntal membershp functons: the number ncludes rules, the shape of each membershp functon. General should meet ε completeness (usually: ε = 0. 5 ), that s a gven value for any one of the nput varables x. There must be a lngustc values A, make μ A ( x) ε establshment. So far, the fuzzy nference system can provde value to the smooth transton from one language to another language values and suffcent overlap. () The hybrd learnng algorthm adjusts the blur f then rules premse / concluson parameters, n each cycle ncludes forward propagaton and backward propagaton two processes (see above). After learnng the rules of lngustc premse does not necessarly guarantee ε completeness, but t may lmt the gradent algorthm to obtan. (3) Output error learnng process untl t reaches a certan tranng sample requrements so far, thereby establshng a whole fuzzy nference system. Sugeno type FIS ANFIS use to descrbe the object model, but also wth a feed-forward neural network to represent the Sugeno type FIS. On the one hand, usng the error back-propagaton algorthm combned gradent steepest descent method for nonlnear parameters Sugeno type FIS wll be to dentfy, on the other hand, took advantage of the least squares method to dentfy Sugeno type FIS current parameters. ANFIS network ncludes specfc antecedent parameters (membershp functon of varous parameters), and the back pece parameters (output model each factor), through some knd of algorthm to tran ANFIS, the ndcators have been specfed values of these parameters to acheve blur the purpose of modelng.

4 ANFIS control mode ANFIS can use a varety of dfferent control methods L (00) reported, ncludng mtaton experence control, nverse control and other control methods: 4. mtate experence control Is to sum up the experence to control essentally mmc the experence of people, then people wth the relevant experence ANFIS mtate, to acheve the controlled object control. 4. Neuron Fuzzy Inverse Control Ths control s dvded nto learnng stage and the applcaton stage. In the learnng phase, random nput set to gve the correspondng controlled object output, ANFIS nverse for learnng controlled object model. In the applcaton stage, the nverse model of the controlled object as a controller drectly on the controlled object. 4.3 Other control methods There are reference model adaptve control, feedback control, fuzzy control, gan schedulng control, stochastc methods to enhance learnng method. Reference Model control s the soluton to nverse control can not ensure that the entre system s the smallest error and proposed an mproved method; lnear and fuzzy feedback control for feedback lnearzaton system can; gan schedulng control for the frst-order Sugeno model; stochastc methods generally for complex control systems; the object model s not clear or s not a very effectve method you can use to enhance learnng S.M.Khazraee (00) reported. Currently, the ANFIS adaptve functon, whch s wdely used for functon approxmaton, pattern recognton, predcton, fuzzy aspects of control, sgnal processng, etc Imad O.Bach (04) reported. 5. Based on ANFIS nverted pendulum control system smulaton 5. Modelng Smulaton Sugeno Fuzzy modelng ncludng structure dentfcaton and parameter dentfcaton of two aspects JANG Jae Hoon (0) reported. Structural dentfcaton to dentfy and dvde the nput space by a grd method or clusterng method, prelmnary desgn blur f then rules (Condtonal expresson after the order Sugeno model); parameter dentfcaton fuzzy rules to determne membershp functon parameters and nput-output lnear parameters (Sugeno fuzzy model polynomal coeffcents) Chu (994) reported. ANFIS model structure dentfcaton s not requred, only the parameters on whch the learnng correcton can be Chen (000) reported. The above-mentoned grd method n 985 proposed by Takag and Sugeno, t wll enter the space s dvded nto a number of unform rectangles (or rectangular), usng the data entered n each dmenson maxmum (small) values nto space, and data dstrbuton nothng to do. Chu proposed fuzzy clusterng and hybrd learnng model based on the rule of recognton n 994. Spatal clusterng method to determne the number by the clusterng radus, coverng all subspace common nput samplng data, can be used to model any number of dmensons Yurkovch, S (996) reported. Sometmes we collected data wth the nose sgnal, then these data may not be representatve of all the typcal features of the system, the modelng wll not acheve good results. Thus, the effectveness of the tranng process and the results of model testng s a very mportant step. We should choose to represent an deal model system both need to smulate, but also wth the tranng data suffcent dfference (to avod unnecessary check) the valdaton data. In addton, n order to avod possble "model redundant" stuaton, should be used to check data set to control the tranng process. All of Matlab ANFIS modelng, the data are dvded nto three groups, the frst group for the tranng model (tranng data), a second set of model checkng (checkng data) for the tranng process, and fnally a set of model results test (testng data). Therefore, based on Matlab ANFIS modelng process Danal BEHNIA (03) reported dvded nto the followng sx steps: () Generates tranng data, test data and test data, data sets s expressed as: trndata, = [ x y], chkdata = [ ], [ ] x, y tesdata = x, y ; () Determne the type of nput varables and the number of membershp functons; the actual modelng, you should select the nput and output data to match the membershp functon to reflect changes n the characterstcs of the obtaned data. Ths paper selects Gaussan membershp functons Shareef Hussan (04) reported. (3) FIS produce ntal structure: for example, can use the functon genfs (grd method), genfs (elmnaton clusterng method) to produce offcal scrpt functon Sugeno type FIS ntal value, the second paper, namely: ( datan, dataout rad) fsmat = genfs, When gven nput data sets and data sets, genfs fuzzy clusterng elmnaton generates a fuzzy nference system. When there s an output, you can use the frst data elmnaton clusterng method, usng genfs

5 899 produce an ntal fuzzy nference system anfs the tranng process. gends usng a fuzzy rule extracton behavor modelng data to complete ths work. datan matrx where each lne contans a data pont nput values. dataout matrx, where each row contans the output value of a data pont. For example, the text data dmenson s 7 (e datan of two, dataout s one), rad = [0.7,0.6,0.5,0.4,0.3,0.,0.] specfy the cluster center n the frst, second,..., seventh-dmensonal space s the scope of the data wdth of 0.7,0.6,..., 0.. If rad s a scalar, then all the data on use of the scalar value dmenson, namely the scope of the center of each cluster s a sphercal neghborhood for a gven radus. (4) ANFIS tranng data set; tranng parameters nclude optons vector trnopt (ncludng number of batches of tranng, tranng error goal, the ntal step length, etc.), dsplay optons vector dspopt (ANFIS result nformaton), model valdaton data set name chkdata, membershp functon tranng methods optmathod (least squares method or the least square method and BP algorthm) and so on. (5) Tranng ANFIS; use anfs functon tranng ANFIS, namely: [ fsmat, error, stepsze, fsmat, error ] = anfs( trndata, fsmt, trnopt, dspopt, chkdata, ) Where: fsmat structure accordng to FIS tranng error crteron obtaned; error and root mean square error error were tranng data and test data; stepsze for the tranng process steps; fsmat FIS structure s based on test error crteron s obtaned. (6) to gve the FIS performance test. FIS structure accordng to the tranng error crteron obtaned usng evalfs (fuzzy nference computng) functon to calculate the output value or predcted and measured values are compared, FIS test the resultng performance. Complete fuzzy nference calculatons are completed by the followng formula: output = evalfs( nput, fsmat) Sugeno fuzzy model s necessary to enter very small space dvson, resultng n a sharp ncrease n the number of rules, systems large and dffcult to understand Yu (05) reported. 5. Double Inverted Pendulum smulaton and real-tme control Double nverted pendulum smulaton process, the tranng data and test data from the sold hgh double nverted pendulum system dagram collecton from, the samplng perod s Matlab default. Among them, the former 00 data as tranng data, ntermedate 50 as the test data, the remanng data as the test data. After the data nput ANFIS, usng the least squares wth BP hybrd algorthm, tranng tmes to 300 tmes. After ANFIS tranng and testng s completed, the smulaton results obtaned the followng: Fgure 5: Tranng error curve (error: ) Fgure 6: Comparng tranng data (average error: 0.05) FIG error curve can be seen n Fgure 5, the man tran 00 steps forward, the smaller the output error dentfcaton of ANFIS nference system closer to the desred output. Fgure 6 can clearly be seen that the output of ANFIS nference system s very close to the deal output we want. Real-tme system control Teng (000) reported. Fgure 7: Real-tme control system Smulnk Fgure 8: Control rule change map You can clearly see the changes n the fuzzy control rules by Fgure 8, the control system wll help us to adjust the rules. Because of double nverted pendulum s hghly nonlnear multvarable system, so durng the tme

6 900 control, we choose Control descendng order of prorty: two pendulum, level swng, car. Output response curve control system s as follows: Fgure 9: ANFIS control double nverted pendulum Fgure 0: system controlled amount of output curve In Fgure 9, the frst response curves that s the poston of the car to the state. The ntal poston s not at zero. After adjustng ANFIS controller, three seconds later met the target requred (zero) poston. The second s that s an nverted pendulum angle curve. As can be seen from the fgure, very small changes n the process of adjustng and super fast response, as the adjustment process s completed n less than 3 seconds. Artcle III of that s secondary angle curve nverted pendulum, ts overshoot relatvely smaller level swng angle, exactly n lne wth our control prorty choce. The three curves n the statc over Chengzhong Jng dfference s almost zero. All control process substantally complete n less than three seconds from Fg. 0, and the statc adjustment process, a small amount of change n control, able to meet the control requrements of the system. References Bach I.O., Abdulrazzaq N., He Z. 04. NEURO FUZZY MODEL FOR PREDICTING THE DYNAMIC CHARACTERISTICS OF BEAMS. Acta Mechanca Solda Snca. Vol. 7. No., pp DOI: 0.06/S (4) Behna D., Ahangar K., Noorzad A., Moenossadat S.R. 03. Predctng crest settlement n concrete face rockfll dams usng adaptve neuro-fuzzy nference system and gene expresson programmng ntellgent methods. Journal of Zhejang Unversty-Scence A (Appled Physcs & Engneerng). Vol. 4. No. 8. pp: DOI: 0.63/jzus.A0030. Chen W.J., Fang L., Cheang S.U., Le K.K., Zhang F.Z Personfed ntellgent control for an nverted pendulum system. Proceedngs of the World Congress on Intellgent Control and Automaton (WCICA). Vol. 3. pp: Chu S Fuzzy model dentfcaton based on cluster estmaton. J of Intell&Fuzzy Syst. Vol., No. 3. pp DOI: 0.333/IFS Hussan S., Safulnzam A.K., Mohd S.H., Mohd M.W. 04. Applcaton of artfcal ntellgent systems for real power transfer allocaton. Journal of Central South Unversty. Vol. 04. No.. pp: DOI: 0.007/s Jang J.H., Kwon S.H., Jeung E.T. 0. Pendulaton reducton on shp-mounted contaner crane va T-S fuzzy model. Journal of Central South Unversty. Vol. 9. No.. pp: DOI: 0.007/s Jang J.S.R ANFIS adaptve-network-based Fuzzy Inference System, Systems, Man and Cybernetcs, IEEE Transactons on, Vol., No. 3, pp DOI: 0.09/.5654 Khazraee S.M., Jahanmr A.H. 00. Composton Estmaton of Reactve Batch Dstllaton by Usng Adaptve Neuro-Fuzzy Inference System. Chnese Journal of Chemcal Engneerng. Vol. 8. No. 4. pp: DOI: 0.06/S (0) Le Y., Zhao D.N., Ca H.B. 05. Predcton of length-of-day usng extreme learnng machne. Geodesy and Geodynamcs. Vol. 6. No.. pp: L H.G., Sun J.H., Zhang B., Zhang Y.S. 00. ANFIS and Its Applcaton n Control System. Journal of Jangsu Unversty of Scence and Technology (Natural Scence Edton). Vol. 5. No. 5. pp: L Z Applcaton of ANFIS predcton of bulk cargo capacty. Dalan Martme Unversty Master Thess. pp: Shng J., Jang R., and Sun C.T Neuro-fuzzy modelng and control. Proceedngs of the IEEE, Vol. 8, No. 3, pp DOI: 0.09/ Takag T. and Sugeno M Fuzzy Identfcaton of Systems and Its Applcaton to Modelng and Control. IEEE Trans. Syst., Man and Cybern. vol. 5, no.. pp: 6-3. Teng F.C Real-tme control usng Matlab Smulnk. Proceedngs of the IEEE Internatonal Conference on Systems, Man and Cybernetcs. Vol 4. pp: DOI: 0.09/ICSMC Yurkovch S., Wdjaja M Fuzzy controller synthess for an nverted pendulum system. Control Engneerng Practce. Vol. 4. No 4. pp: DOI: 0.06/ (96)0006-3