Swing-Free Transporting of Two-Dimensional Overhead Crane Using Sliding Mode Fuzzy Control

Transcription

1 Swng-Free Transportng of Two-Dmensona Overhead Crane Usng Sdng Mode Fuzzy Contro Dantong Lu, Janqang, Dongn Zhao, and We Wang Astract An adaptve sdng mode fuzzy contro approach s proposed for a two-dmensona overhead crane. System nearzaton transforms the two-dmensona system to two ndependent systems: -drecton transport system and -drecton transport system. Both the two systems are wth the same dynamc mode and oth ncude two susystems: postonng susystem and ant-swng susystem. A sdng mode fuzzy contro approach s proposed for oth -drecton transport and -drecton transport, and t comnes SMC s roustness and FLC s ndependence of system mode. Accordng to the nfuences on system dynamc performance, oth of the sope of sdng mode surface and the reatonshp etween susystems are automatcay tuned y rea tme fuzzy nference respectvey. The effectveness of the proposed contro s demonstrated y experments wth a two-dmensona prototype overhead crane. I. INTRODUCTION Overhead crane works as a root n many paces such as workshops and harors to transport a knds of massve goods. It s desred for the overhead crane to transport ts payoads to the requred poston as fast and as accuratey as posse wthout coson wth other equpments. Moreover, the payoad swng ange shoud e kept as sma as posse. Many works have een done n controng the overhead crane. Park [] and Snghose [] adopted nput shapng contro method. But the nput shapng must e pre-cacuated accuratey accordng to the system mode. These approaches acked roustness to externa dsturances and coudn t damp resdua swng we. Moreover, zero nta condton must e satsfed. Lee [3] and Gua [4] proposed feedack contro methods. Besdes needng accurate system mode and onerous matrx computaton, the aove methods were greaty affected y system nearzaton Manuscrpt receved Septemer 6, 003. Dantong Lu s wth Coege of Computer Scence, anta Unversty, anta 64005, Chna. Phone: , ema: [email protected], [email protected]. Dantong Lu, Janqang, Dongn Zhao, and We Wang are wth the Key Laoratory of Compex Systems and Integence Scence, Insttute of Automaton, Chnese Academy of Scences, P. O. Box 78, Bejng, 00080, Chna. Phone: ; fax: ; e-ma: {janqang.y, dongn.zhao, w.wang}@ma.a.ac.cn. and system parameters uncertanty. Fuzzy ogc contro (FLC s ndependent of system mode and has some roustness. Lee [5] used fuzzy ogc ony n ant-swng contro and apped poston servo contro for postonng and swng dampng. Hua [6] ony studed ant-swng contro wth fuzzy ogc and ddn t take postonng contro nto consderaton. Naey [7] adopted fuzzy ogc to oth postonng contro and swng dampng. However, ecause of the arge numer of fuzzy rues, t was dffcut to set oth rues and parameters of the controer ony accordng to experences. Sdng mode contro (SMC s a roust desgn methodoogy usng a systematc scheme ased on a sdng mode surface and Lyapunov staty theorem. The man advantage of SMC s that the system uncertantes and externa dsturances can e handed under the nvarance characterstcs of system s sdng mode state wth guaranteed system staty. Er [8], Kakou [9] and Hasanu [0] used the varae structure contro (VSC wth sdng modes to contro the overhead crane. In [8] and [9], the VSC was used to the postonng contro and hostng contro, ut another state feedack contro scheme must e added for payoad swng dampng contro. In [0], a reference mode was defned to track, and the system mode must e nearzed. A the aove VSC methods have dffcutes n automatcay tunng the reatonshp etween postonng contro and ant-swng contro. Ths paper presents a practca souton to anayze and contro the overhead crane. The payoad swng and crane moton of two transport drectons are consdered. Now that SMC s capae of tackng non-near system wth parameter uncertantes and externa dsturances, and fuzzy ogc contro s ndependent of system mode, crane system mode s ut to anayze system contro characterstcs wthout takng externa dsturances (such as wnds and system parameters varyng (such as dfferent goods nto consderaton. A sdng mode fuzzy contro agorthm s desgned for oth -drecton and -drecton transports of the overhead crane. Comnng SMC s roustness and FLC s ndependence of system mode, the proposed contro aw can guarantee a swng-free transportaton. The remander of ths paper s organzed as foows. In

2 secton, the dynamc mode of -dmensona overhead crane s ut, the nearzed mode s derved and a concuson s otaned that two-dmensona overhead crane can e dvded nto two ndependent transport systems. In secton 3, an adaptve fuzzy sdng mode contro agorthm s proposed for oth -drecton and -drecton transportatons. In secton 4, the proposed agorthm s vadated through experments. Fnay, n secton 5, concusons are drawn. II. DNAMIC MODEL OF OVERHEAD CRANE In ths secton, the system descrpton of two-dmensona overhead crane w e gven and ts dynamc mode w e ut. Then the mode w e transformed y nearzaton and state feedack to a system that s composed of two transport systems wth the same structure. In ths way, the system contro and ts mpementaton are smpfed. A. System Descrpton Fgure shows the coordnate system [] of a two-dmensona overhead crane and ts payoad. Z s the nerta coordnate system, M and M respectvey are the -drecton troey mass and -drecton troey mass ncudng the moment-of-nerta of the gear tran and motors. θ s the swng ange of the payoad n Z space and t has two components θ and θ. θ and θ are the swng ange projected on Z pane and Z pane respectvey. Assume the dynamc mode has the characterstc that the payoad and the troey are connected y a massess, rgd nk. B. System Dynamcs Accordng to Lagrangan equaton: d L L ( = T ( =,,3,4 ( dt q& q where, L = K U, K s system knetc energy, U s system potenta energy, q s generazed coordnate (here s x, y, θ or θ, and T s externa force (here s f or f (T 3 =T 4 =0. The moton equatons of the overhead crane system can e otaned wth respect to the generazed coordnates x, y, and θ. + m && x + m && θ m && θ m & m & θ θ ( m & θ & θ = f D + m & y + m && m & θ θ = f D y& (3 m && θ cos θ + mx && + mg m & θ & θ m && θ + && ym + mg mx && + m & θ (4 (5 θ Fgure Two-Dmenson Overhead Crane where D and D respectvey denote the vscous dampng coeffcents of the crane n the and drectons, f and f are the externa forces on the overhead crane n the and drectons, respectvey. C. System Mode Anays In ndustry, the maxmum acceeraton of the overhead crane s set smaer than the gravtatona acceeraton. For safety consderatons, the rope ength s usuay kept constant when the overhead crane s n moton. For sma swng around the vertca equrum, sn θ θ, sn θ, and. In addton, θ 0, & θ 0, & θ 0 and & θ & θ aso hod 0 for sma swng. The nonnear mode can e smpfed to the foowng nearzed mode: + m && x + m && θ = f D (6 && θ + && x + gθ + m && y + m && θ = f D y y& (7 && θ + && y + gθ In practce, the crane s normay drven y servo-motors. The servomotor has three contro modes: poston contro, speed contro and torque contro. In order to smpfy the system dynamc mode, the speed contro mode s used. Through the foowng state feedack transformatons: u = + m ( f D m && θ (8 u = + m ( f D y& m && θ the system mode can e descred as && x = u θ + u + gθ && y = u θ + u + gθ (9 (0

3 The nearzed dynamc mode conssts of the -drecton transport dynamcs and -drecton transport dynamcs. The -drecton dynamcs and -drecton dynamcs are decouped and wth the same structure. Therefore, the same contro agorthm can e desgned for oth the and drecton transport systems. For ths system mode, the system contro nput s the acceeraton of the overhead crane. Because the vscous dampng and the masses of troeys and payoad maye are not known and the aove transform makes the mode e ndependent of them, the system contro s easy mpemented n practce. NB PS ZO NS -Ф x PB Ф s=0 s x III. CONTROL DESIGN In ths secton, an adaptve sde mode fuzzy controer w e desgned for the two-dmensona overhead crane. Assume the desred state s generazed coordnates orgn. Snce the two-dmensona overhead crane can e decouped nto two ndependent transport systems, a contro agorthm w e desgned for oth of them. Ony the -drecton transport system s consdered eow. A. Sdng mode fuzzy contro (SMFC Consder a second-order system of the form as foows: = x ( = f ( + ( u where, =(x, x s state varae vector, f( and ( are contnuous near or nonnear functons, u s the contro nput. A sdng mode functon can e defned as s = x + λ x ( Very smar to sdng mode contro wth oundary ayer, the contro nput on the two sdes of the sdng mode surface are opposte n sgn and ts magntude s proportona to the dstance etween the state vector and the sdng mode surface. Therefore, the sdng mode fuzzy contro s desgned to: R : IF s IS F THEN u IS U where F s the ngustc vaue of s n the th-fuzzy rue, and U s the ngustc vaue of u n the th-fuzzy rue. The fuzzfcaton of the sdng mode functon s ustrated n Fg. B. Adaptve sdng mode fuzzy contro (ASMFC The -drecton transport systems can e represented as: = x = f ( + ( u (3 3 = x 4 = f ( + ( u 4 where, =(x, x, x 3, x 4 s the state varae vector that ncude crane poston, veocty, payoad swng ange and anguar veocty, f (, f (, ( and ( are contnuous nonnear functons, u s the contro nput. From (3, the -drecton transport system has two couped susystems: postonng susystem and ant-swng susystem. In order to decoupe the system, two sdng mode functons are defned for the two susystems: s = x + λ (4 x x4 + λ x3 s = (5 where λ and λ are postve rea numers. System performance s very senstve to the sope λ (or λ of the sdng mode functon: when the vaue of λ (or λ ecomes arger, the rse-tme w ecome smaer, ut Fgure Fuzzfcaton of sdng mode functon n sdng mode fuzzy contro at the same tme, oth overshoot and tunng-tme w ecome arger, and vce versa. So a aw s desgned to adjust the sope λ (or λ of the sdng mode functon n rea tme: when system state errors are arge, a gger sope of the sdng mode functon s used n order to make the system state approach the sdng mode surface and equrum pont. Ths s ecause the convergence speed on the sdng mode surface s hgh f a arge λ (or λ s used. In mechanca systems, the vaue of λ (or λ s typcay mted y three factors: the frequency of the owest unmodeed structura mode, the argest unmodeed tme deay, and the sampng rate. Accordng to the mechanca system mtaton, the sope of the sdng mode functon s gven for the th susystem (= or y λ = + (6 λ B λ where, λ s the asc vaue of λ, B s the tunng scope of λ, and λ s the tunng varae. The vaue of λ can e otaned accordng to the foowng fuzzy rues: R : IF x - IS A THEN λ IS λ where, R s the th tem of m rues, A s a fuzzy set of nput varae x -, and λ s a fuzzy set of output varae λ. The output sngeton fuzzy sets and the center-of-gravty

4 defuzzfcaton method are used: m λ = ( µ ( x λ / µ ( x (7 A = = µ x s the frng degree of the th rue. ( A The composte sdng mode functon can e defned as s = s + λ s = x + λ x + λ( x + λ 3 (8 m A 4 x where λ s a negatve rea numer. Tunng the coeffcent λ can adjust the functon of the postonng susystem and the ant-swng susystem on the sdng mode functon. When λ ecomes smaer, the postonng susystem s strengthened; and when λ ecomes arger, the ant-swng susystem s strengthened. So another fuzzy nference s desgned: f system state s far from sdng mode surface s =0, a arger vaue of λ s adopted, vce versa. As the tunng of the sdng mode functon sope, et λ = λ + B λ (9 where, λ s the asc vaue of λ, B s the tunng scope of λ, and λ s the tunng varae. The vaue of λ can e otaned accordng to the foowng fuzzy rues: R j : IF s IS F j THEN λ IS λ j where, R j s the jth tem of n rues, F j s a fuzzy set of nput varae s, and λ j s a fuzzy set of output varae λ. The output sngeton fuzzy sets and the center-of-gravty defuzzfcaton method are used: n j λ = ( µ j ( s λ / µ j ( s (0 F j = j = where µ s s the frng degree of the jth rue. j ( F Ony n the ant-swng susystem of the -drecton transport system, the sope of the sdng mode functon s automatcay adjusted y fuzzy nference system, whch s caed as adjustor. The reatonshp etween the postonng susystem and ant-swng susystem are automatcay tuned y another fuzzy nference system, whch s caed as adjustor. For the -drecton transport system of the overhead crane, the composte sdng mode functon s works as the nput to the sdng mode fuzzy contro. The fuzzy rues are: R k : IF s IS F k THEN u f IS U k where, R k s the kth tem of p rues, F k s a fuzzy set of nput varae s, and U k s a fuzzy set of output varae u f. The output sngeton fuzzy sets and the center-of-gravty defuzzfcaton are used: u = p p f F F k k = k = n F k ( µ k ( s U / µ ( s ( where µ (s s the frng degree of the jth rue and uf s the k F output of the adaptve sdng mode fuzzy controer. Now consder the reachng condton of the sdng mode surface that s ased on the fowng Lyapunov functon: s V = ( By takng the tme dervatve of equaton (8 and susttutng nto equaton (, t s easy to otan from equaton (3 ss& = s( f( + λf ( + λx + λλ x4 (3 + ( ( + λλ ( su Form equaton (9, we can see: ( and λ s postve, and ( s negatve. When λ s negatve, the term n equaton (3, + λ ( are aways postve. ( Therefore, ncreasng the contro nput u w resut n decreasng ss& as the sdng mode functon s s negatve, and decreasng the contro nput u w resut n decreasng s& s as the sdng mode functon s s postve. Remark : Accordng to the system dynamc mode that the -drecton transport system and -drecton transport system are decouped and wth the same nearzed mode, the contro agorthm s aso appcae to -drecton transport system. Remark : The proposed contro desgn s ndependence of the nearzed system mode,.e. the contro agorthm s desgned for nonnear overhead crane system. The nearzed mode derved n secton s used to expan the two-dmensona overhead crane consstng of two approxmatey ndependent transport systems. Remark 3: When the nta payoad anges are zeros, oth -drecton and -drecton transportatons w arrve at the goa at the same tme wth the same adaptve sdng mode fuzzy controer. Remark 4: Now that the servomotors adopt speed contro mode whe the output of the controer s acceeraton, actua nput of the servomotors s: u t = c u f dt t0 where t 0 and t c are nta tme and current tme separatey. The contro scheme s ustrated n fgure 3. IV. EPERIMENT RESULTS (4 To confrm the effectveness of the proposed contro agorthm, some experments have een performed wth a two-dmensona prototype overhead crane ustrated n Fgure 4. The prototype conssts of two sets of components that ncude mechanca system, data sampng system, and contro system. For the mechanca system, the troeys are drven y AC servomotors, and the payoad s connected to a cae that s attached to the undersde of the troey, where two precse ange sensors are nstaed to measure the swng anges of -drecton and -drecton. The contro agorthm s mpemented on a Pentum III 800 MHz PC runnng under the Wndows operatng system. In the contro agorthm, the adaptve tunng of the sope of the sdng mode functon s ony used for the ant-swng

5 susystems. The parameters of the controer are as foows: λ =0.5, λ =.8, B=3, λ =-, B=-4.6. The tunngs of λ and λ adopt the same fuzzy rues tae, gven n tae. d/dt The sdng mode fuzzy rues are provded n tae. x u Overhead Adjustor Adjustor ASMFC Crane Tae Rues tae for tunng λ and λ x 3 e 3 or s S M L λ λ or λ λ d/dt Tae Rues tae of sdng mode fuzzy controer s NB NS ZO PS PB u f λ Fgure 3 Contro Scheme Fgure 5 and 6 show the expermenta resuts of transport from poston (-., -0.6 to poston (0, 0 wth zero nta anges. Fgure 7 shows the -drecton transport when the nta ange s not zero. In the aove fgures, veocty s 0.m/s each grd n -axs. Fgure 8, 9 and 0 show the dampng swng experment resuts and the contro s added from the 6th second. Fgure 8 s ange phase pane and fgure 9 and 0 are anges tme responses. From the experment resuts, t s cear that the contro aw can make the -drecton and -drecton transports arrve at goa at the same tme when nta ange s zero, and the contro aw can damp swng ange at goa whether the nta ange s zero. Moreover, the contro aw can damp swng ange n short tme whe keepng poston. V. CONCLUSION In ths paper, a two-dmensona overhead crane s transformed to two ndependent systems that are wth the same dynamc mode n order to smpfy the controer desgn. An adaptve sdng mode fuzzy contro approach has een desgned for oth -drecton transportaton and -drecton transportaton, and ts effectveness has een demonstrated y experments on a two-dmensona prototype overhead crane. The experments have shown the proposed contro aw guarantees oth accurate postonng contro and prompt dampng of payoad swng. The staty and performance of the proposed contro aw are guaranteed n spte of arge nta swng ange. REFERENCES [] Bae-Jeong Park, Keum-Shk Hong, Chang-Do Huh, Tme-Effcent Input Shapng Contro of Contaner Crane Systems, n Proc. Of IEEE Int. Conf. on Contro, 000, pp [] Wam snghose, Lsa Porter, Mchae Kenson etc, Effects of Hostng on the Input Shapng Contro of Gantry Cranes, Contro Engneerng Practce, vo. 8, pp.59-65, 000. [3] Ho-Hoon Lee, Modeng and Contro of a Three-Dmensona Overhead Crane, J. of Dynamc System, Measurement, and Contro, vo. 0, pp , 998. Fgure 4 Two-dmenson Prototype Overhead Crane [4] Aessandro Gua, Cara Seatzu, Gampaoo Usa, Oserver-controer Desgn for Cranes va Lyapunov Equvaence, Automatca, vo. 35, pp , 999. [5] Ho-Hoon Lee and Sung-Kun Cho, A New Fuzzy-Logc Ant-Swng Contro for Industra Three-Dmensona Overhead Cranes, n Proc. of IEEE nt. Conf. on Rootcs & Automaton, 00, pp [6] Keqang Hua, Fuzzy Ant-swng Technoogy for Overhead Crane, J. of cv avaton unversty of Chna, vo. 3, 000, pp.-3. [7] Mchae J.Naey and Mohamed B.Traa, Contro of Overhead Cranes Usng a Fuzzy Logc Controer, J. of Integent and Fuzzy System, vo. 8, pp.-8, 000. [8] M. J. Er, M. Zr and K. L. Lee, Varae Structure Contro of an Overhead Crane, n Proc. of IEEE Int. Conf. on Contro Appcaton, 998, pp [9] Mansour, A.Kakou and Mohamed Zr, Roust Contro Schemes for an Overhead Crane, J. of Vraton and Contro, vo. 7, pp , 00. [0] M. Hasanu Basher, Swng-free Transport Usng Varae Structure Mode Reference Contro, n Proc. of IEEE Southeastcon. 00, pp []. Fang, W. E. Dxon, D. M. Dawson and E. Zergerogu, Nonnear Coupng Contro Laws for a 3-DOF Overhead Crane System, n Proc. of IEEE Conf. on Decson and Contro, 00, pp

6 Poston (m Ange (deg -Ange (deg Veocty Ange Poston Fgure 5 Tme Responses of -drecton Transport -Ange (deg Fgure 8 Phase Pane n Dampng Swng Poston(m Ange (deg -Ange (deg Veocty Ange Poston Fgure 6 Tme Responses of -drecton Transport Poston (m Ange (deg Fgure 9 Tme Response of -drecton Ange Dampng -Ange (deg Veocty Ange Poston Fgure 7 Tme Responses wth Inta Ange Fgure 0 Tme Response of -drecton Ange Dampng