A revsed adaptve fuzzy sldng mode controller for robotc manpulators Xaosong Lu* Department of Systems and Computer Engneerng, Carleton Unversty, 5 Colonel By Drve, Ottawa, Ontaro, Canada E-mal: luxaos@sce.carleton.ca *Correspondng author Howard M. Schwartz Department of Systems and Computer Engneerng, Carleton Unversty, 5 Colonel By Drve, Ottawa, Ontaro, Canada E-mal: schwartz@sce.carleton.ca Abstract: In ths paper, we revsed an adaptve fuzzy sldng mode control algorthm appled to a two-degree-of-freedom robotc manpulator. Stablty of the overall system for the revsed algorthm s proven n a Lyapunov sense. A dscusson on the proof of Lyapunov stablty for dfferent algorthms s presented. Keywords: Adaptve; sldng mode control; fuzzy systems. Bographcal notes: Xaosong Lu was born n Shandong, Chna n 976. He receved hs B.Eng. Degree and M.Eng. degree from Shandong Unversty, Shandong, Chna n 994 and respectvely. He fnshed hs MASc degree from Carleton Unversty, Ottawa, Canada n 7. He s currently workng towards hs Ph.D. n Systems and Computer Engneerng at Carleton Unversty. Hs research nterests nclude adaptve control and fuzzy control. Professor Howard M. Schwartz receved hs B.Eng. degree from McGll Unversty, Montreal, Quebec n June 98 and hs M.S. degree and Ph.D. degree from M.I.T., Cambrdge, Massachusetts n 98 and 987 respectvely. He s currently a Professor n Systems and Computer Engneerng at Carleton Unversty, Ottawa, Canada. Hs research nterests nclude adaptve and ntellgent control systems, robotcs and process control, system modellng, system dentfcaton, system smulaton and computer vson systems.
Introducton Classcal sldng mode control s robust to model uncertantes and external dsturbances. A sldng mode control method wth a swtchng control law guarantees asymptotc stablty of the system, but the addton of the swtchng control law ntroduces chatterng nto the system. One way of attenuatng chatterng s to nsert a saturaton functon (Asada and Slotne, 986) nsde a boundary layer around the sldng surface. Unfortunately, ths addton dsrupts the Lyapunov stablty of the closed loop system. Classcal sldng mode control has dffculty n handlng unstructured model uncertantes. One can overcome ths problem by combnng sldng mode control and fuzzy systems together. Fuzzy rules allow fuzzy systems to approxmate arbtrary contnuous functons (Wang, 997). To approxmate a tme-varyng nonlnear system, a fuzzy system requres a large amount of fuzzy rules. Ths large number of fuzzy rules wll cause a hgh computatonal load. The addton of an adaptaton law to a fuzzy sldng mode controller to onlne tune the parameters of the fuzzy rules n use wll ensure a moderate computatonal load. A number of adaptve fuzzy sldng mode control (AFSMC) algorthms have been proposed n the last decade (Yoo and Ham, 998, ; Wang et al., ; Guo and Woo, 3; Ho et al., 4; Wa et al., 4; Ln and Hsu, 4; Akbarzadeh-T and Shahnaz, 5; Shahnaz et al., 6; Medhaffar et al., 6). The adaptaton laws n these algorthms are desgned based on Lyapunov stablty theory. Asymptotc stablty of the closed loop system for these algorthms s also proved n the sense of Lyapunov. Adaptve fuzzy sldng mode controllers can be classfed nto two categores: ndrect adaptve fuzzy sldng mode controllers and drect adaptve fuzzy sldng mode controllers (Wang, 994). In an ndrect adaptve fuzzy sldng mode control method, the controller s used to estmate the parameters of the system s dynamcs. Yoo and Ham (998) proposed a sngle-nput sngle-output (SISO) fuzzy system to approxmate the unknown functons of a nonlnear system. Based on Yoo and Ham (998) s approach, Medhaffar et al. (6) desgned an ndrect adaptve fuzzy sldng mode control algorthm appled to robotc manpulators. The mult-nput mult-output (MIMO) fuzzy systems used n Medhaffar et al. (6) s approach are appled to estmate the dynamc equatons of the robotc manpulator and the fuzzy rules are reduced by ntroducng sldng surfaces as the nputs. To avod chatterng n Yoo and Ham (998) s algorthm, a fuzzy system s used to substtute for the dscontnuous control term n the algorthms proposed by Medhaffar et al. (6) and Wang et al. (). In a drect adaptve fuzzy sldng mode control method, the controller s used to drectly adust the parameters of the control law wthout estmatng the system s dynamcs. Yoo and Ham () proposed a MIMO fuzzy system to compensate for model uncertantes of a robotc manpulator. Unfortunately, usng a MIMO fuzzy system requres an nordnate number of fuzzy rules whch leads to a hgh computatonal load. Guo and Woo (3) appled a SISO fuzzy system to adust the control gan n the control law for a robotc manpulator whch both decreased the number of fuzzy rules and attenuated chatterng. Dfferent from Guo and Woo (3) s method, Ho et al. (4) appled a PI controller nsde a boundary layer to attenuate chatterng and the parameters of ths PI controller are onlne adusted by adaptaton laws. Wa et al. (4) desgned a AFSMC method to estmate the bound of the approxmaton error for electrcal servo drves. However, algorthms proposed by Yoo and Ham (), Guo and Woo (3), Ho et al. (4) and Wa et al. (4) can only tune the consequence part of the fuzzy rules, whch places the onus of desgnng the premse part upon the desgner. Therefore, Ln and Hsu (4) presented a drect AFSMC method to onlne tune both the premse and consequence parts of fuzzy rules. In Ln and Hsu (4) s algorthm, a fuzzy controller and a compensaton controller are proposed to construct a control law and the bound of the compensaton controller s adusted by adaptaton laws. Snce Ln and Hsu (4) s algorthm s only desgned for nducton servo motor systems, t s not applcable to robotc manpulators. In ths paper, we wll revse the adaptaton laws of the algorthm proposed by Ln and Hsu (4) so that t s sutable for the robotc applcaton. We then prove Lyapunov stablty of the closed loop system for the revsed algorthm. We wll also dscuss stablty ssues for the algorthms proposed by Yoo and Ham (), Guo and Woo (3), Wang et al. () and Medhaffar et al. (6): Lyapunov stablty n a practcal sense for the algorthms proposed by Yoo and Ham () and Guo and Woo (3) and problems on the proof of Lyapunov stablty for the algorthms proposed by Wang et al. () and Medhaffar et al. (6). Ths paper s organzed as follows: an overvew of fuzzy systems and fuzzy rules; the desgn of the revsed algorthms for robotc manpulators; the stablty analyss of four algorthms n the sense of Lyapunov; and, the concluson. Fuzzy Systems The fuzzy rule base of a mult-nput mult-output (MIMO) fuzzy system s comprsed of the followng fuzzy f-then rules R (l) : If x s A l and and x n s A l n, then y s B l and and y m s B l m () where l =,,, M denotes the number of fuzzy fthen rules; x,, x n and y,, y m are the nput and output varables of the fuzzy system. A MIMO fuzzy system can be decomposed nto a collecton of mult-nput sngleoutput(miso) fuzzy systems (Wang, 997). A MISO fuzzy rule base conssts of a collecton of fuzzy IF-THEN rules n the followng form (Wang, 994): R (l) : If x s A l and and x n s A l n, then y s B l where x,, x n are nput varables and y s the output varable. The membershp functons of the fuzzy sets A l and B l are defned as µ A l (x), µ B l(y). The MISO fuzzy systems wth center average defuzzfer, product-nference rule and
sngleton fuzzfer are of the followng form (Wang, 994): y(x) = y l ( n = µ A l (x )) ( n = µ A (x )) l where l =,, M denotes the number of fuzzy f-then rules, =,, n denotes the number of nput varables, and y l s the pont at whch µ B l(y) acheves ts maxmum value (we assume µ B l(y l ) = ). Rewrte () as follows: f(x) = () θ l ξ l (x) = θ T ξ(x) (3) where θ = [θ,, θ M ] T, ξ(x) = [ξ (x),, ξ M (x)] T, and n n ξ l (x) = µ A l (x )/ ( µ A l (x )). (4) = = For a sngle-nput sngle-output(siso) fuzzy system, each fuzzy f-then rule s represented as the form of R (l) : If x s A l, then y s B l (5) The SISO fuzzy system s descrbed as f(x) = θ l ξ l (x) = θ T ξ(x) (6) where θ = [θ,, θ M ] T, ξ(x) = [ξ (x),, ξ M (x)] T, and ξ l (x) = µ A l(x)/ M µ A l(x). In order to use fuzzy systems to estmate nonlnear functons, we ntroduce the followng unversal approxmaton theorem (Wang, 997). Theorem.. For any gven real contnuous functon g(x) on a compact set U R n and arbtrary ɛ >, there exsts a fuzzy logc system f(x) n the form of (3) such that sup f(x) g(x) < ɛ. (7) x U 3 Desgn of adaptve fuzzy sldng mode controller for robotc manpulators The dynamc equaton of an m-lnk robotc manpulator s M(q) q C(q, q) q G(q) = τ (8) where q = [q,, q m ] T s an m vector of ont poston, M(q) s an m m nertal matrx, C(q, q) s an m m matrx of Corols and centrfugal forces, G(q) s an m gravty vector and τ = [τ,, τ m ] T s an m vector of ont torques. 3. Revsed Ln and Hsu s algorthm The adaptve fuzzy sldng mode controller can only tune the consequence part of ther respectve fuzzy rules n the algorthms proposed by Yoo and Ham (998), Yoo and Ham (), Wang et al. (), Guo and Woo (3), Ho et al. (4), Wa et al. (4), Akbarzadeh-T and Shahnaz (5), and Medhaffar et al. (6). Ln and Hsu (4) surpassed ths lmtaton by presentng a number of adaptaton laws that are capable of onlne tunng of both the premse and consequence parts of the fuzzy rules n queston. Snce Ln and Hsu (4) s algorthm s specfcally appled to an nducton servomotor drve, we need to revse ths algorthm for the control of robotc manpulators whch we refer to as the revsed Ln and Hsu s algorthm. We defne the trackng error as e = q q d (9) where q d = [q d,, q md ] T s the desred traectores. The sldng surface s gven as The reference state s gven as The control nput s gven by s = ė λe () q r = q s = q d λ e () q r = q ṡ = q d λ ė. () τ = ˆM q r Ĉ q r Ĝ ˆF (s) F cp (s) (3) where ˆF (s) = [ ˆf (s ),, ˆf m (s m )] T and F cp (s) = [f cp (s ),, f cpm (s m )] T. The fuzzy system ˆf (s )( =,, m) s defned as ˆf (s ) = θ T Φ (s ) (4) where θ = [θ, θ,, θm ]T and Φ (s ) = [Φ (s ), Φ (s ),, Φ M (s )] T. Assume M µ A l (s ) = and (4) becomes θ l µ A (s ) l ˆf (s ) = θ T Φ (s ) = = θ µ l A l (s ) µ A l (s ) (5) where µ A l (s ) = Φ l (s ) = exp [ (σ l (s α l ))] (l =,, M). Defne f such that f = f ˆf (s ) = ˆf (s ) ˆf (s ) = θ T Φ θ T Φ (6) where θ and Φ are the optmal values based on the unversal approxmaton theorem n (7). We defne θ = θ θ, Φ = Φ Φ and (6) s rewrtten as f = (θ θ ) T (Φ Φ ) θ T Φ = θ T Φ θ T Φ θ T Φ (7) We take Taylor seres expanson of Φ around two vectors α and σ where α = [α,, αm ]T and σ = [σ,, σm ]T ( α l and σl are defned n (5)): Φ = Φ Φ α α Φ σ σ h.o.t. (8)
where α = α α, σ = σ σ and h.o.t. denotes the hgher order terms. We rewrte (8) as where Φ = Φ α Φ σ h.o.t. α σ = B α D σ h.o.t. (9) B = D = Φ α Φ α. Φ α M Φ σ Φ σ. Φ σ M We substtute (9) nto (7): Φ α Φ α Φ α M Φ σ Φ σ Φ σ M Φ M α Φ M α Φ M α M Φ M σ Φ M σ Φ M σ M (). () f = θ T (B α D σ h.o.t.) θ T Φ θ T Φ = θ T B α θ T D σ θ T Φ ε () where ε = θ T (h.o.t.) θ T Φ s assumed to be bounded by ε E. E s a constant and the value of E s uncertan to the desgner. We defne E as the real value and the estmaton error s gven by Ẽ = E E (3) We produce an adaptaton law to onlne tune the followng parameters: θ n (4), σ l, αl n (5) and the bound E n (3). The adaptaton laws are gven as θ = η s Φ (4) α = η 3 s B T θ (5) σ = η 4 s D T θ (6) f cp (s ) = E sgn(s ) (7) Ė = η s (8) where η,, η 4 are postve constants, α = [α, α,, αm ]T, σ = [σ, σ,, σm ]T and f cp (s ) s the compensaton term defned n (3). 3. Stablty proof of the revsed Ln and Hsu s algorthm We defne the followng Lyapunov functon canddate: V = st Ms ( Ẽ η θ T θ η αt α η 3 σt σ ) η 4 (9) We take the tme dervatve of V : V = s T Mṡ st Ms αt α σt σ ) η 3 η 4 Ẽ Ẽ η θ T η θ (3) where Ẽ = E E, θ = θ θ, α = α α, σ = σ σ. Snce M C s a skew-symmetrc matrx, we can get s T Mṡ st Ms = s T (Mṡ Cs). From (8) and ()-(3) we get Mṡ Cs = F ˆF (s) F cp (s) (3) where F = M q r C q r G, M = ˆM M, C = Ĉ C and G = Ĝ G. Then V becomes V = = = s (f ˆf (Ẽ Ė θ T θ (s ) f cp ) η η αt α σt σ ) η 3 η 4 s (θ T B α θ T D σ θ T Φ ε f cp ) (Ẽ Ė η [ ( θ T s Φ σ T ẼĖ η θ T θ αt α η θ η ( s D T θ σ ) η 4 ) σt σ ) η 3 α T )] η 4 ( s B T θ α ) η 3 ( s ε s f cp We substtute the adaptaton law (4)-(8) nto (3) and get V = = [ ] s ε s E sgn(s ) Ẽ s [ s ε E s ] (3) [ s ( ε E ) ] (33) where V s negatve semdefnte. We defne V = s ε s E sgn(s ) Ẽ s and rewrte (33) as [ V s (t) ( ε E ) ] (34) From V, we can get s (t) s bounded. We assume s (t) η s and rewrte V s (t) ( ε E ) as s (t) E s (t) ε E V η s E ε E V (35)
Then we take the ntegral on both sdes of (35): t s (ν) dν η s E t η s t E ε dν E (V () V (t)) ε dν E ( V () V (t) ) (36) If ε L, we can get s L from (36). Snce we can prove ṡ s bounded (see proof n (Wang, 994)), we have ṡ L and s s unformly contnuous. Gven that the rght hand sde of (36) s bounded, then by usng Barbalat s lemma, we can get lm t s (t) = and lm t e (t) =. 3.3 Smulaton results The dynamc equaton of a two-lnk robotc manpulator s gven as M(q) q C(q, q) q = τ (37) where M(q) = C(q, q) q = [ ] P P P cos q P P cos q, P P cos q P [ ] P q q sn q P q sn q P q sn ; q P = m l =. and P = m l =. (m, m are mass; lnks and have the same length of l); q and q are ont postons of the lnks and. Snce robotc manpulators cannot follow a step sequence nstantaneously, the desred traectory wll be the output of a fltered sequence of unt steps. We defne the transfer functon of the pre-flter for each ont of the robotc manpulator: W m (s) = 4 s 4s 4. (38) The ntal values of the robotc manpulators ont postons are set to.5 radans. The estmated mass and Corols matrces are gven by [ ˆP ˆM = ˆP ˆP cos q ˆP ˆP ] cos q ˆP ˆP (39) cos q P [ ˆP q Ĉ = sn q ˆP q sn q ˆP ] q sn q (4) ˆP q sn q where ˆP =, ˆP = 4. The number and the type of membershp functons for each nput varable are chosen to be consstent and comparable wth the algorthms proposed by Ln and Hsu (4). We defne fve membershp functons for each nput varable n (5) (l =,, 5). The parameters of the adaptaton laws n (4)-(8) are selected as η = η 3 = η 4 =, η = ( =, ) and σ () = σ () = [,,,, ] T by tral and error. It can be seen from Fgure that the trackng error errors are wth the range [-.,.] radans after 4 seconds. The tunng methodology utlzed n the premse and consequence parts of the fuzzy rules allows the fuzzy system to approxmate the control nput. The drawback s the chatterng phenomenon appearng n Fgure. (b) error(rad) (a) trackng(rad) 4 6 8 4 6 8.5.5 4 6 8 4 6 8 tme(sec) Fgure : (a) Trackng and (b) errors of ont n the revsed Ln and Hsu s algorthm. Dash lne:desred traectory; sold lne: actual traectory. (a) control nput(n*m) 3 5 5 3 4 6 8 4 6 8 (b) sldng surface 4 6 8 4 6 8 tme(sec) Fgure : (a) Control nput and (b) sldng surface of ont n the revsed Ln and Hsu s algorthm. 4 Stablty ssues for adaptve fuzzy sldng mode control algorthms as proposed n the lterature In ths secton, we brefly dscuss some of the dffcultes and drawbacks of the stablty results acheved n much of the publshed lterature on AFSMC. Durng our nvestgaton of the AFSMC algorthms proposed n the lterature, a couple of key stablty ssues arose. One ssue s assocated wth the need to know the bound n the unversal approxmaton theorem n (7) and the other ssue s based on the need to approxmate the dscontnuous sgn functon. The requrement to know the bound n (7) causes a practcal problem n mplementng and desgnng these controllers. The requrement to approxmate the sgn functon results n a theoretcal problem. 4. Case : Lyapunov stablty n a practcal sense In Yoo and Ham () s method, to meet the requrement of Lyapunov stablty, they need to choose a constant K D to make s K D > ω (s ) where ω s defned as the mnmum approxmaton error. Accordng to the unversal approxmaton theorem n (7), there exsts an optmal fuzzy system to estmate the gven functon and the approxmaton error ω s as small as possble. However, n real lfe, we can only defne a fuzzy system wth fnte number of membershp functons and fuzzy f-then rules. Therefore, we cannot fnd an optmal fuzzy system and the mnmum approxmaton error ω whch s as small as possble. The same stuaton happened n Guo and Woo (3) s algorthm where we need to choose a postve constant a
to make a s > γ s ω (s ). Snce ω can not be found n real applcatons, one cannot guarantee that the crteron a s > ω (s ) s satsfed. Therefore, one must desgn the controller by smulaton and expermentaton tral and error. The algorthm proposed n ths paper does not suffer from these practcal problems. 4. Case : Problems on the proof of Lyapunov stablty To prove Lyapunov stablty of the closed loop system, the followng defntons are appled n the algorthms proposed by Wang et al. () and Medhaffar et al. (6). In Wang et al. () s method, a fuzzy system ĥ(s θ h ) = θh T φ(s) s desgned to estmate the swtchng control term u sw and the optmal fuzzy system ĥ(s θ h ) s defned as ĥ(s θ h) = (D η ω max )sgn(s). (4) However, accordng to the unversal approxmaton theorem n (7), we can only fnd an optmal fuzzy system to estmate any gven real contnuous functon. Snce the sgn functon n (4) s a dscontnuous functon, we cannot fnd an optmal fuzzy system ĥ(s θ h ) to estmate the dscontnuous functon (D η ω max )sgn(s). In Medhaffar et al. (6) s algorthm, a fuzzy system ĥ (s θh ) = θh T ξ h (s ) s desgned to estmate the dscontnuous functon (D η )sgn(s ). Snce the unversal approxmaton theorem can only deal wth the real contnuous functon, we cannot fnd the optmal fuzzy system ĥ(s θh ) based on (7). Therefore, the above two algorthms have dffculty provng the convergence of the overall system based on the unversal approxmaton theorem. The method proposed n ths paper does not suffer from these theoretcal dffcultes. 5 Concluson Yoo, B. and Ham, W. (998) Adaptve fuzzy sldng mode control of nonlnear system, IEEE Transactons on Fuzzy Systems, Vol. 6, No., pp.35 3. Yoo, B. and Ham, W. () Adaptve control of robot manpulator usng fuzzy compensator, IEEE Transactons on Fuzzy Systems, Vol. 8, No., pp.86 99. Wang, J., Rad, A.B. and Chan, P.T. () Indrect adaptve fuzzy sldng mode control: Part I: fuzzy swtchng, Fuzzy Sets and Systems, Vol., No., pp. 3. Guo, Y. and Woo, P.Y. (3) An adaptve fuzzy sldng mode controller for robotc manpulators, IEEE Transactons on Systems, Man, and Cybernetcs, Part A: Systems and Humans, Vol. 33, No., pp.49 6. Ho, H.F., Wong, Y.K. and Rad, A.B. (4) Adaptve fuzzy sldng mode control desgn: Lyapunov approach, Proceedngs of the 5th Asan Control Conference, Melbourne, Australa, pp.5 57. Wa, R.J., Ln, C.M. and Hsu, C.F. (4) Adaptve fuzzy sldngmode control for electrcal servo drve, Fuzzy sets and systems, Vol. 43, No., pp.95 3. Ln, C.M. and Hsu, C.F. (4) Adaptve fuzzy sldng mode control for nducton servo motor systems, IEEE Transactons on Energy Converson, Vol. 9, No., pp.36 368. Akbarzadeh-T, M.R. and Shahnaz, R. (5) Drect adaptve fuzzy PI sldng mode control for a class of uncertan nonlnear systems, IEEE Internatonal Conference on Systems, Man and Cybernetcs, Wakoloa, Hawa, pp.548 553. Shahnaz, R., Shanech, H. and Parz, N. (6) Poston Control of Inducton and DC Servomotors: A Novel Adaptve Fuzzy PI Sldng Mode Control, 6 IEEE Power Engneerng Socety General Meetng, Montreal, Canada. Medhaffar, H., Derbel, N. and Damak, T. (6) A decoupled fuzzy ndrect adaptve sldng mode controller wth applcaton to robot manpulator, Internatonal Journal of Modellng, Identfcaton and Control, Vol., No., pp.3 9. Wang, L.X. (994) Adaptve Fuzzy Systems and Control: Desgn and Stablty Analyss, Englewood Clffs, NJ: Prentce Hall. Ln and Hsu (4) created a methodology of learnng both the premse and the consequence part of the fuzzy rules. Snce ths method s only desgned for nducton servomotor systems, we redesgn ths algorthm for robotc manpulators and the Lyapunov stablty for the redesgned algorthm s proved. Smulaton results demonstrate the effectveness of the method. The Lyapunov stablty n a practcal applcaton cannot be guaranteed n the algorthms proposed by Yoo and Ham () and Guo and Woo (3). Accordng to the unversal approxmaton theorem, a dscontnuous functon cannot be estmated by the fuzzy systems defned n the algorthms proposed by Wang et al. () and Medhaffar et al. (6). References Asada, H. and Slotne, J. J. (986) Robot Analyss and Control, New York, NY: Wley-Interscence. Wang, L.X. (997) A Course n Fuzzy Systems and Control, Upper Saddle Rver, NJ: Prentce Hall.