A fractional adaptation law for sliding mode control


 Jonas Lucas
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1 INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING Int. J. Adapt. Control Sgnal Process. 28; 22: Publshed onlne 7 October 28 n Wley InterScence (www.nterscence.wley.com). DOI:.2/acs.62 A fractonal adaptaton law for sldng mode control Mehmet Önder Efe, and Coṣku Kasnako glu Department of Electrcal and Electroncs Engneerng, TOBB Economcs and Technology Unversty, Sögütözü Cad. No. 43, TR656 Sögütözü, Ankara, Turkey SUMMARY Ths paper presents a novel parameter tunng law that forces the emergence of a sldng moton n the behavor of a multnput multoutput nonlnear dynamc system. Adaptve lnear elements are used as controllers. Standard approach to parameter adjustment employs nteger order dervatve or ntegraton operators. In ths paper, the use of fractonal dfferentaton or ntegraton operators for the performance mprovement of adaptve sldng mode control systems s presented. Httng n fnte tme s proved and the assocated condtons wth numercal justfcatons are gven. The proposed technque has been assessed through a set of smulatons consderng the dynamc model of a two degrees of freedom drect drve robot. It s seen that the control system wth the proposed adaptaton scheme provdes () better trackng performance, () suppresson of undesred drfts n parameter evoluton, () a very hgh degree of robustness and mproved nsenstvty to dsturbances and (v) removal of the controller ntalzaton problem. Copyrght q 28 John Wley & Sons, Ltd. Receved 6 December 27; Revsed 25 June 28; Accepted 4 July 28 KEY WORDS: fractonal tunng laws; adaptve sldng mode control; adaptaton; fractonal order control. INTRODUCTION Owng to the lnearty between ts nputs and the output, and the smplcty brought about by ths fact, adaptve lnear element (ADALINE) structure has been used n many applcatons of systems and control engneerng presumably under dfferent names. From ths perspectve, proportonal ntegral dervatve (PID) controllers, state feedback controllers and fnte mpulse response flters are just to name a few of ADALINE applcatons. The output of an ADALINE s a weghted sum of ts nputs and the weght assocated to each nput s adjustable. As dscussed n detal by Haykn [] and Jang et al. [2], ADALINEs are the buldng blocks of neural networks and some Correspondence to: Mehmet Önder Efe, Department of Electrcal and Electroncs Engneerng, TOBB Economcs and Technology Unversty, Sögütözü Cad. No. 43, TR656 Sögütözü, Ankara, Turkey. Emal: Contract/grant sponsor: Turksh Scentfc Councl (TÜBİTAK); contract/grant number: 7E37 Copyrght q 28 John Wley & Sons, Ltd.
2 A FRACTIONAL ADAPTATION LAW FOR SLIDING MODE CONTROL 969 types of fuzzy systems. The drvng force for devsng such complcated archtectures was the fact that ADALINEs were so smple that they could not capture complex nput output relatons wth frozen parameters. Yet t was possble to mplement the ADALINE as an adaptve system, whch can respond approprately by an adaptaton scheme. Gven a task to be accomplshed, the process descrbng the best evoluton of the adjustable parameters s the process of learnng, whch s sometmes called adaptaton, tunng, adjustment or optmzaton, all referrng to the same realty n the context of adaptve systems. Many approaches have been proposed, perceptron learnng rule, gradent descent, Levenberg Marquardt technque, Lyapunovbased technques are just to name a few; a good treatment can be found n [2]. A common feature of all these methods s the fact that the dfferentaton and ntegraton, or shortly dfferntegraton, of quanttes are performed n nteger order,.e. D:=d/dt for the dfferentaton wth respect to t and I=D for ntegraton over t n the usual sense. A sgnfcantly dfferent branch of mathematcs, called fractonal calculus, suggests operators D β wth β R [3, 4] and t becomes possble to wrte D f =D /2 (D /2 f ). Expectedly, Laplace and Fourer transforms n fractonal calculus are avalable to explot n closed loop control system desgn, nvolved wth s β or (jω) β generc terms, respectvely. Fractonal calculus and dynamcs descrbed by fractonal dfferental equatons (FDEs) are becomng more and more popular as the underlyng facts about the dfferentaton and ntegraton s sgnfcantly dfferent from the nteger order counterparts and, beyond ths, many real lfe systems are descrbed better by FDEs, e.g. heat equaton, telegraph equaton and a lossy electrc transmsson lne are all nvolved wth fractonal order dfferntegraton operators. A majorty of works publshed so far has concentrated on the fractonal varants of the PID controller, whch has fractonal order dfferentaton and fractonal order ntegraton, mplemented for the control of lnear dynamc systems, for whch the ssues of parameter selecton, tunng, stablty and performance are rather mature concepts utlzng the results from complex analyss and frequency doman methods of control theory (see [5]) than those nvolvng the nonlnear models (see [6]) and parameter changes n the approaches. Parameter tunng n adaptve control systems s a central part of the overall mechansm allevatng the dffcultes assocated wth the changes n the parameters that nfluence the closed loop performance. Many remarkable studes are reported n the past and the feld of adaptaton has become a blend of technques of dynamcal systems theory, optmzaton, heurstcs (ntellgence) and soft computng. Today, the advent of very hghspeed computers and networked computng facltes, even wthn mcroprocessorbased systems, tunng of system parameters based upon some set of observatons and decsons has greatly been facltated. In [7], an ndepth dscusson for parameter tunng n contnuous and dscrete tme s presented. Partcularly, for gradent descent rule for model reference adaptve control, whch s consdered n the nteger order n [7], has been mplemented n fractonal order by Vnagre et al. [8], where the nteger order ntegraton s replaced wth an ntegraton of fractonal order.25, and by Ladac and Charef [9], where the good performance n nose rejecton s emphaszed. In [], dynamc model of a ground vehcle s gven and an adaptve control law based on the gan adjustment s derved, the adaptaton law s changed to a fractonal order and the benefts of usng ths form are shown through smulatons. Regardng the sldng mode control, Calderón et al. [] descrbes the swtchng functon by a fractonal order PID controller and varants of t. The analyss contnues wth the computaton of the frst dervatve of the swtchng functon and relevant reachng condtons are derved. The method s expermented on a buck converter. In [2], sldng mode control framework s studed. A double ntegrator and the condtons of stablty are descrbed. Modfcaton of the equvalent control s performed so that a fractonally ntegrated sgn term provdes reducton n hghfrequency Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
3 97 M. Ö. EFE AND C. KASNAKO GLU swtchng. In both papers, the stablty has been analyzed through checkng whether sṡ< s satsfed, wth s beng the swtchng functon. The purpose of ths paper s to present an adaptaton approach that yelds () better robustness and nose rejecton capabltes than those utlzng tradtonal nteger order operators, () nondrftng parametrc evoluton when the essental factor drvng the adaptaton scheme s nose, () better trackng capablty and better system response and (v) removal of the controller ntalzaton problem. The four features mentoned above consttute the major results and contrbutons of the paper. Ths paper s organzed as follows: In the followng secton, we gve the Remann Louvlle defntons of fractonal operators used throughout the paper. The sldng mode control n the tradtonal sense s summarzed n the thrd secton. In the fourth secton, a fractonal order adaptaton scheme s ntroduced and the stablty analyss wth condtons for httng n fnte tme s dscussed. In ths secton, the parameter adjustment for the ADALINE s vewed as a supervsed adaptaton scheme. In the ffth secton, the condtons for applyng the scheme as an unsupervsed technque are presented. The dynamcal descrpton of a two degrees of freedom (DOF) SCARA  type drect drve robot s presented n the sxth secton. Smulaton results and the concludng remarks consttute the last part of the paper. 2. FRACTIONAL ORDER DERIVATIVE AND INTEGRAL Gven <β<, Remann Louvlle defnton of the βth order fractonal dervatve operator D β t s gven by f (β) (t) = D β t f (t) d = Γ( β) dt t (t ξ) β f (ξ)dξ () where Γ( ) s the gamma functon generalzng the factoral for nonnteger arguments. Accordng to ths defnton, the dervatve of a tme functon f (t)=t α wth α>,t s evaluated as D β t t α = Γ(α+) Γ(α+ β) tα β (2) Lkewse, Remann Louvlle defnton of the βth order fractonal ntegraton operator I β t gven by s t I β t f (t)= (t ξ) β f (ξ)dξ (3) Γ(β) In addton to these defntons, followng equaltes are helpful n understandng the presented approach. For <β< and a fnte end tme, say t h, the ntegral of the dervatve s evaluated as Selectvely complant artculated robot arm. The gamma functon s defned as Γ(β)= e t t β dt. Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
4 A FRACTIONAL ADAPTATION LAW FOR SLIDING MODE CONTROL 97 gven n the followng equaton [4]: I β t h f (β) = f (t h ) f (β ) () tβ h (4) Γ(β) The ntegral of a constant, say, B wth the same ntegraton lmts s gven as n the followng equaton: I β t h B= Γ(+β) B (5) The materal presented n the sequel s based on the above defntons of fractonal dfferentaton and ntegraton. tβ h 3. AN OVERVIEW OF SLIDING MODE CONTROL Owng to the robustness aganst uncertantes and dsturbances, and the nvarance propertes durng the sldng regme, sldng mode control has become a popular desgn approach that was mplemented successfully for the control of robots [3, 4]. Consder a general dynamc system descrbed by θ (r ) = f (H)+ f (H)+ m (g j (H)+ g j (H))τ j, =,2,...,n (6) j= where H=(θ, θ,...,θ (r ),θ 2, θ 2,...,θ (r 2 ) 2,...,θ n, θ n,...,θ (r n ) n ) T s the state vector of the entre system, r s the order of the th subsystem, f (H) and g j (H) are scalar functons of the state vector descrbng the nomnal (known) part of the dynamcs, f (H) and g j (H) are the bounded uncertantes on these functons and the nput vector T=(τ,τ 2,...,τ n ) T s the manpulated varable. Ths system of equatons can be rewrtten compactly as Ḣ= F(H)+ F(H)+(G(H)+ G(H))T (7) where F(H) and F(H) are n = r dmensonal vectors and G(H) and G(H) are n = r n dmensonal matrces. The desgner has the nomnal plant dynamcs gven by Ḣ= F(H)+G(H)T. Standard approach for the desgn of a sldng mode controller entals a swtchng functon defned as s = (s,s 2,...,s n ) T = K(H H d ) (8) where H d =(θ d,, θ d,,...,θ (r ) d,,θ d,2, θ d,2,...,θ (r 2 ) d,2,...,θ d,n, θ d,n,...,θ (r n ) d,n ) T s the vector of desred states and the locus descrbed by s= corresponds to the sldng manfold or the swtchng hypersurface. The entres of K are chosen such that the th component of the swtchng manfold has the structure ( ) d r s = dt +λ (θ θ d, ), =,2,...,n (9) Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
5 972 M. Ö. EFE AND C. KASNAKO GLU where λ >. Choosng a Lyapunov functon canddate as n () and settng the control vector as gven n (), one gets the equalty n (2) provded that the nverse (KG(H)) exsts: V = 2 st s () s SMC = (KG(H)) K(F(H) Ḣ d ) (KG(H)) Qsgn(s) () ṡ= PQsgn(s)+(P I)K(Ḣ d F(H))+K F(H) (2) where P:=K(G + G)(KG), whch s very close to the dentty matrx, and Q s a postvedefnte dagonal matrx chosen by the desgner. If one sets T:=s SMC, then the system enters the sldng mode after a reachng phase. The expresson n (2) can be nterpreted as follows: If there are no uncertantes,.e. F = and G =, thenwehaveṡ= Qsgn(s), ands T ṡ< s satsfed wth any postvedefnte Q.InthscasewehaveP=I and ths result s straghtforward. If only G =, we obtan ṡ= Qsgn(s)+K F, ands T ṡ< s satsfed f Q s a postvedefnte dagonal matrx and the th entry n the dagonal of Q s greater than the supremum value of the th row of K F. Ths would preserve the sgn of s n the presence of the term K F and the numercal computaton would requre the bounds of the uncertantes. In ths case we have P=I too. In the most general case, where nether of F nor G s zero, the expresson n (2) s obtaned. In ths case, dependng on the uncertantes nfluencng the nput gans ( G), the matrx P s very close to the dentty matrx, and utlzng the uncertanty bounds, the matrx Q can be chosen such that the sgn of s s preserved and s T ṡ< s satsfed. Wth an approprate choce of Q, s T ṡ< can be obtaned for s >, and ths result ndcates that the error vector defned by the dfference H H d s attracted by the subspace characterzed by s= and moves toward the orgn accordng to what s prescrbed by s=. The moton durng s = s called the reachng mode, whereas the moton when s= s called the sldng mode. Durng the latter dynamc mode, the closed loop system exhbts certan degrees of robustness aganst the modelng uncertantes, yet the system s senstve to nose as the sgn of a quantty that s very close to zero determnes the control acton heavly. It s straghtforward to show that a httng to s = occurs and the httng tme (t h, ) for the th subsystem satsfes the nequalty t h, s () /Q. One can refer to [5 7] for an ndepth dscusson on sldng mode control. Our goal wll be to obtan the sldng regme by utlzng an ADALINE structure ntroduced n the followng. 4. SLIDING MODE CONTROL THROUGH A FRACTIONAL ORDER ADAPTATION SCHEME The classcal sldng mode control law gven n () clearly requres F(H) and G(H). Inths secton, we wll focus on an adaptaton law that has the same effect on the closed loop system as () does. Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
6 A FRACTIONAL ADAPTATION LAW FOR SLIDING MODE CONTROL 973 Theorem 4. Let p =(,,,2,...,,r +) T be an adjustable parameter vector and let u =(e,ė,...,e (r ),) T be an nput vector. The nput output relaton of the controller producng τ s gven by the ADALINE τ =p T u, =,2,...,n (3) Denote the response obtaned wth T SMC as the desred response and let τ d, be the control sgnal resultng n the desred response at the th subsystem. Let the bound condtons Γ(+β) Γ(+k)Γ( k +β) (p(β k) ) T u (k) B, (4) k= τ (β) d, B 2, (5) hold true {,2,...,n}. Wth arbtrary μ> andρ>, the tunng law gven by p (β) sgn(u ) = K μ+ρu Tu sgn(σ ) (6) wth σ :=τ τ d, drves the parameters of the th controller to values such that the plant under control enters the sldng mode characterzed by s =, and httng n fnte tme occurs f K >(μ+ρ)(b, +B 2, ) (7) s satsfed. Proof Defne ϒ := k= (Γ(+β)/Γ(+k)Γ( k +β))(p (β k) ) T u (k) σ for every s negatve or not. Wth these expressons, we have σ (β) σ (β) σ = (τ (β) τ (β) d, )σ = ((p (β) ) T u )σ +(ϒ τ (β) d, )σ ( = sgn(u ) K μ+ρu Tu sgn(σ ) ) T u and check whether the quantty σ +(ϒ τ (β) d, )σ sgn(u ) T u = K μ+ρu Tu σ +(ϒ τ (β) d, )σ = K P(u ) σ +(ϒ τ (β) d, )σ K P(u ) σ + ϒ σ + τ (β) d, σ ( K P(u )+B, +B 2, ) σ snce K >(μ+ρ)(b, +B 2, )> B, +B 2, P(u ) where P(u ):=sgn(u ) T u /(μ+ρu Tu )> andmnp(u )=/(μ+ρ). (8) Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
7 974 M. Ö. EFE AND C. KASNAKO GLU Ths proves that the trajectores n the phase space are attracted by the subspace descrbed by σ =. Owng to the defnton n (), clamng σ (β) σ < for stablty s equvalent to the followng: σ (β) (t)σ (t)= σ (t) d t σ (ξ) dξ (9) Γ( β) dt (t ξ) β Obtanng σ (β) (t)σ (t)< can arse n the followng cases. In the frst case, σ (t)> and the ntegral t (σ (ξ)/(t ξ) β )dξ s monotoncally decreasng. In the second case σ (t)< and the ntegral t (σ (ξ)/(t ξ) β )dξ s monotoncally ncreasng. In both cases, the sgnal σ (t) s forced to converge to the orgn faster than t β. A natural consequence of ths s to observe a very fast reachng phase as the sgnal t β s a very steep functon around t. Now we must prove that frst httng to the swtchng functon occurs n fnte tme denoted by t h,.evaluateσ (β) utlzng (6) as gven below: σ (β) sgn(u ) T u = K μ+ρu Tu sgn(σ )+ϒ τ (β) d, (2) Applyng the fractonal ntegraton operator descrbed n (3) wth fnal tme t =t h, to both sdes of (2) one gets σ (t h, ) σ (β ) ( ) Γ(β) = I β sgn(u ) T u t h, K μ+ρu Tu sgn(σ ) () tβ h, + I β t h, (ϒ τ (β) d, ) ( ) = K sgn(σ ()) I β sgn(u ) T u t h, μ+ρu Tu + I β t h, (ϒ τ (β) d, ) = K sgn(σ ()) I β t h, P(u )+ I β t h, (ϒ τ (β) d, ) (2) where P(u ):=sgn(u ) T u /(μ+ρu T u ). Notng that σ (t)= whent =t h,, multplyng both sdes of (2) by sgn(σ ()), wehave σ (β ) ()sgn(σ ()) tβ h, Γ(β) = K I β t h, P(u )+ I β t h, (sgn(σ ())ϒ ) I β t h, (sgn(σ ())τ (β) d, ) (22) Owng to the defnton gven n (3), we have I β t h, (sgn(σ ())ϒ ) I β t h, ϒ I β t h, B, = B, t β h, Γ(+β) (23) Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
8 A FRACTIONAL ADAPTATION LAW FOR SLIDING MODE CONTROL 975 Smlarly, I β t h, (sgn(σ ())τ (β) d, ) = sgn(σ ()) I β t h, τ (β) d, = sgn(σ ()) τ d, (t h, ) τ (β ) d, () tβ h, (24) Γ(β) Snce mnp(u )=/(μ+ρ), we proceed as follows: I β t h, P(u ) I β t h, μ+ρ t β h, = (μ+ρ)γ(+β) Substtutng the results n (23) (25) nto (22), we obtan an nequalty gven as σ (β ) ()sgn(σ ()) tβ β h, Γ(β) K th, (μ+ρ)γ(+β) +B t β h,, Γ(+β) sgn(σ ())τ d, (t h, ) (25) +τ (β ) d, The nequalty above can be rearranged as (K B, (μ+ρ)) (μ+ρ)γ(+β) whch has the form t β h, (σ(β ) ()+τ (β ) ()sgn(σ ()) tβ h, Γ(β) d, ())sgn(σ ()) Γ(β) t β h, sgn(σ ())τ d, (t h, ) (26) σ(β ) () + τ (β ) d, () t β h, + τ d, (t h, ) (27) Γ(β) at β h, btβ h, +c (28) where a,b and c are clear from (27). Clearly, the lefthand sde of (28) starts from zero and ncreases monotoncally as a> and<β<. The rghthand sde, however, s monotoncally decreasng as b> and<β<. The curve descrbed on the rght starts from nfnty when t h, = and converges to c n the lmt. Therefore, the nequalty n (28) always suggests an upper bound. As a specal case, f β= 2,thevalueoft h, can be computed as gven by t h, ( c+ c 2 +4ab 2a ) 2 (29) Remark The tunng law n (6) can be nterpreted as a flterng of the sgnal r := K (sgn(u )/(μ+ρu T u )) sgn(σ ). The flter s a fractonal ntegrator of order β havng a hgher magntude than nteger order Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
9 976 M. Ö. EFE AND C. KASNAKO GLU ntegrator at all frequences except zero. In the nteger order case, the nformaton contaned n the hgh frequences s not exploted as much effcently as n the fractonal order case. The presence of sgn term n r s one evdence of the presence of valuable nformaton n hgh frequences. The specal treatment provded by the fractonal order tunng law therefore extracts a better path toward good parameter values than the nteger order counterpart. 5. CONDITIONS FOR OBTAINING sgn(σ) In the thrd secton, we summarzed the conventonal sldng mode control scheme for multnput multoutput systems of the form (6). On the other hand, f we could know a supervsory sgnal to compute σ, we would use t drectly n the fractonal adaptaton scheme gven n (6). However, the nature of the control systems does not provde such an nformaton; nstead, one has to develop strateges to observe a desred response n the closed loop by utlzng avalable quanttes. Therefore, a crtcally mportant stage of the approach presented n ths paper s to extract an equvalent measure about the sgn of the error on the control sgnal to use n the parameter tunng scheme. In other words, we need to develop a strategy together wth a set of assumptons such that we do not mplement a conventonal sldng mode controller, yet our tunng scheme drves the closed loop system toward the behavor that can be obtaned va the conventonal desgn wthout knowng the system parameters. For ths purpose, denote the response of the ADALINE controllers by T A,whchsn. Snce there are n subsystems, there are n ADALINE controllers. Consder the dfference r = T A T SMC = T A +(KG(H)) K(F(H) Ḣ d )+(KG(H)) Qsgn(s) = Jsgn(s)+H (3) where J:=(KG(H)) Q and H:=T A +(KG(H)) K(F(H) Ḣ d ). Let J be a dagonal matrx where J =J. Let H :=H+(J J ) sgn(s). Wth these defntons, (3) can be paraphrased as r=j sgn(s)+h (3) whose rows can explctly be wrtten as σ =J sgn(s )+H, =,2,...,n (32) The expresson n (32) stpulates that f H <J then sgn(σ )=sgn(s ). In other words, asde from the bound condtons gven n (4) and (5), a thrd one s gven as follows: H <J, =,2,...,n (33) Note that one can obtan nfntely many dfferent desgns of H ncludng those satsfyng the set of nequaltes above. Asde from the components comng from the system dynamcs and the desred response, ths depends also upon K and Q, the choce of whch can change the desred propertes of the sldng mode. Therefore one needs to check whether J s postve or not. Corollary If the nequaltes n (4) and (5) are satsfed, the tunng law n (6) enforces reachng σ = for and ths trggers the emergence of the sldng mode n the tradtonal sense. However, the Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
10 A FRACTIONAL ADAPTATION LAW FOR SLIDING MODE CONTROL 977 condtons derved n ths secton mply a class of plants where such an nducton could be vald. In the followng secton, we gve the dynamcal descrpton of a two DOF robot. 6. DYNAMICS OF THE ROBOT ARM AND THE CONTROL PROBLEM In ths paper, we consder the followng system to vsualze the contrbutons of ths paper. The motvaton for choosng ths system s the nonlnear and coupled nature of dfferental equatons descrbng the behavor. Furthermore, the adverse effects of nose, large ntal condtons and varyng payload condtons make the control problem a challenge for conventonal approaches. The dynamcs of the robot s gven by M(H)Ḧ+C(H, Ḣ) = T L (34) where H=(θ θ 2 ) T s the vector of angular postons n radans and Ḣ=( θ θ2 ) T s the vector of angular veloctes n rad/s. In (34), T=(τ τ 2 ) T s the vector of control nputs (torques) and L=(η η 2 ) T s the vector of frcton forces. The terms n (34) are gven below: M(H)= ( ) p +2p 3 cos(θ 2 ) p 2 + p 3 cos(θ 2 ) p 2 + p 3 cos(θ 2 ) p 2 (35) C(H,Ḣ)= θ 2 (2 θ + θ 2 )p 3 snθ 2 θ 2 (36) p 3 snθ 2 where p = M p, p 2 = M p and p 3 = M p. Here, M p denotes the payload mass. The detals of the plant model can be found n [8, 9]. The constrants regardng the plant dynamcs are τ 245N, τ N, and the frcton terms are η =4.9sgn( θ ) and η 2 =.67sgn( θ 2 ). The control problem s to force the system states to a predefned and dfferentable trajectores wthn the workspace of the robot. More explctly, e =θ θ d,, e 2 =θ 2 θ d,2 and the frst order (nteger) tme dervatves of these error terms are desred to converge to the orgn of the phase space. Accordng to the presented analyss and the model above, we have (KG) =M(H). More explctly, ( ) Q (p +2p 3 cos(θ 2 ))sgn(s ) ( ) Q22 (p + p 3 cos(θ 2 ))sgn(s 2 ) (KG) Q= Q 22 p 3 sgn(s 2 ) + Q (p + p 3 cos(θ 2 ))sgn(s ) (37) The above separaton of terms suggests that J =Q (p +2p 3 cos(θ 2 ))>andj 22 =Q 22 p 3 > for every possble angular state and payload condton. Clearly, the devsed approach s sutable for mechancal systems, robots and systems as they have a postvedefnte nerta matrx. In the Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
11 978 M. Ö. EFE AND C. KASNAKO GLU followng secton, we present the smulaton studes comparatvely wth the nteger order ntegraton scheme n the parameter adaptaton stage. 7. SIMULATION RESULTS The presented approach s mplemented for the plant ntroduced n the second secton. We set β= 2 and the system runs for 2 s of tme for the reference trajectores shown n Fgure. The sold curves represent the reference trajectores, whle the dashed ones stand for the response of the robot. Durng the operaton, a 5 kg of payload s grasped when t =4s and released when t =8s and ths s repeated when the robot s motonless at t =2 and 6s. The manpulator s desred to stay motonless after t =6s. It should be noted that the payload scenaro s a sgnfcant dsturbance changng the dynamcs of the plant suddenly. Another dffculty s the ntal condtons that the ADALINE controllers are supposed to allevate. Intally, θ d, =θ d,2 =, the system s motonless and θ ()= π 3 and θ 2 ()= π 2, whch ndcate large ntal postonal errors to test the performance of the proposed control scheme. Durng the smulatons, we set K =5 and K 2 =. The sldng lnes for both lnks are set by choosng λ = and we set μ= andρ=. Besdes these, n order to avod exctng any undesred chatterng phenomenon assocated tghtly wth the dscontnuous nature of Angular Postons Dashed: Robot Response Sold: Reference Trajectory Angular Veloctes Fgure. Reference trajectores and the response of the robot. Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
12 A FRACTIONAL ADAPTATION LAW FOR SLIDING MODE CONTROL 979 θ θ d,.5.5 θ 2 θ d, d(θ θ d, )/dt.5 d(θ 2 θ d,2 )/dt Fgure 2. State trackng errors. the sgn functon, we choose sgn(σ ) σ /( σ +δ) wth δ beng the parameter determnng the slope around the orgn. Ths paper consders δ=., whch ntroduces a very thn boundary layer and mproves the performance of the control system. If such a smoothng s not used, the fluctuatons n the control sgnals are magnfed and the practcal applcablty of the proposed approach s nfluenced adversely. The dscrepances between the reference trajectores and the system response are depcted n Fgure 2, where an exponental convergence s apparent even n the presence of nose corruptng the observed system states and the changes n the system dynamcs due to the payload varatons. The behavor n the phase space llustrated n Fgure 3 s another evdence of robustness of the control system and nsenstvty to varatons n the plant dynamcs. As mentoned prevously, a very fast reachng phase s followed by the desred sldng mode. In Fgure 4, the appled control sgnals are gven wth the wndow graphs for better vsualzng the ntal transent. As expected, the control efforts durng the frst.2 s have hgher magntudes than what comes later. The adverse effect of the nosy observatons on the control sgnal s another concluson that s worth mentonng. The tme evolutons of the controller parameters, whch are all started from zero, are shown n Fgure 5, where t s clearly vsble that after a fast transent, the parameters multplyng the errors and ther dervatves settle down to constant values, whle the parameters multpled by unty evolve bounded. If we remember the reference profles, the system s desred to be motonless after t>6s; ths means that the tunng actvty durng ths tme s subject to the effects of nose. That s to say, the system s at a desred state but we would lke to fgure out how the parameter Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
13 98 M. Ö. EFE AND C. KASNAKO GLU de /dt de 2 /dt e e 2 Fgure 3. Behavor n the phase space. Output of Base Controller Output of Elbow Controller τ 2..2 τ 4 5 τ τ Fgure 4. Appled control sgnals and ther ntal transents. tunng mechansm functons durng ths perod. The answer to ths queston s n Fgure 5, where any possble undesred drfts n the controller parameters are suppressed approprately. In realzng the tunng law n (6), whch entals the mplementaton of I.5 terms, we choose Crone approxmaton over the bandwdth. Hz, whch s acceptable for a feedback Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
14 A FRACTIONAL ADAPTATION LAW FOR SLIDING MODE CONTROL φ, φ 2, φ,2 φ 2, φ,3 φ 2, Fgure 5. Tme evoluton of the controller parameters for base lnk (left column) and elbow lnk (rght column). control applcaton of ths type. The order s set to 25 and truncated Mclaurn expanson s utlzed n numercal computaton. Wth these settngs, a very good spectral approxmaton to the desred operator s obtaned. For more detals on the numercal realzaton, the reader s referred to [2]. It must be emphaszed that the presented desgn does not utlze the terms seen n the dynamcal descrpton of the plant. The approach adaptvely determnes the controller parameters so that the plant dsplays robustness aganst dsturbances and uncertantes. The smulatons have been repeated wth dfferent values of μ and ρ. We dd not consder ncreasng or decreasng or both as ths corresponds to change n K, nstead of ths, we kept ρ= and resmulated the system wth.,., and as the values of μ. Includng the case wth μ=, the system was observed to respond approprately, yet for the larger values of μ, the sldng mode dsappears, further ncrease causes the loss of trackng capablty totally. A smlar response s observed wth the change n ρ whle μ s kept at unty. As a last ssue, we turned back to μ=ρ=, and the system s run wth β=. Ths has changed the parameter update law gven n (6) as ṗ = K (sgn(u )/(μ+ρu T u ))sgn(σ ).Wehaveϒ := p T u and the dervaton wth the conclusons gven n (8) s seen vald for the nteger order case too. Many dfferent parameter confguratons have been tested and, n most of them, the feedback system has grown nstabltes. Among the tested condtons, the one wth K = 5 has gven the best results that are llustrated n Fgures 6 9. Although the error trends seen n Fgure 6 are promsng, the appled torques resultng n ths observaton are llustrated n Fgure 7. Clearly, Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
15 982 M. Ö. EFE AND C. KASNAKO GLU θ θ d,.5.5 θ 2 θ d, d(θ θ d, )/dt 2 d(θ 2 θ d,2 )/dt Fgure 6. State trackng errors when β= Frst 4 msec Frst 4 msec τ τ Last 2 msec Last 2 msec Fgure 7. Appled control sgnals and ther ntal transents when β =. Copyrght q 28 John Wley & Sons, Ltd. Int. J. Adapt. Control Sgnal Process. 28; 22: DOI:.2/acs
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