A fractional adaptation law for sliding mode control


 Jonas Lucas
 1 years ago
 Views:
Transcription
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
8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationDamage detection in composite laminates using cointap method
Damage detecton n composte lamnates usng contap method S.J. Km Korea Aerospace Research Insttute, 45 EoeunDong, YouseongGu, 35333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The contap test has the
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationAn InterestOriented Network Evolution Mechanism for Online Communities
An InterestOrented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationA Secure PasswordAuthenticated Key Agreement Using Smart Cards
A Secure PasswordAuthentcated Key Agreement Usng Smart Cards Ka Chan 1, WenChung Kuo 2 and JnChou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationMultiple stage amplifiers
Multple stage amplfers Ams: Examne a few common 2transstor amplfers:  Dfferental amplfers  Cascode amplfers  Darlngton pars  current mrrors Introduce formal methods for exactly analysng multple
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationRESEARCH ON DUALSHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.
ICSV4 Carns Australa 9 July, 007 RESEARCH ON DUALSHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationFrequency Selective IQ Phase and IQ Amplitude Imbalance Adjustments for OFDM Direct Conversion Transmitters
Frequency Selectve IQ Phase and IQ Ampltude Imbalance Adjustments for OFDM Drect Converson ransmtters Edmund Coersmeer, Ernst Zelnsk Noka, Meesmannstrasse 103, 44807 Bochum, Germany edmund.coersmeer@noka.com,
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationA frequency decomposition time domain model of broadband frequencydependent absorption: Model II
A frequenc decomposton tme doman model of broadband frequencdependent absorpton: Model II W. Chen Smula Research Laborator, P. O. Box. 134, 135 Lsaker, Norwa (1 Aprl ) (Proect collaborators: A. Bounam,
More informationOnLine Fault Detection in Wind Turbine Transmission System using Adaptive Filter and Robust Statistical Features
OnLne Fault Detecton n Wnd Turbne Transmsson System usng Adaptve Flter and Robust Statstcal Features Ruoyu L Remote Dagnostcs Center SKF USA Inc. 3443 N. Sam Houston Pkwy., Houston TX 77086 Emal: ruoyu.l@skf.com
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationSCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar
SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme,
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationDistributed MultiTarget Tracking In A SelfConfiguring Camera Network
Dstrbuted MultTarget Trackng In A SelfConfgurng Camera Network Crstan Soto, B Song, Amt K. RoyChowdhury Department of Electrcal Engneerng Unversty of Calforna, Rversde {cwlder,bsong,amtrc}@ee.ucr.edu
More informationIMPACT ANALYSIS OF A CELLULAR PHONE
4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng
More informationInterIng 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 1516 November 2007.
InterIng 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 1516 November 2007. UNCERTAINTY REGION SIMULATION FOR A SERIAL ROBOT STRUCTURE MARIUS SEBASTIAN
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationPAS: A Packet Accounting System to Limit the Effects of DoS & DDoS. Debish Fesehaye & Klara Naherstedt University of IllinoisUrbana Champaign
PAS: A Packet Accountng System to Lmt the Effects of DoS & DDoS Debsh Fesehaye & Klara Naherstedt Unversty of IllnosUrbana Champagn DoS and DDoS DDoS attacks are ncreasng threats to our dgtal world. Exstng
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationResearch Article Enhanced TwoStep Method via Relaxed Order of αsatisfactory Degrees for Fuzzy Multiobjective Optimization
Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced TwoStep Method va Relaxed Order of αsatsfactory Degrees for Fuzzy
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationA GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION
A GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION Helena Vasconcelos INESC Porto hvasconcelos@nescportopt J N Fdalgo INESC Porto and FEUP jfdalgo@nescportopt
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCullochPtts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationLaws of Electromagnetism
There are four laws of electromagnetsm: Laws of Electromagnetsm The law of BotSavart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationMultiRobot Tracking of a Moving Object Using Directional Sensors
MultRobot Trackng of a Movng Object Usng Drectonal Sensors Xaomng Hu, Karl H. Johansson, Manuel Mazo Jr., Alberto Speranzon Dept. of Sgnals, Sensors & Systems Royal Insttute of Technology, SE 44 Stockholm,
More informationIDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM
Abstract IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Alca Esparza Pedro Dept. Sstemas y Automátca, Unversdad Poltécnca de Valenca, Span alespe@sa.upv.es The dentfcaton and control of a
More informationAn Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services
An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsnyng Wu b a Professor (Management Scence), Natonal Chao
More informationAn Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems
STANCS73355 I SUSE73013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part
More informationStability, observer design and control of networks using Lyapunov methods
Stablty, observer desgn and control of networks usng Lyapunov methods von Lars Naujok Dssertaton zur Erlangung des Grades enes Doktors der Naturwssenschaften  Dr. rer. nat.  Vorgelegt m Fachberech 3
More informationComparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions
Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationINVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMAHDR NETWORKS
21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!
More informationThe kbinomial Transforms and the Hankel Transform
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 9 (2006, Artcle 06.1.1 The kbnomal Transforms and the Hankel Transform Mchael Z. Spvey Department of Mathematcs and Computer Scence Unversty of Puget
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationNew bounds in BalogSzemerédiGowers theorem
New bounds n BalogSzemerédGowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A
More informationVRT012 User s guide V0.1. Address: Žirmūnų g. 27, Vilnius LT09105, Phone: (3705) 2127472, Fax: (3705) 276 1380, Email: info@teltonika.
VRT012 User s gude V0.1 Thank you for purchasng our product. We hope ths userfrendly devce wll be helpful n realsng your deas and brngng comfort to your lfe. Please take few mnutes to read ths manual
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XIII  Sliding Mode Control  Vadim Utkin
SLIDING MODE CONTROL Vadm Utkn The Oho State Unversty, Columbus,Oho, USA Keywords: Sldng mode, State space, Dscontnuty manfold, Varable structure system, Sldng mode exstence condtons, Lyapunov functon,
More informationMultiplication Algorithms for Radix2 RNCodings and Two s Complement Numbers
Multplcaton Algorthms for Radx RNCodngs and Two s Complement Numbers JeanLuc Beuchat Projet Arénare, LIP, ENS Lyon 46, Allée d Itale F 69364 Lyon Cedex 07 jeanluc.beuchat@enslyon.fr JeanMchel Muller
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationBusiness Process Improvement using Multiobjective Optimisation K. Vergidis 1, A. Tiwari 1 and B. Majeed 2
Busness Process Improvement usng Multobjectve Optmsaton K. Vergds 1, A. Twar 1 and B. Majeed 2 1 Manufacturng Department, School of Industral and Manufacturng Scence, Cranfeld Unversty, Cranfeld, MK43
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationPerformance Management and Evaluation Research to University Students
631 A publcaton of CHEMICAL ENGINEERING TRANSACTIONS VOL. 46, 2015 Guest Edtors: Peyu Ren, Yancang L, Hupng Song Copyrght 2015, AIDIC Servz S.r.l., ISBN 9788895608372; ISSN 22839216 The Italan Assocaton
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages  n "Machnes, Logc and Quantum Physcs"
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationA Performance Analysis of View Maintenance Techniques for Data Warehouses
A Performance Analyss of Vew Mantenance Technques for Data Warehouses Xng Wang Dell Computer Corporaton Round Roc, Texas Le Gruenwald The nversty of Olahoma School of Computer Scence orman, OK 739 Guangtao
More information21 Vectors: The Cross Product & Torque
21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rghthand rule for the cross product of two vectors dscussed n ths chapter or the rghthand rule for somethng curl
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More information1.1 The University may award Higher Doctorate degrees as specified from timetotime in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
More informationDynamic Pricing for Smart Grid with Reinforcement Learning
Dynamc Prcng for Smart Grd wth Renforcement Learnng ByungGook Km, Yu Zhang, Mhaela van der Schaar, and JangWon Lee Samsung Electroncs, Suwon, Korea Department of Electrcal Engneerng, UCLA, Los Angeles,
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More informationA Design Method of Highavailability and Lowopticalloss Optical Aggregation Network Architecture
A Desgn Method of Hghavalablty and Lowoptcalloss Optcal Aggregaton Network Archtecture Takehro Sato, Kuntaka Ashzawa, Kazumasa Tokuhash, Dasuke Ish, Satoru Okamoto and Naoak Yamanaka Dept. of Informaton
More informationLogical Development Of Vogel s Approximation Method (LDVAM): An Approach To Find Basic Feasible Solution Of Transportation Problem
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77866 Logcal Development Of Vogel s Approxmaton Method (LD An Approach To Fnd Basc Feasble Soluton Of Transportaton
More information